^Colorado State Univesity
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Suttmner 1997
Chemical Engineering Education
Volume 30 Number 3 Summer 1997
> DEPARTMENT
146 Colorado State University,
Nazmnul Karim, C.A. Podmore
> LEARNING
152 Toward Technical Understanding: Part 1. Brain Structure and
Function,
J.M. Haile
> CLASSROOM
158 A New Approach to Teaching Dimensional Analysis,
Stuart W. Churchill
168 Teaching Statistics to ChE Students,
Dianne Dorland, K. Karen Yin
180 Use of Computational Tools in Engineering Education:
A Case Study on the Use of Mathcad,
J.N. Harb, A. Jones, R.L. Rowley, W.V. Wilding
194 Use of Spreadsheets in Introductory Statistics and Probability,
Brian S. Mitchell
> SURVEY
166 Engineering Education for the 21st Century: Listen to
Industry!
Ralph A. Buonopane
> CLASS AND HOME PROBLEMS
172 An Introduction to Process Flexibility: Part 1. Heat Exchange,
W.E. Jones, J.A. Wilson
> RANDOM THOUGHTS
178 Objectively Speaking,
Richard M. Fielder, Rebecca Brent
> LEARNING IN INDUSTRY
188 Introducing Graduate Students to the Industrial Perspective,
James B. Riggs, R. Russell Rhinehart
> LABORATORY
192 An Experiment to Characterize a Consolidating Packed Bed,
Mark Gerrard, Mark Hockborn, Jason Glass
> 177 Letter to the Editor
CHEMICAL ENGINEERING EDUCATION (ISSN 00092479) is published quarterly by the Chemical Engineering
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not necessarily those of the ChE Division. ASEE. which body assumes no responsibility for them. Defective copies replaced
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availability. POSTMASTER: Send address changes to CEE, Chemical Engineering Department., University of Florida,
Gainesville, FL 326116005.
rl department
Colorado State
University
NAZMUL KARIM, C.A. PODMORE
Colorado State University Fort Collins, CO 80523
Paradigm: ... a set of rules and regulations
(written and unwritten) that does two things: (1) it
establishes or defines boundaries; and (2) it tells
you how to behave inside the boundaries in order to
be successful.
Joel A. Barker
Paradigms may shift, rattle, and roll, but Judson Harper
possesses no doubts that new boundaries were estab
lished with the advent of chemical engineering at
Colorado State University. Now the University's Vice Presi
dent for Research and Information Technology, Harper
speaks of "fresh cloth" and a "new model" when outlin
ing what he calls the genesis of the chemical engineering
program in 1976.
Part of the Department of Chemical and BioResource
Engineering, chemical engineering at Colorado State started
with the coupling of food and chemical engineering
undergirded by Harper's own work in food extrusion. In
fact, the program was the first one nationally to emphasize
the interface between chemical engineering and the biologi
cal sciences. Two decades later, Colorado State is still one of
the few American universities developing its chemical engi
neering program in such a unique manner.
If the program is unique, it is because of its origins and
evolution; as Harper implies, its genesis is its hallmark
The historic Colorado State University Oval with namely, an integration of interdisciplinary collaboration,
its twin rows of American elms leading to the emerging technology, and applied research.
Administration Building lies just to the east
of the College of Engineering complex. INTERDISCIPLINARY COLLABORATION
Prior to Harper's arrival at Colorado State, a movement
was underway to expand the then Department of Agricul
Copyright ChE Diuvsion o'ASEE 1997 tural Engineering. Rather than go down a traditional road
146 Chemical Engineering Education
such as machinery, however, the powersthatbe de
cided to expand into food engineering. Harper was
hired out of industry to become the department head,
and the evolution of chemical engineering at Colo
rado State began.
The 1970s, with its fuel shortage crises, pushed the
evolutionary process alongemphases merged,
changed, and moved on. Research on the conversion
of biomass to alternative fuels led to broader in
volvement in biochemical engineering, with strong
programs developing in design and control of
bioreactors and, more recently, in environmental bio
technology. By the early 1980s, a new front was
opened with the addition of the emerging area of
advanced materials and its focus on understanding
fundamental chemical principles of hightech thin
film and polymer processing.
These areasbiochemical engineering, environ
mental engineering, and advanced materials
together with the traditional or fundamental chemi
cal engineering areas of thermodynamics, heat and
mass transfer, and process control form the core re
search of the eightmember faculty today. Through
out all of their undertakings, however, the cross or
interdisciplinary hallmark can be found. According
to former department head, Vincent Murphy, "We
only hired people who had an interdisciplinary bent.
Since we were a small group, if we wanted to accom
plish much, we had to form partnerships."
An excellent example can be found in the work of
Kenneth Reardon and the area of bioremediation
that is, the use of biological agents such as microor
ganisms and plants to solve hazardous waste pollu
tion problems. Reardon wants to incorporate
bioremediation concepts into both undergraduate and
graduate engineering courses. He and five other Colo
rado State colleagues, through a $350,000 grant from
the National Science Foundation's Engineering Edu
cation and Centers Division, are developing seven
teaching modules that contain laboratory, video, mini
lecture, and case study components. Two modules
are presently being tested at Colorado State and five
other universities; one module is in a multimedia
(CDROM) format and the other is in a video/paper
format.
EMERGING TECHNOLOGY
David Dandy builds diamonds one atom at a time.
Although Dandy is involved in fundamental research,
its future applications are mindboggling
supercomputers the size of a deck of cards and surgi
Summer 1997
Setting the Scene
Founded in 1870, six years before Colorado gained statehood,
Colorado State University has an enrollment of 22,000 oncampus
students. The College of Engineering has 1600 undergraduates and
500 graduate students.
Fort Collins, a former territorial army fort, has a population of
100,000 and is nestled in the foothills of the Rockies, 65 miles north
of Denver, at an elevation of 5,000 feet. With 300 days of sunshine,
residents freely enjoy all the natural beauty for which Colorado is
known. Camping, skiing, climbing, boating, fishingname your plea
sure and you'll probably find it within easy distance of Fort Collins.
The University itself consists of an 833acre main campus that
houses most of the administration offices and classrooms. The chemi
cal engineering program is located in the 41,200 squarefoot Engi
neering South Building, which was completely renovated in 1984.
The main campus also houses the Lory Student Center, the hub of
student life, which was ranked one of the top ten student unions by the
New York Times columnist Richard Mall in 1986. Pauline Yoshihashi,
writing in the Wall Street Journal in March of 1992, cites the Center
for its studentoriented services: "Scholars can drop by ... to buy a
computer, and also [to] rent skis or hiking boots for a weekend of
work and play." An extensive renovation of the facility was com
pleted recently.
In addition to the main campus, a 1,700acre Foothills Campus is
devoted primarily to research. The Engineering Research Center (ERC)
is located on this campuschemical engineering research programs
in semiconductor processing and groundwater/contaminant transport
use these facilities.
The eightmember chemical engineering faculty focuses on three
applied areas of research: biochemical engineering, environmental
engineering, and advanced materials. This work is complemented by
more basic research in the traditional chemical engineering fields of
thermodynamics, heat and mass transfer, and process control.
Virtually all the chemical engineering research groups interact ac
tively with other departments at Colorado State. These contacts range
from information exchange to joint projects with investigators in
departments such as microbiology, electrical engineering, biochemis
try, civil engineering, and chemistry. The interdepartmental environ
mental engineering program is an exciting new area of interaction for
both teaching and research, bringing together faculty and students
from five engineering fields.
For more information about the program, contact
M. Nazmul Karim, Ph.D., Associate Department Chair
Chemical and BioResource Engineering
Colorado State University
100 Glover Building
Fort Collins, CO 805231370
telephone 9704915252
fax 9704917369
email chembio@engr.colostate.edu
Web site http://www.engr.colostate.edu/depts/chembio
cal blades that never
dull, to name a few.
Considered a world
expert in diamond
fabrication, Dandy
conducted research at
Sandia National
Laboratories in
Livermore, Califor
nia, to develop hair
like diamond slivers
for a major commer
cial application
AT&T's transatlantic
fiber optic telephone
cable. According to
Dandy, growing dia
monds (using a pro
cess called chemical
vapor deposition) is
easy; the real chal
lenge is understand
ing how they grow,
how they interact with
other materials, and
... [Colorado State's] youth is the very factor that has
allowed it to do what more established programs often
cannotchange the boundaries, alter the rules, shift
the paradigm.
Adeyma Arroyo (grad student) records data from an apparatus
used to study biofiltration of offgases from coating operations.
how to make them
behave in an orderly manner. The complexity of issues that
are involved in the process makes any major scientific break
through a future event, but Dandy anticipates being part of
that ultimate breakthrough.
Because his investigations on polymers also fall outside
the realm of basic research, Larry Belfiore considers them
high risk. His overall objective is to understand how a
material's components interact in a mixture and then to use
that information to make chemically compatible systems.
Since his work is not directly marketdriven, Belfiore con
centrates on developing methods that will enhance the
thermal and mechanical properties of materials. On
another project, Belfiore and Allen Rakow are combin
ing stressstrain testing, infrared spectroscopy, and elec
tron microscopy to explore the use of agar, an edible
marine polysaccharide, for packaging and potential bio
mimicry applications.
During the fourteen years she has been on the chemical
engineering faculty, Carol McConica has been a key figure
in the department's advanced materials thrust. The bulk of
her more than $2 million in contracts and grants for inte
grated circuit process research has been spent on graduate
education and a special master's program, which she devel
oped. The program, through twelve months of course work
and some eighteen months of experimental, handson expe
rience, offers students a broad understanding of microelec
tronics while identifying the fundamental principles behind
manufacturing prob
lems that are supplied
from industry. Given
a grounding in those
principles, the stu
dents then develop
new processes.
Before joining the
faculty, McConica
helped Hewlett
Packard design the
chemical processes
necessary to build
more powerful com
puter chips. The ma
jor emphasis of a cur
rent grant from the
National Science
Foundation is to de
velop more environ
mentally benign pro
cesses for that manu
facturing. To
McConica, this uni
versityindustry co
operation not only
fits the landgrant mission of Colorado State ("We are an
educational institution."), but it also prepares chemical
engineering graduates to fit into an everchanging tech
nological landscape.
APPLIED RESEARCH
Much of chemical engineering research at Colorado State
is applied. In fact, that was part of the program's original
mission. This focus spotlights the final characteristic of the
program; that is, its emphasis on reallife systems and coop
eration with the private sector. Whether developing com
puter chips, using biotechnology to extract gold in hardto
mine ores, or developing innovative medical cures, the push
is out into the real world.
This push is clearly exemplified by the work of James
Linden, who holds a joint appointment with the Department
of Microbiology. Linden conducts research into the use of
plant cell cultures for production of valuable medicinal com
pounds such as artimisinin, a possible natural treatment for
malaria. On another project, he is studying another plant cell
culture process that produces taxol, an anticancer drug.
Linden works with private chemical research companies on
both of these projects, indicative of the growing private/
public cooperation.
This cooperative approach is further seen in the work
being conducted under the leadership of Brian Batt at the
Colorado Bioprocessing Center, a statesupported entity that
Chemical Engineering Education
Victor Saucedo (grad student) and Sohana Karim (undergrad)
prepare a computercontrolled bioreactor for a
fermentation experiment.
"Diamond" Dave Dandy admires the fruits
of his latest (thin film) research.
is administered through the Department of Chemical and
BioResource engineering. The Center houses a full spectrum
of pilotscale equipment for cell culture and product recov
ery. Its mission is to assist new biotechnology companies
in proofofconcept and process development studies. Re
searchers at the Center conduct basic and applied re
search to develop highvalue products from genetically
engineered microorganisms.
The Center is an integral part of the Research Experiences
for Undergraduates Program in Bioprocess Engineering, a
project sponsored by the National Science Foundation in
which outstanding undergraduates from across the country
are given an opportunity to participate in a tenweek re
search program during the summer months. It also works
closely with the Colorado Institute for Research in Biotech
nology (CIRB), a universityindustryfederal laboratory net
work that offers seed grants to university researchers to
initiate university/industry projects. In addition, CIRB pro
vides support for graduate students and for students who
intern in Colorado biotechnology companies.
Professor Karim, who has been at Colorado State Univer
sity since the Spring of 1981, is regarded as one of the world
leaders in the research area of process control application to
biotechnology. He is one of the first researchers who has
applied neural network technology to model and optimize
bioprocesses. He is also involved in multivariate statistical
analysis (Principal Component Analysis) for online process
fault detection and diagnosis of chemical and biological
systems. Dr. Karim has researched various microbial spe
cies: bacteria, yeast, fungi, mammalian, and plant cell cul
tures. He has been either an advisor or coadvisor to approxi
mately onethird of the department's PhD graduates. In rec
ognition of his contributions to research and graduate sup
port, the College of Engineering awarded him the Abell
Research Award in May of 1997.
EDUCATIONAL MODEL
"Interdisciplinary," "emerging," and "applied" are simply
adjectival buzzwords if they do not represent a testable real
ity within the academic setting. The quality of research must
impact the quality of education at both the undergraduate
and graduate levels. At Colorado State, that impact exists
with a unique twist that only a smaller program can offer.
The faculty like to think that chemical engineering at Colo
rado State delivers the best of what expensive private schools
offer, small classes and highquality students, plus the best
of a research universityall within the context of a nurtur
ing environment.
Numbers can be impressive: 8 faculty members advise
135 undergraduates and 35 graduate students. Because the
faculty are aggressive in their research efforts, individually
averaging $200,000 a year in contracts and grants, students
have numerous opportunities, starting at the undergraduate
level, to work on research projects. During the last eight
years, 60 BS graduates (25% of the total number) have been
involved in research.
Summer 1997
 FACULTY PROFILES
Judson M. Harper, Professor
Vice President for Research and Information Technology
Ph.D., Iowa State University
Interests: Extrusion processing of foods; manufacturing of lowcost nutritious foods; structural and chemical changes of
polymers during processing.
M. Nazmul Karim, Professor
Associate Department Chair
Ph.D., University of Manchester
Interests: Online adaptive control of solidstate fermentation, use of neural networks, principal component analysis and
genetic algorithms in bioprocess control and optimization; lignin biodegradation and recombinant E. coli fermentation for
ethanol production from xvlose.
Terry G. Lenz, Professor
Ph.D., Iowa State University
Interests: Computational and experimental studies in chemical and biochemical thermodynamics as well as in more applied
areas such as solar cooling systems.
Carol M. McConica, Professor
Ph.D., Stanford
Interests: The use of ultrahigh vacuum as well as in situ Raman spectroscopy techniques to elucidate reaction mechanisms
relevant to integrated chip processing; selective metal deposition processes for threedimensional integration of integrated
circuits.
Vincent G. Murphy, Professor
Ph.D., University of Massachusetts
Interests: Fundamental and applied studies in biochemical and food process engineering; bioremediation of contaminated soils.
Laurance A. Belfiore, Associate Professor
Ph.D., University of Wisconsin
Interests: Phase behavior of polymer blends; polymeric transitionmetal complexes that exhibit synergistic macroscopic physical
properties; applications of solidstate NMR spectroscopy.
David S. Dandy, Associate Professor
Ph.D., California Institute of Technology
Interests: Vapor deposition of diamond, silicon nitride, and cubic boron nitride; threedimensional laminar flows; parallel
numerical algorithm development; application of physical models to process control design.
James C. Linden, Associate Professor, Joint Appointment
Ph.D., Iowa State University
Interests: Biomass refining to provide starting material for ethanol fermentation; cultivation of fungi and bacteria for enzyme
production; plant cell cultures for the production of useful secondary metabolites.
Allen L. Rakow, Associate Professor, Research Appointment
Sc.D., Washington University, St. Louis
Interests: Bioseparations, biorheology, and food engineering
Kenneth F. Reardon, Associate Professor
Ph.D., California Institute of Technology
Interests: Microbial degradation of hazardous organic compounds; the effects of cultivation conditions on genetically modified
bacteria.
Ranil Wickramasinghe, Assistant Professor
Ph.D., University of Minnesota
Interests: Application of the principles of mass transfer and rheology to the development of new separation processes for
biochemical/biomedical systems.
Brian C. Batt, Research Scientist
Director of the Colorado Bioprocessing Center
Ph.D., University of Colorado
Interests: Development of bioprocessing systems involving the use of recombinant microorganisms and mammnalian cell cultures.
150 Chemical Engineering Education
UNDERGRADUATE
PROGRAM
Such handson experience is possible because of the qual
ity of the students. At the undergraduate level, for example,
students have received two AIChE Outstanding Senior awards
and six National Science Foundation fellowships. Over 30%
of the undergraduates are high school valedictorians, and the
average GPA for the 1996 incoming class was 3.8. Alumni
have gone on to graduate programs at MIT, UC Berkeley,
Stanford, Purdue, Wiscon
sin, and Cornell. The
program's AIChE Student
Chapter has received an
Outstanding Chapter
Award from the national
organization in 12 of the
past 14 years.
The faculty have worked
hard to develop a cohesive
curriculum that begins in
the freshman year and in
cludes at least one core
chemical engineering 
course each semester. The
design experience is fully
integrated, beginning with
a projectoriented course in Horsetooth Reservoir, located
State campus, provides a we
the freshman year. Such at State campus, provides a we
tention to the curriculum
simply underlines an overall commitment to students as
persons and professionals. It permeates the whole program
at both the undergraduate and graduate levels.
GRADUATE
PROGRAM
The graduate students, who represent a number of nations
but are primarily American, appear to agree with the under
graduate consensus of an overall departmental commitment.
At a recent roundtable discussion attended by PhDtrack
students, the shared stories emphasized the same program
characteristics noted by the undergraduatessmall num
bers, personal contact, teaching ability of faculty, re
search opportunities, interdisciplinary approach, and real
life challenges.
One student specializing in bioremediation and molecular
biology commented, "The faculty seems determined to make
you step from graduate school into real lifeand do it suc
cessfully." Other students concurred, pointing out the nu
merous ways in which they are not only encouraged, but are
also forced, to stand on their own (research) feet. This in
cludes proposal writing, attending conferences, and present
ing papers, as well as weekly seminars in which they give
research updates to the faculty and regular interdisciplinary
Summer 1997
d af
alth
reviews of projects. As one student spontaneously shared,
"It keeps you alert about what you are doing!"
Asked to support their contention of an interdisciplinary
approach, seven students listed the following areas that they
themselves were incorporating into their chemical engineer
ing programs: microbiology, biochemistry, physics, chemis
try, plant science, and toxicology.
Colorado State alumni are now filling faculty and post
doctoral research positions in chemical engineering
throughout the U.S. and
abroad, including such
institutions as Stevens
Institute of Technology,
the University of
Wyoming, Michigan
State University, and Cal
Tech. Graduates also
are easily found in
companies such as Intel,
HewlettPackard,
Sandoz, Internation
al Paper, Genentech,
HoffmanLaRoche, J.D.
Searle, and Exxon.
PARADIGM
ew miles west of the Colorado OF REALITY
of recreational opportunities.
No one denies how
young the chemical en
gineering program at Colorado State is. In fact, its youth is
the very factor that has allowed it to do what more estab
lished programs often cannotchange the boundaries, alter
the rules, shift the paradigm.
As Harper notes, "We created a program out of dust into
something substantial that is recognized today both nation
ally and internationally. We crossed disciplines to create a
new model, one more reflective of reality . maybe you
could call it a paradigm of reality."
McConica expands, "Not only were we created out of
dust, but also outside the boundaries. With the advent of
emerging technologies and the collaboration of industry and
academia. we were already out there, waiting. You could say
that the paradigm engulted us."
Joel A. Barker, quoted at the opening of this article, con
tends in his book Paradigms: The Business of Discovering
the Future (Harper, 1992) that with any paradigm shift, a
new game begins with a new set of rules. The way to mea
sure your ability to be successful, he maintains, is by your
ability to solve problems.
By any measure, chemical engineering at Colorado State
is not only in the game, but it also continues to make new
rules. 1
Learning
TOWARD
TECHNICAL UNDERSTANDING*
Part 1. Brain Structure and Function
J.M. HAILE
Clemson University Clemson, SC 296340909
One must acquire many different ways to understand.
Minsky'"
4 'X ou know, Prof, my grades on the quizzes don't
really reflect my understanding of the material."
I ... "When you talk to Clarence about the mate
rial, it's evident he understands a lot about it, so why can't
he do the homework?" ... "I know the class understands this
concept, we've been through it many times, so why can't
they apply it when they need it? Why can't students access
what they know?"
These kinds of comments, from students and colleagues,
are familiar to any of us who have spent time in education.
They signal frustration in various guises, and often a voiced
frustration is but a symptom of deeper dissatisfaction and
perplexity. In pondering such comments, I've concluded
that many of them spring from a common basis: confusion
and misconception about what we mean by an understanding
of technical material. Such confusion should not be dis
missed lightly, for it can hamper our attempts to help others
learn; and so it seems worthwhile to try to clarify what it
means to understand. But to unravel such confusion is no
small task. The word understanding is itself obscured by a
J.M. Haile, a professor of chemical engineering at Clemson University,
is the author of Molecular Simulation, Published by John Wiley & Sons
in 1992.
* EDITOR'S NOTE: This is the first of three installments. The
second installment, "Elementary Levels," will appear in our Fall
1997 issue, and the third installment, "Advanced Levels," will be
published in our Winter 1998 issue.
vagueness that approaches the enigmatic. For once, Webster
fails us, merely offering as synonyms "perception," "com
prehension," "appreciation," and "mental grasp." These move
us no closer to the root of the matter.
The ambiguity arises because there is not one understand
ing, or even just a few. There are, in fact, manymany
kinds of understanding and many ways to reach them. It is
one thing to recognize you have a problem, another thing to
articulate the problem, yet another to identify what is needed
to solve it, still another to carry out the solution, and even
another to appreciate what the solution means. Given those
many goals and the many paths to each, it is no wonder we
have difficulty articulating general rulesor even rules of
thumbthat will consistently lead us to an understanding.
But what is difficult in general may be manageable in par
ticular. Perhaps by restricting our attention to particular realms
of knowledge, such as those embodied by engineering and
the physical sciences, we can clarify what it means to under
standat least for those restricted realms. That is the thesis
for the papers in this series.
Having recognized that there are many ways to understand
technical material, we then ask how those ways can be
organized. One appealing organization is a hierarchy be
cause hierarchies identify levels, and this usage coincides
with commonly used, but illdefined, ideas concerning lev
els of understanding. In addition, a hierarchy provides a
systematic progression that can serve as the basis for helping
people learn. For example, a hierarchy of understanding can
Copyright ChE Division ofASEE 1997
Chemical Engineering Education
help us identify the current stage in a student's study of a
topic, it can help us show the student what must be done to
reach the next stage, and it can help us determine when a
transition between stages has been successful. This series
of papers is primarily concerned with presenting and dis
cussing a hierarchy for understanding technical material.
But the job of fostering understanding can also be clarified
if we know something about how people
learnthat is, how the human mind as
similates new information and integrates
it with old information. Over the past Since 1
ten years we have seen significant create
progress in neuroscience, especially in
neurobiology, psychology, and artificial StructuI
intelligence. As educators, we should brain by
take advantage of that progress, recog
nizing that the next ten years will bring existing ,
still more progress. By clarifying how learning
the brain functions, we can obtain clues
as to how to improve learning. We there begin fr(
fore will use the rest of this paper to the s1
review, in an elementary way, relevant
aspects of brain structure and function. already
These discussions will support the hier Thi
archy of understanding presented in the
second and third papers in this series. implicate
BRAIN whether
STRUCTURE should
AND FUNCTION top
topi
The human brain is not a single entity,
but rather a composite of several brains. (deduct
The top of the spinal cord forms the bottle
medulla, which supervises basic motor
functions, including heart beat, respira inducec
tion, and digestion. Behind the medulla
lies the cerebellum, which coordinates
body position, movement, and balance. Above the cerebel
lum we find the limbic system, which includes the pituitary
gland, the hypothalamus, the hippocampus, and other struc
tures. The pituitary makes hormones that control the func
tion of most other glands in the body; its action is controlled
by the hypothalamus. More generally, the hypothalamus
regulates all lifesupport functions, including heart rate, body
temperature, chemical balances in the body, hunger, thirst,
and emotional responses to threats for survival. The hippoc
ampus apparently participates in the formation of longterm
memories; this will be discussed in the third of this series of
papers. Atop the limbic systems sits the cerebrum, which is
devoted to all higher mental activities, including language,
conscious awareness, and abstract thought.
Of all these structures, the cerebrum is by far the largest,
yet most highlevel mental activities are confined to its sur
Summer 1997
facethe cerebral cortexa layer only 2mm thick and con
voluted into folds to increase its surface area within a con
fined volume. The cortex contains a significant portion of
the brain's gray matterthe little gray cells so favored by
Hercule Poirot. The volume enclosed by the cerebral cortex
is filled largely with white matter: the strands and fila
ments that connect brain cells. That is, much of the hu
man brain is mere wiring.
The following sections of this paper
present an elementary overview of the
rning functioning of the neuron and of the huge
iew number of neurons that form the cerebral
cortex. These functions allow us to draw
in the certain conclusions about the nature of
edifying learning. For a more detailed introduction
to brain structure and function, see refer
Ictures, ences 26; the illustrations by Macaulay'"l
In Only are particularly instructive.
things THE NEURON
ent The basic unit of mental activity is a
noWS, single nerve cellthe neuron. Function
ally, a neuron in the cortex collects signals
as from other neurons, integrates them into a
is as to single signal, and then either suppresses
the signal or forwards it to other neurons.
tOpic Structurally, a neuron is composed of three
taught principal parts: a cell body, which con
tains the nucleus and performs the life
wn support functions common to any biologi
ly) or cal cell; a treelike array of branches called
dendrites that carry signals from other neu
'UP rons to the cell body; and an axon, a single
ely). strand that carries the signal from the cell
body to other neurons.
More generally, neurons are the primary
functional elements of the nervous system; for example, a
nerve is a bundle of axons. Axons vary in length from
millimeters in the cortex to about a meter in the case of the
axons that connect the toes to the spinal cord. Variations in
geometry provide a means for classifying neurons by struc
ture;'1 those in the cerebral cortex are called pyramidal
neurons because of the distinctive shape of the cell body.
The function of an individual neuron is illustrated sche
matically in Figure 1 (next page). From contacts with other
neurons, a dendrite carries a signal, as a voltage difference,
to the cell body. At the cell body, signals from all dendrites
are combined into a single voltage that propagates to the
head of the axon. If this output voltage exceeds a certain
threshold, the neuron is said to fire, and a pulse voltage
propagates down the axon. The end of the axon divides into
branches, providing hundreds of terminals to other neurons.
eal
es 1
*es
mo
stru
ca
Dm
tud
y ki
she
ion
ra
be i
doi
ive
Miv
tiv
Each axonal terminal is separated from another neuron by a
microscopic gap called a synapse. When a voltage pulse
reaches an axonal terminal, it causes vesicles in the terminal
region to fuse onto the wall (the presynaptic membrane) of
the neural cell. This, in turn, causes the vesicles to open,
releasing a few thousand molecules of a neurotransmitter
into the synapse. These molecules diffuse across the synapse
to a dendrite or cell body of another neuron. If the neu
rotransmitter can find an appropriate receptora pro
teinembedded in the postsynaptic mem
brane, then the signal is successfully passed
from one neuron to the other.
The voltage difference propagating from den
drite through cell body and down an axon is not
carried electrically, but chemically; that is, it is
not carried by free electrons but by sodium ions.
Therefore, the speed of signal propagation is of
the order of milliseconds, which is slow relative to
the speed of electrical conduction in metal wires.
Further, an axon does not conduct the signal
by a current propagating axially, as in a wire.
Instead, a local voltage difference in one part of
an axon relative to a neighboring part induces
opening and closing of molecular gates on chan
nels within the cell membrane; these channels
allow flow of ions between the interior and exte
rior of the neuron, changing the voltage in one
region of the neuron relative to an adjacent
region. Thus, sequential radial flow of ions
through cell walls produces the effect of a
voltage propagating axially.
The activity induced by the voltage reaching a
synapse amounts to a keylockgate scenario. If
a neurotransmitter (the key) can find its recep
tor (the lock), then a molecular gate opens,
allowing ions to enter the dendrite, creating a
local voltage difference.
A few dozen neurotransmitters have been iden
tified, and more probably remain to be discov
ered."11 The common ones include glutamate,
dopamine, acetylcholine, and yaminobutyric
acid (GABA). Certain drugs, including the opi
ates, nicotine, and the antipsychotics, are known
to either mimic or block the actions of certain
neurotransmitters.151 This is possible because
neurotransmitters activate receptors by match
ing physical structures, so any molecular frag
ment that matches the receptor structure might
activate that receptor; not only will a key open a
lock, but so too will a skeleton key.
Synaptic connections are of two general types.
Excitatory synapses tend to promote firing of
impulse
(
the neuron by activating receptors that allow sodium ions to
enter the neuron through the postsynaptic membrane. These
connections typically occur on dendritic branches, with the
common excitatory neurotransmitter being glutamate. In con
trast, inhibitory synapses tend to suppress firing by acti
vating receptors that allow chlorine ions to enter through
the postsynaptic membrane. These typically occur di
rectly on the cell body, with the common inhibitory neu
rotransmitter being GABA.
dendrites cell body
( axor
Impulse Propagation Synaptic Transmission
Na+ Na+
from  
cell + +
body 
Na+1 Na
 impulse
axon /\ c.
vesicle
neurotransmitter (*)
synaptic gap
receptors
Na+ Na+
from I \ to
cell + + synapse
body Na
Na+ Na+
dendrite
Figure 1. (top) Principal parts of a single neuron, including synaptic connection to
the dendrite of another neuron. (lower left) Exploded view of an axonal segment;
radial flow of sodium ions between axon and extracellular fluid propagates a
nerve impulse from the cell body to the synapse. (lower right) Exploded view of an
axonal terminal and synapse. At the terminal, a nerve impulse stimulates vesicles
to fuse with the presynaptic membrane, releasing neurotransmitters into the syn
aptic gap. The bottom views are adaptations of drawings by Macaulay.1 '
Chemical Engineering Education
THE NEURAL NETWORK
Although the functioning of a single neuron is a fascinat
ing electrochemical process, the really astonishing functions
occur not at the molecular or cellular levels, but in the
collective behavior of large numbers of connected neurons.
The number of neurons in the human brain is estimated at
between 20 and 100 billion. Further, the average pyramidal
neuron makes roughly 1000 connections to other neurons; so
the total number of connections may well be about 1014.13'61
Most connections are among neighboring neurons, but many
axons connect neurons that lie in very different regions of
the cortex. In principle, any neuron can influence the
firing of any other neuron.
By itself, the firing of a single neuron is essentially mean
ingless. Meaning only arises when a pattern is established by
the simultaneous or sequential firings of many neurons. We
do not know how firing patterns encode meanings (that is
part of the puzzle), but the following metaphor may capture
Figure 2. Schematic representation of sequences of neural firings in t
Grid represents an array of neurons; path represents the firing seque
represent repeated firings of patterns of neurons that have meaning, su
a sight or formation of a coherent thought. Path segments between sha
the search for meaning. In studies of dynamic systems, such as in p
statistical mechanics, the grid is interpreted as a phase space and the
of the system. Then the shaded regions are local attractors that order
patterns. Based on a figure in Calvin171 and a phasespace plot in Freem
taken from a rat's olfactory bulb.
Sununer 1997
the essence. Imagine an array of lights forming a scoreboard.
The array itself has no meaning; if none of the lights are
activated, we have no meaning, and if all of the lights are
activated, we still have no meaning. Informative meaning
occurs only when some lights are activated while others are
not. Moreover, the meaning is in the pattern, not in any
particular lights; that is, meaning is encoded in the spatial
and temporal relations among the lights that are activated
and those that are not. For example, meaning is preserved
even when the pattern scrolls across the array. Now reread
the previous four sentences, everywhere replacing "light"
with "neuron" and "array" with "neural net." This metaphor
suggests one reason for having both excitatory and inhibi
tory synapses for in this way, not only can any one
neuron participate in any pattern, but in addition, when
the same pattern is replayed at different times, a neuron
can participate by sometimes being activated and other
times being quiescent.
Neural activity in the cerebral cortex distinguishes brain
from mind; that is, brain is the struc
ture and mind is the function. As
Minsky has written,'" "Minds are what
brains do." But what is it that minds
do? In particular, what does the cere
bral cortex do?
There is always at least a baseline of
neural activity in the cortex. It can be
ne.ual grid seen crudely on an electroencephalogram
(EEG). But that minimal activity can be
driven to more active modes by stimula
tion, either from the external world
through the senses or from the internal
World through other parts of the brain.
The response appears to be a search for
meaningan attempt to find a pattern
that interprets or makes sense of the
stimulus. In other words, the baseline
firing of a huge number of interconnected
neurons amounts to a chaotic dynam
icsnot random, but apparently random
with some underlying order.'791 Such dy
namics are, by definition, sensitive to
small disturbances, so even a small stimu
lation of the cortex can produce a quali
tative change in the character of the fir
ing pattern. Some changes in the dynam
he cerebral cortex. ics take the form of convergence to a
nce. Shaded areas
as recognition of local attractora firing pattern that is
ch as recognition of
led areas represent recreated and sustained whenever the fir
rocess control and ing trajectory passes sufficiently "close"
path is a trajectory to neurons that activate the pattern. Such
Sthe trajectory into
an'i from EEG data attractors constitute meaning to the or
ganism. This is illustrated in Figure 2.
search for
meaning
ghost of
/ an idea
coherent
thoughts
We may become conscious of attractorsthat is, con
scious of thought patternswhen we encounter ambiguity.
An ambiguous stimulus causes neural firing patterns to bi
furcate into conflicts or competition between two or more
attractors; the result is mental confusion. In such situa
tions, the mind contrives more than one pattern that is
consistent with the data, and the conflict can only be
resolved with more data.
For example, consider the object shown in Figure 3, which
presents a conflict between foreground and background. If
you focus to bring the shading to the foreground of the
figure, you see the letter E. But if you shift your focus
slightly, the shading can be pushed to the background, and
you see the characters L and 3.
Your recognition of the E is an attractor produced by one
assembly of neurons, while recognition of the L and the 3
is a second attractor produced by another assembly of
neurons. Both attractors are consistent with the data and
additional visual cues would be needed for one attractor
to dominate.
The interpretation of thought as a dynamic process driven
to local attractors, as shown schematically in Figure 2, is
appealing, but it is likely an oversimplification for at least a
couple of reasons. First, the coding of meaningful patterns is
probably not just in the relative positions
of firing neurons; it may also involve
firing rates and sequences. That is, mean
ing may involve both positional codes
and temporal codes.1'"0 Second, the ac
tion of the cortex appears to involve dis
tributed processes, in which multiple sub
processes are performed simulta
neously."'"" For example, visual recog
nition of an object involves perception
of contours, depth, and colorthree ac
tivities that are performed simultaneously Figure 3. M
but in separate regions of the cortex. sion can ari
brain devise
More complex functions appear to flicting patte
progress through hierarchies of distributed same inform
processes, which may explain why we like figure can b1
to use hierarchies for organizing societies, because it d
institutions, and problemsolving tasks.111 vide enough
Distributed processes make efficient use of low one to
neural networks because the same assem foreground
blies of neurons can be used for the same ground. FocL
the shaded
kinds of tasks in different situations. foreg nd
foreground
Although we do not yet know the details character, bu
for how meanings are assigned to patterns, push the s,
we at least know what is being done: minds into the ba
veals two di
are what brains do, and the search for mean
acters.
ing is what the cortex does.
MODIFICATION OF
BRAIN STRUCTURE: LEARNING
The association of meanings with particular stimuli con
stitutes one aspect of learning. For example, if the stimulus
is a right triangle, then part of the associated meaning would
be the Pythagorean theorem. But meaning is connected to a
stimulus through neural firing patterns created in the cere
bral cortex, so to learn new meanings, we must create new
firing patterns. For these new patterns to be accessible over
long times, the mind must bias connections among neurons
so that the new pattern is recreated whenever an appropriate
stimulus is encountered. That is, to make longlasting changes
in function (the mind), we must make changes in structure
(the brain).
We do not yet know much about how learning modifies
the brain, though many observations are suggestive and the
more obvious possibilities are listed below. But we do know
one mechanism that is not usedthe brain does not modify
itself by growing new neurons. The number of neurons is
constant through adolescence, and then neurons begin to
degenerate and die over the remaining lifetime. Estimates
usually put the average total loss at 10% of the original
number.131 During youth, the brain grows in complexity by
forming and pruning dendritic trees and axonal branches,
that is, by increasing and refining connections among exist
ing neurons.1361
Here are some ways by which learning may
change brain structure and function:'51
Changes in the size and shape of axon
terminals coupled with changes in the
number of presynaptic vesicles (which
hold the neurotransmitter) to increase or
decrease the amounts of neurotransmitter
released
Changes in the size and shape of postsyn
1 confu aptic receptors and channels to change
'hen the the level of activation voltages created
wo con
from the when a receptor opens
n. This Increases and decreases in the number of
'biguous receptors at certain synapses
not pro
ns to al Changes in the number and location of
s to al
tinguish synapses
n back Sprouting of new axonal terminals
ito bring Remolding of terminal bulbs on dendrites.
in to the
'als one
using to IMPLICATIONS FOR EDUCATION
ed area The brain, then, is a selfmodifying neural
iund re network. The processes carried out by that net
nt char work constitute the mind, and the function of a
part of that mindthe cerebral cortexis to
Chemical Engineering Education
enta
se w
es ti
mrns
nation
e am
oes
Scum
dist
fror
sing
irea
reve
itfoc
hade
*kgrc
ffere
ascribe meanings or interpretations to both external and in
ternal stimuli. These observations lead to certain implica
tions about the nature of education, including
> Learning is a natural activity of the human mind.
Learning is not storage and retrieval of information; the
brain does not store information.18'"l It only develops a
propensity to reproduce neural firing patterns that have
been found beneficial; that is, what we call memory is
actually a recreation of information. To be able to use
what they know, students must learn cues that recreate
useful patterns.
Since learning creates new structures in the brain by
modifying existing structures, learning can only begin
from things the student already knows. This has impli
cations as to whether a topic should be taught topdown
deductivelyy) or bottomup (inductively).
The brain apparently modifies neural connections as
part of its response in those neurons that are activated to
form a pattern. Thus, learning new things amounts to a
perturbation of things already known; but if the pertur
bation is too large, then no related neural firing pattern
can be created and no learning takes place. Thus, stu
dents must be led to new knowledge in small chunks of
information that allow the brain to modify existing neu
ral networks. Repetition is then needed to strengthen
new neural connections. The importance of repetition is
addressed in the third of this series of papers.
> Experts in a topic have highly interconnected constella
tions of neurons that can be activated by stimulating any
of many different nodes.1"l These elaborate networks
allow experts to quickly learn new things because their
vast networks offer numerous nodes that can be easily
modified to assimilate new information. In contrast,
students generally have few networks related to techni
cal material; the networks they do have tend to be mea
ger and largely fragmented. The assimilation of new
information into those networks is often a laborious task
because a small addition to a small structure can require
a large change in the structure.
We often witness instances at which disjointed neu
ral assemblies finally become fused into a coherent
network. It happens to those students who struggle with
a topic for several weeks, laboriously piecing together
several disjoined networks. Then, about midterm or
shortly thereafter, one more piece of new information
perturbs the entire system sufficiently that those several
disjointed networks become unitedrevelation! The stu
dent understands.
Recognition is easier than recall."'71 Recognition forms
a meaningful pattern in response to an external stimu
lus, while recall forms a meaningful pattern in response
to an internal stimulus. To test recognition, we might
pose a question such as "What quantity is defined by
Re=upd/pT?" But to test recall, we would ask, "What
equation defines the Reynolds number?" Recall is more
difficult because we must not only create the pattern,
but we must also generate the stimulus that produces the
pattern. Because of this difference in difficulty, students
generally prefer certain kinds of quizzes over others.
> Learning is easier than unlearning. Unlearning refers to
correcting misunderstandings from earlier learning. Dur
ing learning, we modify a neural network to create a
new net; but during unlearning, we not only create a
new network, but we must also suppress formation of
the old erroneous pattern.
To learn, students must actively participate in their own
education. Only the individual can modify its own syn
apses, dendritic trees, and axonal terminals. No instruc
tor can do this for the student.
N Quickness of mind (a commonly used indicator of intel
ligence) decouples from the ability to think."12J A quick
mind is one that moves immediately and decisively to a
local attractor. But the Latin root for intelligent is inter
+ legere, which means to select. And to select implies a
consideration of alternatives; that is, intelligent thinking
involves the identification of alternative attractors and
choosing from among them. This cannot necessarily be
done quickly.
REFERENCES
1. Minsky, M., The Society of Mind, Simon and Schuster, New
York, NY (1986)
2. Restak, R., The Brain, Bantam Books, New York, NY (1984)
3. Restak, R., Brainscapes, Hyperion, New York, NY (1995)
4. Calvin, W.H., and G.A. Ojemann, Conversations with Neil's
Brain, AddisonWesley, Reading, MA (1994)
5. Fischbach, G.D., "Mind and Brain," Sci. Am., 267(3), 48
(1992)
6. Ornstein, R., and R.F. Thompson, The Amazing Brain, il
lustrated by D. Macaulay, Houghton Mifflin, Boston, MA
(1984)
7. Calvin, W.H., The Cerebral Code, MIT Press, Cambridge,
MA (1996)
8. Freeman, W.J., "Role of Chaotic Dynamics in Neural
Plasicity," Progress in Brain Research, 102, 319 (1994)
9. Kauffman, S.A., The Origins of Order, Oxford University
Press, New York, NY (1993)
10. Singer, W., "Synchronization of Cortical Activity and Its
Putative Role in Information Processing and Learning,"
Ann. Rev. Physiol., 55, 349 (1993)
11. Searle, J., The Rediscovery of the Mind, MIT Press, Cam
bridge, MA (1992)
12. de Bono, E., de Bono's Thinking Course, Facts on File Publi
cations, New York, NY (1985) O
Summer 1997
classroom
A NEW APPROACH TO TEACHING
DIMENSIONAL ANALYSIS
STUART W. CHURCHILL
University of Pennsylvania Philadelphia, PA 19104
Rayleigh,1'l in the opening lines of a short note on the
subject of this article, wrote "I have long been im
pressed by the scanty attention paid by the original
workers in physics to the great principle of similitude. It
happens not infrequently that results in the form of 'laws'
are put forward as novelties on the basis of elaborate experi
ments, which might have been predicated a priori after a few
minutes consideration." The full power of dimensional analy
sis (which Rayleigh referred to as "similitude") has, in the
intervening eighty years continued to be underestimated and
underutilized by engineers as well as by physicists.
Dimensional analysis is most powerful when it is applied
to a complete mathematical model in algebraic (differential
and/or integral) form, but it is remarkably productive even
when a complete model is unknown or unwieldy and the
analysis must be applied to a simple list of the relevant
variables. The utility of dimensional analysis when applied
to either a model or a list of variables may often be greatly
enhanced by the collateral use of speculative and asymp
totic analyses. It is this combined use that first suggested
to the author the new educational approach described in
this paper in which dimensional analysis itself is inter
preted as a speculative process.
Undergraduate students in chemical engineering are usu
ally first exposed to dimensional analysis in a course in fluid
mechanics, transport phenomena, or unit operations. In that
first exposure, the application of dimensional analysis is
ordinarily and appropriately limited to a list of variables.
Expression of relationships and correlations in terms of di
mensionless groups is invariably implied in subsequent
courses. The application of dimensional analysis to a math
ematical model is usually illustrated in a higherlevel under
graduate course, but too superficially to impart a working
knowledge. The subject of dimensional analysis may or may
not be reintroduced and reinforced in the graduate program.
In this article, attention will be confined to the application of
dimensional analysis to a list of variables. A companion
Copyright ChE Division ofASEE 1997
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 ofMichi
gan, where he also taught from 19501967.
continued to teach and carry out research on
Since his formal retirement in 1990, he has
heat transfer and combustion.
article using the speculative approach in connection with
mathematical models is in preparation.
Dimensional analysis of a list of variables that define a
physical, chemical, or biological process is invaluable in
guiding the correlation of experimental data or numerically
computed values, but its contribution to understanding may
be even more important for students. The development of
skill and confidence in applying this methodology and in
interpreting the results should be recognized as an essential
element of undergraduate education in chemical engineer
ing. Unfortunately, the exposure of most undergraduates to
dimensional analysis in the previously mentioned context is
brief, incomplete, and unsatisfying. The students ordinarily
learn the mechanics of the algebraic method of dimensional
analysis and are usually convinced that they should always
express experimental results, either graphically or algebra
ically, in terms of dimensionless groups. They are, however,
invariably put off by the mysterious and seemingly arbitrary
choice of the appropriate dimensional variables. Further
more, they generally gain only a hazy idea of the signifi
cance, scope, and possible nonuniqueness of the results.
Since these uncertainties and misunderstandings are not ex
cised in subsequent courses, they graduate with insufficient
confidence in dimensional analysis, and in particular in the
choice of variables, to apply it to some entirely new problem
that they encounter in practice. As a consequence, they will
probably not use dimensional analysis in their entire career,
at least on their own initiative.
The commonly used textbooks on heat transfer and mass
Chemical Engineering Education
transfer, as well as on the subjects mentioned The utility
above, are almost all adequate in terms of analysis w
describing the mechanics of dimensional either a m
analysis, but as a rule are deficient in their variable,
discussion of alternative groupings, limiting greatly eI
cases, the significance of the groupings, and colla
most of all, the topic of greatest concern to
the studentsthe selection of the proper set specu
of variables. On the other hand, the mono asymptE
graphs and books on dimensional analysis, It is this
similitude, and related topics are too lengthy thatfirst s
and cover too wide a range of subject matter auth.
to be wedged into an undergraduate course education
such as fluid mechanics. In any event, these describe.
specialized books generally sidestep, by start in which
ing with a mathematical model, the problem analy
of selection of an appropriate set of variables, inter
If such a model is known, dimensional analy specula
sis is more effectively applied thereon in
stead of using the model merely to compile a
list of variables. The inclusion or omission of terms from a
model is analogous to the inclusion or omission of variables in
a list. The material presented in this paper is intended as a
compromise in the form of a brief supplement to the coverage
in the current elementary textbooks on fluid mechanics, etc.
The description herein of the process of dimensional analy
sis as applied to a list of variables differs only marginally
from the standard treatment, but that slight difference has
been found to have a great, positive impact on the compre
hension and understanding of this process by students. The
novelty of the exposition is in the presentation and interpre
tation of dimensional analysis as a speculative process. This
concept has three distinct but closely related aspects. First,
the mere designation of the process as speculative greatly
relieves student's frustration at the lack of guidance by their
teacher and their textbook in the selection of variables. They
suspect, quite justifiably, that this omission is not accidental,
but rather that it reflects a lack of knowledge in this respect
by both the teacher and the authors) of the textbook. The
term speculation implies that the choice is tentative and must
be tested with experimental data. This is an acceptable state
of affairs for the student. The choice by their teacher or the
authors) is now recognized as being based on experience
with the result of the analysis, not omniscience. They now
have, at least in principle, a criterion for evaluation of any
particular choice of variables.
The second aspect of speculation as applied to dimen
sional analysis consists of the methodical reduction of the
original, presumably complete, set of variables one at a time,
then two at a time, etc., in the hope of determining useful
asymptotic or limiting relationships. Of course, the range of
validity, if any, of these reduced representations must also
Summer 1997
Sof dimensional
vhen applied to
odel or a list of
s may often be
Ihanced by the
,eral use of
native and
otic analyses.
combined use
suggested to the
or the new
nal approach
d in this paper
dimensional
sis itself is
reted as a
tive process.
I
be tested experimentally or by comparison
with the theoretical solutions that may be
possible for the simplified cases. Most of the
reduced expressions will not be found to
have any physical validity, but some may
prove to be useful and enlightening. As a
generic example, elimination of the density
of the fluid may result in a valid expression
for the limiting case of purely viscous flow,
while elimination of the viscosity may result
in a valid expression for the other extreme of
purely inertial flow. For flow through a cir
cular sharpedged orifice, both of these lim
iting expressions have physical validity as
confirmed by experiments and theory,
whereas for flow through a smooth round
pipe the result for purely viscous flow has
validity but not that for purely inertial flow,
as evidenced by the everdecreasing value of
the friction factor for a smooth pipe as the
Reynolds number increases.
A third aspect of speculation as applied to dimensional
analysis is associated with the inverse process, namely the
determination of the variable or variables that must be elimi
nated from a listing in order to obtain a known experimental
or theoretical result. This procedure, which has seldom been
employed, often provides unexpected insights. As an ex
ample, consider the identification of those variables that lead
by dimensional analysis to the Reynolds analogy.
The above introductory and expository section is intended
primarily for teachers, while the details that follow are in
tended primarily for students. For that reason, some limited
repetition has been considered necessary.
SPECULATIVE DIMENSIONAL ANALYSIS
The following description and examples of dimensional
analysis differ from previous ones in the literature by virtue
of the treatment of this process as a speculative one. In
particular, the choice of variables is considered to be tenta
tive and subject to experimental confirmation or refutation.
General Principles
Dimensional analysis is based on the simple principle that
all additive or equated terms of a complete relationship,
whether known or unknown, between the variables must
have the same net dimensions. As described herein, dimen
sional analysis starts with the preparation of a list of the
individual dimensional variables (dependent, independent,
and parametric) that are presumed to define the behavior of
interest. As will be shown, the performance of dimensional
analysis in this context is reasonably simple and straightfor
ward; the principal difficulties and uncertainties arise from
159
identification of the variables to be included in or excluded
from the listing. If one or more important variables are
inadvertently omitted, the reduced description achieved by
dimensional analysis will be incomplete and inadequate as a
guide for the correlation of a full range of experimental data
or computed values. The familiar band of plotted values in
many graphical correlations is more often a consequence of
the omission of one or more variables than of inaccurate
measurements. If, on the other hand, one or more irrelevant
or unimportant variables are included in the listing, the con
sequently reduced description achieved by dimensional analy
sis will result in one or more unessential dimensionless
groups. Such excessive dimensionless groups are generally
less troublesome than missing ones because the redundancy
will ordinarily be revealed by the process of correlation.
Excessive groups may, however, suggest unnecessary ex
perimental work or computations, or result in misleading
correlations. For example, real experimental scatter may
inadvertently and incorrectly be correlated in all or in part
with the variance of the excessive grouping.
The selection of a necessary and sufficient set of dimen
sional variables may require knowledge gained from experi
ence, and hence is particularly difficult and uncertain when
dealing with a new or unfamiliar aspect of behavior. A
complete and certain mathematical model is an ideal
source for identification of the variables, but in that event
the analysis might more profitably be applied to the model
itself rather than to a listing, even though the latter is
exact. Of course, if significant terms are inadvertently
omitted from the model or if unnecessary terms are in
cluded, a list of variables based on the model may also be
incomplete or redundant.
In consideration of the inherent uncertainty in selecting
the appropriate variables for dimensional analysis, it is rec
ommended that this process always be interpreted as specu
lative, that is, as tentative and subject to correction on the
basis of experimental data or other information. Speculation
may also be used as a formal technique to identify the effect
of eliminating a variable or of combining two or more. The
latter aspect of speculation, which may be applied either to
the original listing of dimensional variables or to the result
ing set of dimensionless groups, is often of great utility in
identifying possible limiting behavior or dimensionless groups
of marginal significance.
Speculation in this context is to be distinguished from
conjecture, in which a decision (here the inclusion or omis
sion of a variable) is based on some mechanistic rationale.
The failure of a speculation carries no burden of explanation
or guilt while that of conjecture may.
The systematic speculative elimination of all but the most
certain variables, one at a time, two at a time, etc., followed
by regrouping, is recommended as a general practice. The
additional effort as compared with the original dimensional
160
analysis is minimal, but the possible return is very high. A
general discussion of this process is presented by Churchill[21
and further illustrations are provided by Churchill1341 as well
as by the examples of dimensional analysis that follow herein.
Determination of the Minimum Number
of Required Independent Dimensionless Groups
The minimum number of independent dimensionless
groups, i, that are required to describe the fundamental and
parametric behavior was stated by Buckingham"51 to be
i=nm (1)
where n is the number of variables and m is the number of
fundamental dimensions such as mass M, length L, time 6,
and temperature T that are introduced by the variables. The
inclusion of redundant dimensions such as force F and en
ergy E that may be expressed in terms of mass, length, time,
and temperature is at the expense of added complexity and is
to be avoided. (Of course, mass could be replaced by force
or temperature by energy as alternative fundamental dimen
sions.)
Van Driest[61 has since shown that in some rare cases, i is
actually greater than nm and that a more exact expression is
i=n k (2)
where k is the maximum number of the chosen variables that
cannot be combined to form a dimensionless group.
On the other hand, Kline17' has shown that an excessive
number of dimensionless groups may be predicted by Eqs.
(2) and (3) if necessary relationships between some of the
variables are not recognized. As an example, an excessive
number of dimensionless groups is predicted for the descrip
tion of the performance of a doublepipe heat exchanger if
the two mass rates of flow, w, and w2, and the corresponding
specific heat capacities, c, and c2, are all four considered to
be independent variables. The recognition that these vari
ables enter the description of the process only in the form of
the products wc, and w2c2 leads to the correct prediction.
Equations (1) and (2) are analogous to the Gibbs Phase
Rule in one respectthey appear to be very simple, but
actually are quite difficult to apply because of the uncer
tainty in determining the components of the equation, in this
instance the number of variables n.
Determination of the minimum number of dimensionless
groups is helpful if the groups are to be chosen by inspec
tion, but is unessential if the algebraic procedure described
below is used to determine the groups, since the number is
then obvious from the final result.
Form of the Results of Dimensional Analysis
The particular minimal set of dimensionless groups is
arbitrary in the sense that two or more of the groups may be
multiplied together to any positive, negative, or fractional
Chemical Engineering Education
power as long as the number of independent groups is
unchanged. For example, if the result of a dimensional
analysis is
P{X.Y.Z}= 0 (3)
where X, Y, and Z are independent dimensionless groups, an
equally valid expression is
XY 2,Z / Y, Z =0 (4)
Dimensional analysis itself does not provide any insight as
to the best choice of equivalent dimensionless groupings,
such as between those of Eqs. (3) and (4). But isolation of
each of the variables that are presumed to be the most impor
tant in a separate group may be convenient in terms of
interpretation and correlation. Another possible criterion
in choosing between alternative groupings may be the
relative invariance of a particular one as demonstrated by
experimental data.
The functional relationship provided by Eq. (3) may equally
well be expressed as
X = {Y.Z} (5)
where X is now implied to be the dependent grouping and Y
and Z to be independent or parametric groupings.
Methods of Dimensional Analysis
Three primary methods of determining a minimal set of
dimensionless variables will be described: 1) by inspection;
2) by combination of the residual variables, one at a time,
with a set of chosen variables that cannot be combined to
obtain a dimensionless group; and 3) by an algebraic proce
dure. It is more expeditious to illustrate these three methods
for a number of specific examples than to describe them in
general terms.
ILLUSTRATIVE EXAMPLES
Example 1
Fully Developed Flow of Water through a
Smooth Round Pipe.
Choice of variables The shear stress ,, on the wall of
the pipe may be postulated to be a function of the density p
and the dynamic viscosity p of the water, the inside diam
eter D of the pipe, and the spacemean u,,, of the timemean
velocity. The limitation to fully developed flow is equivalent
to a postulate of independence from distance x in the direc
tion of flow, and the specification of a smooth pipe is equiva
lent to the postulate of independence from the roughness e of
the wall. The choice of T, rather than the pressure drop per
unit length dP/dx avoids the need to include the accelera
tion due to gravity g and the elevation z as variables. The
choice of um rather than the volumetric rate of flow V, the
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mass rate of flow w, or the mass rate of flow per unit area G,
is arbitrary, but has some important consequences as noted
below. This speculatively postulated dependence may be
expressed functionally as
{T w, p., D.u, }= 0 (6)
or
Tw=0{p,..D,un} (7)
Tabulation It is convenient as a first step in all instances
and with all procedures to prepare a tabular listing of the
variables and their dimensions. In this example, that tabula
tion takes the following form:
T, p p D u,,
M 1 1 1 0 0
L 1 3 1 1 1
0 2 0 1 0 1
T 0 0 0 0 0
Minimal Number of Groups The number of postulated
variables is 5. Since the temperature does not occur as a
dimension for any of the variables, the number of fundamen
tal dimensions is 3. From Eq. (1), the minimal number of
dimensionless groups is 5 3 = 2. From inspection of the
above tabulation, a dimensionless group cannot be formed
from as many as three variables such as D, V, and p. Hence,
Eq. (2) also indicates that i = 5 3 = 2.
Method of Inspection By inspection of the tabulation or
by trial and error it is evident that only two independent
dimensionless groups may be formed. One such set is
0 T, Dup 0 (8)
_ = 0 (8)
Method of Combination The residual variables T, and
p may be combined in turn with the noncombining variables
p, D, and u,,, to obtain two groups such as those of Eq. (8).
Algebraic Method The algebraic method makes formal
use of the postulate that the functional relationship between
the variables may in general be represented by a power
series. In this example, such a power series may be ex
pressed as
N
= Aip b DC um, (9)
where the coefficients A, are dimensionless. Each additive
term on the righthand side of Eq. (9) must have the same net
dimensions as T Hence, for the purposes of dimensional
analysis, only the first term need be considered and the
indices may be dropped. The resulting highly restricted ex
pression is
T, = Ap b Dc Um (10'
Substituting the dimensions for the variables in Eq. (10)
gives
M aM ar b d(L
A LL} L Lc) (11
Equating the sum of the exponents of M, L, and 0 on the
righthand side of Eq. (11) with those of the lefthand side
produces the following three simultaneous linear algebraic
equations:
l=a+b (12)
1=3ab+c+d (13)
2= bd (14)
These three equations may be solved for a, c, and d in terms
of b to obtain
a =1b (15)
c = b (16)
d=2b (17
Substituting in Eq. (10) from Eqs. (15) through (17) gives
T, =Aplb b b 2b (18)
Equation (18) may be regrouped as
=b A t (19)
pum Dump
Since the simplification of Eq. (9) to Eq. (10) was for the
purpose of dimensional analysis only, Eq. (19) should not be
interpreted as implying that / pu2 is necessarily propor
tional to some power of g./Dump. Rather, Eq. (19) should be
inferred to be equivalent to Eq. (8) in every respect. The
misinterpretation of Eq. (19) and its analogs for other pro
cesses as implying a powerdependence between the dimen
sionless groups is the most common and serious error in
applying dimensional analysis.
Speculative Reductions Eliminating p as a variable in
Eqs. (6) and (7) on speculative grounds leads by each of the
above three methods of dimensional analysis to
1wD = 0 (20)
or its equivalent
T =D A (21)
,um
where A is a dimensionless number. Equation (21) with
A=8 is actually the exact solution for the laminar regime
Dump/ g < 1800. A relationship that does not include p may
alternatively be derived directly from Eq. (8) as follows:
first, p is eliminated from one group, say /pu 2, by multi
plying it with DumP/ to obtain
SwD Dump =0 (22
[hum 9 J
The remaining group containing p is then simply dropped,
162
resulting in Eq. (20). Had Eq. (8) been composed of three
independent groups, each containing p, that variable
would have to be eliminated from two of them before
) dropping the third one.
The relationships that are obtained by the speculative elimi
nation of i, D, and ur, one at a time, do not appear, on the
basis of experimental data, to have any range of physical
validity. Furthermore, if w or G had been chosen as the
independent variable rather than un, the limiting relationship
for the laminar regime would not have been obtained by the
elimination of p.
The postulated attainment of a state of fully developed
flow as the length of the pipe increases indefinitely might be
considered to be speculative. The inclusion of the distance x
from the inlet as a variable would result in the appearance of
another dimensionless group such as x/D in Eq. (8). The
dependence, if any, on this group and hence on x would then
S need to be tested experimentally.
Alternative Forms Equation (8) may also be expressed
in an infinity of other forms such as Eq. (22) and
,TwD2p Dup (23
) If T, is considered to be the principal dependent variable,
and um the principal independent variable, Eq. (23) is prefer
able to Eqs. (8) and (22) in that these two quantities do not
then appear in the same grouping. On the other hand, if D is
considered to be the principal independent variable, Eq. (8)
is preferable to Eqs. (22) and (23). The variance of w, /pu2
is known on the basis of experimental data to be less than
that of TwD/gium, and TD2 /g2 in the turbulent regime,
while that of TwD/gpu, is known from a theoretical solution
to be zero in the laminar regime. Such considerations may be
important in devising convenient graphical correlations.181
Alternative Notations Equations (8), (22), and (23) are
more commonly expressed as
{f, Re} =0 (24)
)(4
,f ReRe= 0 (25)
SfRe2 Re= 0 (26)
where f=2r, /pu2 is the Fanning friction factor and
Re=Dump/g is the Reynolds number. The more detailed
form of Eqs. (8), (22), and (23) is, however, to be pre
ferred for purposes of interpretation or correlation be
cause of the explicit appearance of the individual, physi
cally measurable variables.
Addition of a Variable The above results may readily be
extended to incorporate the roughness e of the pipe as a
variable. If two variables have the same dimensions, they
Chemical Engineering Education
will always appear as a dimensionless group in the form of a
ratio. In this case, e appears most simply as e/D. Thus, for a
rough pipe, Eq. (8) becomes
,,w Dump e}=0 (27)
9 2 = 0 (27)
Surprisingly, as contrasted with Eq. (22), the speculative
elimination of u and hence of the group Dump/l from Eq.
(27) is found on experimental grounds to result in a valid
asymptote for (DumPp/)oO and all finite values of e/D,
namely
{ 1 =0o (28)
Although it is not apparent from dimensional analysis, Eq.
(21) is found experimentally to remain valid for the laminar
regime for e/D<
Example 2
Velocity Distribution in Fully Developed Flow
of Water in a Smooth Round Tube
This example is closely related to the prior one, but their
interrelationship is instructive. The description of the process
of solution is abbreviated from that above as appropriate.
The timemean local velocity u might be postulated to be a
function of y, the radial distance from the wall, as well as of
Tz, p, pi, D, and ur. In Example 1, however, r, was
postulated to be a function of p, i, D, and um. Hence, one of
these variables is redundant. The most reasonable candidate
for elimination on this basis is u, since it may be determined
by integrating u over the crosssection of the pipe. Hence,
the following relationship will be used as an improved specu
lative starting point:
u=0( {T,p,uI,D,y} (29)
The corresponding tabulation is
u z, p D
0 1 1 1 0
1 1 3 1 1
1 2 0 1 0
The number of variables is six, the number of fundamental
dimensions is three, and the maximum number of variables,
such as g, p, and D, that cannot be combined to obtain a
dimensionless group is three. Hence from both Eqs. (1) and
(2), the minimal number of independent dimensionless groups
is three.
Since y and D have the same dimensions, the ratio y/D
will constitute one dimensionless group and y may be ex
cluded from the continuing process of analysis. By inspec
tion or by reference to Eq. (8), the remaining variables may
Summer 1997
be grouped and then combined with y/D to obtain
p 2 Dup y
2 y = 0 ( 3 0 ',
On the other hand, the residual variables u and T, may be
combined, one at a time, with p,g, and D to obtain the first
two groups of
uDpT = 0 (31]
Obviously, the first two groups in Eq. (31) may be combined
with one another to obtain Eq. (30) or vice versa.
With y withheld from the algebraic process, the analog of
Eq. (10) is
u=Apa b DC zd (32:
Substituting the dimensions in Eq. (32), equating the expo
nents of M, L, and e and then solving the resulting set of
simultaneous equations for a, b, and c in terms of d leads to
uDp (twD2pp)d
Dp= A d (33:
which, with the inclusion of y/D is the equivalent of Eq.
(31).
The following three rearrangements of Eqs. (30) and (31)
prove to be convenient in considering speculative limiting
cases:
u( p2 ,i y(T, p)2 D(,wp)w2
w)P P.
up y(Cwp)2 y
up y(T w)2
TwD D
Speculative Reductions The principal supplemental value
of Example 2 with respect to Example 1 is in the demonstra
tion of the great usefulness of speculative reductions. For
example, eliminating D reduces Eq. (34) to
2( {y(WP)2 (37)
which is found experimentally to provide such a good ap
proximation near the wall (y<0.2D) that it is called the "law
of the wall." On the other hand, eliminating V as a variable
reduces Eq. (35) to
163
( y D{ (38
which is found experimentally to be a very good approxima
tion for the balance of the crosssection (0.2
ing p as a variable reduces Eq. (36) to
up = (39)
TD L D
which is known on theoretical grounds to be an exact result
for the laminar regime. Eliminating p from Eq. (37) or D
from Eq. (39), or both p and D from Eq. (34) results in
g A (40)
yEw
which with A= 1 is found experimentally and computationally
to be a valid asymptote for both the laminar and turbulent
regimes very near the wall, such that
Including e/D in Eq. (34) and then eliminating both D and
p leads to
U"' Pe (41)
which proves experimentally to be a valid expression for
e<
moderate roughness, Eq. (37) remains valid for that regime.
Example 3
Flow Through a Horizontal Circular,
SharpEdged Orifice
It may be postulated speculatively that
AP= {uo,p,l4.,D,Dp} (42)
where here AP is the pressure drop across the orifice, uo is
the mean, linear velocity of the fluid, D,, is the diameter of
the orifice, and Dp is the diameter of the pipe in which the
orifice is installed. The value of the pressure drop depends
on the exact location of the pressure tap, but those locations
need not be identified for the purposes of this example. The
density and viscosity are implied to be constant. The corre
sponding tabulation is
AP u_ Ip D D
M 1 0 1 1 0 0
L 1 1 3 1 1 1
e 2 1 0 1 0 0
The number of variables is six and the number of indepen
dent dimensions is three, as is the number of variables such
as D,, u,, and p that cannot be combined to form a dimen
sionless group. Hence, the minimal number of dimension
less groups is 63=3.
The following acceptable set of dimensionless groups may
be derived by any of the procedures illustrated in Example 1:
AP Doup Dp (43p
puo [ p Do
Elimination of D, reduces Eq. (43) to
AP D= up (44i
4PU0{= (44
The further elimination of p leads to
(AP)D A(
= A (45'
whereas the elimination of p leads to
AP
= B (46:
PUo
Equation (44) is found experimentally to be valid for D,<
and Eqs. (45) and (46) to be valid on both experimental and
theoretical grounds for D,<
spectively.
Example 4
Free Convection from a Vertical Isothermal Plate
The behavior may be postulated to be represented by
h=({g,p,T, T,,x,P,p,C, k} (47;
where g is the acceleration due to gravity, p is the volumet
ric coefficient of expansion with temperature, T_ is the
unperturbed temperature of the fluid, and x is the vertical
distance along the plate. The corresponding tabulation is
h g p T,T, x A p Cp k
M 1 0 0 0 0 1 1 0 1
L 0 1 0 0 1 1 3 2 1
e 3 2 0 0 0 1 0 2 3
T 1 0 1 1 0 0 0 1 1
The minimal number of dimensionless groups given by both
Eqs. (1) and (2) is 94=5. A satisfactory set of dimensionless
groups, as found by any of the methods described in Ex
ample 1, is
hx Pp gx Cp2 p2x
0P! , P(T T ),CA(T T (48)
k p k p w P p
It may be reasoned that the buoyant force that generates the
convective motion must be proportional to pgp(T, T,), hence
g in the first term on the righthand side of Eq. (48) must be
Chemical Engineering Education
multiplied by p(T,T,). Thereby, Eq. (48) becomes
hx Ip 2 g (T, T _)x' CP T C )px
k 2 k ),e ( (
(49)
The effect of expansion other than on the buoyancy is now
represented by P(T,T.) and the effect of viscous dissipa
tion by Cp(T,T_)(px/g)2. Both effects are negligible for all
practical circumstances. Hence, Eq. (49) may be reduced to
hx_ p2gp(T. T)x3 Cp (50)
k [ k
or
Nu ,={Gr,,Pr} (51)
where Nu, = hx/k and Grx =(p g(T,T.)x)/p2 is the Grashof
number.
Elimination of x speculatively reduces Eq. (60) to
k 2 )x Pr} (52)
turbulent regimes.
Eliminating x for Eqs. (54) and (56) speculatively results
in, respectively
Nux= A(GrPr) = A(Ra Pr)3 (58'
and
Nu =B(Grx Pr)3 = B(Rax)3 (59:
Equation (58) appears to be a valid asymptote for Pr>0 and
Gr, + and a reasonable approximation for very small val
ues of Pr in the turbulent regime, while Eq. (59) is well
confirmed as a valid asymptote for Pr> and Grx oo and
as a good approximation for moderate and large values of Pr
over the entire turbulent regime. The expressions in terms of
Gr, are somewhat more complicated than those in terms of
Ra,, but are to be preferred since Gr, is known to character
ize the transition from laminar to turbulent motion in natural
convection just as ReD does in forced flow in a channel.
The power of speculation combined with dimensional
analysis is well demonstrated by this example in which valid
asymptotes are thereby attained for almost every regime.
SUMMARY
Nu =Gr3 {pPr} (53
Equation (53) appears to be a valid asymptote for Gr, 4*
and a good approximation for the entire turbulent regime.
Eliminating g speculatively from Eq. (50) results in
hx {p2C gp(T ,T,)x (
k k (54
or
Nu, ={Gr, Pr2} (55
Equation (55) appears to be a valid asymptote for Pr0 for
all Gr,, that is for both the laminar and the turbulent regimes.
The development of a valid asymptote for large values of
Pr requires more subtle reasoning. First, Cpi/k is rewritten
as t/pa where a=k/pCp. Then p is eliminated specula
tively except as it occurs in pgp(T,T,) and k/pCp. The
result is
Nu,=({Rax
where Rax=(Cpp2gp(TXT,)x3)/pk=GrPr is the Rayleigh
number. Equation (57) appears to be a valid asymptote for
Proo and a reasonable approximation for even moderate
values of Pr for all Grx, that is, for both the laminar and
Summer 1997
The mechanics of dimensional analysis are relatively
S simple, but the choice of the appropriate variables provides
an uncertain and intimidating task for students. The concept
of speculation provides a rational approach to both the initial
choice of variables and their elimination to attain asymptotic
forms. The necessity of testing the results with experimental
or computed values introduces no difficulty, at least in con
) cept.
REFERENCES
) 1. Rayleigh, Lord (J.W. Strutt), "The Principle of Similitude,"
Nature, 45 (No. 2368), 66 (1915)
2. Churchill, S.W., "The Use of Speculation and Analysis in
the Development of Correlations," Chem. Eng. Commun., 9,
19 (1981)
3. Churchill, S.W., "An Analysis of Heat and Component Trans
fer by Turbulent Free Convection from a Vertical Plate,"
Proc. Chemica '70, Butterworths, Sydney, Vol 6A (1970)
4. Churchill, S.W., "A Theoretical Structure and Correlating
Equation for the Motion of Bubbles," Chem. Eng. Process,
26, 269 (1989) and 27, 66 (1990)
5. Buckingham, E., "On Physically Similar Systems: Illustra
)tions of the Use of Dimensional Equations," Phys. Rev., Ser
2, 4(4), 345 (1914)
6. Van Driest, E.R., "On Dimensional Analysis and the Pre
sentation of Data in Fluid Flow Problems," J. Appl. Mech.,
S13,A34(1946)
7. Kline, J., Similitude and Approximation Theory, McGraw
Hill Book Co., New York, NY, pp. 28, 63 (1965)
8. Churchill, S.W., The Interpretation and Use of Rate Data:
The Rate Process Concept, revised printing, Hemisphere
Publishing Corp., Washington, DC, Chapter 10, Figures 18
21 (1979) J
hx OC,pPgp(T,T )x3
k pk
m %survey
ENGINEERING EDUCATION
FOR THE 21ST CENTURY
Listen to Industry!
RALPH A. BUONOPANE
Northeastern University Boston, MA 02115
As chemical engineering education approaches the
complete implementation of ABET's Engineering
Criteria 2000 in the next few years, much of the
focus in our colleges and universities has turned to the mean
ing and measurement of outcomes assessment. Successful
employment in the chemical industries of the 21st century
will require that graduates of our chemical engineering pro
grams meet their needs with realworld work experience.
Graduating engineers need to quickly develop an under
standing of how their work contributes to the business re
sults of a company. The way our graduates speak and present
ideas and the way they write to communicate results are
becoming assessable knowledge bases that industry looks
for. Besides deriving or finding the proper chemical engi
neering equation and cranking out a computer solution to
provide the numerical answer to a problem, industry also
wants chemical engineers to consider the impact of their
work on the environmentboth on people and on business.
Communication skills, both oral and written, are not em
phasized enough in today's typical chemical engineering
curriculum. Chemical engineers need to prepare for the fact
that they will spend much more of their time communicating
with others than they will spend determining an answer to
some engineering problem. It's one thing to get the correct
answer to an exam problem, but it's another to design some
thing useful that others are going to be involved with and
that people are going to use.
What evidence has been provided to substantiate this call
for change in the education of our chemical engineering
graduates? The Council for Chemical Research's (CCR)
Education Committee has developed the following "Chal
lenge" statement:
The general view is that the scientific research and technical
development expertise of U.S. graduates is excellent in both
science and engineering. However, there exists in today's
rapidly changing global economy a need for a shift of focus for
the "knowledge workers" to an emphasis on traits and skills
and not just content.
With this in mind, the Education Committee recommends that
CCR adopt the following positions
New chemists and chemical engineers must have the
opportunity for a broader exposure to other areas of science
and engineering to foster interdisciplinary and collaborative
research. Examples include biological science, polymer
science, catalysis, physics, environmental science, etc. The
desired benefit would be that, in future employment, the
successful graduate has a greater appreciation for issues
broader than the pursuit of pure research goals.
So called "soft" skills must be more strongly incorporated
into the graduate curriculum to allow for the success of the
graduate. It has been repeatedly emphasized that failure in
industrial positions is more often related to these areas than
to technical expertise. This is true in academic positions as
well. These areas include a basic understanding of ethics,
the environment, team working, economics, patents, and
corporate/university culture.
Communication skills of all typesoral, written, computer,
and group dynamicsmust be more heavily stressed.
Foreign language skills have gained renewed importance in
this respect.
Breadth of training must be reemphasized to produce a
graduate capable of handling the diverse and rapidly
changing global world of industry.
Awareness of the goals, products, competitors, and areas of
emphasis in industry should be increased in all graduates to
better facilitate university/industry interactions as well as to
better prepare those students interested in industrial
positions.
Although this statement was prepared for the Council for
Chemical Research and thus it contains references to gradu
Ralph Buonopane is Associate Professor and Chair of the Department of
Chemical Engineering at Northeastern University. He is a Chemical Engi
neering Program Evaluator for AIChE/ABET and currently chairs the AIChE
Admissions Committee. He is also active in CCR andASEE. Ralph received
his BSChE, MSChE, and PhD degrees from Northeastern University.
Copyright ChE Divtsion ofASEE 1997
Chemical Engineering Education
ate research and education, it also has significant application
directly to undergraduate chemical engineering education.
In addition to the recent reports from ABET, ASEE, NAE,
NAS, and NSF that provided much information for this CCR
Education Committee statement, a survey of two large em
players of chemical engineers and chem
ists was conducted several years ago.
Dr. Norman N. Hochgraf of Exxon
Chemical Company (now retired) con
ducted a survey of technical employ
ees with less than five years experi
ence at Exxon and Dow Chemical. The
survey basis included 138 people iden
tified as campus technical hires in R&D
at Exxon and 427 individuals as tech
nical hires in central R&D at Dow. In
addition, 102 supervisors at Exxon
were surveyed. The results of the sur
vey were presented in a session at the
17th Annual Meeting of CCR (Pitts
burgh) in October of 1995. The survey
response was 68% for Exxon Chemi
cal and 82% for the Dow employees
and included chemistry, chemical en
gineering, and other engineering re
lated personnel at all degree levels.
Although the Exxon group included
52% BS, 25% MS, and 23% PhD de
grees and the Dow group had 38% BS,
15% MS, and 47% PhD degrees, no
significant differences were found be
tween the companies, degree levels,
disciplines, new hires and supervisors,
or researchers and plant technical
people.
The 26 survey questions were the
same for all and were grouped into the
six skill categories: Technical Knowl
edge (6), Application of Knowledge
(5), Work (5), Communication (3),
Team/Interaction (4), and Independence (3). The survey re
spondents were asked to evaluate both the importance of a
particular skill to their job and the degree of their preparation
upon entering the workplace.The response measures were
values (15) from "none" to "utmost" for Importance, and
from "very little" to "very great" for Preparation; 22 of the
26 skills rated importance greater than preparation (denoted
by > in Table 1). Table 1 summarizes the survey results.
Several openended questions were also included in the
survey. Responses to the question "What experiences or
activities in academia were most helpful?" were mostly in
accordance with the responses for the importance/prepara
tion responses. Team interactions to solve complex prob
lems, working together, technical communications course,
Summer 1997
TABLE
Survey Res
Technical Knowledge
science/engineering concepts
plant operations and control
computing/computers
statistical experimental design
use of technical/patent literature
environmental/safety requirements
Application of Knowledge
gathering necessary information
defining the problem
applying concepts to obtain solution
applying concepts to business valui
reaching workable results
Work Skills
judging proper time
acquire and retain information
setting priorities/developing plans
judging "perfection" needed
meeting schedule dates
Communication Skills
explaining ideas and concepts
preparing and delivering presentati
writing effective reports
Team/Interaction Skills
effectiveness in a team
learning through consulting
getting cooperation
supervising others
Independence
originating own projects
working independently
continuing to learn/develop
mentoring responsibilities that recognize the longterm impact
of early job experiences, and to recognize and solve problems
before they develop.
Increase industrial experience opportunities for faculty experi
ences in industry and for student coop and intern experiences.
Improve overall effectiveness by sacrificing shortterm costs to
improve longterm effectiveness.
As we in chemical engineering education move toward
full implementation of ABET Engineering Criteria 2000, the
changes that are sought appear to be in line with and sub
stantiated by the industrial views presented here. Let us not
lose sight of the results we seek to achieve as we focus on the
process of providing relevant chemical engineering educa
tion for the 21st century. 0
167
technical writing, speaking to an audience, presenting pa
pers, and literature searching were most frequently cited as
helpful. In another openended question 99% of the supervi
sors responded that prior work experience is desirable. The
new hires valued prior work experience as valuable in relat
ing theories and concepts to reallife prob
1 lems, in solving problems with incom
ults plete data, and teaching the "art" of hu
Importance Preparaion man interactions. One new hire, a coop
student, offered that coop experience was
very = great more valuable than the degree program.
very > some Very few responders offered excellence,
moderate > little breadth, or depth of technical course work
moderate = some as being helpful.
moderate > little
Dr. Hochgraf concluded that the results
of the survey suggest that colleges and
very > some
very > some universities should
very > some
ns very > great Change the ways in which learning takes
S moderate > little place.
very > some
Provide more emphasis and opportunity
for solving incompletely specified prob
very > some lems, deciding the data needed, working in
moderate = some and leading problemsolving teams, and
very > some presenting and defending results.
very > some Build in more "realworld" experience by
very > some seeking and creating opportunities for in
dustrial input, expecting faculty to obtain
very > some industrial experience, and supporting the
ons very > some value of coop and intern experiences.
very > some Sacrifice some science/engineering knowl
edge to achieve a greater overall effective
ness.
very > some
very > some Dr. Hochgraf also concluded that the
very > some survey results suggest that industry should
moderate > little
Provide more early training in environ
mental and safety requirements and prob
very > some lems, in reaching and making decisions, in
very = great
very => rme industrial structure relating to competition
________ and economics, and in working in teams.
Increase supervisor training to provide
= classroom
)
TEACHING STATISTICS
TO ChE STUDENTS
DIANNE DORLAND, K. KAREN YIN
University of Minnesota Duluth Duluth, MN 55812
here is a general consensus that statistics is needed in
chemical engineering curricula,11] and recently ABET
added the requirement, "Students must demonstrate
knowledge of the applications of probability and statistics to
engineering problems." We must prepare our customers
future engineers and researchersto effectively use statisti
cal methods in product development and improvement as
well as in scientific research.
With the dramatic shift in management attitudes toward
using quality measures for competitive advantage and sur
vival in the challenging world market, statistics has been
recognized as a useful tool for decision making and quality
improvement. We as educators must respond to the needs of
industry and provide both technical support and wellequipped
personnel in a timely fashion.
Realizing the importance of statistics in engineering edu
cation, the Chemical Engineering Department at the Univer
sity of Minnesota Duluth has, since 1986, included in its
required curriculum a threecredit course, "Experimental
Design for Chemical Engineering." Our main objective has
been to help the students comprehend the omnipresence of
variability in the real world and to equip them with neces
sary statistical tools to deal with it.
COURSE CONTENT
The course content was determined by the needs of our
students and the demands of industry. Over the past ten
years, most of our graduates have taken entrylevel jobs in
diverse industries, while others pursued advanced studies. A
number of the junior and senior students have also partici
pated in coop or intern programs with various industries.
In a concerted effort to collaborate with industry, our
faculty consults and has offered different short courses (in
cluding applied statistics) for local industries. The feedback
from students, the input from industry, and our own experi
ences have not only confirmed our belief in the necessity of
the course, but also offered us better insights into what
should be taught. We keep track of the most frequent
questions and the subjects of most interest to industries,
and we have been making continuous efforts to tailor the
course to address them.
The course is designed to show the kind of problems that
call for the use of statistics, when and where they may arise,
and how they can be solved. By carefully explaining these
questions, we are able to help the students understand the
rationale and usefulness of statistics and to develop "sta
tistical thinking."
One of the challenges we face in teaching such a course is
to strike a proper balance between statistical foundation and
its application. Because of the already fully packed chemical
engineering curricula, it does not seem feasible to require a
prerequisite course in probability and statistics. Therefore
Dianne Dorland is Professor of Chemical Engi
neering at the University of Minnesota Duluth,
where she has taught since 1986. She received
her BS and MS degrees from the South Dakota
School of Mines and Technology and her PhD
from West Virginia University, all in chemical
engineering. Her research interests include in
dustrial wastewater treatment and hazardous
waste management.
Karen Yin is Associate Professor of Chemical
Engineering at the University of Minnesota Duluth,
where she has taught since 1991. She received
her MS degree in Mathematical Statistics and
her PhD in Chemical Engineering from the Uni
versity of Maryland, College Park. Her major re
search interests include system identification, fault
detection and diagnosis, and process control.
Copyright ChE Division ofASEE 1997
Chemical Engineering Education
We must prepare our customers
future engineers and researchersto effectively
use statistical methods in product development and improvement
as well as in scientific research .... We as educators must respond to the
needs of industry and provide both technical support and
wellequipped personnel in a timely fashion ...
only the firstyear calculus is desig TA
nated as a prerequisite. Cour
Given the main objective of this c
1. BasicTheory
course and the background of the Probability ai
majority of students taking it, we have Distributions
been "application oriented" and have
allotted very little class time for 2. Comparing Two
Procedure
proofs and mathematical derivation Randomizati
of probability and statistical theory. Hypothesis te
Rather, we emphasize understanding
the essence of statistical inference 3. Comparing Multi
ANOVA
from the engineer's point of view. Between and
The course is based on the follow
. 4. Factorial Design a
ing objective: To efficiently teach the 4. Factorial Design
Analysis
statistical methods in experimental Applications
design, data analysis, and statistical Software
process control that are most useful
5. Statistical Process
in chemical engineering. The key Basic tools
components of the course are closely Noisy data
related, proceeding from the easy con Control chart
cepts to the more difficult ones and Multivariate
from the simple case studies to the 6. Regression Analy
more complicated applications. Linear model
The course is taught in a lecture Software ana
format with 25% of the classes in a 7. Applications
computer laboratory. The course out
line is shown in Table 1, and a brief
description of each of the compo
nents follows.
( Basic Theory
This component provides students with some fundamental
knowledge of probability and statistics. The use of several
introductory realworld examples enables the students to
differentiate between populations and samples and helps
them understand that extracting information from data, an
everyday task in research or in industry, often requires using
sample statistics or estimates of population parameters. The
emphasis on the omnipresence of variability highlights the
necessity of using statistics.
BLE 1
se Outline
nd statistics
Averages
on and blocking
sting
pie Averages
within treatment variations
it Two Levels
Control
s
SPC
sis
s from data
lysis
After the introduction of certain
basics in probability, we discuss ran
dom variables and the most fre
quently used probability distribu
tions, including the normal, t, X2,
and F distributions. Without spend
ing too much time on mathemati
cal details, we strive for an under
standing of the meaning of distri
bution and probability and the use
of various tables.
The central limit theorem is elabo
rated following the discussion of the
common knowledge that averages
are usually better estimates than in
dividual observations. That, in turn,
leads to the use of the sample mean
as an estimate of the population
mean, the assessment of the quality
of the estimate by the confidence
interval approach, and the determi
nation of sample size to give a good
estimate with desired accuracy with
a certain confidence levelone of
the most frequently encountered
questions in reality.
Estimating population variances
with sample variances, evaluating the
closeness of the estimate to the 'true' parameter, and deter
mining the required sample size are also addressed, in that
order. Our experience has shown that students have little
trouble following the logic of these methods or applying
them to case studies.
Comparing Two Averages
This topic addresses the comparison of two thingstwo
methods, two materials, two operating conditions, etc. A
class discussion can be a good review of the general proce
dure needed: determining the sample size, planning an ex
periment, conducting the experiment to collect data, and
finally, analyzing the data for a conclusion. We then use
Summer 1997
several examples to convince the students that "conclusions
are easily drawn from a welldesigned experiment, even
when rather elementary methods of analysis are employed,"
and conversely, "even the most sophisticated statistical analy
sis cannot salvage a badly designed experiment."[21
The two important strategies in experimental design, ran
domization and blocking, are then dis
cussed in detail, followed by the ratio
nale of hypothesis testing, the procedure The court
of t tests for simple comparison, and the
method of handling paired data. Our ap o0
proach is to use indepth discussion of followinA
realworld examples to show the students To efficient
the philosophy of experimentation and
data analysis and the role that statistics statistical
plays in it. experime
data an
S Comparing statistic
Multiple Averages control th
A discussion of the comparison of more useful it
than two averages naturally follows the engine
previous topic. The analysis of variance The key
(ANOVA) methodology is introduced
and used. It is very important to make Of the count
sure that the students comprehend the related, j
essence of ANOVAthe estimation and from the e
evaluation of variations caused by vari
to the mo
ous sources.
We normally use several relatively ones an
simple examples to show the calculation simple cai
of different sums of squares and mean the more i
squares for the between and within treat
ment variations. More complicated and appli
practical cases are then solved using com
puters. There are many statistical soft
ware packages available, and data entry and use of the pack
ages are usually straightforward and easy to follow. There
fore we spend most of the time interpreting and elaborating
on the results.
Our experience has been that further discussion of certain
questions will greatly enhance the students' understanding.
Some of the questions we have used include: Why do we
need to use a t (or F) test? When can we use it? What is the
meaning of t (or F)? What does a big t (or F) mean? What is
the effect of the sample size on the result? How do you select
a? Why should you be concerned about p ?
Factorial Design at Two Levels)
The concepts of factorial design are introduced next. We
choose a relatively simple example to demonstrate how to
plan a twolevel factorial design as well as how to calculate
and evaluate the effects. Graphic descriptions are used when
ever possible so that the class can "visualize" the design and
the idea behind the calculation. More complicated case stud
ies are again solved by using computers.
The homework assignments in this com
ponent require solutions obtained by both
s based using and not using computer software
packages, enabling the students not only
to understand and carry out design and
jective: analysis correctly, but also to do it effec
teach the tively.
thods in
Sdesin Statistical
design,
Process Control
is, and
process To improve process productivity and
ire most product quality, statistical process control
(SPC) has been widely used in industry,
although at different levels of sophistica
ing. tion. In this component, we aim to expose
ponents the students to SPC in industry and to
Sc equip them with such basic tools as con
re closely trol charts. For these purposes, a brief
:eeding review is given of the current situation in
concepts industry, the roles of control engineers
f t and statisticians, and the importance of a
iicult chemical engineer's ability not only to
im the handle process dynamics using first prin
tudies to ciples and process knowledge, but also to
cad use statistical tools to deal with noisy data.
plicated
The most commonly used control charts,
ons. such as i charts and R charts, and their
rationales and use are discussed in detail
with several industrial examples. A brief
introduction to some more advanced techniques, such as
multivariate SPC, is also given to broaden students' fields of
vision.
( Regression Analysis
Building linear models from data is the subject of this
component. Probabilistic models, a new concept to many of
the students, are introduced first, followed by the idea of the
leastsquares estimation and its role in regression analysis.
There are many software packages available for this purpose
whose uses are similar to each other and which are not
difficult to learn.
Our practice has been to spend most of the lecture time on
illustration and elaboration of the assumptions, things needed
Chemical Engineering Education
se i
the
Sob
tly
me
ntal
alys
alp
at a
ch
leer
com1
se a.
proc
asy
pre
dfra
se si
com
cati
170
for model validation, as well as on introducing the most
commonly used transformations. No attempts at theoretical
derivation are made. Rather, examples and graphic descrip
tions are presented whenever possible to help explain impor
tant but fairly sophisticated concepts such as noise correla
tion and equal variances. One of the purposes here is to
remind the students that it is not sufficient for them to be
merely aware of available techniques or methods for their
purposes, but that they must also be able to use them cor
rectly under specific conditions.
We have used as our required textbooks Statistics for
Experimenters by Box, Hunter, and Hunter, and Engineer
ing Statistics by Hogg and Ledoter. Each has its own strengths.
We also include current literature and examples from pro
cess industries for their practical relevance.
Applications
Experimental design techniques are heavily integrated into
undergraduate research projects. A typical example involved
the removal of mercury during incineration and wastewater
treatment. Because a high percentage of the mercury was
associated with particulates in the scrubbing water, their
removal was key to mercury removal.
A twolevel factorial experimental design examined mer
cury removal efficiency as a function of inorganic coagu
lants, polymers, combinations of coagulants and polymers,
reaction conditions, and dosages. The experimental work
economically provided information on trends and parameter
interactions so that the optimum parameters could be se
lected for mercury removal. Plantscale tests were performed
and the industrial process was permanently changed to pro
vide the mercury removal demonstrated by this project.131
STUDENT RESPONSE
The course evaluation system at the University of Minne
sota Duluth is similar to that used in many other universities.
Students rank various aspects of the course on a scale of one
to seven at the end of the quarter. Among the 27 questions
asked, 5 are related to course design and course content. The
average ranking of these questions in the last three years was
5.0. Most students were satisfied with its orientation toward
application, and some of them even took additional courses
in statistics out of interest or perhaps a newly developed
realization of its importance.
We should mention that a similar syllabus was adopted for
an intensive workshop offered at a local industry. The main
difference between the industrial workshop and the regular
undergraduate course is that in the former, more applied
examples from the plant were included and discussed. We
also put particular emphasis on correct methodology for
each example at the host's request. The inclusion of famil
iar examples never failed to generate lively discussions and
it also was conducive to a better understanding of the con
cepts and methods.
Most of the workshop attendees had not taken any prob
ability and statistics courses before. However, the majority
of them had little trouble following the lectures and showed
strong interest and appreciation for them. Among the largely
favorable evaluations we received, many students indicated
that the strongest point of the course was its practical na
ture. The inclusion of more class discussions was suggested,
which, we believe, should be adopted in the future under
graduate teaching as well.
CONCLUSIONS
We have developed and taught a course in applied statis
tics to chemical engineering undergraduate students. Our
intent has been to show the relevance and importance of
statistics in engineering practice and to provide students
with the necessary statistical tools, such as experimental
design, statistical process control, and model building.
The main feature of the course is its practical emphasis.
The students are frequently reminded of the role that statis
tics plays in the real world, and their motivation is thereby
sustained throughout the entire course. We hope they leave
this course with a clear sense of use and a firm grip on how
to apply theory to practice, rather than possessing merely a
collection of looseknit, halfunderstood topics or theorems.
We believe this course is well suited to chemical engineer
ing undergraduates. The universal use of the computer has
made it possible to include practical problems that are more
complicated and require more computation than traditional
textbook examples. Our experience in teaching this course
has been interesting, rewarding and enjoyable.
To foster the students' ability to use statistics correctly and
effectively is not an easy task. To train and to bring up a
"statistically minded generation"14' requires much more than a
single course. The integration of statistics into our curriculum
is but the first step; it remains for us to make continuing efforts
to improve it to meet the growing challenge in the real world.
REFERENCES
1. Eckert, R.E., "Applied Statistics: Are ChE Educators Meet
ing the Challenge?" Chem. Eng. Ed., 22(2), p. 122 (1996)
2. Box, G.E.P., W.G. Hunter, and J.S. Hunter, Statistics for
Experimenters, John Wiley & Sons, New York, NY (1978)
3. Dorland, D., and J. Stepun, "Mercury Behavior in a Waste
water Sludge Incinerator," inAirWater Mass Transfer, S.C.
Wilhelms and J.S. Gulliver, Eds., Selected Papers from the
Second International Symposium on Gas Transfer at Water
Surfaces, ASCE, New York, NY, p 742 (1991)
4. Shewhart, W.A., and W.E. Deming, Statistical Methods from
the Viewpoint of Quality Control, Dept. of Agriculture, Wash
ington, DC (1939)
Summer 1997
L= class and home problems
AN INTRODUCTION TO
PROCESS FLEXIBILITY
Part 1. Heat Exchange
W.E. JONES, J.A. WILSON
University of Nottingham University Park Nottingham NG7 2RD England
Process plants need to be flexible to cope with changes
in production rates, product specifications, feedstock,
catalyst deactivation, and heat exchanger fouling. Tra
ditionally, once the process structure has been decided, vari
ous operating cases are evaluated and one is chosen as the
basis for detailed design. However, selection of the design
case is not straightforward. Effectively dealing with all the
highly interrelated issues during design is a formidable prob
lem. Hence, engineers often resort to the application of rule
ofthumb safety factors during equipment design (e.g., add
ing 10% extra area to a heat exchanger) in an effort to ensure
flexibility. Following this strategy, an experienced engineer
would hope to develop a design that is operable across the
anticipated process range, but there is no guarantee that the
required flexibility will be achieved."'4 As the problem pre
sented here clearly illustrates, different plant operating modes
can easily lead to equipment design situations that are not
covered by a simple safety factor.
The above comments explain why no substantial coverage
of flexibility is found in any of the standard undergraduate
design textbooks, apart from a few remarks on safety fac
tors. Despite these difficulties, we feel the topic is very
important, particularly because of the highly integrated plants
being built today, and that the basic ideas should be intro
duced to all students.
Some students will encounter flexibility problems as part
of their finalyear design project. These projects are nor
mally simplified from industrial reality, considering only
one feedstock and, at worst, a catalyst deactivation or heat
Warren Jones holds BSc and PhD degrees in
chemical engineering from the University of
Nottingham and is a registered Chartered Engi
neer. He has a wideranging interest in both front
end processes and detailed plant design, devel
oped initially through nine years of experience
with a major engineering and construction com
pany. Teaching responsibilities include several
design courses, process economics, and engi
neering thermodynamics.
Tony Wilson holds BSc and PhD degrees in
chemical engineering from the University of
Nottingham. With industrial and consulting ex
perience in process control and batch process
engineering, and with active research in both
fields, he coordinates the department's research
in computeraided process engineering and is
responsible for process control teaching at the
undergraduate level.
A,
Copyright ChE Division ofASEE 1997
Chemical Engineering Education
The object of this column is to enhance our readers' collections 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 Professor James O. Wilkes (email:
wilkes@engin.umich.edu) or Mark A. Burs (email: maburns@engin.umich.edu), Chemical
Engineering Department, University of Michigan, Ann Arbor, MI 481092136.
exchanger fouling cycle. Nevertheless, to design an operable
plant, thought must be given at an early stage to conditions
under which each item of equipment is expected to operate
and to the processcontrol scheme to be used. Unless the
project supervisor is alert, many students will simply size
equipment for the conditions implied by the design mass and
heat balances without considering flexibility.
This and a subsequent article will attempt to illustrate how
selected aspects of flexibility can be introduced through
interesting examples. In particular, the heatexchange prob
lem developed here may be used directly in a design course,
while the reactor recycle loop featured in the second article
could form the basis for project work, or for a discussion
question in reactor design, or simply to indicate to supervi
sors an area worth discussing and developing in future de
sign projects.
BACKGROUND
Most students, if asked, will suggest adjusting steam pres
sure or hot oil flow rate to a heat exchanger in order to
maintain exit temperature in the face of process flow change.
Slightly less obvious would be the suggestion to alter the
condensate level in the heatexchanger shell, thereby cover
ing/exposing more heat transfer area for steam condensa
tion. The important point is that steam and hot oil are utili
ties, and changing their consumption does not disturb the
process.
Difficulty is immediately encountered when considering
heat exchange between two process streams; changing the
flow rate of one will certainly affect the exit temperature of
the other. Unfortunately, interfering with a process stream
flowrate immediately upsets the plant mass balance, which
is undesirable. The difficulty is overcome by using a bypass
(see Figure 1) that does not affect the total flow rate but
changes the proportion actually passing through the heat
exchanger and hence the heat transferred. The problem pre
sented here is concerned with heat exchanger bypass ar
rangements to ensure satisfactory operation, in the face of
aging catalyst, of a reactor at both beginningofrun (BOR)
and endofrun (EOR).
T1 Bypass i T2
(iC p)H I
I I
Figure 1. Heat exchanger with bypass.
T4  T3
Figure 1. Heat exchanger with bypass.
Students are familiar with using heat balances to calculate
the heat load, Q,
Q=(mncp)H(TI T2)= M (T 4T3) (1e
and the rate equation
[(T, T)(T2 T1)]
T TT
T T3
to determine heat transfer area A, knowing the overall heat
transfer coefficient U. Here, (rcp)H and (rimcp) are the
products of flow rate and specific heat capacity for the hot
and cold streams. Temperatures T, to T4 are identified in
Figure 1, and for the moment we assume the bypass is
closed. But design to take into account flexibility implies not
only calculating A, but also looking at the implications for
other operating conditions, and this is where Eqs. (3) and (4)
become useful.5'61 They are derived for Eqs. (1) and (2) in
order to permit the writing of explicit temperature equations:
(1 RB)T, + R(B 1)T3 +(R 1)T =0 (3'
(1 RB)T4 + B(R 1)T3 + (B 1)T = 0 (4'
where R=(mcp) /(ricp), and B=exp[(UA/(mCp)J(R1)].
To illustrate, for a given heat exchanger (A specified) and
known stream properties ((imcp)c, (rhcp) and U specified),
we can easily calculate the effect of an inlet temperature (T,
and T,) change on both exit temperatures (T2 and T4) using
Eqs. (3) and (4). It is a very simple extension to apply
sequentially the above pair of equations to a heat exchanger
network, thereby evaluating the new temperatures through
out the network.[6]
Bearing in mind that various operating modes are to be
accommodated, it is likely that the heat exchanger sized for
the most severe case will be too large for the other modes.
As hinted earlier, this difficulty is overcome by opening the
bypass. We assume the heat exchanger will operate with the
same UA (see the Appendix), but the effective UA for the
heat exchanger plus the partially open bypass (as indicated
by the dotted box in Figure 1) is reduced. Increasing bypass
flow progressively reduces the effective UA, whereas maxi
mum heat transfer is achieved when the bypass is closed.
Good engineering practice would maintain a minimum flow
rate of 510% through the bypass.
Equations (1) through (4) are written for the case of sen
sible heating and sensible cooling of process streams.
Special cases result for Eqs. (3) and (4) when one side of
the heat exchanger operates isothermally. If the coldside
operates with isothermal vaporization at T3, then Eq. (3)
Summer 1997
reduced to
TI BT2+(B1)T3 = 0 (5)
where B=exp(UA/(rmcp) ).
If the hotside operates with isothermal condensation at
T,, then Eq. (4) reduces to
(B1)TI BT3 + T4 = 0 (6)
where B=exp( UA/(rimCp) C
For the special case of (mer,) = (mcp)c (i.e., constant tem
perature driving force, AT, throughout the heat exchanger),
it is easy to show that
SA +1P + Tl T3 (7)
II P J(7)
In the following problem, Eqs. (5) and (7) are more imme
diately useful than the general Eqs. (3) and (4).
Finally, the hydraulic interaction between the heat ex
changer and control valve in the bypass line is important.
Selection of controlvalve type and size is crucial to ensure it
remains operable over the range of bypass flows expected.
Difficulty occurs because transferring flow from the heat
exchanger to the bypass results in a reduced pressure drop
across the heat exchanger. The control valve experiences the
same pressure drop and so must accommodate the largest
flowrate at the lowest pressure drop. (To achieve steady
state bypass flowrates in excess of 3035%, if the minimum
is 5%, requires an unrealistically large controlvalve size,
and it is better to use two synchronized valves, the second
being in series with the heat exchanger and compensating
for the decreasing pressure drop.t71)
Luyben[81 summarizes the important properties of control
valves. Volume flowrate, q(m3 / sec), through a control valve
depends on the pressure drop, AP(bar), controlvalve size,
Cv, fluid density, p(kg/m3), and valve opening, x. The rel
evant equation, if we assume AP is small compared to the
operating pressure is
q = Cf(x)(AP/p)0.5
or, for mass flowrate
m= Cf(x)(pAP)0"5 (8
where f(x) defines the controlvalve characteristic in terms
of valve opening. For this problem, two valve characteristics
are important:
1) Linear
f(x) = x
2) Equal percentage f(x) = ax1 (typically, a=50)
Note particularly that Cv must have dimensions consistent
with the other variables in Eq. (8). The units used here are
chosen for convenience in the rest of the problem rather than
to agree with engineering practice; hence, the Cv values
cannot be compared directly with, say, control valve
manufacturer's data.
PROBLEM STATEMENT
Figure 2 shows the basic flowsheet for the heat exchangers
surrounding a catalytic reactor operating at EOR conditions.
The hot reactor effluent is cooled first by boiling water at
2000C and then by preheating the reactor feed. The process
operates entirely in the gas phase, and you may assume a
constant specific heat capacity of 2.5 kJ/kgK.
At BOR, the catalyst is much more active, requiring a
reactor inlet temperature of only 1850C. The corresponding
process flowrate and reactor effluent temperature are 22.5
kg/sec and 296.1 C.
a) Calculate the UA requirements for both heat exchangers
implied by EOR operation. Investigate the feasibility of
BOR operation using the UA values just determined if
no flexibility is added to the flowsheet.
b) Determine UA requirements for BOR operation if the
temperature to product recovery is to be maintained at
I 2000C
Steam
BFW
Interchanger
100 c
Reactor Feed
25 kg/s
Figure 2. EOR flowsheet.
To Product
Recovery
Chemical Engineering Education
125'C. Which operating mode sets the design for
1) the interchanger?
2) the boiler?
c) If the minimum flowrate through the bypass is 5% of
the main flowrate, determine the design UA require
ments for both heat exchangers. What percentage of the
main flowrate should pass through the bypass to permit
the alternative operating mode?
d) Add the bypasses to the flowsheet and indicate how you
would configure the temperature control loops. How
would the plant be operated with your control scheme?
e) The interchanger has a coldside pressure drop of 0.6 bar
calculated for EOR flowrate and no bypassing. Select a
suitable control valve from the following range of valves
with linear characteristics:
C, = 1.0, 1.75, 2.5
In steadystate operation, a control valve should operate
with an opening between 0.2 and 0.8. You may assume a
constant gas density of 20 kg/m' and neglect piping
friction losses.
f) The boiler has a hotside pressure drop of 0.4 bar calcu
lated for EOR flowrate with no bypassing. Why would
an equal percentage valve with a =50 be more suitable
for this service than a linear one?
185 OC
Steam
BFW
Interchanger
100 o
Reactor Feed
22.5 kg/s
To Product
Recovery
Figure 3. BOR flowsheet.
SOLUTION
a) Interchanger duty
Temperature difference
Interchanger UA
Boiler duty
Log mean temp. difference
Boiler UA
= 25 x 2.5 x 100 = 6250 kW
= 25 K
= 6250/25 = 250 kW/K
= 25 x 2.5 x 75 =4687.5 kW
=54.1 K
= 4687.5/54.1 = 86.65 kW/K
To investigate the feasibility of BOR operation with the
above UA values, it is probably best to start with the
boiler. Using Eq. (5),
B=exp (86.65/(22.5 x 2.5)) = 4.667
and the process exit temperature
S296.1 + (4.667 1)200 22
= 220.60C
4.667
We can now calculate the two exit temperatures from the
interchanger. Using Eq. (7),
UA /(rmcp) = 250/(22.5 x 2.5) = 4.444
and the temperature difference is
(220.6 100) /(4.444 + 1) = 22.15 K
Hence the preheated reactor feed will be at
(220.6 22.15) = 198.45C
and the stream to product recovery is slightly too cold at
122.150C (but may be acceptable). The preheated reac
tor feed, however, is certainly far too hot at 198.450C.
Conclusion: heat exchangers sized for EOR operation
will not function satisfactorily at BOR.
b) To determine the UA requirements for BOR, it is advis
able to redraw the flowsheet to reflect BOR operation, as
shown in Figure 3. The reactor feed must be heated from
1000C to 1850C, hence the crossover temperature of the
reactor effluent from the boiler to the interchanger is
2100C to maintain a temperature of 1250C to product
recovery.
Interchanger duty
Temperature difference
Interchanger UA
= 22.5 x 2.5 x 85 = 4781.3 kW
= 25 K
= 4781.3/25 = 191.25 kW/K
which is 23.5% less than the UA required for EOR;
hence, EOR operation will set the design of this
item of equipment.
Boiler duty = 22.5 x 2.5 x 86.1 = 4843.1 kW
Log mean temp. difference = 38.05 K
Boiler UA = 4843.1/38.05 = 127.28 kW/K
Summer 1997
which is 46.9% more than the UA required for
EOR; hence BOR operation will set the design of
this item of equipment.
c) We consider the interchanger case in detail here. Figure
4A shows the actual interchanger exit temperature for
the reactor feed at EOR to be T*; after blending with the
bypass, the required preheated temperature of 200'C is
achieved. Our first concern is to calculate T*.
200 = (0.05 x 100) + (0.95 x T*)
T* = 205.26C
and this implies a logmean temperature difference of
22.27 K, giving a UA of 280.65 kW/K, i.e., this UA
should be installed to give an effective UA of 250 kW/K
for the combination of heat exchanger plus 5% bypass.
Figure 4B shows the fraction bypassed during BOR
operation to be Z. The determination Z is a trialand
error calculation, requiring
1) Guess Z
2) Calculate T* to give mix temperature of 185C
3) Calculate logmean temperature difference
4) Calculate UA and compare with installed value
of 280.65 kW/K. If agreement is not achieved,
return to 1) and repeat calculations until conver
gence is obtained.
The converged solution is
Z = 0.1415, i.e., well below maximum bypass of 0.3
T*= 199.01C
Logmean temperature difference = 17.05 K
UA = 280.5 kW/K, close enough to installed value of
280.65 kW/K
A similar procedure is used for the boiler, for which the
following key values are calculated:
2000C
185 C
125 CC
25 kg/s
A: EOR Operation
125 C
22.5 kg/s
B: BOR Operation
1) installed UA = 153.2 kW/K based on BOR and
5% bypass
2) Z = 0.216 for EOR operation
d) Figure 5 shows the flowsheet with bypasses and tem
peraturecontrol loops added. The setpoint for TC 1 may
be set at 1250C for all operating modes, but the operator
will need to increase the setpoint of TC2 from time to
time to compensate for the fall off in reactor catalyst
activity.
e) If a control valve with C, = 1.75 is selected, the follow
ing f(x) values are calculated using Eq. (8):
1) EOR 0.05 x 25 = 1.75 f(x) (0.542 x 20)05
f(x) = 0.217
where 0.542 bar has been estimated by 0.952 x 0.6
2) BOR 0.1415 x 22.5 = 1.75 f(x)(0.358 x 20)0s
f(x) = 0.68
where 0.358 bar has been estimated by
(0.8585 x 22.5)2
25 ) x0.6
S25 9
The valve is known to have linear characteristics; hence,
f(x) translates to openings of 0.217 and 0.68. These are
acceptable, lying within the specified range of 0.2 to 0.8.
Using either of the other valves gives unacceptable or
impossible solutions. C, = 1.0 means the valve is not
large enough to handle BOR and C, = 2.5 is too large,
Reactor Feed
Steam
BFW
To Product
Recovery
Figure 4. Temperature and flowrate details for the interchanger.
Figure 5. Flowsheet showing bypasses and
temperature control loops.
Chemical Engineering Education
such that the opening for EOR is below 0.2.
f) Trialanderror calculations soon show that it is impos
sible to find a C, for a linear valve that spans the BOR
EOR flow range, giving both openings between 0.2 and
0.8.
Equal percentage characteristics enable a wider range of
bypass flowrates to be accommodated; i.e., calculation
soon demonstrates that a C,, of 5.32 gives an opening of
0.8 for EOR and 0.404 for BOR.
APPENDIX
Strictly, the assumption of constant UA is inaccurate be
cause opening the bypass reduces flow through the heat
exchanger, which affects the film coefficient and hence the
overall coefficient. The lower real UA means slightly less
would have to be bypassed, hence the bypass flow calcu
lated on the assumption of constant UA is an upper bound.
For the problem presented, you would expect the bypass to
change from 14.15% to around 13% if account is taken for
the reduced U. Thus for design, provided that the most
severe case and associated flows have been identified, the
small "errors" for the alternative modes are easily accommo
dated by the control loop.
ACKNOWLEDGMENTS
Thanks to Carl Pulford for drawing the figures and John Dos
Santos for helpful discussions on heat recovery.
REFERENCES
1. Grossman, I.E., and M. Morari, "Operability, Resiliency,
and FlexibilityProcess Design Objectives for a Changing
World" in Proc. Sec. Int. Conf., "Foundations of Computer
Aided Process Design," CACHE (1983)
2. Morari, M., Comp. and Chem. Eng., 7, 423 (1983)
3. Linnhoff, B., E. Kotjabasakis, and R. Smith, AIChE Annual
Meeting, Washington, DC, Paper 79d (1988)
4. Perkins, J.D., (ed), IFAC Symposium on Interactions be
tween Process Design and Process Control, Pergamon, Ox
ford, England (1992)
5. Kay, J.M., An Introduction to Fluid Mechanics and Heat
Transfer, p. 313, Cambridge University Press, Cambridge,
England (1963)
6. Kotjabasakis, E., and B. Linnhoff, Chem. Eng. Res. Des., 64,
197(1986)
7. Marlin, T.E., Process Control: Designing Processes and Con
trol Systems for Dynamic Performance, p. 801, McGraw
Hill, New York, NY (1995)
8. Luyben, W.L., Process Modeling, Simulation, and Control
for Chemical Engineers, 2nd ed., p. 213, McGrawHill, New
York, NY (1990) O
W letter to the editor
Dear Editor
The universities with the eight topranked doctoral pro
grams in chemical engineering in 1982, as rated by the
National Academy of Sciences, were listed in Changing
Times. In a 1991 letter to the editor of this journal (Vol. 25,
page 181), I pointed out that an analysis of this ranking
revealed that 63.4% of the faculty members in these eight
"elite" programs had obtained their doctoral degrees from
one of the same eight topranked schools. I suggested that
these programs had maintained and enhanced their reputa
tions by hiring their own and one another's graduates.
Doctoral programs in chemical engineering were ranked
more than one decade later by the National Research Coun
cil (Chronicle of Higher Education, 42(4), 1995). I studied
this report to find 1) the extent to which the eight chemical
engineering programs that ranked highest in 1982 retained
their high rankings in 1995, and 2) the extent to which these
programs persisted in hiring their own and one another's
graduates.
The eight universities and their respective ranking in 1982
and 1995 are: Minnesota, 1, 1; Wisconsin, 2, 4; California,
Berkeley, 3, 3; California Institute of Technology 3, 6;
Stanford, 4, 7; Delaware, 5, 8; Massachusetts Institute of
Technology, 6, 2; Illinois, 7, 5. Each of the eight programs
that ranked highest in 1982 was again ranked among the top
eight programs in 1995.
The names and alma maters of the fulltime faculty mem
bers in these eight programs in 1982 and 1995 were obtained
from the Internet. The median percentage of faculty mem
bers who had obtained their doctoral degrees from their own
school or from one of the other seven topranked schools in
1995 was 70.8% (range, 50.0% to 91.7%). This is very
similar to the 67.4% (range, 50.0% to 75.0%) figure in 1982.
Sincerely,
Jeffrey H. Bair
Division of Sociology
Emporia State University
Emporia KS 66801
Simmer 1997
Random Thoughts...
OBJECTIVELY SPEAKING
RICHARD M. FIELDER, REBECCA BRENT
North Carolina State University Raleigh, North Carolina 27695
Student A: "Buffo'sfirst test is next Monday. I haven't had
him beforecan you just plug into formulas on his exams
or does he make you do derivations and stuff?"
Student B: "There's no tellinglast fall most of his ques
tions were straight substitution, but a couple of times he
threw in things I never saw in the lectures."
Student C: "Yeah, and if you ask him what you're respon
sible for on the test he just gets mad and gives you a
sermon on how bad your attitude is ... we had a 600
page textbook and according to Buffo we were supposed
to know everything in it."
Student A: "Forget thatno time. I'll just go through the
homework problems and hope it's enough."
You can often hear conversations like that in the student
lounge, and if you step across the hall to the faculty lounge
you'll hear their counterparts.
Professor X: "All these students can do is memorizegive
them a problem that makes them think a little and they're
helpless."
Professor Y: "I don't know how most of them got to be
sophomores. After my last exam some of them went to the
department head to complain that I was testing them on
things I never taught, even though the chapter we just
covered had everything they needed to know."
Professor Z: "It's this whole spoiled generationthey want
the grades but don't want to work for them!"
Things are clearly not going quite the way either group
would like. Many students believe that their primary task is
to guess what their professors want them to know, and if
they guess wrong they resent the professors for being
unreasonably demanding, tricky, or obscure. Professors
then conclude that the students are unmotivated, lazy, or
just plain dumb.
There is another way things can go. Suppose you hand
your students a preview of the kinds of problems they will be
expected to solve, including some that require real under
standing, and then include such problems on homework
assignments and tests. Since they will know up front the
things you want them to do and will have had practice in
doing them, most of them will be able to do them on the
testswhich means they will have learned what you wanted
them to know. Some professors might regard this process as
"spoonfeeding" or "coddling." As long as you maintain
high expectations, it is neither. It is successful teaching.
INSTRUCTIONAL OBJECTIVES
An effective way to communicate your expectations is by
giving your students instructional objectives, statements of
specific observable actions they should be able to perform if
they have mastered the course material. An instructional
objective has one of the following stems:
At the end of this [course, chapter, week, lecture], you
should be able to ***
To do well on the next exam, you should be able to ***
where *** is a phrase that begins with an action verb (e.g.,
list, calculate, solve, estimate, describe, explain, predict,
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 institu
tions. He is coauthor of the text Elementary
Principles of Chemical Processes (Wiley, 1986).
Rebecca Brent is an educational consultant and
Adjunct Professor of Education at East Carolina
University. She received her BA from Millsaps
College, her MEd from Missssippi State Univer
sity, and her EdD from Auburn University. Her
research interests include applications of simula
tion in teacher education and writing across the
curriculum. Until 1997 she was an Associate Pro
fessor of Education at East Carolina University,
where she received the 1994 East Carolina Uni
versity Outstanding Teacher Award.
Copyright ChE Dtivsion of ASEE 1997
Chemical Engineering Education
model, design, optimize,...). Here are some examples of
phrases that might follow the stem of an instructional objec
tive, grouped in six categories according to the levels of
thinking they require."1
1. Knowledge (repeating verbatim): list [the first ten
alkanes]; state [the steps in the procedure for cali
brating a gas chromatograph].
2. Comprehension (demonstrating understanding of
terms and concepts): explain [in your own words
the concept of vapor pressure]; interpret [the output
from an ASPEN simulation].
3. Application (applying learned information to solve
a problem): calculate [the pump brake horsepower
required for a specified process line and fluid
throughput]; solve [the compressibility factor equa
tion of state for P, T, or V from given values of the
other two].
4. Analysis (breaking things down into their elements,
formulating theoretical explanations or mathemati
cal or logical models for observed phenomena):
derive [Poiseuille's law for laminar Newtonian flow
from a force balance]; explain [why we feel warm
in 70'F air and cold in 700F water].
5. Synthesis (creating something, combining elements
in novel ways): formulate [a modelbased alterna
tive to the PID control scheme presented in
Wednesday's lecture]; make up [a homework prob
lem involving material we covered in class this
week].
6. Evaluation (choosing from among alternatives and
justifying the choice using specified criteria): de
termine [which of several specified heat exchanger
configurations is better, and explain your reason
ing]; select [from among available options for ex
panding production capacity, and justify your
choice].
WHY BOTHER?
Wellformulated instructional objectives are more than
just an advance warning system for your students. They can
help you to prepare lecture and assignment schedules and to
spot course material that the students can do little with but
memorize and repeat. They also facilitate construction of in
' The six given categories are the levels of Bloom's Taxonomy of
Educational Objectives [B.S. Bloom, Taxonomy of Educational
Objectives. 1. Cognitive Domain. Longman, New York, NY 1984].
The last three categoriesanalysis, synthesis, and evaluation
are often referred to as the higher level thinking skills.
class activities, outofclass assignments, and tests; you sim
ply ask the students to do what the objectives say they
should be able to do. A set of objectives prepared by an
experienced instructor can be invaluable to someone about
to teach the course for the first time and can help instructors
of subsequent courses know what their students should have
learned previously. If objectives are assembled for every
course in a curriculum, a departmental review committee
can easily identify both unwanted duplication and gaps in
topical coverage, and the collected set makes a very impres
sive display for accreditation visitors.
TIPS ON WRITING OBJECTIVES
Try to write instructional objectivesfor every topic
in every course you teach. Take a gradual approach,
howeveryou don't have to write them all in a
single course offering.
Include some objectives at the levels of analysis,
synthesis, and evaluation. They are not that hard to
write, even in undergraduate courses,121 but if you
don't consciously set out to write them you prob
ably won't.
Avoid four leading verbs in instructional objec
tives: know, learn, appreciate, and understand. You
certainly want your students to do those things but
they are not valid instructional objectives, since
you cannot directly see whether they have been
done. Think of what you will ask the students to do
to demonstrate their knowledge, learning, apprecia
tion, or understanding, and make those activities
the instructional objectives for that topic.
Formulating detailed instructional objectives for a course
or even for a single course topic takes effort, but it pays off.
When we have asked alumni of our teaching workshops to
rate the usefulness of the instructional methods we discussed,
instructional objectives ranked second only to cooperative
learning. Many professors testified that once they wrote
objectives for a coursesometimes one they had taught for
yearsthe course became more interesting and more chal
lenging to the students and more enjoyable for them to
teach.13'
2 Examples of higherlevel questions are given by Felder ["On
Creating Creative Engineers," Engr. Edu., 77, 222 (1987)] and
Brent and Felder ["Writing AssignmentsPathways to Connec
tions, Clarity, Creativity,"] College Teaching, 40(2), 43 (1992)].
SFor more information about objectives, see N.E. Gronlund, How
To Write and Use Instructional Objectives, 4th ed., Macmillan,
New York (1991). For examples of their use in engineering educa
tion, see J.E. Stice, "A First Step Toward Improved Teaching,"
Eng. Ed., 66(5), 394 (1976).
All of the Random Thoughts columns are now available on the World Wide Web at
http://www2.ncsu.edu/effectiveteaching/
Summer 1997
, 1classroom
USE OF COMPUTATIONAL TOOLS
IN ENGINEERING EDUCATION
A Case Study on the Use of MathcacP
J.N. HARB, A. JONES, R.L. ROWLEY, W. V. WILDING
Brigham Young University Provo, UT 84602
With the widespread use of personal computers, a
variety of powerful computational tools are now
available that permit engineers to routinely per
form calculations and analyses that were once very difficult.
The introduction of these tools into the undergraduate engi
neering curriculum can facilitate student computations, but
can the use of these tools also foster student learning and
understanding of engineering principles?
The purpose of this paper is to examine issues related to
incorporating an equationsolving package into the under
graduate curriculum in chemical engineering. Although the
content of the paper is based on our experience with
Mathcad, our observations and conclusions are equally ap
plicable to other equationsolving packages. Of particular
interest is the impact that using equation solvers has on
student learning and ways in which such software might be
used to enhance learning.
HISTORY OF MATHCAD AT BYU
Five years ago, most undergraduate student work in our
department was done with calculators, with some of the
more demanding problems requiring spreadsheets or FOR
TRAN programming. At that time we felt that equation
solving packages had been developed to the point where
they represented a potentially useful tool for our students
(see Table 1). After reviewing several different numerical
and symbolicbased equation solvers, the faculty voted to
incorporate Mathcad into the undergraduate curriculum. It
was our opinion that the department should adopt a single
package and that Mathcad was the package best suited to our
needs. In particular, Mathcad's graphical interface was ap
pealing from an educational point of view since students
could manipulate equations that looked like those found in
their textbooks or presented in class.
Mathcad was initially implemented in our sophomore ma
terial balance course taught from the Felder and Rousseau
180
text'". We required the students to use the software and
instructed them on its use, providing instructions throughout
the course when new material suggested the need for new
computational techniques. We also prepared a short tutorial
to help beginning students. As a general rule, students did
not have easy access to Mathcad manuals since relatively
few were available in the student computer labs. Therefore,
students generally depended upon online help, the tutorial,
inclass examples, the instructors, and the TA for assistance.
In addition to the material balance course, Mathcad was
heavily used in two upper division undergraduate courses
and was incorporated to a lesser degree into several other
undergraduate and graduate courses.
Four years have passed since the initial implementation of
Mathcad. Since that time we have observed a significant
change in the way in which our students approach and solve
engineering problems, and some of the change is directly
related to use of the math software. In addition, we have
gained experience with the software and have learned ways
John N. Harb is Associate Professor of Chemical Engineering at Brigham
Young University. He received his BS from Brigham Young University
(1983) and his MS and PhD from the University of Illinois, Urbana (1985
and 1988, respectively), all in chemical engineering. His research interests
include electrochemical engineering and mathematical modeling of com
plex physical systems.
Angela Jones is currently a graduate student in Botany at Brigham Young
University. She received her BS in chemical engineering from Brigham
Young University in 1996.
Richard L. Rowley is the James J. Christensen Professor of Chemical
Engineering at Brigham Young University. He received his BS from Brigham
Young University and his PhD from Michigan State University. He has
taught at Brigham Young University for the past thirteen years and taught
at Rice University for five years before that. His research interests are
molecular simulations and prediction, estimation, and measurement of
thermophysical properties.
W. Vincent Wilding is Associate Professor of Chemical Engineering at
Brigham Young University. He obtained his BS from Brigham Young
University and his PhD from Rice University. He worked at Wiltec Re
search Company, Inc., for nine years before joining the BYU faculty three
years ago. His research is in thermodynamics and environmental engi
neering.
0 Copyright ChE Diision of ASEE 1997
Chemical Engineering Education
in which it can be used to enrich learning. We have also been
able to observe the impact of adapting such software on our
overall undergraduate program in chemical engineering.
BENEFITS FROM USING THE PACKAGE
To illustrate the benefits listed in Table 1, we will examine
the solution of an engineering problem with Mathcad. Note
that the benefits listed in the table apply to equationsolving
packages in general, although specific capabilities will vary
from package to package.
An important part of engineering is the use of mathemati
cal expressions to describe and model the behavior of physi
cal systems. Solution of the resulting equations can be very
difficult and time consuming. Therefore, solutions are often
limited in scope, based on simplifying assumptions, and
dependent on a restricted set of variables in order to be
computationally tractable. With use of a tool such as
Mathcad, students can perform calculations more rigorously
and broadly, obtaining a deeper and more complete under
standing of the problem. The following solution of an
undergraduatelevel thermodynamics problem illustrates the
use of Mathcad to solve an engineering problem.
TABLE 1
Potential Benefits of an EquationSolving Package in an
Undergraduate Engineering Program
1. It enables more efficient solution of problems that had previously
been tedious or difficult to solve.
2. It enables solution of more realistic engineering problems that had
previously been impossible or impractical for undergraduate
students.
3. Use of the new capabilities provided by the tool could enrich and
enhance student learning.
Problem Set Up
Constants
T 3 Define bar bar 10o
STL =310 K
PL = 20 bar CO2 Gas constant Rg 0.08
Tank
T CO, Properties
compressor Critical constants Tc 304.1 K
P2 'L Acentnc factor o( 0 239
Ideal gas heat capacity.
Equation to solve: f(T2)AH P2 V2=0 joule
A 19.80 B = 0 0734
Solve for tank temperature, T2 mole K
Initial guess: T2 r 400K o
C 5 602105 D 1.715
moleK3
Cp id(T) A BT CT2 + D T3
Figure 1. Input section of Mathcad sheet for example pr
Sumner 1997
Problem Statement A cylinder is to be filled with CO,
from a compressor that discharges the CO, at 20 bar and
310K. Initially, the cylinder (also at 31 OK) is evacuated (see
Figure 1). Assuming no heat transfer to the cylinder, what is
the final temperature of the CO, in the cylinder when the
cylinder reaches the compressor line pressure?
Solution One can show from the first law of thermody
namics for this situation that the final temperature is found
by solving the following equation:
AHP,V,=0 (1)
where AH is the difference between the final gas enthalpy
and the enthalpy of the gas in the filling line, P, is the final
gas pressure in the cylinder, and V, is the final molar volume
of gas in the cylinder. All terms in this equation can either be
evaluated in terms of known values or be represented as a
function of the unknown final gas temperature (T,). The
enthalpy of the gas in the filling line is a function of the
filling line pressure (P,) and temperature (T,), both of which
are known. The final gas enthalpy must be evaluated at the
known final pressure P, and the unknown T,. The volume
(V,) is related to P, and T, through an equation of state.
Figure 1 is the input section of a Mathcad sheet for the
above problem. Mathcad allows text to be inserted alongside
the computation lines so that the calculation is easily docu
mented. In this example, text is shown in "Arial italics"
whereas the active Mathcad equations are in "Times Ro
man." Note that units can be used with variables and Mathcad
will take care of all unit conversions. Additional units, not
internal to Mathcad, can be defined, as has been done here
with bar. The newly defined unit can then be used through
out the worksheet. Known values are defined for T, and P,.
The gas constant is also specified, along with physical data
for CO,. The heat capacity is specified as a function of
temperature to be used by numerical routines.
Also included in the input section of the sheet is a specifi
cation of an initial guess for the final tempera
ture. The equation to be solved is also noted for
convenience as text; this is not an active
pa Mathcad equation. The equation has been writ
liter bar ten explicitly in the form f(T,) = 0 to empha
8314 
mole K size that we seek a root that satisfies Eq. (1).
Two separate solutions to the problem are il
Pc 73 8 bar lustrated in Figure 2 and discussed in the
following sections. These sections are intended
to illustrate the first two benefits listed in Table
joule 1. Such benefits are commonly perceived and
mole K2 have been documented by others.le' 21 How
0 joule ever, we often fall short of achieving the full
mole K4 educational benefits that the software makes
possible. Therefore, the last section of the ex
ample attempts to illustrate how the capabili
ties of Mathcad can be used to further enhance
oblem. learning (Benefit #3).
181
Using Mathcad for Efficient Problem Solut
(Benefit #1)
The first solution, which assumes an ideal gas, is
the type of solution required of students in the
enthalpy difference is expressed as an integral ove
capacity, which is substituted into the equation i
The Mathcad "root" function is then used to sob
desired temperature T2. The solution was complete
lines with minimal algebra. In contrast, a calcula
have been used to obtain the same solution. To d
would first perform the integration, then substitu
sulting polynomial into Eq. (1), and finally, pro
final function of T2 into the calculator. In this case,
provides a more efficient solution to a problem
traditionally been solved by other methods. Mat]
makes it easier to correct a mistake made during th
tion. Independent of how the solution is obtained,
the student arrives at the answer with little
physical insight. For example, how accurate is
the ideal gas assumption?
Using Mathcad to Solve More
Realistic Problems
(Benefit #2)
Figure 2 also presents a more realistic solu
tion to the problem, which uses the Soave equa
tion of state in place of the ideal gas assump
tion. (A full description of the equations used
in the solution is beyond the scope of this
paperinterested readers should refer to a text
on thermodynamics for more information.)
Mathcad provides an elegant solution to a com
plex mathematical problem that would be dif
ficult and/or impractical for undergraduate stu
dents to solve using either a calculator or a
spreadsheet. For example, P2V2 is required in
the equation, but the Soave equation cannot be
solved explicitly for V. In Mathcad, V is ob
tained as a function of temperature and pres
sure by solving the equation of state using a
root function. This volume is then used in the
calculation of the enthalpy departure function,
which is in turn used as part of the expression
that is solved iteratively for T2.
A realistic solution of the problem can thus
be achieved in a straightforward manner. Such
a solution would not have been practical for
undergraduates without the use of software such
as Mathcad.
What have we gained from the more realistic
solution of the problem? Because of the time
saved in the simplified solution of the problem,
it is feasible to ask students to solve the prob F
lem both ways and to compare the answers.
182
The fact that the answers are different is significant, but the
physical insight gained from the comparison is still quite
limited since the student is only comparing two numbers.
Note that students also gain experience in using realistic
equations of state by performing the more complicated cal
culations. Using a tool such as Mathcad to expose students
to models and methods actually used in industry better pre
pares graduates for realistic engineering calculations. In con
trast, using inappropriately simple models for computational
convenience may give students a false impression of the
applicability of such models.
Using Mathcad as a Teaching Tool
(Benefit #3)
There is considerably more that can be done in using the
software to improve learning. For example, learning can be
enhanced by extending the problem statement itself. In this
Solution for Ideal Gas with Cp as a function of temperature
T2
AH id(T2) Cp id(T) dt
*TT.
f(T2) AHid(T2) Rg.T2
Note: For ideal gas, P2 V2=Rg.T2
T2 id root(f(T2),T2)
T2 id=392.874K Answer for Ideal gas
Solution for Soave Equation of State
1) Define equation of state in terms of known parameters
0 42747 RgTe2 b 0.08664RgTc
Pc Pc
/ [2i )
a(T)= 1 + 1
m = 0.48508 + 1.55171 0.15613o2
F(T,PV) =P RgT a(T)
V b V.(V+b)
Soave equation of state
2) Set up expression to calculate V for a given T and P from the equation of state
Initialguess" V = 0.5liter
Expression: V(T,P) root(F(T,P,V),V)
3) Define enthalpy departure function (Hid H) for Soave equation of state
T
K(T) : ma (T)
STca(T)
AH(TP) Rg.T.(I .T b.Rg l + )J
T,P) = RT P V(T,P) aa(T) KT) In I+ b
\ Rg T bRgT \ V(T,P) j
4) Define and solve final expression for T2
AHreI(T2,P) = AH(TL,P) AH(T2,P) + AHid(T2)
f(T2,P2) = AH real(T2,P2) P2.V(T2,P2)
T2 al(P2) root(f(T2,P2),T2)
T2 real(P2) =380 36K Answer for real gas
igure 2. Mathcad solution assuming ideal gas (top) and using the Soave
equation of state (bottom).
Chemical Engineering Education
problem, students might be asked to calculate the final tem
perature in the cylinder for a range of compressor line pres
sures and to plot the results. This extension is easily achieved
with Mathcad by adding a couple of assignment state
ments and creating a graph. The resulting Mathcad fig
ure, Figure 3, provides physical insight not previously
available. The student can see that the two solutions
approach each other as the pressure is decreased and CO2
behaves more ideally. It is also apparent that the two
solutions diverge as the pressure is increased, and that
the ideal gas solution does not change with pressure.
To be still more useful, the extended problem statement
should include qualitative questions that ask the students to
explain, justify, and generalize their observations. For ex
ample, why do the two curves diverge as the pressure is
increased? Or, why is the ideal gas solution independent of
pressure? It is also useful to have them evaluate the practical
importance of the result by asking a question such as what
significance would the error introduced by the ideal gas
assumption for CO2 have on the design of ... ?
The extended problem helps students gain insight into the
effect of pressure on the nonideal behavior of CO2, but what
about other gases? When would they need to worry about
nonidealities? To teach this, one might have the students
repeat the solution for a different gas. As the equations are
identical, the solution is readily obtained by simply chang
ing the parameters at the top of the sheet. You may want to
ask the students to predict the results for the second gas
prior to performing the calculation. For example, "Would
you expect N2 to behave more or less like an ideal gas than
CO2? and "Why?" Because the selected temperature range
is so far above the critical point of N2, the students should
expect more ideal behavior from N2. Figure 4 shows the
results for N2 indicating that this is indeed the case. Students
could be asked to compare the solutions and rationalize their
observations. For example, "Why is one gas more ideal than
0 10 20 30 40 5C
Final Tank Pressure (bar)
Ideal
 Real
Figure 3. Temperature of the tank as a function of the
CO, final pressure.
Summer 1997
another at these conditions?" Other questions related to the
properties of the two gases might also be included, such as
"Which gas has the higher heat capacity, and why?"
The above example illustrates several capabilities of
Mathcad relevant to engineering education. (As mentioned
previously, these benefits also apply to other equationsolv
ing packages.) First, the software enabled efficient solution
of engineering problems as shown in the initial "ideal" solu
tion to the problem. Solution of the same problem using the
Soave equation of state demonstrated a realistic solution that
was not previously tractable, and use of both the ideal and
real solutions allowed comparison of the two. Mathcad also
permitted calculation of the temperature for a range of pres
sures and visualization of the results, providing physical
insight that was previously unavailable. Qualitative ques
tions were used to help develop this insight. Finally, calcula
tions were easily repeated for a different gas, and students
were asked to rationalize and evaluate the differences, illus
trating the use of Mathcad to answer "What if...?" questions.
In short, Mathcad permitted extension of the original prob
lem, at relatively little overhead, to enhance learning. This
type of extension, however, requires effort on the part of the
instructor; it does not follow automatically with the use of
Mathcad. Using the "same old problem statements" with the
new tool will do relatively little to enhance student learning.
In fact, in some cases, it may even reduce learning.
There are other ways to use equationsolving packages to
enhance the education of engineering students. The combi
nation of powerful computational and graphical capabilities
is ideal for exploring tradeoffs between opposing factors in
order to reach an optimal solution. Students might be asked
to evaluate and recommend options based on economic con
siderations. Important physical insight can be gained by
using software such as Mathcad to identify the most impor
tant variables or factors and then offering an explanation for
Filling of N2 Tank
(D
0.
E
C)
45C I I I I
45
400
350d I I I
0 10 20 30 40 50
Final Tank Pressure (bar)
Ideal
 Real
Figure 4. Temperature of the tank as a function of the
final pressure for N2.
Filling of C02 Tank
their relative importance. Students might be asked to solve a
problem and then use the software to explore an issue of
their own choice related to that problem. These and other
opportunities are made possible by the tool's capabilities.
Relationship to Learning Styles
We can use the tool's flexibility to teach to a variety of
learning styles. Students have a variety of learning prefer
ences or styles, and the needs of the individual learners are
best met with a variety of learning activities.13 ' Activities
such as those described in the example problem can be used
to help address different learning styles and to improve the
learning of all students. For example, Mathcad provides an
excellent opportunity for students to visualize information, to
actively experiment with the concepts and ideas, to explore
new ideas, and to solve realistic problems. As such, it repre
sents another important addition to our instructional toolbox.
The following sections of the paper document our experi
ence with the equationsolving package.
FACULTY PERSPECTIVE
The response of our faculty to the introduction of Mathcad
into the curriculum varied from one person to another, but
was generally positive. Individual responses were correlated
with the level of awareness: awareness of the tool's features,
awareness of students' capabilities with the tool, and aware
ness of how the tool can impact learning in and out of the
classroom. Generally, the more familiar the faculty member
was with the tool, the more positive the reaction. Our obser
vations regarding the faculty have been divided into the
following seven areas.
Faculty Involvement
Mathcad was initially introduced into the curriculum by a
core group of two faculty members. Although the entire
faculty voted in support of the program, there were no re
quirements for using Mathcad in specific courses other than
the initial sophomore course. Consequently, only some of
the faculty have learned the software and adapted their teach
ing to take advantage of the opportunities it provides. In fact,
less than half (five out of twelve) of our faculty currently use
the software, although some who have not used it them
selves have required their students to use it in specific courses.
The number of "users" appears to be growing as other fac
ulty members have recently shown interest. We believe that
more faculty will adopt the software as they become aware
of the significant positive impact it can have on their courses
and the value that students obtain from those courses.
Student's New Capabilities
Using Mathcad has enabled our students to efficiently
solve a variety of engineering problems. Some faculty have
been surprised by the new capabilities of our students. One
faculty member teaching a juniorlevel class in our depart
184
ment commented that he didn't know exactly what we had
changed in our sophomore class (where Mathcad is intro
duced), but "we are certainly doing something right!" He
believes that the students understand concepts better and
compute significantly better than previous classes. This same
faculty member began using Mathcad because of the capa
bilities he saw in his students. A faculty member from an
other department found that our students were able to easily
and quickly complete his traditional assignments. The stu
dents had asked his permission to use Mathcad on the as
signments beforehand, and he had agreed. He was delighted
and amazed at the ease with which they completed the work.
Math Skills
Some faculty members have sensed a decline in the math
skills of some students, presumably due to their dependence
on the software to do the math. They mention that students
do not have the same "by hand" math skills possessed by
previous groups of students. This is, in a real sense, both a
practical and a philosophical issue. Are these math skills a
means to an end, or do they have intrinsic worth to the
students? This question is certainly not a new one, having
been raised, for example, with regard to calculators. In fact,
many "by hand" math skills once thought essential have
been forgotten with the advent of the slide rule and the
calculator. Similarly, the use of equation solvers may lead to
another shift in the skills required of engineers and engineer
ing students. It seems to us that the key issue is not the
ability of students to manipulate equations, but their ability
to solve engineering problems.
Exam Procedures
One issue of practical concern among the faculty has been
the effect of Mathcad on exam performance. Exams have
been influenced in several ways by the use of an equation
solver. First, a couple of instructors have given exams that
require students to use Mathcad. They have been given
primarily as "take home" exams with a specified time limit.
Additionally, the exam content has changed in some classes
to focus more on problem definition and setup rather than on
solution methods. Qualitative problems, similar to those dis
cussed previously in conjunction with the Mathcad example,
have also been used effectively on exams.
The exams in many of our undergraduate classes, how
ever, remain unchanged. Also, for logistic reasons, most of
our students have not had access to Mathcad during exams.
In fact, use of the equation solver on exams has been largely
restricted to a few upperdivision courses. The unavailability
of Mathcad has caused difficulties for a few students who
felt it hindered their performance on exams. For example,
they might develop a problem to the point of one or more
equations and then state, "I could now use Mathcad to solve
this problem." Such responses have raised a concern that the
software may have a negative impact on students who have
Chemical Engineering Education
lost (or never developed) the ability to simplify problems,
make approximations, or even solve relatively simple math
ematical problems.
We view these concerns as a manifestation of the math
skills issue discussed in the previous section where students
have developed skills that differ from those expected by the
instructor. This mismatch between the students' skills and
those expected by faculty can be avoided if faculty members
are aware of the tools the students are using and if they
clearly communicate their expectations to the students. If we
expect students to work a certain type of problem by hand
(e.g., on an exam), we should tell them so, encourage them
to practice solving it that way, and reinforce our expectation
by giving them problems of that type to solve.
Our need to explicitly communicate our expectations has
increased as alternate methods for solving problems have
been developed. As a general rule, students will solve prob
lems in the way they deem most efficient (usually with the
equation solver) unless they are required to do otherwise.
Impact on Course Content
Incorporation of an equation solver into our undergraduate
curriculum has had, and continues to have, a significant
impact on the content of several courses. For example, our
separations course was recently restructured to take advan
tage of both Mathcad and a process simulator. We have also
found that we spend much less time teaching numerical meth
ods and solution procedures than was previously necessary,
especially in courses that are computationally intensive.
Using an equation solver has also had significant impact
on course content by providing the means for students to
solve difficult, realistic engineering problems. The choice of
problems is now less restricted by the means available for
their solution, and this has allowed problems to be selected
for their content and connection to real engineering rather
than for their solvability.
Impact on Learning Concepts
Several faculty members have used Mathcad to help stu
dents learn concepts, with very positive results. It is their
perception that the software has helped students to develop a
better conceptual understanding of the material.
Other members of the faculty, however (particularly those
who have not used the software), feel that the software has
had an adverse effect on student learning. For example, one
faculty member recently expressed a concern that students
were losing touch with the physics behind problems. He
expected students to use physical insights to make assump
tions that simplified problems for solution on a calculator.
Many of the students, however, simply solved complete sets
of complex equations in Mathcad without thinking about the
physics. In this case, the problems were designed to maxi
mize learning when worked by hand and were therefore not
Summer 1997
ideally suited for advanced computational tools. Conse
quently, the educational value of the problems was short
circuited by the ability of the students to solve the complete
set of equations without simplification.
This example illustrates the fact that faculty members
should know what tools the students are using and should
adjust instruction and/or requirements to maximize learning.
In the above case, the professor could have required the
students to work the problems by hand in order to achieve
the desired results. Alternatively, the problems could have
been restructured to take advantage of the available tools
and still achieve (or exceed) the desired educational objec
tives. For example, students could have been required to calcu
late the magnitude of individual terms in the equations and to
comment on their physical and numerical significance.
Group Assignments
There has been a movement in recent years toward more
collaborative learning and teamwork. Consequently, we need
to consider the effects on learning as we use computational
tools in group assignments. We have observed, for example,
that students in a group at a single computer do not derive
equal benefit from an assignment. In particular, the indi
vidual actually at the keyboard appears to get more out of the
exercise. The literature discusses creative ways around this
problem." 6 For example, each member of a group might
be given instruction on a particular feature of the software and
then be asked to teach that feature to their group. Additionally,
the group might be asked to solve a complicated problem that
requires the "skills" of all the group members to complete.
Summation
In summary, faculty who have used Mathcad have gener
ally been pleased with the software and found it to be a
valuable education tool. In spite of this, less than half of our
faculty use Mathcad. It was also the perception of the faculty
that a majority of the students used the tool, even when the
teacher did not anticipate (or desire) its use. Some teacher
and student frustration has been observed in courses taught
as they had been in the past, with no modification of assign
ments or instruction to take advantage of Mathcad. This
frustration can be avoided, however, by clearly communicat
ing expectations to the students and/or adapting problems to
take advantage of the educational possibilities that a math
tool such as Mathcad makes available.
STUDENT FEEDBACK
A survey was conducted in order to evaluate the equation
solver from the students' perspective. The survey consisted
of ten questions, and we have chosen to discuss five of these
(shown in Table 2). Three of the remaining five questions
were actually subsets of the questions chosen for discussion;
the responses to these questions have been grouped with the
questions shown in Table 2. One of the last two questions
185
was specific to our institution and was therefore not in
cluded. The last question was discarded because it was poorly
phrased and did not yield any useful information. Responses
were received from sophomores, juniors, and seniors. First
year students were not surveyed since Mathcad is not used as
part of the curriculum in the first year.
Question #1: Overall Opinion A combined total of just over a
hundred responses from sophomore, juniors, and seniors was re
ceived for the question that asked students for their overall opinion
of the Mathcad package. Of these, the responses from the seniors
were overwhelmingly positivein fact, none of the seniors re
sponded negatively.
The response from the juniors was also very positive, with only
about ten percent being negative. Several of the positive responses
were also qualified with statements such as "I like it but I'm just not
all that familiar with it," or "Great timesaving tool if you know
how to use it." Responses from the sophomores were similar to
those from the juniors.
It was evident from the responses that the seniors were more
comfortable with the software package than either the juniors or
sophomores. Incidentally, the positive opinions expressed by the
students are strongly supported by the fact that they used the
software in almost all of their courses, even when it was not
required or expected by the instructor. In our experience, this has
never been the case with FORTRAN programming.
Question #2: Benefits Students noted several benefits in using
Mathcad. The most frequently mentioned benefit was that using the
software typically reduced the time required to solve a particular
problem. Time savings were attributed to the package's ability to
solve equations and to perform unit conversions. Students also
noted that mistakes made early in the sequence of calculations
could be easily fixed without having to "repunch" the numbers.
The students also recognized that Mathcad allowed them to solve
problems that otherwise would not have been assigned. In addition,
students noted that the software made it easy to see the equations
and to follow the logic of the solution. Others mentioned that
Mathcad allowed them to spend more time learning concepts and
less time with tedious calculations. The graphing capability of the
package was also noted as a benefit by students.
Question #3: Conceptual Understanding The survey asked
students to evaluate how the use of Mathcad to complete home
work assignments affected their ability to understand the concepts
presented in class. Most of the student responses to this question
could be classified into three groups. Responses in the first group
were from students who got lost in the solution procedure and felt
that Mathcad actually had a negative influence on their conceptual
understanding. Additional instruction would undoubtedly help these
students. Some of the students who struggled were transfer students
who did not have the same exposure and experience with Mathcad
as students who began their studies in our department.
It was the opinion of the second group that Mathcad was merely
a computational tool that did not have any effect, positive or nega
tive, on their ability to understand concepts. The third and largest
group felt that Mathcad enhanced their conceptual understanding
by allowing them to focus on the concepts rather than on the math
or mechanics of the solution and by providing them with tools to
help increase their understanding.
TABLE 2
Sample Survey Questions
1. What is your overall opinion of Mathcad?
2. What are the benefits of using Mathcad?
3. The purpose of homework is to help you understand concepts.
How has using Mathcad on homework affected your ability to
understand the material presented in class or in the text?
4. Mathcad is typically not available on exams. How did this affect
your performance on exams?
5. How can the Chemical Engineering Department improve the use
of Mathcad in undergraduate education.
Student comments indicated that the software had a significant
positive impact on their education. As far as overall distribution is
concerned, just over half (42 of 83) of the students whose responses
fit into one of these three groups felt that Mathcad was beneficial to
their conceptual understanding (group 3).
Question #4: Influence on Exams The survey also asked
students to evaluate the impact that incorporation of an equation
solver into the curriculum has had on their exam performance.
Mathcad is not typically available for use on exams in our depart
ment so we expected this issue to be a sore spot with students. The
response to this question was also divided. Some students felt that
using Mathcad on homework assignments had a negative impact on
their exam performance. In fact, several students referred to a
dependency on Mathcad that weakened their math skills and al
tered their approach to problems, which was detrimental to good
performance on exams. There were also students who considered it
unfair not to allow use of the software on exams. In contrast, it was
the opinion of a number of students that using Mathcad on home
work had no substantial effect on their exam performance.
Finally, there was another group of students who thought the
exams were actually easier because they had been able to solve
more difficult, complicated problems as homework on Mathcad.
These students felt they had a deeper understanding and a better
command of the material than they would have had if they had not
used the software.
Of the 76 responses that fit into one of these three groups, just
over 40% thought that using the equation solver did not have a
significant impact on exam performance. The remaining responses
were split evenly between those who felt that Mathcad had either a
positive or a negative impact on their exam performance. There
was a substantial number of students (about 30) who provided
responses that could not be classified. Most of them failed to
address the question that asked them to evaluate the impact of the
software on exam performance. For example, many of these stu
dents commented on whether or not they thought the equation
solver should be used on exams, without evaluating the impact of
the current implementation on their exam performance.
Question 5: Suggestions Most students felt the department
could improve the use of Mathcad by teaching the program earlier
and/or better. Apparently many students perceive that Mathcad is
difficult to learn, contrary to our expectations prior to implementa
tion of the software. Students recommended that Mathcad be taught
either in the freshman year or in the sophomorelevel programming
Chemical Engineering Education
course that currently covers only FORTRAN. Other suggestions
included integration of the software into more courses in the cur
riculum and making it available on exams. Additional tutorials and
inclass examples using Mathcad were also suggested.
In response to student suggestions, our traditional FOR
TRAN programming class was expanded this past year to
include parallel teaching of FORTRAN, spreadsheets, and
Mathcad, with an emphasis on tool strengths for different
types of problems. Toward the end of the semester, the
students were asked to compare the strengths and weak
nesses of the packages and to identify their preferences.
Student preferences were: Mathcad, 64%; spreadsheets, 25%;
and FORTRAN, 11%. Most students were perceptive enough
to see the utility of all three programs for different applica
tions. Many of them felt that seeing the problem in math
ematical terms in Mathcad facilitated their understanding
and solving of the problem. There was also a feeling that
FORTRAN was out of date and that Mathcad is a more
powerful and flexible tool than the others. The positive
perceptions of these students toward Mathcad and the
sense of ability to solve problems would be expected to
enhance student learning.
SUMMARY AND CONCLUSIONS
This article examined the use of an equation solver,
Mathcad, in the undergraduate chemical engineering cur
riculum. An example illustrated some of the capabilities of
this tool and, more importantly, ways in which it can be used
to enhance learning. As demonstrated by the example, equa
tionsolving packages such as Mathcad are powerful and
flexible tools that can be used to develop a variety of instruc
tional activities designed to promote learning.
Students responded positively to the tool and used it in
most of their classes, even when it was not required. Most
students felt that using Mathcad enhanced their conceptual
understanding of the material. Specifically, it allowed them
to focus on the concepts rather than on the solution proce
dure and gave them a tool to "experiment" with the material.
But other students struggled with the software and felt that it
had a negative impact on their learning. Additional instruction
and practice is needed to help these students. In general, stu
dents felt that Mathcad was difficult to learn at first and that
formal instruction should be provided to beginning students.
Faculty who have used Mathcad have generally been
pleased with the software; in spite of this, it is currently used
by less than half of our faculty. Occasional frustration has
been experienced in courses where no modification of as
signments or instruction has been made to take advantage of
Mathcad. These frustrations can be avoided by clearly com
municating expectations to the students and/or adapting prob
lems to take advantage of the educational possibilities that a
math tool such as Mathcad makes available.
In conclusion, we have found that Mathcad can be used
Summer 1997
effectively by innovative teachers to promote learning, to
deepen student understanding of concepts, and to provide
experience with more realistic engineering problems. But it
is not a panacea in this regard, and it can actually be detri
mental to learning if used inappropriately. We have found
that many students enjoy solving problems in Mathcad and
therefore use it as a tool of choice, often in ways the faculty
had not anticipated.
RECOMMENDATIONS
Based on our experience, we offer the following recom
mendations to departments considering implementation of
Mathcad or some other similar package into their under
graduate curriculum:
1. It is important that the faculty learn and use the software.
Therefore, some method of training and encouraging the
use of the software by faculty members should be imple
mented.
2. Students perceive the need for formal instruction on using
the software. We recommend that the instruction be pro
vided early, in the first or second year, and that at least a
portion of a course should be dedicated to learning the
software.
3. Implementation of such a package requires substantial num
bers of adequately equipped computers. Such facilities should
be available before implementing the software as a required
part of the curriculum.
4. It is important that the software be used consistently through
out the curriculum in order for students to develop and
maintain proficiency. We recommend that most, if not all, of
the core courses actively incorporate the software. This will
help students to be sufficiently comfortable with the soft
ware to go beyond the mechanics and use it as a learning
tool.
5. Opportunities should be provided for faculty to share their
experiences and ideas on using the software to promote
learning. This could be part of a faculty meeting, a users
group, etc.
6. If possible, facilities should be provided so students can
have access to the software during exams.
REFERENCES
1. Felder, R.M., and R.W. Rousseau, Elementary Principles of
Chemical Processes, John Wiley & Sons, New York, NY
(1986)
2. Sandler, S.I., "Spreadsheets for Thermodynamics Instruc
tion: Another Point of View," Chem. Eng. Ed., 31, 18 (1997)
3. Felder, R.M., and L.K. Silverman, "Learning and Teaching
Styles in Engineering Education ," Eng. Ed., 78, 674 (1988)
4. Stice, J.E., "Using Kolb's Learning Cycle to Improve Stu
dent Learning," Eng. Ed., 77(5), 291 (1987)
5. Harb, J.N., S.O. Durrant, and R.E. Terry, "Use of the Kolb
Learning Cycle and the 4MAT System in Engineering Edu
cation," J. of Eng. Ed., 82(2), April (1993)
6. Johnson, D.W., R.T. Johnson, and K.A. Smith, Active Learn
ing: Cooperation in the College Classroom, Interaction Book
Company, Edina, MN (1991) J
L=" learning in industry
This column provides examples of cases in which students have gained knowledge, insight, and
experience in the practice of chemical engineering while in an industrial setting. Summer interns and
coop assignments typify such experiences; however, reports of more unusual cases are also welcome.
Description of analytical tools used and the skills developed during the project should be emphasized.
These examples should stimulate innovative approaches to bring real world tools and experiences
back to campus for integration into the curriculum. Please submit manuscripts to Professor W. J.
Koros, Chemical Engineering Department, University of Texas, Austin, Texas 78712.
INTRODUCING
GRADUATE STUDENTS
TO THE INDUSTRIAL PERSPECTIVE
JAMES B. RIGGS, R. RUSSELL RHINEHART
Texas Tech University Lubbock, TX 794093121
t has long been recognized that there is a wide gap
between industrial practice and most university research
and that this gap seems considerably pronounced in the
field of chemical process control. In the extremes, academic
researchers deal almost completely in applied mathematics,
emphasizing proofs and theorems, while it is likely that a
process control engineer has had no formal training in pro
cess control and his knowledge has come largely from on
thejob training. It is not surprising that there is almost no
exchange of ideas between these two groups since they do
James B. Riggs is Professor of Chemical Engi
neering at Texas Tech University. He holds a
PhD from the University of California, Berkeley,
and an MS and BS from the University of Texas at
Austin. His primary research areas are process
control and online optimization. He has published
over fifty papers in technical journals and is the
author of a numerical methods text for chemical
engineers. He also has over five years industrial
experience.
R. Russell Rhinehart is a Professor in the ChE
Department at Texas Tech University. After re
ceiving his BS in chemical engineering and his
MS in Nuclear Engineering from the University of
Maryland, he worked for thirteen years in industry
before obtaining his PhD in chemical engineering
from North Carolina State University. His research
interests are in using computers to automate pro
cess management decisions.
Copyright ChE Division of ASEE 1997
not speak the same "language" and neither group, in general,
appreciates the value of the other. The academic feels that
the process control engineer does not understand the funda
mentals of process control, which is generally true, while the
process control engineer feels that the academician does not
understand how to make process control work in an indus
trial setting, which is also accurate.
The truth of the matter is that both the academic and the
process control engineer could greatly benefit from the knowl
edge held by the other. Process control engineers could
better perform their job if they had a fundamental under
standing of the principles of process control. Likewise, the
developments produced by academics would be more likely
to have an impact on the practice of process control if they
understood the key issues associated with the industrial imple
mentation of process control.
In the Department of Chemical Engineering at Texas Tech
University, the process control program is aimed at the in
dustrially relevant study of advanced methods in chemical
process control. Moreover, we strive to instill an industrial
perspective of process control in our graduate students so
they are prepared to perform industrially relevant research
and upon graduation have the knowledge and skills to effec
tively perform in an industrial setting. This paper addresses
approaches that we have found useful; these approaches
Chemical Engineering Education
involve a combination of classroom instruction, research
activities, interaction with industrial control engineers, and
summer internships.
ELEMENTS OF THE
INDUSTRIAL PERSPECTIVE Teach
The practice of process control is based on students I
issues such as process knowledge, controller principles
reliability and maintainability, and operator obvious
acceptance. A key distinguishing character but
istic of chemical process control engineers is student
their reliance on process knowledge in the where th
application of process control. This knowl tofunctii
edge is based on understanding the opera an indu
an indu
tional objectives and constraints of a process also
unit and understanding how the units com
prising the process are interconnected. With We hav
this knowledge, the control engineer can industrial
implement control approaches that are cor be effect
patible with and even enhance the overall into th
operational objectives of the process, program.
As a simple example, consider the prob classroom
lem of tuning a level controller. Without re acti
gard to the rest of the process, the level con inter
troller can be tuned to provide very tight in
level control. If, for example, the outflow repr
from the level controller serves as the feed to
a reactor, tight level control would likely be
undesirable since it would result in sharp changes in the flow
rate to the reactor. On the other hand, for certain cases tight
level control may be required. Therefore, knowing how the
controller fits into the rest of the process is a prerequisite to
properly implementing control. In other words, overall pro
cess knowledge is important to insure that a cost effective
solution to the control problem has been obtained.
Process knowledge is valuable as a consistency check for
a process control application to ensure that the application
will effectively fit into the overall process scheme. Research
ers have recently shown that process understanding is very
useful in designing control systems for complex units.[!I
Reliability is a major issue for industrial process control.
A controller may perform well 99% of the time, but if it
upsets the entire plant even 1% of the time, it is unaccept
able. Online factors of 95% are considered good, but it is
not acceptable for a controller to create major upsets
even occasionally.
The amount of effort required to keep a controller online
with an acceptable service factor relates to the maintainabil
ity of the controller. Maintainability is a key issue with
industrial process control. For example, a highmaintenance
controller may function well as long as the control engineer
who implemented it is maintaining it. But as soon as the
original control engineer moves on to a new assignment, the
controller will fall into disuse. Industrially, the KISS prin
ciple (Keep It Simple, Stupid!) is well established from a
maintainability standpoint.
ing graduate
the fundamental
Sof technology is
sly important,
cutting these
's in a position
ey are prepared
on effectively in
strial setting is
important.
Found that the
perspective can
lively integrated
ie oncampus
m through the
, research group
vities, and
actions with
idustrial
esentatives.
pecially when dealing with operators.
APPROACH
There are a variety of ways to convey to graduate students
the sense of an industrially relevant approach to process
control. They include
Classroom and laboratory material and exercises
Research projects
Research group meetings
Interactions with industrial process control engineers
Summer internships in industry
The classroom is an ideal venue for introducing the ele
ments of the industrial process control pointofview. At
Texas Tech, we offer three graduate courses in process con
trol: a first course, a course on modelbased control, and a
laboratory course. There are no specific lectures on the topic
of industrial process control practice; instead, the topic is
interspersed throughout each course. That is, as each topic
and method is presented, a discussion of how it is addressed
in industry is added. In addition, certain simulator software
is used to give the student the feel of a dynamic process for
controller implementation and tuning. To effectively inte
grate the industrial pointofview into the classroom, it is
essential to have instructors with significant industrial ex
Summer 1997
Finally, interpersonal skills have a tre
mendous effect on the effectiveness of a
process control engineer. Being able to
work effectively in a group setting is es
sential in today's chemical processing in
dustry. There is no place where interper
sonal skills are more important for a pro
cess control engineer than in dealing with
operators. If an operator does not trust the
ability of a process control engineer or
does not like the engineer's attitude, the
engineer's controller will not be used even
when it works well. Control engineers who
are effective are able to get the opera
tors to assume "ownership" of the con
trol project. This is usually done by
soliciting information from the opera
tors and listening to their input. Telling
the operators what they should be do
ing or how they should do it is the fast
est route to failure. Although we do not
attempt to teach interpersonal skills di
rectly, we do make sure that our stu
dents are aware of their importance, es
perience. This experience can come from fulltime em
ployment with industry or by working directly with in
dustrial plant operations.
As the students perform their research, there are always a
number of issues that come up in the pursuit of maintaining
industrially relevant research. On many occasions, our weekly
research group meetings involve discussions of research pa
pers from other groups. During those times, a number of
interesting issues arise that are related to industrial relevance.
One of the best ways to convey the industrial perspective
is to promote direct interaction between the student and the
industrial process control engineer. We have semiannual
consortium meetings (discussed later) during which the lead
ing control engineers from our member companies visit our
campus. In addition, a number of our research projects re
quire frequent input from industrial experts.
While the previous approaches have largely occurred on
campus, the summer internship approach requires the stu
dent to join the company for a summer and to work as a
control engineer in a plant. Obviously, for the students to be
effective as summer interns, they must have been exposed to
the elements of the industrial process control perspective on
campus. The summer internship demonstrates firsthand the
various elements of the industrial perspective, which greatly
increases the student's understanding of these important is
sues. In addition, in several cases, a graduate student has
gone onsite for periods of just one to two weeks to install
control software (the result of their research projects).
CONSORTIUM
The Texas Tech Process Control and Optimization Con
sortium was established in 1992 as a mutually beneficial
arrangement between industry and the Chemical Engineer
ing Department. We currently have eleven industrial mem
bers representing the major sectors of the chemical process
ing industry: the leading refining companies, chemical com
panies, and control consulting companies.
The Consortium is an indispensable element in our effort
to expose students to the industrial practice of process con
trol. During the semiannual meetings, the industrial repre
sentatives of our member companies review and guide our
advanced control research to keep it industrially relevant. In
addition, they serve as a major source of research ideas and
as a source of industrial data. During the fall Consortium
meeting, graduate students present posters on their research
and the industrial visitors have an opportunity to discuss the
research approach and results directly with the students. The
students are often inspired by the opportunity to interact
with process control experts from these leading companies.
ESTABLISHING A SUMMER INTERNSHIP
The key to establishing a successful summer internship
experience is ensuring that all parties involved benefit, i.e.,
190
the student, the company, and the university. The first
step is to establish an industrial contact who will sell the
summer internship idea within the company. We have
used our Consortium contacts and other professional con
tacts for this purpose.
Many companies have a summer internship program in
place, but they are typically used for recruiting undergradu
ates. As a result, the industrial contact usually only has to
convince a midlevel manager to use one of the allotted
summer internship positions for a graduate student.
An advantage for the company is that with a graduate
student they are more likely to have a summer intern who
will actually accomplish something significant. Typically,
undergraduate summer interns are not trusted with much
responsibility. For example, an undergraduate is usually as
signed to an engineer and given small tasks such as tracing
utility lines or completing backlogged paperwork.
In order for graduate student summer interns to be suc
cessful, the company must be willing to give them enough
responsibility so they can have a chance to make signifi
cant contributions during their threemonth tenure. Be
fore giving the student such responsibilities, the supervi
sor must have confidence that the student has the skills to
be successful. Students are usually watched closely until
they prove themselves.
It is best if at least a portion of the summer internship
assignments directly relate to the student's university re
search. This is not always possible, but it is certainly desir
able. It is usually easier to establish a summer internship for
a graduate student if the student is a U.S. citizen.
The advantages to each of the parties involved for the
summer internship program described above are
The Company The company is exposed to a highly qualified
potential employee and to technology that it may not have.
The Students Students are exposed to industry and industrial
operations directly related to their research area.
The University Projects of this sort enhance the university's
interaction with industry, providing industrial results for con
trol approaches, and in certain cases, industrial data on pro
cesses of interest. In addition, working directly with industry
can lead to the identification offimndamental problems that can
be assimilated into the university's research program.
SUMMER INTERNSHIP EXPERIENCES
> Student A
Student A was an undergraduate who joined our graduate
program after his graduation in the fall semester. During the
spring semester we made arrangements for a twosummer
internship for Student A. In addition, he took our first pro
cess control graduate course in the spring semester. The
chemical company understood that his master's research
project involved using neural networks for distillation con
Chemical Engineering Education
trol and that he would spend a portion of his time with them
implementing the controller on one of their columns.
In discussions with supervisors from the chemical plant,
the decision was made that the student would spend the first
summer identifying the test column, closing material and
energy balances around the column, benchmarking a steady
state simulator on the column, tuning the flow and level
controllers on the column, and evaluating the online prod
uct analyzers. The first summer went as planned, with Stu
dent A spending approximately 20% of his time on the tasks
relating to his control project and the remainder of his time
on various plant projects (controller tuning, troubleshoot
ing, etc.). The student required about one month to become
comfortable with the company's distributed control system.
The company was pleased with the student's performance
since he was handling the typical responsibilities of a pro
cess control engineer for most of the summer. During the
following school year, Student A took classes and worked
with one of our PhD candidates on developing and testing
the neural network controller.
Student A returned to the chemical plant for the second
summer and installed and tested the neural network control
ler. During the second summer the student spent approxi
mately 50% of his time on his control project. The controller
used a steadystate neuralnetwork model of the column
combined with a simple lineardynamic model. The indus
trial column exhibited much more complex dynamic behav
ior; therefore, even though the neural network controller
understood the steadystate nonlinearity of the process, the
complex dynamics of the industrial column undermined the
effectiveness of the controller.
Although this control project did not result in a successful
industrial controller, the student did identify a fundamental
problem with the way that we were applying neural net
works for control. As a result, we are currently studying
neural network controllers that can handle highly com
plex dynamics behavior.
It is noteworthy to point out that the portion of the plant
where Student A was doing most of his work had some of
the oldest operators in the plant. In fact, this group of opera
tors was well known for being difficult to work with. Through
patient efforts, however, Student A was able to build a
positive relationship with the operators so that by the
second summer they were doing all they could to help
him with his control project.
Finally, it should also be pointed out that upon graduation
with his master's degree, Student A took a process control
position with the same chemical company where he had
done his summer internship.
 Student B
One of the industrial representatives from our Consortium
invited one of the authors (JBR) to attend a meeting on
Summer 1997
developing a companywide approach for controlling C3
splitters (propylenepropane separation). Since Student B's
PhD research involved C3 splitters, he also went to the meet
ing. We both made presentations on our research to the group
and participated in the discussions that followed. An approach
for controlling all the company's C3 splitters was developed
and endorsed by the membership of the committee.
Before we left, I proposed to the chairman of this commit
tee that Student B be hired by the company for a summer
internship so he could demonstrate the proposed control
approach in their plant. About six weeks later, after a couple
of followup calls, the student received an offer for a sum
mer job with the company.
When Student B joined the company for his summer em
ployment, he was assigned to an ethylene plant that con
tained a C3 splitter. During the summer, he served as a
control engineer for this ethylene plant and was able to
implement the new control approach on the C, splitter. The
new controller was successfully commissioned and then
turned over to operations. The new controller remains in
service while providing enhanced propylene recovery and
reduced variability in the propylene product.
In addition, Student B worked with the temperature con
trols for the ethylene furnaces. His efforts resulted in re
duced outages due to furnace coking. The company deter
mined that the benefits of the C3 splitter and ethylene fur
nace work is saving them in excess of one million dollars per
year. Once again, a major reason for the student's successful
summer internship was the positive relationship that he de
veloped with the operators. The company offered the student
employment for the following summer.
CONCLUSIONS
Teaching graduate students the fundamental principles of
technology is obviously important, but putting these stu
dents in a position where they are prepared to function
effectively in an industrial setting is also important. We have
found that the industrial perspective can be effectively inte
grated into the oncampus program through the classroom,
research group activities, and interactions with industrial
representatives. A summer internship experience for our
graduate students preceded by the oncampus prepara
tions provides students with a comprehensive prepara
tion for working in industry. Moreover, when the stu
dents return to the university from an internship experi
ence, they typically become disciples of the industrial
perspective and greatly facilitate the transfer and accep
tance of the principles of industrial practice.
REFERENCES
1. Luyben, M.L., B.D. Tyreus, W.L. Luyben, "A PlantWide
Control Design Procedure," presented at the AIChE annual
meeting, Chicago, IL, November 1996. 3
Ef laboratory
AN EXPERIMENT
TO CHARACTERIZE A
CONSOLIDATING PACKED BED
MARK GERRARD, MARK HOCKBORN, JASON GLASS
University of Teesside Middlesbrough TS1 3BA England
Packed beds are much used as contractors for interphase
mass transfer and chemical reaction. An interesting
example involves their use in the biofiltrationI' of air
streams laden with volatile organic compounds (VOC). In
this instance, the bed may consist of naturally occurring
materials such as soil, heather, peat, or compost. The resi
dent (or seeded) microorganisms then digest the VOC in
the incoming stream. With natural packing materials, how
ever, the bed can collapse in time, leading to increased
pressure drop and higher operating costs etc. In this short
article we will describe an inexpensive modification to a
standard undergraduate laboratory experiment that studies
such consolidating behavior.
THEORY
The pressure drop in fixed beds can be predicted using the
wellknown Kozeny''231 equation for low gas flowrates:
Ap=5a2(1 )2 lhv/E' (1'
A.M. Gerrard is Reader in Chemical Engineering
at the University of Teesside in northeast En
gland. His research and teaching interests in
clude process economics and optimization.
Mark Hockborn (left)
and Jason Glass (right)
work for international
chemical companies
and are parttime stu
dents currently complet 
ing a Higher National
Diploma program in
chemical engineering.
Copyright ChE Divtsion ofASEE 1997
where the symbols are defined in the nomenclature. As these
natural beds tend to collapse and compress, so the height, h,
voidage, e, and specific surface, a, change from the original
values, h0, o, ao, and hence the bed pressure drop increases.
If the structure of the bed is assumed not to alter, then
hA(1E,)h,,A (1Eo)ho
a= 1 (2
hA h
a= ah (3
h
Putting these back into Eq. (1), we find
5a5 h (1 Eo)v 2
Ap= lo(4
{h(1ro)ho}
kv
(h G)
(h G)3
where
k=5ao h0(lEo)2
G=(lEo)ho (7
Thus, for Eq. (5) we would predict an inverse cubic rela
tionship between pressure drop and bed height for a given
bed velocity. So a small change in bed height can have a
profound effect on Ap.
Rearranging Eq. (5), we have
V )0.333
v
=k0333 Gk333
=k h Gk
So, a plot of (v/Ap)0.333 versus h will yield a straight line
whose slope is k0'333 and the intercept is Gk 0.333. From the
intercept, we find G and then e, from Eq. (7). Finally, Eq.
(6) leads directly to ao.
We can extend the analysis by calculating the mean equiva
Chemical Engineering Education
t nel article diam r
4000 p
SD 6(1 E) (9)
D (9)
a
Sand can also check to see if the Reynold's number is in the right
i range.[3]
S 3000 EXPERIMENTAL
I We used an existing 6inch diameter glass column, complete
with manometer and rotameter, which is usually used as a sandair
tI fluidized bed. The natural packing material was a commercial
S I peatmossbased potting soil purchased from a local plant nursery.
An original packed height of some 0.61m gave reasonable results.
% 2000
2 The pressure dropflowrate curve for the empty vessel was deter
Smined, then the column was filled with the compost, and the
P measurements were repeated. The contents were gently compressed
by hand and the net pressure drops were again found. Some five
compressions were applied, leading to Figure 1.
,2 1000 The gradients from Figure 1 allow us to calculate (v/Ap), which
When raised to the onethird power, can be plotted against the bed
height, h, to give Figure 2, which is an excellent straight line.
0 Finally, we compute the original voidage and specific surface to be
0.6 and 17000m', respectively
000 oo0 002 003 004 00 006 007 CONCLUSIONS
3 Velocty, m/s
Sh061m ,055. =o m Existing equipment can be used with unconventional packing
to measure the effect of bed consolidation. The experiment gives
the student an unconventional use of the Kozeny equation, to
0050 gether with an interesting opportunity to linearize Eq. (5). It is also
] an extremely inexpensive addition to the undergraduate lab.
NOMENCLATURE
Q 0.045 a specific surface area (variable)
U A bed crosssectional area
D mean particle size
G constant defined by Eq. (7)
0040 h bed height (variable)
k constant defined by Eq. (6)
v superficial bed velocity
0
Ap pressure drop
t P e voidage (variable)
S o.0 s gp gas viscosity
3 Subscripts
0 3
3 0 indicates original value before compressions
S o0o ACKNOWLEDGMENT
SThis work is part of an EUfunded project on the purification of
Waste gases.
S 0025 REFERENCES
S1. Bohn, H., "Consider Biofiltration for Decontaminating Gases,"
S/ Chem. Eng. Prog., 88, 34 (1992)
2. Coulson, J.M., and J.F. Richardson, Chemical Engineering, Vol. 1,
0020 Pergamon, Oxford, England (1990)
040 045 050 h 055 60 065 3. Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenom
__________ ena, Wiley & Sons, New York, NY (1960) 0
Summer 1997 193
M= classroom
USE OF SPREADSHEETS
in Introductory Statistics and Probability
BRIAN S. MITCHELL
Tulane University New Orleans, LA 70118
Statistical software packages such as Minitab, Statistica,
or SAS are extensively used by engineers to provide
descriptive statistics (mean, median, coefficient of
variation, etc.) for a data set, to generate values to the vari
ous probability distributions, and to perform a linear regres
sion or a Student's ttest. The use of these programs may be
taught in various undergraduate chemical engineering courses,
but their use in the chemical engineering curriculum is not
standardized and there is no consensus on which, if any, of
the programs is preferable.
Spreadsheets such as Microsoft Excel, Corel Quattro Pro,
and Lotus 123 provide a convenient and standardized way
to teach the fundamentals of statistical analysis and prob
ability to undergraduate students. In addition to being able to
execute the statistical analyses listed above, they can be used
for everything from calculating numbers of combinations to
constructing control charts for average (x charts). The stan
dard user interface also reduces the time required to learn
each new concept, and documents and features are virtually
identical across not only platforms (IBM to Macintosh), but
also across software vendors (Microsoft to Lotus).
The concept of spreadsheets in the classroom is by no
means new. There are some very excellent examples of
using spreadsheets for everything from general applications
for firstyear chemistry labs,' 1 or teaching regression analy
sis,'12 to very elaborate applications such as calculation of X
Brian S. Mitchell is Assistant Professor of Chemi
cal Engineering at Tulane University. He received
his BS in chemical engineering from the Univer
sity of IllinoisUrbana in 1986 and his MS and PhD
degrees in chemical engineering from the Univer
sity of WisconsinMadison in 1987 and 1991. His
research interests are in fiber technology and com
posites engineering, and in the use of alternative
teaching techniques in the classroom.
Copyright ChE Dvwsion ofASEE 1997
ray diffraction patterns from crystallographic information,131
but little information exists on how to bring spreadsheets
into an introductory course in statistics and probability.
Chemical engineering students at Tulane University are
introduced to a variety of spreadsheet applications in their
freshman course on the chemical engineering profession.
The skills they develop there are used to a greater extent in
the first semester of their sophomore year in a "Chemical
Engineering Design I" course. This course, taught concur
rently with stoichiometry, introduces the students to statisti
cal analysis, probability, reliability, quality control, and en
gineering economics.
The book Statistics and Probability for Scientists and
Engineers"4' by Mendenhall and Sincich is used in the course.
It provides numerous example problems from all engineer
ing disciplines, and although a software package is supplied
with the textbook, we have opted to solve the problems
using spreadsheets. This article describes how the spread
sheet can be used in a course such as this and presents
example problems and solutions to illustrate the ease with
which this can be accomplished.
We are fortunate to have an electronic classroom in our
department that allows us to work example problems in
class. With the use of a projection system that displays the
screen from a laptop computer, the instructor can "walk
through" the example while students perform the same func
tions at their own computer. The examples shown here are
from Mendenhall and Sincich (hereafter referred to as MS)
unless indicated otherwise, and all have been used either in
the electronic classroom or assigned as homework problems.
All spreadsheet solutions are from Microsoft Excel, version
7.0 for Windows 95.
DESCRIPTIVE STATISTICS
College sophomores should certainly be able to calculate a
mean, median, and mode from a data set, but it's always best
Chemical Engineering Education
to begin at the beginning. Ad
ditionally, a simple problem
that the students can easily
check by hand helps introduce
them to the use of a spread
sheet if they are not yet famil
iar with it.
Example 1 is such a prob
lem, complete with the Excel
solution. Descriptive statistics
are generated by selecting the
data set (in this case, A2..A51)
and choosing "Descriptive Sta
tistics" in the Analysis Toolpak
under the "Tools" menu. (The
Analysis Toolpak is included
with Excel, but is not a default
item and must be installed us
ing the "Add Ins..." command
under the "Tools" menu.) The
confidence level for these cal
culations can be specified. This
example illustrates not only
how to generate the desired
analysis, but also how a great
deal of time can be saved with
large data sets. In this example,
even more time is saved be
cause the data set is available
on the textbook diskette in
ASCII text format, so it can be
readily imported into the
spreadsheet. For most example
problems, the data set is placed
on a server that the students
can retrieve using File Trans
fer Protocol (FTP). This saves
a substantial amount of class
time.
Close inspection of the data
set in Example 1 points out one
of the limitations of the spread
sheet; namely, that only one
mode is specified for multi
modal data sets. The example
data set contains two modes:
128 and 131 (three values
each). The spreadsheet displays
the first mode it finds, in this
case 128. This example also
points out a "quirk" of Excel
data can be analyzed only in
columns or rows. That is, con
Spreadsheets ... provide a convenient and standardized way to teach the
fundamentals of statistical analysis and probability to undergraduate
students.... This article describes how the spreadsheet can be
used. ., and presents example problems and solutions to
illustrate the ease with which this can be accomplished.
Example 1 Descriptive Statistics (MS 2.48)
A B C
Industrial engineers periodically
conduct "work measurement" analyses
to determine the time required to
produce a single unit of output. At a
large processing plant, the number of
total workerhours required per day to
perform a certain task was recorded for
50 days. Compute the mean, median,
and mode of the data set. Find the
range, variance, and standard deviation
of the data set.
Workerhours
Descriptive Statistics
2 128
3 113 Mean 117.82
4 146]Standard Error 2.122896
5 124 Median 117.5
6 100 Mode 128
7 119 Standard Deviation 15.01114
8 109 Sample Variance 225.3343
9 128 Kurtosis 0.69123
10 131 Skewness 0.00906
11 112 Range 62
12 95 Minimum 88
13 124 Maximum 150
14 103 Sum 5891
15 133 Count 50
16 111 Confidence Level(95.0%) 4.266116
17 97
18 132
19 135
20 131
21 150
22 124
23 97
24 114
25 88
26 117
27 128
28 138
29 109
30 118
31 122
32 142
33 133
34 100
35 116
36 97
37 98
38 136
39 111
40 98
41 116
42 108
43 120
44 131
45 112
46 92
47 120
48 112
49 113
50 138
51 122
Summer 1997
tiguous columns cannot be analyzed as one set, even if the
data in them belong together. Quattro Pro, on the other hand,
does allow for such analysis.
Histograms are also readily generated in Excel. In ad
to the input data, the number and size of bins mt
specified (see column B in Example 2, a modified fo
MS problem 2.18). The histogram function generate
absolute frequency for each bin, as shown in columns
D. A histogram is then generated automatically, but i
problem, a relative frequency diagram is requested.
user intervention is required here (this is a good thing si
makes the students think about what they are doing).
tive frequency is calculated in column E by using th(
from column D and the formula
shown. Each of these calculations
requires entering one formula
only, which can then be copied to
the remaining cells. Formula cell
references automatically adjust, Each year, U.S. N
survey included a i
except for those "locked" with include each schod
include each scho,
dollar signs ($). A relative fre total enrollment,
quency distribution chart can then acceptance rate. C
be generated using Excel's graph and interpret the r
ing tool.
PROBABILITY
The fundamentals of probabil
ity are usually introduced using
examples with decks of cards and
rolls of the dice. Spreadsheets
aren't a great deal of help here.
They can be used at the next level,
howeverparticularly for simple
things like permutations and com
binations. The number of permu
tations is the number of ways to
put y elements in y distinct posi
tions from a single net of n dif
ferent elements, or
# of permutations
(n y)!
(1)
The number of combinations of n
distinct items taken y at a time is
given by the binomial coefficient
n! (2)
Sy!(n y)!
Neither of these relationships
is very complex, and either can
be calculated fairly easily by sim
ply typing the formula into a
spreadsheet cell. All spreadsheets are able to calculate facto
rials, in most cases with the FACT function. Example 3
illustrates the use of builtin functions to accomplish
the same task more easily, using the PERMUT and
COMBIN functions of Excel. Two arguments are specified
for each function: the total number of values to choose from
(n, number) and the number of values taken at a time
(y, number_chosen).
PROBABILITY DISTRIBUTIONS
Probability distributions are a vital component in the intro
duction of reliability analysis. There are many forms of
distributions, though, and students often get hung up trying
Example 2: Relative Frequency Histograms (MS 2.18)
ews and World Report surveys America's best graduate schools. The 1993
list of the top 25 graduate programs of engineering. The accompanying data
old's overall score (based on a weighted average of rankings in five areas),
dollar amount awarded for research, doctoral studenttofaculty ratio, and
construct a relative frequency diagram for the doctoral student/faculty ratio
results.
Chemical Engineering Education
A B C D E
Doctoral
Student/Faculty Relative
1 Ratio Bin Bin Frequency Frequency
2 2.52 1 1 0 =D2/$D$10
3 5.24 2 2 6 =D3/$D$10
4 2.82 3 3 12 =D4/$D$10
5 4.66 4 4 3 =D5/$D$10
6 2.36 5 5 2 =D6/$D$10
7 2.63 6 6 1 =D7/$D$10
8 3.27 7 7 1 =D8/$D$10
9 3.3 More 0 =D9/$D$10
10 2.32 Sum =SUM(D2:D9) =SUM(E2:E9)
11 2.23
12 3.7
13 2.51
14 1.75 Relative Frequency
15 2.04
16 1.96 0.5
17 2.38 0.45
18 4.88 S 0.4
19 6.35  o.3
20 2.41. 0.25
0.25
21 2.46 >= 0.2
22 1.27 0.15
23 1.42 0.1
24 2.7 0.os
25 1.85 1 2 3 4 5 6 7
26 1.5 Ph.D. StudentlFaculty Ratio
27
28 
to memorize the formulas for five or six distributions rather
than concentrating on the appropriate applications for each
distribution. While a spreadsheet cannot totally eliminate
this problem, it can reduce the anxiety of distributions a bit,
and certainly eliminates the need for distribution tables,
which can vary in terminology from book to book.
Most spreadsheets contain a number of distributions. Ex
Example 3: Probability Distributions (MS 3.40 and 3.42)
A security alarm system is activated and deactivated by correctly entering the appropriate three
digit numerical code in the proper sequence on a digital panel. Compute the total number of
possible code combinations if no digit may be used twice.
A B C
1 Function Description Formula Numerical Result
2 PERMUT(number, number_chosen) =PERMUT(10,3) 720
Suppose you need to replace 5 gaskets in a nuclearpowered device. If you have a box of 20
gaskets from which to make the selection, how many different choices are possible, i.e., how
many different samples of 5 gaskets can be selectedfrom the 20?
A B C
4 Function Description Formula Numerical Result
5 COMBIN(number, number_chosen) =COMBIN(20,5) 15504
Example 4: Binomial Distribution
Consider a sample of 15 valves. The probability that a given valve fails is 0.18. a) Calculate
the probability offailure of 015 valves. b) Calculate the probability that most five valves will
fail.
A B C D E F G
p.
n, total probability
number of single
of valve P.D.F. C D.F.
1 valves failing y P.D.F., P(y) (formula) (values) C.D.F. (formula) (values)
2 15 0.18 0 =BINOMDIST(C2,$A$2,$B$2,FALSE) 0.05095 =BINOMDIST(C2,$A2,$2,B$2,TRUE) 0.05095
3 1 =BINOMDIST(C3,$A$2,$B$2,FALSE) 0.16778 =BINOMDIST(C3,$A$2,$B$2,TRUE) 0.21874
4 2 =BINOMDIST(C4,$A$2,$B$2,FALSE) 0.25781 =BINOMDIST(C4,$A$2,$B$2,TRUE) 0.47656
5 3 =BINOMDIST(C5,$A$2,$B$2,FALSE) 0.24524 =BINOMDIST(C5,$A$2,$B$2,TRUE) 0.72180
6 4 =BINOMDIST(C6,$A$2,$B$2,FALSE) 0.16150=BINOMDIST(C6,$A$2,$B$2,TRUE) 0.88330
7 5 =BINOMDIST(C7,$A$2,$B$2,FALSE) 0.07799 =BINOMDIST(C7,$A$2,$B$2,TRUE) 0.96129
8 6 =BINOMDIST(C8,$A$2,$B$2,FALSE) 0.02853 =BINOMDIST(C8,$A$2,$B$2,TRUE) 0.98983
9 7 =BINOMDIST(C9,$A$2,$B$2,FALSE) 0.00805 =BINOMDIST(C9,$A$2,$B$2,TRUE) 0.99788
10 8 =BINOMDIST(C10,$A$2,$B$2,FALSE) 0.00176 =BINOMDIST(C10,$A$2,$B$2,TRUE) 0.99965
11 9 =BINOMDIST(C11,$A$2,$B$2,FALSE) 0.00030 =BINOMDIST(C11,$A$2,$B$2,TRUE) 0.99995
12 10 =BINOMDIST(C12,$A$2,$B$2,FALSE) 0.00003 =BINOMDIST(C12,$A$2,$B$2,TRUE) 0.99999
13 11 =BINOMDIST(C13,$A$2,$SB2,FALSE) 0.00000 =BINOMDIST(C13,$A$2,$B$2,TRUE) 0.99999
14 12 =BINOMDIST(C14,$A$2,$B$2,FALSE) 0.00000 =BINOMDIST(C14,$A$2,$B$2,TRUE) 0.99999
15 13 =BINOMDIST(C15,$A$2,$B$2,FALSE) O.O000O =BINOMDIST(C15,$A$2,$B$2,TRUE) 0.99999
16 14 =BINOMDIST(C16,$A$2,$B$2,FALSE) 0.00000 =BINOMDIST(C16,$A$2,$B$2,TRUE) 0.99999
17 __15 =BINOMDIST(C17,$A$2,$B$2,FALSE) 0.00000 =BINOMDIST(C17,$A$2,$B$2,TRUE) 1
cel includes the binomial and
Poisson distributions, among
others. Generating values from
these distributions is a simple
matter of using a function with
anywhere from two to four ar
guments. An example of the
binomial distribution is given
in Example 4, where the bino
mial distribution is given by
P(y) n! ) y p)(ny)
P( y!(n y)!
(3)
Here P(y) is the probability
that a known outcome occurs
y times out of n trials, and p is
the probability that an isolated
event of the given outcome
will occur. The example prob
lem asks not only for the prob
ability of 115 valves failing,
which is just the binomial dis
tribution evaluated at a single
value of y in each case, but
also for the cumulative prob
ability of failure for 05
valves. First, values for n and
p are entered in cells A2 and
B2, respectively. For the first
part of the question, all pos
sible values of y are entered
in cells C2..C17. This is eas
ily accomplished by placing
the value of zero in cell C2
and using the "Fill" command
in the "Edit" pulldown menu
to fill in the rest of the series
up to 15. The values of the
binomial distribution for each
value of y are generated in
column D using the
BINOMDIST function. (Nor
mally, this would be displayed
as a number, but for instruc
tional purposes, both the for
mula in column D and the
number it returns in column
E are shown.) The
BINOMDIST function re
Summer 1997
quires four arguments, the first three of which are the values
of y, n, and p, in that order, all of which can be referenced to
their respective cells. Note that the value of y changes as it
should, whereas n and p are fixed by the problem statement.
The final argument is a "switch" that allows the cumulative
distribution to be calculated. This argument is set to FALSE
in column D so that singular values of the distribution can be
calculated. Column E generates the answers to part a) of the
statement. The cumulative distribution function is exactly
what is required to solve part b) of Example 4, so the switch
is set to TRUE in column F. Again, values returned by this
function are shown in the following column, and the answer
to the problem is shown in cell G7; the cumulative probabil
ity of zero through five valves failing is 0.961.
HYPOTHESIS TESTING
There are a number of useful nullhypothesis tests, includ
ing the Chisquare test, Ftest, and Student's ttest, all three
of which are covered in Design I. Most spreadsheets are
capable of performing at least some of these tests, although
much more interpretation of the results is required than for
the previous examples. In particular, the final decision as to
whether or not the null hypothesis has been verified is left up
to the student.
An Ftest is used here as an
illustration of how nullhypoth Example 5:
esis tests can be performed, at
least in part, using a spreadsheet. A simplified analyticalpr
Recall that an Ftest compares determine not only whether
the variances, s of two data means ofa duplicate set at
as the current test.
sets
F= s (4)
s
The nullhypothesis is that the
two variances are statistically
equivalent at some specified
confidence level, typically 95%.
The value of F is calculated us
ing Eq. (4), and compared with
a value in a table at the speci
fied confidence level and ap
propriate degrees of freedom for
each data set. If the Fvalue in
the table is greater than the cal
culated value of F, the null hy
pothesis is substantiated, and the
two variances can be considered
statistically equivalent.
A wellknown problem from
Peters and Timmerhauss15 is used
here as an illustration of an F
test applied to a chemical engi
neering problem and how the results from the spreadsheet
manipulation must be interpreted (see Example 5). The data
are entered in cells A2..A8, and B2..B6 for the revised and
current analytical methods, respectively. The "Ftest Two
Sample for Variances" function is selected from the "Data
Analysis..." option under the "Tools" menu in Excel. The
cell indexes for both data sets must be provided, as must the
cell assignment for the output, and the value of a, which
determines the confidence level. The resulting table is shown
in cells A12..C19. The calculated value of F appears in cell
B17 and is the ratio of cells B14 to C14. In this instance, no
formulae are present in the tablesonly numerical values
appear at the end of the analysis. The tabulated value of F for
the specified value of a =0.5 is given in cell B19. The final
step is left to the student. In this case, the calculated value of
F is greater than the tabulated value, indicating that the two
analytical procedures may not be equivalent. Once again, the
spreadsheet is helpful, but the user must have a knowledge
of the underlying principles to correctly interpret the results.
ANALYSIS OF VARIANCE
Analysis of variance (ANOVA) is a statistical analysis
tool that provides a smooth transition from Ftests into a
Ftest (Peters and Timmerhaus Chpt. 17, Example 10)
ocedure is proposed for a routine laboratory test. It is necessary to
the new procedure gives the same results as the old, i.e., whether the
e the same, but also whether the precision of the new test is as good
Chemical Engineering Education
A B C
1 Revised Method Current Method
2 79.2 79.7
3 79.7 79.5
4 79.5 79.6
5 79.4 79.5
6 80 79.7
8 79.8
1 
9
10 FTest TwoSample for Variances
11
12 Variable 1 Variable 2
13 Mean 79.6 79.6
14 Variance 0.07 0.01
15 Observations _7 5
16 df 6 4
17 F 6.999999998
18 P(F<=f) onetail 0.040280016
19 F Critical onetail 6.163134003
discussion of linear regression. Recall that an ANOVA table
compares N similar data points from k different treat
ments by essentially performing an Ftest on the variance
between treatments (mean square treatment, or MST)
and the variance within the treatments (mean square er
ror, or MSE). The MST and MSE are calculated from the
sum of squares (SST and SSE, respectively) and corre
sponding degrees of freedom for each type of error, as
shown in Table 1.
TABLE 1
Analysis of Variance Table
Source
Treatment
Error
Total
Sum of Squares
n,(Y.Y),
k n
n Y,Y )2
Degrees of
Freedom
kI
Mean Square
SST/k MS
Nk SSE/Nk
Nl
Example 6 shows how an ANOVA table can be generated
on the spreadsheet. In this case, there are k=4 treatments
(locations, i=l through 4), each with six data points (j=l
through 6), for a total of N=24 data points. By selecting the
"ANOVA:Single Factor" function in the "Data Analysis..."
option of the "Tools" pulldown menu, the input range can
be specified, which in this example is A3..D8. The data is
this case are grouped by columns, and this radio button must
be selected on the menu. Once again, the desired confidence
level can be specified, here as ca=0.05, and the
location for the resulting ANOVA table specified in
the "Output Range" box. The ANOVA table ap
pears in cells A10..G15. Additionally, this table not
only shows the calculated Fvalue, but also the criti
cal Fvalue for the specified degrees of freedom and
F confidence level. This makes comparison of the two
Fvalues particularly easy for the student, who must
once again arrive at the final evaluation. In this case,
T/MSE there is sufficient evidence to suggest that the ozone
contents differ statistically for the different loca
tions.
CONTROL CHARTS
The final example of spreadsheets in Design I
deals with quality control. In the chemical process
industry, this usually means control charts. Control
charts for average (xchart), range (Rchart), pro
Example 6: Analysis of Variance (MS 14.27)
An excessive amount of ozone in the air is indicative of air pollution. Six air samples were
collected from each offour locations in the industrial Midwest and measured for their content
of ozone. Construct an analysis of variance table for the data. Do the data provide sufficient
evidence to indicate differences in the mean ozone content among the four locations? Use a =
0.05.
A B C D E F G
1 Location
2 1 2 3 4
3 0.08 0.15 0.13 0.05
4 0.1 0.09 0.1 0.11
5 0.09 0.11 0.15 0.07
6 0.07 0.1 0.09 0.09
7 0.09 0.08 0.09 0.11
8 0.06 0.13 0.17 0.08
9
10 ANOVA
11 Source of Variation SS df MS F Pvalue F crit
12 Between Groups 0.006779 3 0.00226 3.498925 0.034527 3.098393
13 Within Groups 0.012917 20 0.000646
14
15 Total 0.019696 23
Summer 1997
portion defects (pcharts), and
defects per unit (ccharts) are
introduced in this course. Ex
ample 7 shows how Excel can
be used to generate a pchart.
In this case, the formulae for
the lower control limit (LCL),
upper control limit (UCL),
and centerline must be speci
fied as
LCL=p3 (5)
n
fp(lp)
UCL=p+3(I ) (6)
n
Here
p= centerline = average
fraction detectives =
(d, + d, +...dk)/nk, cell C24
n = sample size, cell B2
k = number of sample periods.
There is nothing particu
larly involved about this ap
plication, but it does allow the
199
introduction of IF/THENtype statements into spreadsheet
calculations. In this case, the calculated LCL may have a
negative value, in which case it must be replaced by zero.
Cell C27 shows how the IF statement can be used to com
pare the calculated LCL to zero and place the appropriate
value in F3. The LCL is copied down column F (in this case
it is zero), as are the UCL and centerline values down col
umns D and E, respectively, to facilitate plotting. The p
chart is generated by plotting the data in column B (shown as
triangles), UCL, LCL, and centerline vs. the sample ID in
column A. The resulting plot is shown at the bottom of the
spreadsheet in Example 7, and the process appears to be in
control. Time allowing, this example also provides an excel
lent opportunity to introduce the concept of spreadsheet
macro commands that can automate the production of x
charts every time a new set of data is generated.
CONCLUSION
Spreadsheets continue to grow in popularity and avail
ability, and their utility in solving everyday engineering
problems develops with each new version. Some ex
amples have been presented on how to incorporate spread
Example 7: Control Chart (MS 16.44)
Highlevel computer technology has developed bitsized microprocessorsfor use in operating
industrial "robots". To monitor the fraction of defective microprocessors produced by a
manufacturing process, 50 microprocessors are sampled each hour. The results for 20 hours of
sampling are provided Construct a control chart for the proportion of defective
microprocessors. Locate the center line and upper and lower control limits on the chart. Does
the process appear to be in control?
sheet use into a sophomore
level chemical engineering
course on statistics and prob
ability. Hopefully, these ex
amples will inspire more of
us to use spreadsheets to il
lustrate chemical engineering
fundamentals in the class
room.
ACKNOWLEDGMENT
Support for this work comes
in part from the Camille and
Henry Dreyfus Special Grant
Program in chemical sciences.
REFERENCES
1. Birk, J.P., "FirstYear Chem
istry Laboratory Calculations
on a Spreadsheet," J. Chem.
Ed., 69(8), 648 (1992)
2. Wood, W.C., and S.L. O'Hare,
"A Spreadsheet Model for
Teaching Regression Analy
sis," J. Educ. Bus., May/April,
233(1992)
3. Shapiro, F., "The Calculation
of Crystal Diffraction Patterns
Using a Spreadsheet," J.
Mater. Ed., 14, 93 (1992)
4. Mendenhall, W., and T.S.
Sincich, Statistics for Engi
neering and the Sciences, 4th
ed., PrenticeHall, Upper
Saddle River, NJ (1995)
5. Peters, M.S., and K.D.
Timmerhaus, Plant Design
and Economics for Chemical
Engineers, 4th ed., McGraw
Hill, New York, NY (1991) O
Chemical Engineering Education
