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Toward technical understanding: Part 1. Brain structure and function ( PDF ) A new approach to teaching dimensional analysis ( PDF ) Engineering education for the 21st century: Listen to industry! ( PDF ) Teaching statistics to ChE students ( PDF ) An introduction to process flexibility: Part 1. Heat exchange ( PDF ) Objectively speaking ( PDF ) Use of computational tools in engineering education: A case study on the use of Mathcad ( PDF ) Introducing graduate students to the industrial perspective ( PDF ) An experiment to characterize a consolidating packed bed ( PDF ) Use of spreadsheets in introductory statistics and probability ( PDF ) ( XML ) 
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^Colorado State Univesity AUTHOR GUIDELINES This guide is offered to aid authors in preparing manuscripts for Chemical Engineering Education (CEE), a quarterly journal published by the Chemical Engineering Division of the American Society for Engineering Education (ASEE). CEE publishes papers in the broad field of chemical engineering education. Papers generally describe a course, a laboratory, a ChE department, a ChE educator, a ChE curriculum, research program, machine computation, special instructional programs, or give views and opinions on various topics of interest to the profession. SSpecific suggestions on preparing papers * TITLE Use specific and informative titles. They should be as brief as possible, consistent with the need for defining the subject area covered by the paper. AUTHORSHIP Be consistent in authorship designation. Use first name, second initial, and surname. Give complete mailing address of place where work was conducted. If current address is different, include it in a footnote on title page. TEXT We request that manuscripts not exceed twelve doublespaced typewritten pages in length. Longer manuscripts may be returned to the authors) for revision/shortening before being reviewed. Assume your reader is not a novice in the field. Include only as much history as is needed to provide background for the particular material covered in your paper. Sectionalize the article and insert brief appropriate headings. TABLES Avoid tables and graphs which involve duplication or superfluous data. If you can use a graph, do not include a table. If the reader needs the table, omit the graph. Substitute a few typical results for lengthy tables when practical. Avoid computer printouts. NOMENCLATURE Follow nomenclature style of Chemical Abstracts; avoid trivial names. If trade names are used, define at point of first use. Trade names should carry an initial capital only, with no accompanying footnote. Use consistent units of measurement and give dimensions for all terms. Write all equations and formulas clearly, and number important equations consecutively. ACKNOWLEDGMENT Include in acknowledgment only such credits as are essential. LITERATURE CITED References should be numbered and listed on a separate sheet in the order occurring in the text. COPY REQUIREMENTS Send two legible copies of the typed (doublespaced) manuscript on standard lettersize paper. Submit original drawings (or clear prints) of graphs and diagrams on separate sheets of paper, and include clear glossy prints of any photographs that will be used. Choose graph papers with blue cross sectional lines; other colors interfere with good reproduction. Label ordinates and abscissas of graphs along the axes and outside the graph proper. Figure captions and legends will be set in type and need not be lettered on the drawings. Number all illustrations consecutively. Supply all captions and legends typed on a separate page. State in cover letter if drawings or photographs are to be returned. Authors should also include brief biographi cal sketches and recent photographs with the manuscript. Send your manuscript to Chemical Engineering Education, c/o Chemical Engineering Department University of Florida, Gainesville, FL 326116005 EDITORIAL AND BUSINESS ADDRESS: Chemical Engineering Education Department of Chemical Engineering University of Florida Gainesville, FL 32611 PHONE and FAX: 3523920861 email: cee@che.ufl.edu EDITOR T. J. Anderson ASSOCIATE EDITOR Phillip C. Wankat MANAGING EDITOR Carole Yocum PROBLEM EDITORS James O. Wilkes and Mark A. Burns University of Michigan LEARNING IN INDUSTRY EDITOR William J. Koros University of Texas, Austin PUBLICATIONS BOARD CHAIRMAN * E. Dendy Sloan, Jr. Colorado School of Mines PAST CHAIRMEN * Gary Poehlein Georgia Institute of Technology Klaus Timmerhaus University of Colorado MEMBERS Dianne Dorland University of Minnesota, Diluth Thomas F. Edgar University of Texas at Austin Richard M. Felder North Carolina State University Bruce A. Finlayson University of Washington H. Scott Fogler University of Michigan David F. Ollis North Carolina State Universit" Angelo J. Perna New Jersey Institute of Technology Ronald W. Rousseau Georgia Institute of Technology Stanley I Sandler University of Delaware Richard C. Seagrave Iowa State University M. Sami Selim Colorado School of Mines James E. Stice University of Texas at Austin DonaldR. Woods McMaster University 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 Division, American Society for Engineering Education, and is edited at the University of Florida. Correspondence regarding editorial matter, circulation, antd changes of address should be sent to CEE, Chemical Engineering Department, University of Florida, Gainesville, FL 326116005. Copyright 0 1997 by the Chemical Engineering Division, American Societyfor Engineering Education. The statements and opinions expressed in this periodical are those of the writers and not necessarily those of the ChE Division. ASEE. which body assumes no responsibility for them. Defective copies replaced if notified within 120 days of publication. Write for information on subscription costs and for hack copy costs and 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 Summer 1997 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 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< 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,< theoretical grounds for D,< 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 
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