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Front Cover  
Table of Contents  
University of Virginia  
Book reviews  
Phillip C. Wankat, of Purdue...  
Confirming thermodynamic stability:...  
Three problems in fluid mechan...  
Book reviews  
What do they know, anyway?  
A course sequence for instrumentation...  
The effect of agitation on oxygen...  
"Product in the way" processes  
A statistical look at significant...  
Add some flavor to your agitation...  
Molecular enrichment of the core...  
Design of CSTRs in tandem...  
Book reviews  
Back Cover 
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Front Cover 1 Front Cover 2 Table of Contents Page 113 University of Virginia Page 114 Page 115 Page 116 Page 117 Page 118 Book reviews Page 119 Phillip C. Wankat, of Purdue University Page 120 Page 121 Page 122 Page 123 Confirming thermodynamic stability: A classroom example Page 124 Page 125 Page 126 Page 127 Page 128 Page 129 Three problems in fluid mechanics Page 130 Page 131 Page 132 Book reviews Page 133 What do they know, anyway? Page 134 Page 135 A course sequence for instrumentation and control Page 136 Page 137 Page 138 Page 139 Page 140 Page 1401 Page 1402 Page 1403 Page 1404 Page 1405 Page 1406 Page 1407 Page 1408 Page 1409 Page 14010 Page 14011 Page 14012 Page 141 The effect of agitation on oxygen mass transfer in a fermentor Page 142 Page 143 Page 144 Page 145 "Product in the way" processes Page 146 Page 147 Page 148 Page 149 Page 150 Page 151 A statistical look at significant figures Page 152 Page 153 Page 154 Page 155 Add some flavor to your agitation experiment Page 156 Page 157 Page 158 Page 159 Molecular enrichment of the core curriculum Page 160 Page 161 Page 162 Page 163 Design of CSTRs in tandem revisited Page 164 Page 165 Page 166 Page 167 Book reviews Page 168 Back Cover Back Cover 1 Back Cover 2 

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      I ILJ I University Microfilms International Please send additional information for Name (name of publication) Institution Street City State 300 North Zeeb Road Dept. PR. Ann Arbor, Mi. 48106 U.S.A. 3032 Mortimer Street Dept. P.R. London WIN 7RA England EDITORIAL AND BUSINESS ADDRESS: Chemical Engineering Education Department of Chemical Engineering University of Florida Gainesville, FL 32611 FAX 9043920861 EDITOR Ray W. Fahien (904) 3920857 ASSOCIATE EDITOR T. J. Anderson (904) 3922591 CONSULTING EDITOR Mack Tyner MANAGING EDITOR Carole Yocum (904) 3920861 PROBLEM EDITORS James 0. Wilkes and Mark A. Burns University of Michigan 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 George Burnet Iowa State University Anthony T. DiBenedetto University of Connecticut Thomas F. Edgar University of Texas at Austin Richard M. Felder North Carolina State University Bruce A. Finlayson University of Washington H. Scott Fogler University of Michigan J. David Hellums Rice University Carol M. McConica Colorado State University Angelo I. Perna New Jersey Institute of Technology Stanley I Sandier University of Delaware Richard C. Seagrave Iowa State University M. Sami Selim Colorado School of Mines James E. Stice University of Texas at Austin Phillip C. Wankat Purdue University Donald R. Woods McMaster University Summer 1992 Chemical Engineering Education Volume 26 Number 3 Summer 1992 DEPARTMENT 114 University of Virginia, Peter T. Cummings, Roseanne M. Ford, John P. O'Connell EDUCATOR 120 Phillip C. Wankat, of Purdue University, Frank Oreovicz CLASSROOM 122 Confirming Thermodynamic Stability: A Classroom Example, Kenneth R. Jolls, Jeffrey L. Butterbaugh 146 "Product in the Way" Processes, Noel de Nevers 152 A Statistical Look at Significant Figures, Park M. Reilly 164 Design of CSTRs in Tandem Revisited, A. A. Adesina CLASS AND HOME PROBLEMS 130 Three Problems in Fluid Mechanics, James 0. Wilkes, Stacy G. Bike CURRICULUM 136 A Course Sequence for Instrumentation and Control, Carlos A. Smith, Richard A. Gilbert 160 Molecular Enrichment of the Core Curriculum, Henry A. McGee, Jr. LABORATORY 142 The Effect of Agitation on Oxygen Mass Transfer in a Fermentor, Ronnie S. Roberts, James R. Kastner, Maqsood Ahmad, D. William Tedder 156 Add Some Flavor to Your Agitation Experiment, M. Elizabeth Sensel, Kevin J. Myers 134 RANDOM THOUGHTS What Do They Know, Anyway? Richard M. Felder 119, 133, 168 Book Reviews 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, and changes of address should be sent to CEE, Chemical Engineering Department, University ofFlorida, Gainesville, FL 32611. Copyright 1992 by the Chemical Engineering Division, American SocietyforEngineering 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 back copy costs and availability. POSTMASTER: Send address changes to CEE, Chemical Engineering Department., University of Florida, Gainesville, FL 32611. artment UNIVERSITY OF VIRGINIA 7.. l. A The beauty of Mr Jefferson's handi work lives on: top photo is the his toric "academical village" and bot tom photo frames and reflects the elegance of eschool building. ~ ~ The Center for Bioprocess Develop stateoftheart instruments View of the new (Phase 1) chemical engineering/biotechnology building. PETER T. CUMMINGS, ROSEANNE M. FORD, JOHN P. O'CONNELL University of Virginia Charlottesville, VA 229032442 A after retiring from public life, Thomas Jefferson's preoccupation was the founding of the University of Virginia as the first truly public university in the United States. He formulated its first curriculum, recruited the first faculty, and designed and supervised the construction of all the original buildings. Jefferson had a singular vision for the university: faculty and students would live together in the "academical village," an environment of unparalleled beauty where they, as equals, would pursue and expand knowledge. As he wrote to William Roscoe in 1820, This institution will be based on the illimitable freedom of the human mind, for here we are not afraid to follow truth wherever it may lead, nor to tolerate any error so long as reason is left free to combat it. His legacy to the students and faculty at UVa includes the historic campusknown as The Groundscontaining perhaps the most famous and beautiful university buildings in the United States: the Lawn and the Rotunda. Jeffer son also left UVa's students and faculty a unique and rich educational tradition. This article will attempt to convey how the Department of Chemical Engineering at the University of Virginia strives to fulfill Jefferson's educational vision. The department is a blend of old and new. The "old" is UVa's long history of engineering in general and chemical engineering in particular. Jefferson had a strong personal ment with its interest in science and the "mechanical arts." The earliest nation. Copyright ChE Division ofASEE 1992 Chemical Engineering Education ChEde .. fulfilling Thomas Jefferson's vision The educational philosophy of the department reflects a commitment to continuing the Jeffersonian ideal of students and faculty as equal partners in the pursuit of knowledge. curricular plans for the university included instruction in mili tary and civil architecture. Engineering courses were offered in 1827, about one year after the university's opening, and the School of Engineering and Applied Science was established in 1836. This makes it the oldest universitybased engineering school in America. The Department of Chemical Engineering was established by several faculty from the Chemistry Department in 1908, the same year that the American Institute of Chemical Engi neers (AIChE) was founded. The Masters program began in 1949 and the first PhD was awarded in 1961. Both our under graduate and graduate alumni have distinguished themselves through outstanding contributions in many branches of indus try and in academia. The "new" is a recent significant change in personnel and facilities. Of the eleven fulltime and one halftime faculty, four have joined in the last five years and six are new in the last ten years. John O'Connell joined the department as Chair in 1988 after twentytwo years at the University of Florida, while the other new faculty all began their academic careers at UVa. Phase I of a new 50,000 ft2 building was completed in March of this year. It houses  7: the faculty offices and half of the chemical engineering re search laboratories. The other research laboratories, all in volved in various aspects of biotechnology, are currently  located in 15,000 ft2 of nearby space that was renovated in 1986. All faculty have active, funded research programs, with over $3M in current sponsored research grants. There are more than fifty graduate students, mostly PhDs, using stateoftheart Two views of the new ChE building. Part of the lobby (dedicated in honor of laboratory and computational Charles Brown, alumnus and a former CEO of AT&T) above, and the Mobil equipment for advancedlevel classroom, used for essentially all undergraduate and graduate chemical engi research in contemporary neering courses, below. Itfeatures a stateoftheart lighting system and seats 72, chemical engineering, with handicapfacilitiesfor both listeners and speakers. Summer 1992 11 EDUCATIONAL PHILOSOPHY The educational philosophy of the department re flects a commitment to continuing the Jeffersonian ideal of students and faculty as equal partners in the pursuit of knowledge. Jefferson's academical village began with sixtyeight students and ten faculty drawn together in an atmosphere of close inter action and learning. This continues today in the close relationships our faculty develop both with undergraduate and graduate students, leading to friendships that continue many years beyond gradu ation. In keeping with the idea of equal partnership in the educational process, academic titles are not used for faculty; they are addressed by the students as Mr., Mrs., or Ms. THE UNDERGRADUATE PROGRAM The engineering school at UVa is renowned for the quality of its undergraduate students. Several published surveys indicate that the average SAT scores of our engineering undergraduates are the highest of any public university in the country. Only one out of every six applicants is admitted, and a significant fraction is from out of the state. All of the courses offered by the School of Engi neering and Applied Science are taught by faculty none by teaching assistants alone. This is only possible because UVa has the lowest studentto faculty ratio (9 to 1) of any US public engineering school. Faculty are expected to maintain an "open door" policy of being available during all normal working hours (not just during posted office hours) to assist students in every aspect of their personal and scholastic development. The undergraduate program is typical of ABET accredited programs, with preparatory courses in mathematics, physics, chemistry, and computer sci ence and engineering fundamentals, followed by traditional and modern chemical engineering courses. In addition, the engineering school has its own Humanities Division which focuses on technical reading, writing, and presentation. In recent years the department has graduated fifteen to fifty BSChEs per year. The faculty recognizes exceptional graduating students with a variety of academic and leadership awards. In the final year, each undergraduate student is required to write a senior thesis with both technical and Humanities Division advisors. Students often use original research done in our laboratories for their thesis topics. Besides producing a detailed project plan and a final written document, students orally defend their thesis proposal and summarize 116 their findings to their Humanities class. Currently, the best eight theses are presented to a panel of industrial and faculty judges in an Undergraduate Research and Design Symposium who select the win ners of research awards. Chemical engineers have been prominent in these design symposia. Students are very active in professional service and social organizations. The focus of chemical engi Table 1 Faculty and Research Interests Giorgio Carta, Associate Professor PhD (ChE), Delaware, 1984 separation technologies hioseparations adsorption and ion exchange Peter Cummings, Professor PhD (Math), Melbourne, Australia, 1980 physical properties and phase equilibria optimization and synthesis of chemical processes modeling of bacterial migration Bob Davis, Assistant Professor PhD (ChE), Stanford. 1989 heterogeneous catalysis kinetic studies of selected probe reactions Erik Fernandez, Assistant Professor PhD (ChE), UC Berkeley. 1989 Nuclear magnetic resonance (NMR) characterization of biochemi cal reactors and mammalian tissues NMR imaging offlow in porous media Roseanne Ford, Du Pont Assistant Professor PhD (ChE), Pennsylvania, 1989 application of chemical engineering principles to microbial ecology bacterial chemotaxis bioremediation Elmer Gaden, Wills Johnson Professor PhD (ChE), Columbia, 1949 biotechnology and bioprocesses social impact of technological development John Gainer, Professor PhD (ChE), Delaware, 1964 immobilized biocatalysts twophase aqueous extraction oxygen transport in living systems Jack Hudson, Wills Johnson Professor PhD (ChE), Northwestern, 1962 dynamic behavior of chemically reactive systems stability, periodic oscillations, and chemical chaos electrochemical engineering Don Kirwan, Professor PhD (ChE), Delaware, 1967 biochemical engineering mass transfer, crystallization * Director of Center for Bioprocess Development Doug LeVan, Professor PhD (ChE), UC Berkeley, 1976 fixedbed adsorption thermodynamics of adsorption equilibria * modeling offixedbed adsorption systems computeraided design Lem Lilleleht, Associate Professor PhD (ChE), Illinois, 1962 nucleation of refractory vapors in microgravity environments * utilization of solar and other alternative energy resources John O'Connell, Professor and Chair PhD (ChE), UC Berkeley, 1967 applied molecular theory strongly nonideal liquids surfactant solutions Chemical Engineering Education neering involvement is the student chapter of the AIChE, advised by John O'Connell. Last year the group was selected as a national Chapter of Excel lence (no more than ten percent of all chapters are cited for this award). The major activities of the chapter include presentations during the semester by industrial speakers, a symposium on graduate school, the securing of industrial sponsorship to sup port attendance of about ten members at annual AIChE meetings and regional student chapter con ferences. There are several social events for both faculty and students, and the chapter also has a novel outreach program called "Wahoo Wizards" (Wa hoos being one of the nicknames for UVa sports teams) where students visit local elementary and middle schools to perform experiments that stimu late children's interest in science and engineering. THE GRADUATE PROGRAM The closeness that exists between undergradu ates and faculty is paralleled in the relationships of graduate students and faculty. Graduate students undoubtedly benefit from the opendoor policy at least as much as the undergraduates do. The graduate program offers MS, ME, and PhD degrees. The two Masters degrees include a group of five core courses in the fundamentals of chemical engineering. The primary difference between them is that the MS requires a research thesis. There is also a special program for highly qualified students with degrees other than in chemical engineering. It begins with an intensive summer program covering the undergraduatelevel material, followed by en trance into the regular graduate program. The ME degree can be taken as a nonresident BIoproce un tilzatic. Engineering Physical Propeitta tDiffuilon and S Reaction Millu ranks fer Separations S Engineering ineerin n en s and Reaction Electrochterr.i K. nl c Engineering heterogeneous Process Synthesis, Engineering E Optimization and Applications and Fundamentals control E Technologies Figure 1. Schematic illustration of faculty research inter ests divided into engineering fundamentals and applica tions. Placement of fundamentals at the center and appli cations at the periphery indicates how applications oriented research depends on advances in fundamentals. Summer 1992 ...research interests range from applied chemistry, biotechnology, and chemical technology to mathematical modeling, molecular and process simulation, and design. through the Virginia Cooperative Graduate Engi neering Program (VCGEP). Each term, one of the regular graduate courses is broadcast via satellite throughout Virginia and the U.S. using oneway video and twoway audio capabilities. Many companies have established classrooms at their industrial loca tions in order to allow their employees to participate in this program. There are now more than thirty nonresident students enrolled in the ME ChE pro gram through VCGEP, and more than a dozen ME degrees have been awarded in the last three years. The chemical engineering faculty view research as an integral part of graduate education. The re search interests range from applied chemistry, bio technology, and chemical technology to mathemati cal modeling, molecular and process simulation, and design. Figure 1 illustrates how the ongoing research programs at Virginia cover the two broad categories of fundamentals and applications. The major funda mental research programs are: Diffusion and mass transfer Gasliquid and solidliquid systems * transport processes in biological systems homogeneous nucle ation Thermodynamics, physical properties, and adsorption Statisti cal thermodynamics prediction of physical properties fluid phase equilibria adsorption equilibria ion exchange solubil ity of biochemicals Fluid mechanics Low Reynolds number flow surfacetension driven flow multiphase flow and stability Reaction kinetics Oscillations and chaotic behavior; heterogeneous catalysis The programs in applications and technologies in clude: Separations technology Fixed bed adsorption ion exchange and chromatography precipitation and crystallization extraction * air pollution control Bioprocess technology Immobilized enzymes, microorganisms, and cells aeration and oxygen transfer bioseparations * bioremediation Biochemical engineering Modeling of metabolic processes sec ondary metabolite regulation Biomedical engineering Mammalian cell biocatalysis metabo lism in diseased tissues enhanced oxygenation in living sys tems NMR spectroscopy Reaction engineering Bioreactors multiphase reactors flow reactors Electrochemical engineering Corrosion dynamics of electro chemical reactions Heterogeneous catalysis Structural characterization of metal clus ters acidbase properties of solid oxides Process synthesis, design, and control Mathematical modeling 117 and simulation computer control computeraided process design. Solar energy utilization Thermal energy conversion and storage * photovoltaics While faculty direct independent research pro grams, there is considerable collaboration both within and outside of the department. Many of our faculty are involved in multidisciplinary research efforts at UVa. Six faculty members from chemical engineering (Carta, Fernandez, Ford, Gaden, Gainer, and Kirwan) participate in the Center for Bioprocess Develop ment along with several faculty from the Medical School and Department of Biology. Founded in 1987 under the sponsorship of Virginia's multiprogram Center for Innovative Technology (CIT), the Bioprocess Center conducts research in biotechnol ogy applications such as largescale use of biological catalysts (microbes, cells, and enzymes) and novel processing for producing valuable products in medi cine, agriculture, and the food, energy, and chemical industries. The center's annual budget is over $1M from federal agencies (including NSF and NIH), state funds through CIT, and fifteen companies. Four faculty (Cummings, Ford, Fernandez, and Gainer) are involved in UVa's Biophysics Program, an interdisciplinary degree program with over forty other faculty members from the Medical School, the School of Engineering and Applied Science (Biomedi cal and Chemical), and the College of Arts and Sci ences (Biology, Chemistry, and Physics). Jack Hudson participates in the Center for Electrochemical Sci ences and Engineering which draws together faculty from chemical engineering, materials science, and nuclear engineering around the themes of corrosion, electrochemical reactions, and electrochemical phe nomena. Its research support of several million dol lars annually comes from a variety of federal agen cies, CIT, and industry. ENVIRONMENT The university is located in the historic city of Charlottesville in beautiful Albemarle County, nestled in the foothills of the Blue Ridge Mountains. The area's mild climate, historical significance (three presidentsJefferson, Madison, Monroeall resided nearby), academic stature, and physical beauty attract a wealthy and culturally diverse popula tion of about 120,000. The locale combines the ameni ties and attractions of a city with the charm and ambience of rural America. Furthermore, Washing ton, DC, and Richmond are both less than two hours away by car, while the Skyline Drive can be reached in less than onehalf hour. The Charlottesville/Albemarle airport is serviced by several major airlines. UVa is often regarded as one of the finest univer sities in the US, with outstanding undergraduate and graduate programs in the arts, law, medicine, business, and engineering. For the last two years it has been ranked by the New York Times' "Selec tive Guide to Colleges" as one of the three best universities in the U.S. In a recent U.S. News and World Report article surveying American universities, UVa was the only public university ranked in the top twenty. The current enrollment is 17,000 students, with 7,000 of them in our graduate and professional pro grams. Thus, it is one of the smallest PhDgranting state universities in the country. Combined with the historical buildings and grounds, this gives the uni versity the look and feel of a small private school. Much of chemical engineering is now housed in the first phase of a new building designed specifi cally for chemical and biochemical engineering re search. Its dedication was on April 25, 1992, with John M. Prausnitz of UC Berkeley giving the princi pal address. In keeping with Jefferson's view of close faculty and student interactions, the building's fac ulty offices are dispersed throughout the building, generally across the hall from their laboratories and graduate student offices. The second phase of this 50,000 ft2 facility will house the bioprocessing re search laboratories, the undergraduate laboratory and mechanical shop. Funding for both phases does not include any stateallocated money. The generos ity of alumni, industry, and philanthropic founda tions to complete this $11M building demonstrates how highly our program is regarded by these groups. CONCLUSION The goal of our department is to make measur able and distinctive contributions toward enhancing the quality of life consistent with the rich educa tional tradition entrusted to us by Thomas Jefferson. In doing so our objectives are to satisfy our public and private benefactors and stimulate growth in all of our students. Our energetic faculty members are strongly committed to both teaching and innovative research, deeply involved with undergraduates and graduates in a community of shared learning and research, and housed in a new building of architec tural beauty that promotes scholarly and profes sional interactions in much the same way as Tho mas Jefferson's beloved Lawn. We invite you to visit us and experience firsthand the ways in which chemical engineering at UVa is meeting its objec tives and fulfilling Thomas Jefferson's vision. J Chemical Engineering Education 1 book review ) PLANT DESIGN AND ECONOMICS FOR CHEMICAL ENGINEERS 4th Edition by M. S. Peters and K.D. Timmerhaus; McGraw Hill, Inc., New York, NY 10020; 910 pages, $60.45 (1990) Reviewed by Reena Chakraborty, Martin C. Hawley Michigan State University The fourth edition of this classic text retains the same layout and philosophy as its predecessors. The first third of the book covers the principles of process design development and plant design and safety in four chapters: 1Introduction, 2Process Design De velopment, 3General Design Considerations, and 4ComputerAided Design. The second third of the book, economics, is covered in the next five chapters: 5Cost and Asset Accounting, 6Cost Estimation, 7Interest and Investment Costs, 8Taxes and In surance, 9Depreciation, and 10Profitability, Alternative Investments, and Replacements. The remaining third deals with the technical design problem: 11Optimum Design and Design Strategy, 12Materials Selection and Equipment Fabrication, 13The Design Report, 14Materials Transfer, Han dling, and Treatment Equipment: Design and Costs, 15Heat Transfer Equipment: Design and Costs, 16Mass Transfer Equipment: Design and Costs, and 17Statistical Analysis in Design. Chapters 11, 17, and 13 could be dealt with, in that order, as a sequence in setting up and representing the solution to a design problem. Most of the solved problems and most of the charts still use engineering units despite the stress on SI. Surely the sample problems could have at least incorporated both sets of units, since this was a problem with the third edition and the authors have had ten years to work on it! To the authors' credit, many of the unsolved problems at the end of each chapter are in SI. The extensive lists of references at the end of each chapter that added to the utility of the third edition have been deleted in this edition. Perhaps these will be compiled and added as an appendix in a future printing. They will be sorely missed in this one. The first two chapters remain unchanged from the third edition, with no new problems. The third chapter has been greatly expanded with much new Summer 1992 material on health and safety hazards, including sources of exposure, exposure evaluation, exposure hazard control, fire and explosion hazards, person nel safety, and safety regulation. In each of the sub sections, relevant material on measures and stan dards of hazards, measures and standards of safety, and pertinent references to codes and regulations and the agencies which administer them have been made. A new section on HAZOP studies has been added which is clear, concise, and comprehensive, and the material on environmental protection and pollution control has been updated. The rest of the chapter is generally unchanged, with the exception of the section on patents, which has been streamlined. There are twentyfive new problems at the end of this chapter on hazard pre vention which are simple, but instructive. Many of them have been adapted from Safety, Health, and Loss Prevention in Chemical Processes: Problems for Undergraduate Engineering Curricula (copyrighted by the AIChE in 1990). ComputerAided Design, Chapter 4, is a concise, comprehensive, well written and referenced section on the various aspects of computeraided design and covers everything from the use of spreadsheets in material and energy balance calculations to flowsheeting software. The eight problems at the end of the chapter vary from fairly straightforward material and energy balances to fairly complex evalu ations of alternative process designs. The material in the economics section of the book is virtually unchanged, with the following excep tions: the section on evaluating interest has been expanded into a chapter with fifteen elementary but illustrative problems at the end, and one new prob lem each has been included at the end of Chapter 6 and Chapter 8 (neither is complex). A section on the accelerated cost recovery system (ACRS) and the modified accelerated cost recovery system (MACRS) has been included in Chapter 9, along with three related unsolved problems and a solved sample prob lem. Some of the unsolved problems at the end of the chapters in the third edition have been deleted here. All the cost data and charts have been updated. Some of the charts for costing are now smaller than their predecessors and thereby are harder to inter polate. The example on reactor design has been up dated and includes programs in BASIC, FORTRAN, and PASCAL. There are no new problems in this section of the book. Five new unsolved practice ses sion problems have been included in Appendix C. Continued on page 155. educator PHILLIP C. WANKAT of Purdue University FRANK OREOVICZ Purdue University . West Lafayette, IN 47907 F or many young people, the choice of a career is a trial anderror process that lasts for years. But not for Phil Wankat. His father sug gested that he become a chemical engineer, Phil said "OK" ... and that was that. His father, an ana lytical chemist, was actu ally making an informed suggestion, knowing as he did the engineering world and Phil's penchant for sci ence and math. Phil's other major deci sionto become a profes soralso came early and easily. However, Phil chose a roundabout way of achieving his goals by de ciding to take advantage of the opportunity for get ting an education in one of the military academies. His first and second choices, the naval and air force academies, were unavailable to him, so he went to West Point. It didn't take long for him to realize that military life wasn't for him (roughly one month, to be exact!) but he felt he had to give it a fair shake and he stayed two years. To those of us who have never seen him without his beard, the image of "Gen eral Phil" is intriguing, to say the least. He has no regrets, however, and claims that three valuable lessons came out of his Academy ex perience. First, he learned that you could want some thing very badly, work very hard for itand still not attain it. This proved to be equally true for his expectations of the military and for his hopes for fluency in French. Second, he learned discipline (a valu able but difficult lesson for many college students to learn), and he became accus tomed to "being yelled at." And third, he decided in his second year that he wanted to earn a PhD and become a Professor; through some in formal tutoring he was do ing in math, chemistry and physics, he had discovered he greatly enjoyed working with people and explaining things to them. Phil transferred to Purdue after two years at West Point, and eventually graduated first in his class (in 1966). His interest in separations was kindled by Lowell Koppel's lectures on distillation. He also contin ued tutoring, nurturing his interest in teaching. When he applied for graduate school he made note of his interest in teaching, which (it turned out) qualified him for a National Defense Education Act Scholar ship. Princeton offered him one and he accepted. At Princeton his interest in separations was side tracked for a while since the only person doing re search in separations was Dick Wilhelm (who was doing parametric pumping). Unfortunately, Phil "couldn't understand the research at all, the way it was presented," so he went to work with someone else, in the area of Monte Carlo simulation in ther modynamics. But when his advisor was denied ten ure, Phil had to switch again! This time he decided to talk only to full professors, and he eventually went to work for Bill Schowalter, who not only taught @ Copyright ChE Division of ASEE 1992 Chemical Engineering Education him how to do research but also served as Phil's model of how to guide grad students. His research was on hydrodynamic stability analysis, specifically on the Bernard problem. Although it was a far cry from separations, his experience taught him some thing he still strongly believes today: that the area of one's PhD is not all that important since the real purpose of grad school is to train one how to do research and how to formulate problems. It was in graduate school that Phil learned how to ask questions. In his own research and in his relationships with his grad students, he still regards questioning to be one of the main purposes of grad school. A student at the Master's level may need to be given a problem, but he or she should have the freedom to work out the solution, with the advisor acting only as a coach along the way. At the PhD level, a student should be exposed to a broad area and then guided toward framing important questions. RESEARCH During Phil's first semester as an assistant pro fessor at Purdue, Robert Greenkorn (department head at the time) suggested that he go to Puerto Rico for an AIChE meeting. Although Phil wasn't very interested in attending, money was availableso he went. At a conference session on separations, he talked at length with Norm Sweed, who suggested that since Phil was interested in separations he should do some parametric pumping. The result: he did just that ... for the next ten years. One of Phil's major research interests has been in developing new operating cycles for adsorption and chromatography. Early work in this area fo cused on parametric pumping and cycling zone ad sorption. However, since industry saw little advan tage in operating in this mode, Phil later switched to Pressure Swing Adsorption (PSA) and chromatogra phy, where there is considerable industrial interest. Industrial research is usually driven by the need to solve a specific separation problem, and this need often results in interesting and novel ideas. Working in a university atmosphere, Phil has been able to define his problems in more abstract and general terms, without having to tie the research to a specific problem. Although this approach can lead to sterile solutions, it can also lead to solu tions that are different from (but every bit as useful as) industrial solutions. In PSA, Phil's re search led the way to multicomponent separations and showed the importance of pressure drop in ordi ...his [grad school] experience taught him something he still strongly believes today: that the area of one's PhD is not all that important since the real purpose of grad school is to train one how to do research and how to formulate problems. An early (1965, Phil's junior year) experiment in tensile strength? nary operations. Both of these advances have since been adopted by industry. In chromatography, Phil has been interested in developing generic operational methods, treating chromatography as a unit operation. One example is moving withdrawal chromatography which can in crease throughput by one or two orders of magni tude and which is generally applicable to migration chromatographic systems. This approach is easy to scale up and easy to generalize, contrasting with the biotechnological approach that looks at each separa tion problem as chemically unique. The combination of these approaches will eventually make chroma tography a standard separation method. Phil has always mixed independent and collabo rative research. He is currently working with a cross disciplinary team of chromatographers in an NSF sponsored minicenter. This group includes George Tsao and Linda Wang (chemical engineering), Mike Ladisch (agricultural engineering), and Fred Regnier (chemistry). In addition, he has a longterm collabo ration with Martin Okos (agricultural engineering) on combined fermentation and separation, which has resulted in a patent. He has also worked with Alden Emery and Dave Kessler (chemical engineer ing) on separations, and with Fritz Friedlander (elec trical engineering) on high gradient magnetic sepa ration. Even further afield, he has coauthored pa pers with Bud Homsy of Stanford (an outgrowth of a Summer 1992 sabbatical at Berkeley), with Rich Noble of the Uni versity of Colorado, with Daniel Tondeur of ENSIC in Nancy, France (from another sabbatical), and with Renato Rota of the University of Milan. Research, scholarship, and teaching are inextri cably linked for Phil. His research in separations led to the development of his course on advanced sepa rations; this course fed into his work on adsorption and chromatography, ion exchange, and membrane separation; this work in turn led to modifications in the course, out of which came his book on rate controlled separations. Clearly, this book has a solid 0 Neal Houze, Phil, and Ron Barile making a philo sophical statement. (1976) background in research. Among his projects, Phil considers his intensification work on adsorption and chromatography, as well as his chromato graphic research on developing largescale systems, to be the most significant. TEACHING His first experiences in teaching taught him that he didn't know what he was doing. He wanted to give the students in a sophomore distillation class a strong, abstract, theoretical base of separations, us ing a deductive, topdown approach. The result was a disaster. The material was entirely over their heads. Although his years of study had enabled him to distill the knowledge for himself, his students didn't have that preliminary study to build upon. After that experience, he "became more concrete in his teaching"he talked about equipment and took stu dents to the unit ops lab so they could see the equip ment, enabling them to visualize it in the future. Another experience occurred later, during his re search. When he was a grad student, he studied the Thomas method for adsorption, but the material was covered only in lectures and he had never solved any problems with it. Later, while doing research, Phil came across the Thomas method again, but he had no memory of the earlier encounter. He studied it and figured it outbut it wasn't until two years later that he came across some old notes and real 122 ized that he had studied the method during his gradstudent days! Apart from making conclu sions on the quality of Phil's short and longterm memory, we can appreciate the import of his real ization that "the incident convinced me that you have to make students do things. Lecturing isn't enough. They have to do problems and derivations, and make connections." Phil learned why that earlier class hadn't worked when, in 1972, he took a course entitled "Educa tional Psychology for College Teachers," taught by John Feldhusen at Purdue. The course opened up whole new ideas about teaching, and it became the seed out of which his Masters degree in counseling and his own course on teaching grew. After his unhappy experience with the separa tions class, Phil converted his class to a selfpaced format. Anyone who has ever been involved in in structional development knows what a risk it is for a young assistant professor to make this kind of class change. (The phrase "professional suicide" comes to mind.) Such conversions consume a great deal of timetime regarded by primary committees as bet ter spent on research and becoming promotable. But Phil admits that he was immodest (or cocky?) in his belief that he could do it all. Events have validated that decision, of course, but that first semester re quired about thirty hours a week simply to develop test problems and study guides. His students had to meet an absolute standard, but if they didn't meet it, the only "penalty" was a retest. In this way students determined their own grades and did not finish until they showed evidence of mastery. The class was taught this way until 1982, when it ended after a curriculum revision. Phil felt that new faculty members could be helped greatly in their first few years as professors if they had some prior knowledge of what being a teacher entailed. His idea was to teach a graduate course on educational methods, and the idea resulted in a course that has been taught biennially since 1983. In 199091, with an NSF curriculum development grant, the course was expanded to include all of engineering, and a workshop was conducted in July of 1991 for professors from ten major research uni versities. A book has been developed for this course, and it should be published in 1992. Phil used the ideas first encountered in the teach ing class and extended them to engineering, relying along the way on a number of noted educators in engineering as well as in science and education. In addition to those already mentioned: Rich Felder and Karl Smith, for getting away from lecturing and Chemical Engineering Education One thing which makes Phil unique ... is that during a time when he was a full professor he also became a student again and earned a Masters in counseling [in 1982] .... he was only the second professor at Purdue who was allowed to enroll as a grad student... towards student learning; Jim Stice, for teaching methods; Rich Noble and Don Woods, for problem solving; Dick Hackney and Janine Bernard, who helped foster his early ideas on education; and Dendy Sloan, for combining caring and professionalism. COUNSELOR One thing which makes Phil unique (or odd, de pending on your point of view) is that during a time when he was a full professor he also became a stu dent again and earned a Masters in counseling. The incident that triggered this move occurred in 1975 when he had a student who was, by all accounts, very "strange." Exasperated and completely at a loss as to what to do with the student, Phil finally told him that his behavior was abnormal and bizarre and that he needed helpbut Phil had no idea where to send him for such help. "It devastated me, having to tell that to a student. It left me in a state of shock." Then, a friend of Phil's in Purdue's counseling program told him about a course that covered the basics of counseling. (Actually, it was quite an intro duction: five credits with about twenty hours of work per week.) During that course, Phil discovered other interests in the area and wound up taking another course the following semester. Then, after a sabbati cal, he applied for degree status in the program, but found himself confronting a bureaucracy that wor ried about conflicts of interests when faculty in one area wanted to study for a degree in another area. Phil argued that no conflict of interest existed since he was already a full professor and that he simply wanted to improve his teaching and counseling abili ties. He finally got his wish, and he became only the second professor at Purdue who was allowed to en roll as a grad student. He finished the Masters in Education in Counseling in 1982. Because of his interest in people and his desire to gain some practical counseling experience, he volun teered to work at the Crisis Center in Lafayette. Over the span of seven years he estimates that he gained at least a year's worth of valuable experience as a working counselor through this volunteer activ ity. An added dividend from this work was meeting a lady named Dot, who became his wife in 1980. AUTHOR It is not unusual for a professor to entertain thoughts of writing a book. The reasons, both peda Summer 1992 gogical and personal, are many and varied. For Phil, it was simply a "huge desire to write a book." Ever since 1975 he had wanted to write a book, but there were already a number of good texts on equilibrium staged separations in the marketplace. With his back ground, however, he felt he could write a book that was pedagogically sound, with an emphasis on what helps people to learn. For example, he would give specifics, be concrete, build up to a general argu ment, give detailed example problems, and follow a specific problemsolving strategy (based on the ideas of Don Woods). He would present the strategy in the first chapter and then use it in all of the example problems, giving the student a clear method to follow. He would also try to have at least one homework problem drawing on each and every section of the book, so that profes sors could have a choice. And he would list objectives for each chapter and provide numerous figures. The result, after ten years of intermittent labor, was EquilibriumStaged Separations. In addition to over one hundred and twenty pub lications, he is the author of four books: Large Scale Adsorption and Chromatography (CRC Press, 1986); EquilibriumStaged Separations (Elsevier, 1988); RateControlled Separations (Elsevier, 1990); and, with Frank Oreovicz, Teaching Engineering (McGrawHill, in press). He was a coeditor of Ad sorption and Ion Exchange: Fundamentals and Ap plications (AIChE Symposium Series) and is Editor inChief of the journal Separations and Purification Methods. His publications on education and teach ing number more than thirty articles and include coauthors such as Ron Barile, Alden Emery, Neal Houze, and Frank Oreovicz. ADMINISTRATOR At one time, Phil swore he would never be an administrator, but his Dean looked at Phil's resume (after being prodded by Nick Peppas) and thought Phil would be a perfect match for Head of Freshman Engineering. The more Phil thought about heading up the freshman engineering program at Purdue, the more intrigued he became. For one thing, he had been a professor for seventeen years and felt he could benefit from a new challenge (while still main taining his research and teaching). For another, the new program was clearly focused on students, with a Continued on page 159. classroom CONFIRMING THERMODYNAMIC STABILITY A Classroom Example KENNETH R. JOLLS, JEFFREY L. BUTTERBAUGH1 Iowa State University Ames, IA 50011 Stability theory is a topic that has begun to appear more frequently in modern thermody namics courses. The recent chemical engineer ing texts by Sandler[l] and Kyle[2] and the mechani cal engineering text by Bejan[31 discuss the subject briefly, and the more advanced monographs by Glansdorff and Prigogine,[4] Callen,[51 and Modell and Reid161 treat it in greater depth. Stability consid erations underlie much of the thinking in classical thermodynamics and are essential to any thorough understanding of processes involving phase change. Teaching stability theory, however, is hampered by a lack of practical examples. Students are accus tomed to systems that are presumed to be in stable equilibrium, and they have little experience with states removed from that condition. While one may memorize stability precepts formallyentropy maximization under isolation or other potential minimizations under corresponding constraints translating such notions into an understanding of their significance is not easy. Thermodynamics has its share of skeptics among students, both in regard to its content and to its conventional pedagogy. Ex pecting such individuals to give serious thought to a difficult theory describing states rarely (or never) observed is naive at best. But everyday examples certainly exist. Common among these are supersaturated solutions and su perheated liquids, both predictably metastable and both primed to revert to the more stable, twophase conditions in response to nucleating stimuli. Experi ments involving superheated liquids have been per formed by PatrickYeboah and Reid,171 and class room demonstrations have been developed by Jolls and Prausnitz.181 Still more deeply metastable liquid states were discussed in the interesting articles by Address: Procter and Gamble Company, Winton Hill Technical Center, Cincinnati, OH 452241797 Hayward[9l and Scholander[i01 on "negative pressure" and "tensile water." In the absence of such experiments, however, one must develop descriptions of stabilityrelated phe nomena that are sufficiently concrete to be convinc ing. Students must be persuaded that these ideas are real and merit the same level of attention as the more tangible aspects of thermodynamic analysis. INTRODUCTION We have found a way to reinforce stability con cepts that both satisfies the pragmatist and retains theoretical rigor. We use the tabulations of stable and metastable states for water and steam found in the wellknown Steam Tables by Keenan, Keyes, Hill, and Moore (KKHME111). Students use these data to locate pairs of matched states in stable metastable combinations. Then, using the appro priate thermodynamic potential for a given pair, they compare the stability levels of the states and confirm the rankings. In this paper we show a typical set of comparisons for the five potentials customarily applied to pure fluids. Before proceeding, however, we give a brief review of the principles of stability analysis, pre sented in the style of Modell and Reid.[61 Kenneth R. Jolls has undergraduate degrees from Duke and North Carolina State and gradu ate degrees from the University of Illinois. His specialties include electronic instrumentation, thermodynamics, and the use of computer visu alization in chemical engineering research and practice. Jeffrey L. Butterbaugh received his BS and ME from Iowa State University. His graduate minor was in Business Administrative Sciences, and he is presently working as a process development F engineer for Procter and Gamble Research and Development. @ Copyright ChE Division ofASEE 1992 Chemical Engineering Education The following inequalities are associated with the stable equilibrium states of vari ously constrained systems: ASM <0 AUM >0 AAT,V,M > 0 All >0 AHS,P,M > 0 AG > 0 T,P,M (2a) (3a) In these expressions the underbar signifies an extensive property, subscripts specify the constraints imposed for a given comparison, and the symbolism is conventional. Inequality (la) expresses the entropy maximum principle: for an isolated system, unconstrained internally and in a stable equilibrium state, the entropy decreases in response to all perturbations that preserve isolation. One tests this idea through a so called "thought" experiment in which an iso lated system in a stable equilibrium state is partitioned into two contiguous sections and perturbed by means of a "virtual" process. The two (intensively) identical parts pro vide the mutual "give" needed for the ther modynamic properties to change locally while overall isolation is maintained. Kyle's discussion[2] provides a helpful background for conceptualizing such processes. Inequality (2a) reflects the duality of the entropy and energy representations of the fundamental equationthe entropy maxi mum implies an energy minimum.* Inequali ties (3a), (4a), and (5a) follow from (2a) and are derived by contriving a subsystem within the (S,Y,M)fixed composite whose tempera ture and/or pressure may be held constant during the perturbation process. Paraphrasing inequality (3a), for example, we note that any extensive state, known to represent a stable equilibrium condition, will possess a value for the Helmholtz energy less than that of a second state, remote from the first but having the same temperature, mass, and total volume. Analogous state ments characterize the comparisons implied by the other four constrained inequalities. For singlephase states well removed from any phasechange boundary, verifying in * Page 126 of Reference 6. Summer 1992 Paraphrasing inequality, we note that any extensive state, known to represent a stable equilibrium condition, will possess a value for the Helmholtz energy less than that of a second state, remote from the first but having the same temperature, mass, and total volume. /'B (T/T) = 0.85 p = G = A,q + (P*)(' V1qd) D= Avapor + (P*)(V p ,) ^ Slope at any point energy Slpatp A, slope of,  mass Ithe common Aaor tangent ine, P* I liquid volumes of the vapor I coexisting phases 0 1.0 3.0 5.0 7.0 9.0 V (multiples of V.), L3M Figure 1. Subcritical isotherm of the Helmholtz energy from the PengRobinson equation (P* denotes vapor pressure). qualities (la) through (5a) can be carried out only through statistical reasoningexamination of molecular configurations that are allowed but less probable. (Balzhisert12] presents simple but effective examples of these.) For twophase conditions, how ever, particularly where the equilibrium state lies only a short distance inside the coexistence boundary, more tangible com parisons are possible. In Figure 1 we show a typical isotherm of the specific Helmholtz energy* for a temperature below the fluidphase critical point in a pure system. Intrinsic stability is guaranteed for states where (a2A/aV2), is positive (volumes to the left of point C and to the right of point D). The line tangent to the two lobes of the curve identifies states B and E that have the same temperature, pres sure, and chemical potential. These states can thus coexist in equilibrium at any fraction vapor so as to yield (through lever rule proportioning) values of A and V for the twophase mixture intermediate to those for the individual states. The dotted line in Figure 1 intersects the metastable, single phase state I and the stable, twophase state IIeach at the same temperature and volume but with the latter possessing a lower value of A to coincide with its more stable condition. Similar arguments applied to a subcritical isobar on HS coordi nates show that the more stable state has a lower value of H when entropy and pressure are the constraints. * Based on a cubic equation of state. In the following sections we will quantify these comparisons for inequalities (la) through (4a) and show analogous stable/metastable pairs for inequality (5a). THE KKHM TABLES The data tabulated in the Steam Tables[111 are based on an empirically determined expression for the Helmholtz energy per unit mass of water sub stance A(T,V).* The form of this function is shown in the Appendix of the Steam Tables, which also in cludes tabulated values of the sixtyone constants used and discussions of the supporting experimental data. In addition to the usual coverage of properties in the fully stable regions, data for the metastable, singlephase states that lie just inside the satura tion curves are given for both the liquid and vapor phases. In compiling the Steam Tables this informa tion was generated by extrapolating the fundamen tal equation inside the coexistence boundary and tabulating property values as continuous extensions of the fully stable isolines outside. (Italics are used to designate metastable conditions.) While no ex perimental values were used to control these exten sions, the authors refer to them as "reasonable" and as providing "the best values available" (ca. 1978). * Given in the Steam Tables as TY(T,p). In Table 1 we reproduce a small portion of the data for saturated and superheated steam (regular face) and for subcooled steam (italic) in the vicinity of the normal condensation point for a pressure of 0.38 MPa. The actual condensation temperatures are shown in parentheses, and a dashed line is drawn in each column to separate data for the two kinds of stability. It is thus possible to identify properties and prop erty changes in the metastable regions and also to expect that conventional thermodynamic operations will be borne out using those data. For example, one might use numerical differentiation to verify the following Maxwell relation: pST \ Tlp Substituting property values centered around the metastable condition t = 95C, P = 0.38 MPa, into the finitedifference approximations of the deriva tives, we obtain ( (6.59726.6600)(103) APT 0.04 5710(m/kgK (408.9424.5)(103) (Y)p 10 156x 103(m3 /kg.K) The small error results from simple finite differencing TABLE 1 VaporPhase Data (Steam Tables,'11 pages 2829) P(t Sat.) 0.36 (139.87) 0.38 (141.79) 0.40 (143.63) t VX103 U H S Vx103 U H S Vx103 U H S Sat 510.6 2549.9 2733.7 6.9311 485.3 2551.8 2736.2 6.9130 462.5 2553.6 2738.6 6.8959 75 407.9 2429.2 2576.1 6.5148 384.1 2425.9 2571.9 6.4800 362.6 2422.6 2567.6 6.4463 80 416.7 2439.4 2589.4 6.5528 392.5 2436.4 2585.5 6.5189 370.8 2433.2 2581.5 6.4860 85 425.2 2449.5 2602.5 6.5897 400.8 2446.6 2598.9 6.5564 378.7 2443.7 2595.2 6.5243 90 433.6 2459.3 2615.4 6.6254 408.9 2456.6 2612.0 6.5928 386.6 2453.9 2608.5 6.5614 95 441.8 2469.0 2628.1 6.6600 416.8 2466.5 2624.9 6.6280 394.2 2464.0 2621.6 6.5972 100 449.9 2478.5 2640.5 6.6935 424.5 2476.2 2637.5 6.6621 401.7 2473.8 2634.5 6.6319 110 465.7 2497.1 2664.8 6.7578 439.7 2495.1 2662.2 6.7273 416.3 2493.0 2659.5 6.6981 120 481.0 2515.2 2688.4 6.8187 454.4 2513.4 2686.1 6.7890 430.4 2511.6 2683.8 6.7605 130 496.0 2532.9 2711.5 6.8765 468.8 2531.3 2709.4 6.8475 444.2 2529.7 2707.3 6.8197 140 510.8 2550.1 2734.0 6.9318 482.8 2548.7 2732.2 6.9033 457.6 2547.3 2730.3 6.8761 150 525.2 2567.0 2756.1 6.9847 496.6 2565.8 2754.5 6.9567 470.8 2564.5 2752.8 6.9299 160 539.4 2583.7 2777.9 7.0355 510.2 2582.6 2776.4 7.0079 483.8 2581.4 2774.9 6.9815 170 553.5 2600.1 2799.4 7.0846 523.6 2599.1 2798.0 7.0573 496.6 2598.1 2796.7 7.0312 180 567.4 2616.3 2820.6 7.1320 536.8 2615.4 2819.4 7.1049 509.3 2614.5 2818.2 7.0792 190 581.2 2632.4 2841.7 7.1779 550.0 2631.6 2840.6 7.1511 521.8 2630.7 2839.5 7.1256 Properties at conditions below the saturation temperature for each pressure are italicized and correspond to metastable (subcooled) states. Symbol and Meaning Units Symbol and Meaning Units Symbol and Meaning Units P pressure MPa V specific volume m'/kg H specific enthalpy kJ/kg t temperature C U specific energy kJ/kg S specific entropy kJ/kg.K Energy and entropy are each taken to be zero for the saturated liquid phase at the triple point. .26 Chemical Engineering Education of these nonlinear functions. Because the Steam Tables present data on a per unitmass basis, we need not be concerned with the fixedmass constraint. Thus we can express each of the conditions we wish to confirm in the simpler, doubly subscripted form involving specific properties only: ASu,v < 0 (Ib) AUsv > 0 (2b) AAT, > 0 (3b) AHSP > 0 (4b) AGT,P > 0 (5b) Inequalities (3b), (4b), and (5b) are the easiest to verify because one or both constraints conform to the temperature and pressure indexing of the data.[111 We show these calculations first. Helmholtz Energy To confirm inequality (3b), we must find two states at the same temperature and volumeone that rep resents stable equilibrium conditions and the other metastable. In Figure 2 we show the twophase re gion in the vicinity of the 950C isotherm on pressure volume coordinates. (Note the difference in volume scales for liquid and vapor.) Let us consider the metastable (subcooled) vapor state at 950C, 0.38 MPa, as our base point. From Table 1 we have the follow ing properties: V= 0.4168 m3 /kg U= 2466.5 kJ/kg S = 6.6280 kJ/(kg.K) Liquid water and steam at 95C will coexist in Figure 2. Pressurevolume diagram, lowpressure range (volume scales differ for liquid and vapor). Summer 1992 stable equilibrium under a vapor pressure of 0.08455 MPa1 and with any fraction vapor. To find a tem peraturevolume match for the base point, we need only compute the fraction vapor in the twophase state that gives the same overall specific volume as in the metastable state (0.4168 m3/kg). Noting that the saturatedstate liquid and vapor volumes at this temperature are 1.0397 x 103 and 1.9819 m3/kg, respectively, we solve for the desired fraction. x = 416.8 10397 0.2099 19819 0397 Thus, the stable, twophase state at 950C (0.08455 MPa) and with vapor fraction 0.2099 has the same temperature and volume as the metastable, single phase state at 95C and 0.38 MPa, and we can pro ceed to compare the Helmholtz energies. metastable state: A = U TS = 2466.5 (95 + 273.15)(6.6280) = 26.4 kJ/kg stable state: saturated phases at 95C liquid vapor U 397.88 2500.6 S 1.2500 7.4159 from which A = x[U TS]ap,r + (1 x)[U TS]iquid = 97.4 kJ/kg (6) Stable < Ametastable Enthalpy Confirming inequality (4b) requires a stablemeta stable pair at the same entropy and pressure. Figure 3 shows the twophase region in the vicinity of the 0.38 MPa isobar on t S coordinates. Again we choose the metastable (subcooled vapor) base state along the 0.38 MPa isobar at 950C [where S = 6.6280 kJ/(kg.K) and H = 2624.9 kJ/kg]. From entropymatching calculations analogous to the volumematching calculations in the previous case, we find that the stable, twophase state at 0.38 MPa (t = 141.790C) and vapor fraction x = 0.9447 has =the same entropy.2 With satura SI tion properties 3400 Hiuid = 596.83 and 1 Reference 11, page 3. 2 Reference 11, page 10. 0.50 S 95TC ' 0.38 0.38 MPa base 0.30 point saturated vapor point saturated vapor 0.30 95C (metastable vap P, MPa 0.20 V=416.8 saturated liquid 0.10 0.08455 MPa (1x)=0.7901 0.00 ' ' 1.02 1.07 400 V, m'/kg x 103 _1 I I I or) Hvapor = 2736.2 the enthalpy of the twophase state is given by Hstable = xHvapor +(1 x)Hliquid = 2617.94 kJ / kg (7) and Stable < Hmetastable Gibbs Energy Stablemetastable pairs that confirm inequality (5b) involve liquid states that are either stably com pressed above or metastably expanded below satu ration. The limited amount of data for such states in the Steam Tables requires that we move the region of interest to a higher temperature. Figure 4 shows the liquid and vapor branches of the 2600C isotherm with the saturation points (at 4.688 MPa) dividing each branch into stable and metastable sections. Numerical data appear in Table 2. Two Gibbs energy comparisons are possible from these data. At 2.5 MPa the liquid state (0) is metastable: G = H TS = 1134.8 (533.15)(2.8898) = 405.9 kJ / kg whereas for the stable (superheated) va por at this pressure (o) G = 2907.4(533.15)(6.4601) = 536.8 kJ/ kg and Stable < Gmetastable At 5.1 MPa the situation is reversed. For the metastable (supercompressed or subcooled) vapor (A) G = 2770.4 (533.15)(5.9214) = 386.6 and for the stable (compressed or subcooled) liquid (0) G = 1134.3 (533.15)(2.8827) = 402.6 Again the inequality is confirmed. Re eating the calculation at the exact saturation pres sure confirms the equality of G for phases coexisting at equilibrium.* Glquid= 403.1 G va = 403.0 vapor Entropy and Energy Inequalities (Ib) and (2b) are more difficult to confirm because the twophase state in each case must satisfy two nonindexed constraints. For the entropy comparison one must find a stablemeta stable pair with the same volume and energy. Re * Properties of coexisting states are obtained from the fundamen tal equation at the observed vapor pressure for a given tempera turethus the insignificant difference in Gvalues for saturated liquid and vapor phases (Ref. 11, p. 135). 165  . / 141.79C '0 =05 t,C S66280 saturated vapor 115 \ 950 base point 95 / saturated liquid 0 38 MPa (metastable) 65 1 i l l1 l L I I I 0.5 1.0 1.5 2.0 2.5 6.0 6.5 7.0 7.5 8.0 S, kJ/kgK Figure 3. Temperatureentropy diagram. saturated vapor 5.1 MPa 5.0 4.688 MPa P,MPa 4.0  saturated liquid 260C (metastable liquid) 260C (stable vapor) 3.0 2.5MPa 2.0 1.1 1.4 20 100 V, m3/kg x 103 Figure 4. Pressurevolume diagram, highpressure range (volume scales differ.) Chemical Engineering Education turning to Figure 2 and to the original metastable base point, we search for a twophase state with a temperature and fraction vapor such that xUvapr (t) + (1 X)Uiquid (t) = Umetstble =2466.5 (8) and xVvapor (t) + (1 X)Viquid (t) = Vmetastble = 0.4168 (9) Trialanderror solution of these equations (using linear interpolation for saturation properties between tabulated points) yields t= 145.84 C x = 0.9541 (P = 0.4251MPa) with saturation values at this temperature* U 103V S liquid 613.78 1.0859 1.7992 vapor 2555.7 436.8 6.8756 Thus, the entropy of the stable, twophase state is Stable = xSvaor +(1 X)Sliquid = 6.6426 This exceeds the entropy of the metastable state (6.6280), and we conclude Stable > Smetatable For the energy comparison we must match the volume and entropy of the base state. We replace Eq. (8) with the analogous expression for entropy xSvapor (t) + (1 X)Sliquid (t) = Smetastable = 6.6280 (10) and solve as before through linear interpolation t = 145.71C x = 0.9510 (P = 0.4236 MPa) with saturation properties liquid vapor S 1.7979 6.8768 103V 1.0858 438.3 U 613.24 2555.5 From these data we determine Ustable = 2460.3 and stable < Umetastable The twophase states for these latter comparisons are indistinguishable at the scale of Figure 2, and we represent both on the single horizontal line t'. CLOSING REMARKS The significance of these comparisons must be explained carefully. They must not be characterized as any form of proof of the validity of stability theory. Indeed, they are not that at all. The principles of stability theory are no more capable of proof than are the Laws of Thermodynamics themselves. Given our acceptance of the Laws (or of the Postulates that * Reference 11, page 4. Summer 1992 underlie the Laws in the neoGibbsian tradition[13]), stability criteria follow logically. Thus, material be havior in violation of any criterion of thermodynamic stability is in de facto violation of the Second Law. These examples do, however, offer a tangible link between real systems and abstract models. They reveal the thermodynamic consistency of an empiri cally fashioned fundamental equation (the basis for the KKHM Steam Tables), and they give support to the too often rotelearned precepts of entropy maxi mization and Gibbs energy minimization. We offer these exercises as a way to give students a semiquantitative feeling for concepts usually rel egated to pure abstraction. They may be understood even better when accompanied by descriptions of "familiar" metastablestable transitionscrystalli zation from a supersaturated solution or the explo sive (and potentially dangerous) vaporization of a superheated liquid. Reid's series on the latter sub ject114] provides excellent background reading. ACKNOWLEDGMENTS Tabular data were reprinted by permission of John Wiley and Sons. Financial support came from Iowa State University and from the Camille and Henry Dreyfus Foundation. Marcia Pierson edited this paper and managed its production through the Office of Editorial Services in the College of Engi neering. David Sauke used Aldus Freehand to pre pare the figures. LITERATURE CITED 1. Sandler, S.I., Chemical and Engineering Thermodynamics, 2nd ed., Chap. 5, John Wiley and Sons, Inc., New York, NY (1989) 2. Kyle, B.G., Chemical and Process Thermodynamics, Chap. 7, Prentice Hall, Inc., Englewood Cliffs, NJ (1984) 3. Bejan, A., Advanced Engineering Thermodynamics, Chap. 6, John Wiley and Sons, New York, NY (1988) 4. Glansdorff, P., and I. Prigogine, Thermodynamic Theory of Struc ture, Stability and Fluctuations, Chap. IV, WileyInterscience, Lon don, England (1971) 5. Callen, H.B., Thermodynamics and an Introduction to Thermosta tistics, 2nd ed., Chap. 8, John Wiley and Sons, New York, NY (1985) 6. Modell, M., and R.C. Reid, Thermodynamics and Its Applications, 2nd ed., Chaps. 6, 9, PrenticeHall, Englewood Cliffs, NJ (1983) 7. PatrickYeboah, J.R., and R.C. Reid, Ind. and Eng. Chem., Funds., 20,4,315(1981) 8. Jolls, K.R., and J.M. Prausnitz, "Laboratory Demonstrations for Teaching Chemical Thermodynamics," paper presented at the An nual Meeting of AIChE, Washington, DC, November (1983) 9. Hayward, A.T.J., "Negative Pressure in Liquids: Can It Be Har nessed to Serve Man?" Amer. Sci., 59, 434 (1971) 10. Scholander, P.F., "Tensile Water," Amer. Sci., 60, 584 (1972) 11. Keenan, J.H., F.G. Keyes, P.G. Hill, and J.G. Moore, Steam Tables (International System of UnitsS.I.), John Wiley and Sons, New York, NY (1978) 12. Balzhiser, R.E., and M.R. Samuels, Engineering Thermodynamics, Appendix G, PrenticeHall, Inc., Englewood Cliffs, NJ (1977) 13. Tisza, L., Generalized Thermodynamics, The M.I.T. Press, Cam bridge, MA (1966) 14. Reid, R.C., "Superheated Liquids: A Laboratory Curiosity and, Pos sibly, an Industrial Curse," Parts 1 and 2, Chem. Eng. Ed., 1, 60, 108 (1978) 0 O class and home problems The object of this column is to enhance our readers' collection of interesting and novel problems in chemical engineering. Problems of the type that can be used to motivate the student by presenting a particular principle in class, or in a new light, or that can be assigned as a novel home problem, are requested, as well as those that are more traditional in nature and which elucidate difficult concepts. Please submit them to Professors James O. Wilkes and Mark A. Burns, Chemical Engineering Department, Univer sity of Michigan, Ann Arbor, MI 481092136. THREE PROBLEMS IN FLUID MECHANICS JAMES O. WILKES, STACY G. BIKE University of Michigan Ann Arbor, MI 481092136 We present here (and solve) two homework problems that we have developed in the undergraduate chemical engineering fluid mechanics course at the University of Michigan. The first problem involves a fundamental principle of hydrostatics and requires thoughtful but simple rea soning for its solution, while the second problem is a good illustration of the application of potentialflow principles. A third problem is also presented, but is left for the reader to solve. The course is our second required undergraduate course, taken in the second term of the sophomore year, after thermodynamics (mass and energy balances). After trying a few text books, we (and our students) have opted instead for an extensive set of course notes that we have written and typeset. We always attempt to set problems that apply the principles of fluid mechanics to practical situations, albeit simplified in some cases. The authors are both faculty members in the Department of Chemical Engineering at the University of Michigan. James 0. Wilkes, who is also Assistant Dean of the College of Engineering, has current research inter ests in the flow of paint films and injectionmolding of polymer composites. Stacey G. Bike received her PhD from Camegie Mellon University in 1988, and conducts research in the area of colloid science, including the fluid mechanics of colloid al dispersions and the theological char 1 acterization of coat ings. Copyright ChE Division of ASEE 1992 1 2 3 H h t h M  1  Figure 1. Ship moving through locks. PROBLEM 1 Water Supply for a Ship Moving Through Locks A ship of mass M travels uphill through a series of identical rectangular locks, each of equal superficial (birdseye view) area, A, and elevation increase, h. The steps involved in moving from one lock to the next (1 to 2, for example) are shown as ABC in Figure 1. The lock at the top of the hill is supplied by a naturally occurring source of water of density p. Initially (A), the ship is isolated in lock 1, which has a depth of water H. The gate between locks 1 and 2 is then opened (B), equalizing the depths of water in the two locks. Finally (C), the ship moves into lock 2 and the gate is closed behind it. 1. Derive an expression for the increase in mass of water in lock 1 for the sequence shown, in terms Chemical Engineering Education H t H+h (a) Uphill (b) fD n Dh t^T D __h_ (c) Downhill Figure 2. Ship moving from one lock to the next. of some or all of the variables M, H, h, A, p, and g. 2. If, after reaching the top of the hill, the ship descends through a similar series of locks to its original elevation, again derive an expression for the mass of water gained by a lock from the lock immediately above it. In this case, the initial depth in the uppermost lock will be D (greater than H). 3. Does the mass of water to be supplied depend on the mass of the ship if: (a) the ship travels only uphill, (b) the ship travels uphill, then downhill? Explain your answer. SOLUTION 1. First, examine the ship as it travels uphill. As it passes from one lock to the next (say, from lock 1 to lock 2), the new depth of water in lock 2 must be Hexactly the same as it was in lock 1. The depth of the water remaining in lock 1 is therefore H + h. Figure 2 shows two appearances of lock 1: (a) first, when the ship is still in it, and (b) after the ship has moved into lock 2. Now examine the corresponding masses of water in lock 1 under these two conditions: (a) From Archimedes' law, the weight of the wa ter displaced by the floating ship is the weight of the ship itself, namely Mg. Therefore, when the ship is still in the lock, the mass of water displaced by the ship is M, so the mass of water in the lock is pAH M. (b) After the ship has moved out of lock 1, the lock subsequently contains a mass of water pA(H + h). Hence, the mass of water to be supplied is the difference between these two quantities: pA(H + h)(pAHM)= pAh+M (1) 2. When the ship is proceeding downhill, as shown in Figures 2(c) and (d), the amount of water lost from the higher lock is likewise (pAD M)pA(D h)= pAh M (2) 3. In conclusion, we observe from the above that 0 The amount of water to be supplied is pAh M, depending on whether the ship is proceed ing uphill or downhill, respectively. > Thus, the amount of water does depend on the mass of the ship, and is different for motion uphill or downhill. 0 If the ship navigates both uphill and down hillas in traversing the Panama Canal, for examplethe total water supply needed is 2pAh, which is independent of the ship's mass. Thus, whether the Queen Mary or a rowboat is involved, the total water supply required is the same. PROBLEM 2 GroundWater Seepage Figure 3 shows the seepage of water through the ground under a dam, caused by the excess pressure P (beyond that naturally occurring in the absence of the impounded water) that arises from the buildup of water behind the dam, which has (underground) a semicircular base of radius rD. 1. Verify the following relation, which has been pro posed for the (excess) pressure in the ground: p=P(1a) (3) 2. Determine the streamlines for the flow. 3. Between points A and B, a large amount of cop perimpregnated soil has been detected, with the possibility that some of this toxic metal may leach out and have adverse effects downstream of the dam. To help assess the extent of this danger, 6=ir Figure 3. Seepage of water under a dam. Summer 1992 derive an expression for the volumetric flow rate, Q, of water between A and B (per unit depth in the zdirection, normal to the plane of the dia gram), in terms of P, K (the permeability of the ground), i (the viscosity of water), rA, and rB. SOLUTION 1. Start by observing that the flow of water in the ground is governed by D'Arcy's law v KVp (4) in which v is the (vector) superficial velocity and p is the pressure. By applying the continuity equa tion Sv = 0 (5) and assuming constant permeability K and vis cosity T1, we find that the pressure obeys Laplace's equation V2p = 0 (6) We are now reminded that the problem is essen tially one of potential flow; indeed, the flow is irrotational, because the vorticity of a velocity that is proportional to the gradient of a scalar is zero, as may be checked by expanding V x v = V x Vp and discovering that it is a vector with three zero components. Now examine the proposed pressure distribution by checking to see if it satisfies the following constraints: (a) The conditions on pressure at the ground level. For 0 = 0 and i, Eq. (3) gives p = P and p = 0, respectively, confirming the known pressures both upstream and downstream of the dam. (b) Laplace's equation, V2p = 0, in cylindrical (r/0/z) coordinates, in which all z derivatives are zero, is 1 (P 1 2 po V2p =p a 2 = 0 (7) r 2r rW The first term on the righthand side of Eq. (7) is zero, because the proposed expression for p is independent of r. The second term is also zero, because p is only a linear function of 0. Thus, Laplace's equation is satisfied. (c) Zero radial flow at the base of the dam. It will soon be seen that the radial velocity vr is proportional to ap/ar, which is zero. Thus, all constraints are satisfied by the pro posed solution, which indicates that the pressure decreases linearly with the angle 0 between the groundlevel upstream and downstream values of P and zero, respectively. 2. The r and 0 components of Vp in cylindrical (r/0/z) coordinates are _Dp 1 ap (Vp), = and (Vp)e = (8) from which it follows (in conjunction with D'Arcy's law) that the radial and angular velocity compo nents are V K p v T =0 T ) j r and ve = KP Te I r T rr The corresponding expressions in terms of the stream function w are known to be 1 ra Vr r as and v= a 0 ar which, because of the minus sign in D'Arcy's law, are the negatives of those usually encountered. Substitution of the known values for vr and v, from Eq. (9) into Eq. (10), and integrating, gives S= f(r) and = KPnr + g(0) (11) nil in which f(r) and g(0) are functions of integration. The two expressions for the stream function in Eq. (11) must be compatible, so that f(r) = (KP/hri) In r and g(0) isat mosta constant, which may be taken as zero, giving S= KP Inr (12) Since the streamlines are contours of constant y, they must also be curves of constant rthat is, semicircles, as shown in Figure 3. The isobars (or equipotentials) are, from Eq. (3), lines of constant 0 and are orthogonal to the streamlines. If both streamlines and isobars were drawn, they would appear as the circular arcs and radial lines of a spider's web. 3. The flowrate between A and B (per unit depth, normal to the plane of the diagram) can be ob tained in two ways. First, by definition of the stream function, it is simply the difference be tween 4A and ~B Q = B A = nr n rA) = KP nrB (13) M = 1A rA Second, the same result can be obtained by inte grating the velocity between the two points: B B Q = dr = KPdr = nB (14) A AJ nilr M rA A A Chemical Engineering Education PROBLEM 3 Bubble Rise We leave the reader with an intriguing problem that originated with our colleague, Professor (now Emeritus) M. Rasin Tek. As shown in Figure 4, a hollow vertical cylin der with rigid walls and of height H B 0 is closed at both ends and is filled with a volume, V, of an incompress ible and nonvolatile oil of density p H at a uniform temperature T. A gauge registers the pressure at the top of the cylinder. A When a small spherical bubble Figure 4. of volume v initially adheres by sur Bubble rising in face tension to point A at the bot liquid in a tom of the cylinder, the absolute closed cylinder, pressure at the top of the cylinder is po. The gas in the bubble is ideal, and has a molecular weight of M. The bubble is liberated by tapping on the cylinder and rises to point B at the top. Derive an expression in terms of any or all of the specified variables for the new absolute pressure pi at the top of the cylinder. Ex plain your answer carefully! We leave the reader in suspense, requesting that he or she solve this problem. It is instructive for a fluid mechanics class because it shows that if you proceed methodically, the answer is deceptively simple. And, if you find it too easy, try it for the case when the oil is slightly compressible, with an iso thermal compressibility p. 0 book review CHEMICAL AND PROCESS THERMODYNAMICS 2nd Edition by B.G. Kyle Prentice Hall, Inc., Englewood Cliffs, NJ 07632 Reviewed by E. Dendy Sloan Colorado School of Mines In this useful second edition, the author has avoided an encyclopedic "drinkofwaterfromafirehydrant" approach to thermodynamics in favor of pedagogical digestion. The examples and problems with each chap ter are well conceived, and a complete solutions manual is available. The text nomenclature and topic ordering will seem familiar to professors teaching the topic. Modern aspects of the book involve applications of classically stated fundamentals to environmental con trol, electrolytes, biochemicals, and electronic materi als. Material on Jacobians, stability, and complex chemical equilibria go beyond topics found in many undergraduate texts. A major asset of the book is its treatment of fluid properties. The author has eschewed the use of three parameter corresponding states, providing graphic visualization of changes in compressibility and re sidual properties as a function of reduced tempera ture and pressure. An IBMcompatible floppy disk program (ca. 4000 lines) in the endpapers enables more accurate calculation of pure fluid properties (other than vapor pressure) through the Peng Robinson equation of state (EOS). A second major asset is the treatment of phase equilibria. After a brief treatment of principles, the author goes straight to applications, with advanced topics relegated to a later chapter. For example, in the first chapter on phase equilibria principles the author gives examples of activity coefficient hand cal culations to optimize van Laar and Margules param eters, but a floppy disk program (ca. 5000 lines) is provided to either optimize or use Wilson equation parameters. Regular solution and UNIFAC treatments are delayed until the third chapter on phase equilib ra. The floppy disk programs represent one approach to introduce students to the foundations of ubiquitous flowsheeting programs in the profession. As such, a reader might wish for the unifying device of a Peng Robinson EOS program applied to mixtures so that, for example, students could observe relative inaccu racies of an EOS versus activity coefficients for mix tures of alcohol+water or those of hydrocarbons. Summing up, this book is a welcome addition for students learning undergraduate thermodynamics. If the author included an extension to molecular exposi tion and a final chapter on statistical thermodynam ics, the book might also be a foundation to address the dearth of introductory graduate texts. O Summer 1992 Random Thoughts... WHAT DO THEY KNOW, ANYWAY? RICHARD M. FIELDER North Carolina State University Raleigh, NC 276957905 S ooner or later, the conversation at the commit tee meeting or in the faculty lounge turns to student ratings of instructors. It's a sure bet that within six seconds, someone will announce that ratings are meaninglessstudents don't know enough to evaluate the quality of their instruction. Others agree: one grumbles that the high ratings always go to the easy graders and entertainers; an other adds with complete assurance that the rigor ous instructors who are really the best teachers may get low ratings now but in later years their students will come to appreciate them. What is interesting is that these assertions are invariably offered without a scrap of evidence by individuals with welldeserved reputations for ana lytical thinking. If someone offered such unsupported arguments in a research seminar, most of us would dismiss both the arguments and the arguer out of hand. In discussions of teaching, however, we routinely suspend the rules of logical inference without a second thought. It's not as if data on student ratings are lacking. Cashin[11 notes the existence of 1300 articles and books dealing with research on the subject; Feldman[21 sees Cashin and raises him to 2000! So, for the record and in case you happen to find your Richard M. Felder is Hoechst Celanese Profes sor of Chemical Engineering at North Carolina 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). i Copyright ChE Division ofASEE 1992 self on a committee where student ratings come up, here are some facts to throw into the conversation. MYTH: Students lack the wisdom and experience to evaluate the effectiveness of their current in structors. Those who give instructors low rat ings at the end of a course will in future years appreciate those instructors. FACT: High correlations exist between courseend ratings and ratings by those who presum ably have the required wisdom and experi encepeers,[31 administrators,[4] alumni,[571 and graduating seniors.s8'91 If professors in your department who know how you teach rated your effectiveness, the results would probably not differ all that much from your student ratings. If students rate you highly now, they'll probably still do so when they look back in future years; if they dislike you now, the chances are that in their later wis dom they won't decide you were really a gem. MYTH: Student evaluations are just popularity contests. Easy teachers/easy graders get the highest ratings. FACT: Teachers who assign more work and more difficult work tend to be rated as most effec tive.[3,9,101 Some studies show no effect of grad ing practices on overall student ratings,[11,12] others find tendencies for teachers giving higher grades to get higher ratings. The lat ter result does not invalidate the ratings, however: as McKeachie[111 observes, if stu dents learn more from a teacher, one would expect both their grades and their ratings to be higher. Chemical Engineering Education ... research shows that student evaluations of an instructor provide a reliable, valid assessment of that instructor's teaching effectiveness, especially if they reflect the views of many students in several different course offerings .... next time someone says that there's no good way to evaluate teaching, quietly mention that one or two thousand research studies suggest otherwise. MYTH: Even if student evaluations have some valid ity, there's no value in the multiplechoice forms used to collect most of them. You've got to interview students and ask openended questions for the results to mean anything. FACT: Comparisons have been run on student rat ings collected in three different ways: objec tive questionnaire items, written responses to openended questions, and group inter views. The average correlation among the rating methods was 0.86.[131 MYTH: Teachers who get high ratings aren't really doing a better job of teaching. FACT: Teachers rated as effective by students tend to be those whose students perform best on achievement tests.[31 Classes in which stu dents give instructors higher ratings when multiple sections are taught tend to be those in which the students score higher on com mon external exams.[P] Good teaching also motivates interest and desire to learn; stu dents in courses taught by highlyrated teach ers are subsequently more likely to elect ad vanced courses in the same subjects[14] and to major in those subjects.1151 MYTH: Student evaluations don't improve teaching. FACT: Students of instructors who got student feed back scored higher on achievement tests and assessments of motivation for learning than students of instructors who got no feed back.[161 In short, the research shows that student evalua tions of an instructor provide a reliable, valid assess ment of that instructor's teaching effectiveness, es pecially if they reflect the views of many students in several different course offerings. So, next time some one says that there's no good way to evaluate teach ing, quietly mention that one or two thousand re search studies on the topic suggest otherwise. You may not change anyone's mind on the spot, but it might raise the discussion to a higher level than it usually occupies. It remains to consider how evaluations can be structured to have the maximum impact on teaching effectiveness. That's another column. REFERENCES 1. Cashin. W.E., "Student Ratings of Teaching: A Summary of the Research," Center for Faculty Evaluation and Develop ment, Kansas State University, September (1988) 2. Feldman, K., quoted in The Teaching Professor, p. 5, De cember (1990) 3. Marsh, H.W., Students' Evaluations of University Teaching: Research Findings, Methodological Issues, and Directions for Future Research, Pergamon Press, Elmsford, NY (1987) 4. Kulik, J.A., and W.J. McKeachie, "The Evaluation of Teach ers in Higher Education," in Review of Research in Educa tion, Vol. 3, F.N. Kerlinger, ed., Itasca, IL, F.E. Peacock, p. 210 (1975) 5. Centra, J.A., "The Relationship Between Student and Alumni Ratings of Teachers," Educational and Psychological Mea surement, 34(2), 321 (1974) 6. Drucker, A.J., and H.H. Remmers, "Do Alumni and Stu dents Differ in Their Attitudes Toward Instructors?" J. Ed. Psych., 42, 129 (1980) 7. Overall, J.U., and H.W. Marsh, "Students' Evaluations of Instruction: A Longitudinal Study of Their Stability," J. Ed. Psych., 72, 321 (1980) 8. Aleamoni, L.M., "Student Ratings of Instruction," in Hand book of Teacher Evaluation, J. Millman, Ed., Sage, Beverly Hills, CA, p. 110 (1981) 9. Marsh, H.W., "Students' Evaluations of University Teach ing: Dimensionality, Reliability, Potential Biases, and Util ity," J. Ed. Psych., 76, 707 (1984) 10. "Applicability Paradigm: Students' Evaluations of Teaching Effectiveness in Different Countries," J. Ed. Psych., 78(1), 465 (1986) 11. McKeachie, W.J., Teaching Tips, D.C. Heath, Lexington, MA (1986) 12. Palmer, J., G. Carliner, and T. Romer, "Leniency, Learning and Evaluation," J. Ed. Psych., 70(5), 855 (1978) 13. Ory, J.C., L.A. Braskamp, and D.M. Pieper, "Congruency of Student Evaluative Information Collected by Three Meth ods," J. Ed. Psych., 72, 181 (1980) 14. Marsh, H.W., and D. Solomon, "Student Ratings of Instruc tors: A Validity Study," J. Ed. Research, 51, 379 (1958) 15. Sullivan, A.M., and G.R. Skanes, "Validity of Student Evalu ation of Teaching and the Characteristics of Successful In structors," J. Ed. Psych., 66, 584 (1974) 16. Overall, J.U., and H.W. Marsh, "Midterm Feedback from Students: Its Relationship to Instructional Improvement and Students' Cognitive and Affective Outcomes," J. Ed. Psych., 71,856 (1979) 0 Summer 1992 Curriculum A COURSE SEQUENCE FOR INSTRUMENTATION AND CONTROL CARLOS A. SMITH, RICHARD A. GILBERT University of South Florida Tampa, FL 33620 At the University of South Florida we believe that a controls and instrumentation back ground is vital for new chemical engineers. Our curriculum specifies a required threesemester course sequence in the area, with an additional elective course also available. The three required courses are Mechanical Engineering Laboratory I (MELab), 3 credits Instrument Systems I (ISys), 4 credits Automatic Process Control I (APCI), 3 credits and the elective course is Automatic Process Control II (APCII),3 credits The first course is taught by the mechanical engi neering department, and all the other courses are taught by the chemical engineering department. The first two courses are required for all mechanical engineering and chemical engineering students. This paper will describe the subject matter and the goals the authors have set for each course. The principal goals for the complete sequence are to pro vide the students with the technical background to immediately begin productive careers in any industrial controls group to understand the fundamental principles involved in data acquisition for experimentation and process control Carlos A. Smith is professor of chemical engi neering at the University of South Florida, where he has been since 1972. He received his BSChE from the University of Florida and his MSChE and PhD from LSU. He is involved in continuing education courses and consultancy and is coau thor of the text Principles and Practice of Auto matic Process Control (Wiley, 1985). Richard Gilbert is an associate professor with interests in instrumentation for experiment analy sis and process control. Current funded research includes the investigation of optical sensors and techniques for growth control in MBE, the appli cation of programmable controllerbased control systems in Florida's produce packing industry, and the use of electrofusion and electroporation techniques in cancer research and treatment. to be able to keep up with future developments in this very dynamic field Because of the nature of the goals, all of the courses are practiceoriented and heavily dependent upon laboratory and project work. Mechanical Engineering Laboratory I (MELab) This onesemester course consists of one twohour lecture and one threehour laboratory period per week. The goals of the course are to introduce the student to various physical variable measurement devices and techniques and to develop proper report writing skills (Table 1 lists the contents of the course). Initial lecture times are devoted to laboratory safety issues, required elements in laboratory reports and presentations, and the properties of measurement and experimental procedure errors. In the remaining weeks of lecture, students are introduced to measurements of fundamental inter est to the process industry (i.e., temperature, pres sure, and flow) as well as measurements that relate to the monitoring and control of mechanical systems (i.e., strain, torque, and displacement). Laboratory work during this phase of the course involves setting up and taking measurements with sensors similar to the ones discussed in class. The students provide written reports for each of the ten experiments and must include a summary of any required engineer ing calculations. They must also contain descrip tions of the experimental arrangement and any spe cial analog signal conditioning required to complete the measurement, i.e., any temperature compensa tion technique used for a thermocouple measure ment, the type of measurement bridge used with the strain gauge, and the flow element analog output signal manipulation employed to obtain measure ment signals proportional to engineering flow units. Instrument Systems I (ISys) This onesemester course consists of three one hour lectures and one threehour laboratory period per week. Since MELab is a corequisite and/or a prerequisite for this course, knowledge of various types of sensors and their analog output signal char Copyright ChE Division ofASEE 1992 Chemical Engineering Education acteristics is expected of the students. The course introduces the various ways digital technology in corporates a sensor into an instrument system used for process monitoring and control. Table 2 summa rizes the topics presented in the course. Although the digital skills learned in ISys are applicable in any researchoriented application, it is convenient to present the course by using process control examples. At this stage in their education, the students are quick to appreciate how the sensor and its digital interface can be used in an industrial control situation. They also understand the useful ness of a computer or digital controller as the center element of a control loop. By contrast, it is difficult to find examples in their chosen fields of study that are within the experience level of both the chemical and mechanical engineering students in the course. Finally, they are able to relate the digital technology to monitor and control projects of immediate inter TABLE 1 Course Content: Mechanical Engineering Laboratory I TOPICS: 1. Laboratory safety 5. Force and torque 2. Technical reports measurement 3. Experimental errors 6. Flow measurement 4. Length, displacement, 7. Temperature measurement and strain measurement 8. Pressure measurement 9. Humidity measurement TABLE 2 Course Content: Instrument Systems I WEEK CONTENT 1 Components of a control loop Properties of analog and digital signals 2 Properties of logic operators Use of logic operations for discrete element process control (start, stop, alarm, etc.) 3 Binary, octal, and hexadecimalbased counters Codes and decoders Application of totalizer in process con trol 4 Analogtodigital conversion Digitaltoanalog conver sion 5 Signal conditioning Common industrial process sig nals 6 Multiplex concepts Latches and flip flops Memory devices 7 Structure of an 8080 singleboard microcomputer 8 8080 and Z80 Assembly code as it relates to I/O opera tions 9 I/O ports and their bus connections to the CPU; Sensor to A/D to I/O port interfacing 10 Intel 8250 series devices and their operational literature 11 DAS 8 dataacquisition board for IBMtype PCs 12 Sensor interface boards for IBMtype PCs (i.e., RTDs, thermocouples, strain gauges, etc.) 13 Interrupt I/O strategies Memory map I/O 14 Parallel interfaces IEEE 488 interface boards Summer 1992 est to themselves, i.e., digital display of wind speed on a sailboard, a variety of home comfort and secu rity interlocks, etc. Three aspects of control and data acquisition sys tem interfacing discrete component technology software drivers dedicated computer interface boards are covered. Lecture and laboratory time are divided equally among them. Component technology is dis cussed first while the selection/operation of interface boards is covered in the last four weeks. The discrete component technology section of the course covers TTL devices and their use as inter faces between a process sensor, a controller, and the final control element. Presenting the discrete digital technology concepts first has the effect of delaying discussions about analogtodigital conversion and other aspects of interfacing an analog sensor output to a digital system until those students taking MELab as a corequisite with ISys have been exposed to a few examples of analog sensors. In this first part of the course, students explore simple control schemes that can be handled by an arrangement of discrete components. Examples include elementary on/off in terlock logic, event counting, and timedelay situa tions. The function of open collector and threestate devices as interfaces to traditional industrial control voltages and computer circuits is also examined. At this point in the course, our intent is to make students feel comfortable working with control TTL circuits. Example A (next page) summarizes a project given in APCIthe same assignment is also given as an ISys lab. The problem is scaled down to facili tate the time base when discrete counters are used, but the logic portion of the assignment is the same. The full effect of the learning experience is not ap preciated until the controls course is taken, but the lesson is not lost with time. Once the students solve the problem by both methods, they can appreciate the value and place for each. Another reason for presenting discrete component information is to show students the interrelation ship between operation of the components and the function of a complete interface subsystem. This pre sentation is not an abstract academic exerciseit is an opportunity for the student to develop skills that can be employed in real instrumentation trouble shooting scenarios. In a research environment these troubleshooting skills might be used to determine and maintain a specific arrangement of triggering and then multi plexing the signals from several analog experimen 137 tal measurements into a single A/Dbased data log ger. In an industrial setting, troubleshooting situa tions inevitably involve problem diagnosis of a sen sor, a controller, and a final control element in a loop with no initial certainty of which loop element is at fault. Understanding a loop's digital components fa cilitates the distinction between the analog and digi tal aspects of the control scheme. This helps the engineer isolate the portion of the loop that is not performing properly. Because control loops are so intimately related to the process under control, the person responsible for that part of the process usu ally directs the troubleshooting "mission." More of ten than not, that person is a chemical engineer. The course's software driver section focuses on data transfer to and from a digital system. For this purpose a software driver is defined as the subrou tine responsible for the data transfer operation. Stu dents learn to write code that directs interactions between a CPU and its I/O ports. The intent is to make them aware of the I/O port structure, how those ports transfer sensor information to the CPU, and the immediate operations required of a CPU to accomplish that task. Once these operations are un derstood, an engineer can alter the computer inter face of an existing control system to meet the new or modified needs of the project. Laboratory experiences for the software driver portion of the course include exercises in writing code to operate digital I/O ports as well as A/D and D/A converters. Students learn to write 8080 and then Z80 code. They may crossassemble their pro grams on available IBMtype computers. Completed programs are run on singleboard computers that the student has connected to the ports of interest. Singleboard computers are used in the labora tory because they are simple to operate, they sup port a variety of INTEL "smart chips," and they are easy for the department to maintain. Because of the simplicity of these computers, students are forced to learn what the role of the I/O driver is and what must be done to complete the interface. They also develop valuable software debugging techniques that can be used with any higherlevel language. The last section of the course covers the function and operation of dedicated computer peripheral in terface boards. By this time students are expected to understand the technical information provided by the manufacturer of such boards. Lecture and labo ratory material cover A/D boards, thermocouple and RTD interface boards, and how to use them. Although some of the interface boards used in the lab come with manufacturersupplied menudriven software packages, the students are directed to de velop small routines that control the board. This approach shows them that they understand what the technical manual is trying to say, that they can write code that actually takes a measurement, and how to get maximum flexibility out of the board. Students develop their programs in Quick BASIC and use the program debugging skills they devel oped during the second phase of the course. One laboratory assignment at this point in the course is based on an RS232 serial port card and a serial printer. Requesting that the student create a program that alters the font of the printer is identi cal in practice to setting up a RS232 supporting multimeter to take and store various research ex periment related voltage, current, and resistance measurements. The same computer interfacing skills are required to set up a PID temperature controller so that all of its control parameters (i.e., set point, gain, reset time, etc.) are under computer control. The RS232 card has to be set up to handle the selected protocol. The correct command string has to be sent to the instrument's control register, and EXAMPLE A Consider a tank where a dilution process takes place. The tank and accessories are shown in the figure. A concentrated solution enters the tank where water is sprayed at a significant rate to promote mixing. There are four water valves feeding the tank. Because of process constraints not all valves can be open at the same time; maximum of two are permitted at the same time. It has been proposed to open the valves in the following sequence: Concentrated Solution from PLC Irom PLC V V2 V Valves Time Duration, sec. 1 2 3 4 ON OFF ON OFF 5  to PLC OFF ON ON OFF 10 ON OFF OFF ON 10 OFF ON OFF ON 5 Start/stop PBs are available to start/stop the sequence. In addition, a dilution sensor (conductivity sensor/transmitter) is available to measure the amount of dilution. If the solution becomes too diluted (conductivity switch goes high) the valves must close until the sensor indicates (conductivity switch goes low) to resume the valves sequence. Desirn the logic for this process. develop the logic diagram, and program the PLC to accomplish this control strategy. .38 Chemical Engineering Education proper delimiters must be used to frame the mes sages. Finally, status information about the instru ment and the process must be imported through the RS232 interface back to the computer. Using a printer instead of another instrument in this experiment has several advantages: printers are cheap, and even lowpriced ones give several protocol and delimiter options; students enjoy learn ing how to write their own software to drive a printer; and finally, students have no trouble transferring what they have learned to the more sophisticated, more expensive, and more difficult to repair RS232 driven measurement instruments. Collectively, the device, the driver, and the dedi cated board portions of this course provide a sound background for all phases of digital interfacing. A detailed note set, a study guide, and a collection of over twentyfive different subjectrelated books on library reserve provide additional independent learn ing resources for the students. The practical nature of the course materials, to gether with the handson concept of reinforcing labo ratory exercises, make the course immediately re warding for most of the participants. Those who do not enjoy it usually complain about the lack of spe cific guidance during the laboratory periods. How ever, it is a conscious decision on our part not to provide stepbystep laboratory projects. Several lab periods are provided so that each student can work with the isolated concepts. During this time, the problems students have in working with example circuits, programs, or subsystems are explored. They learn alternative routes to diagnose the problems, TABLE 3 Lecture: Automatic Process Control I TOPICS 1. Introduction 2. Firstorder and higherorder systems a. Modeling b. Transfer functions c. Block diagrams d. Response to different forcing functions 3. Valves and transmitters 4. Feedback controllers a. Types b. Process identification and tuning 5. Analysis of closedloop control systems a. Modeling b. Block diagrams and transfer functions c. Stability Routh test Direct substitution d. Effect of loop parameters and types of controllers on stability 6. Case studies Summer 1992 but little time is spent on telling them exactly how to fix their problem. This approach of suggesting more things to try instead of offering corrective instruc tion is just not what some students want. Automatic Process Control I (APCI) This is a onesemester course consisting of two hours of lecture and three hours of laboratory per week. It is directed at the process industries and includes several examples of environmental and ma terial handling processes. The textbook is Principles and Practice ofAutomatic Process Control. [1 Table 3 presents the lecture content. Note that the subject of Laplace transforms is not discussed it is assumed that the student has "learned" this subject in differential equations. To test this as sumption and to help the students review the mate rial, a test is given during the second week of class. Questions about temperature, pressure, level, and flow sensors are also asked during the test as an aid in helping the students review their MELab course. The lecture begins with a presentation on system dynamics, block diagrams, and transfer functions. This material is usually perceived as very theoreti cal, mathematical, and in general boring. Therefore, a deliberate, complete discussion on why the mate rial is necessary and how it is useful is presented on the first day of lecture. We stress mathematical mod eling, the physical significance of gain, time con stant, dead time, how the response of systems in series provides dead time, and the importance and significance ofnonlinearities. We also provide a physi cal explanation of what the mathematics indicate. Once the foregoing material is learned, we present a discussion on controllers: the action of controllers, the mathematics, the physical significance, and the different types of controllers are discussed in detail. Process identification, by loworder models, and con troller tuning are discussed in the lecture and prac ticed in the laboratory. The identification method discussed is the step, or bump, testing procedure. Students are organized into groups of two in the laboratory, and each group is asked to tune the controllers for two different simulated processes. Each simulated process is connected to real control lers. The laboratory equipment consists of a Honeywell TDC 2000[2] distributedcontrol system, two CLC002 Bailey,E[3 and two Yokogawa[4] stand alone controllers. The topic of stability starts with a presentation of the development of the mathematical model, block diagram, and the closedloop transfer functions of a closedloop control system. We continue with a defi nition of the characteristic equation and its relation 139 ship to stability. Using the Routh Test and direct substitution to study stability is then presented. Finally, we demonstrate the effect on the stability of the loop if any of its parameters change, i.e., if a transmitter with a different span is used, if a faster or slower sensor is used, or if the dead time changes, etc. We also discuss the effect of the reset and de rivative actions on the loop stability. It is important to notice that only the simple techniques of the Routh Test and direct substitution are used in this presen tation (Pade approximations are used for dead times). The idea is to show in simple, but effective, language the effect of the different terms on the stability of the loop and that these techniques do the job. In addition to the lecture material and the home work, we ask the students to complete about three simulation projects during the course. The simula tion software package used is TUTSIM.151 (If a stu dent wants to work with another simulation pack age, or with straight FORTRAN or BASIC, he/she is allowed to do so.) The TUTSIM simulation package is a PCbased system and is considered to be one of the easiest packages for this purpose. The TUTSIM package is highlighted in the recently published sec ond edition of the textbook Process Systems Analysis and Control.[6] The projects usually involve the de velopment of the model for a process system, the simulation, and the tuning of the control system. Toward the end of the semester two design projects are assigned. They provide an opportunity for the students to design control systems for complete pro cesses. The students usually like these projects. The laboratory presents another interesting part of the courseTable 4 shows the weekly schedule. The first week is used to stress the importance of the course. We use this period to motivate the student to learn the subject. We then present the TUTSIM software package in the second week. This presenta tion usually takes about an hour and a half, and once it is done a project is assigned. During the next seven weeks we introduce the subject of discrete or sequential logic. Using this logic has become an important tool for process con trol engineers. The complexity of new processes, the increasing use of batch processes, the increasing concern with safety, and the development of the new tools to implement this logic such as program mable logic controllers (PLCs)[71 and distributed con trol systems (DCSs)have precipitated the impor tance of this logic. These new capabilities provide an easy way to introduce logic statements in continuous control schemes. Until now, the design and imple mentation of logic systems have been principally the 140 domain of electrical engineers, but now chemical engineers must also become acquainted with them. The projects are all actual industrial cases that have been collected over the years. Example A shows a typical weekly project. The laboratory contains ten AllenBradley SLC 100[8] PLCs. Each is connected to a set of start/stop pushbuttons and to switches that represent field signals. Lights are connected to the outputs and act as final control elements. These systems provide a realistic environment where the students can test their control logic design. The PLCs are available during lab periods as well as during the hours that the College of Engineering's PC labs are open. In the first week PLC relay logic (AND, OR, latches, etc.) and instructions on how to develop logic and ladder diagrams are presented. This is usually accomplished in about one and onehalf hours, and then a project for the following week is assigned. In week two, timers are covered and a project is assigned. Counters are presented in the third week, along with another project assignment. Finally, in the fourth week, students read about sequencers, and a new project is assigned. During the fifth week of discrete logic lab, the final project (exam) is presented. During the past TABLE 4 Automatic Process Control I Laboratory WEEKS) CONTENT 1 Introduction to process control 2 Introduction to TUTSIM 39 Discrete and sequential logic 10 Valves 1114 Tuning of feedback controllers 15 Wrapup TABLE 5 Automatic Process Control II: Lecture TOPICS 1. Control loop stability a. Root locus b. Frequency response c. Pulse testing 2. Cascade control 3. Ratio control 4. Feedforward control 5. Selective, override, and constraint control 6. Multivariable control 7. Digital control a. Ztransforms, sampling, and stability b. PID discrete controllers c. Deadtime compensation Smithprediction Dahlin's controller d. Filtering Chemical Engineering Education EDITORIAL NOTE The following detachable pages describe some indus trial employment opportunities for graduating chemical engineers. Please post the information in a conspicuous place for the benefit of your students, or distribute the pages to students who may be interested. These companies have expressed a definite interest in hiring chemical engineers in the areas described, and we strongly encourage students seeking employment to respond as indicated. Ray W. Fahien Editor Chemical Engineering Education L __ DOW USA University Relations Box 1713CH Midland, MI 48674 GENERAL INFORMATION Dow manufactures and markets chemicals, plastics, metals, consumer prod ucts, and specialty products and services. Dow USA has over 2600 chemical engineers working in all functions and geographic locations. CITIZENSHIP REQUIREMENTS: Only U.S. citizens, aliens who have a legal right to work and remain permanently in the U.S. or aliens who qualify as "Intending Citizens" under the Immigration Reform and Control Act of 1986 are eligible for employment. REGIONS WHERE BS/MS CAMPUS RECRUITING IS CONDUCTED: Nationwide HOW TO APPLY IF UNABLE TO SCHEDULE A CAMPUS INTERVIEW: Send resume and letter to the above address, stating your job interests and geographic preferences ENTRY LEVEL OPPORTUNITIES FOR CHEMICAL ENGINEERS BS / MS Functional Area Degree Level Major Hiring Locations Design BS,MS Michigan, Texas, Louisiana, Ohio, Cali Process Engineering BS,MS Michigan, Texas, Louisiana, Ohio, Cali Manufacturing BS,MS Michigan, Texas, Louisiana, Ohio, Cali Research and Development BS,MS Michigan, Texas, Louisiana, Ohio, Cali: Sales BS,MS Offices in over thirty major cities * (  fornia fornia fornia fornia t Fields of Special Interest Math Modeling Polymer Processing Polymer Characterization Catalysis Tech Center Locations Michigan, Texas, California Michigan, Texas, California, Ohio Michigan, Texas, California, Louisiana Michigan, Texas An Equal Opportunity Employer Advertisement published in Chemical Engineering Education, Volume 26, No 3 (1992) PhD MERCK & CO., INC. P.O. Box 2000 Rahway, NJ 07065 GENERAL INFORMATION Merck & Co. is a worldwide, research intensive health products company that discovers, develops, produces, and markets human and animal health products and specialty chemicals. The company has 37,700 employees with sales of over $8.6 billion in 1991. CITIZENSHIP REQUIREMENTS: U.S. citizen, lawful permanent resident of the U.S., or otherwise authorized to work in the U.S. REGIONS WHERE BS/MS CAMPUS RECRUITING IS CONDUCTED: We recruit on campuses nationwide (U.S.) HOW TO APPLY IF UNABLE TO SCHEDULE A CAMPUS INTERVIEW: Please submit resume or application which clearly states educational background, objectives, and work experience to: Office of College Relations Merck & Co., Inc. P.O. Box 2000 Rahway, NJ 07065 ENTRY LEVEL OPPORTUNITIES FOR CHEMICAL ENGINEERS BS / MS I iviiv M inr. I4n 1 , s niviDnn De ree Level Mainj Hfirinn L tni Merck Res Merck Manufacturii ~' t c. ons ,,. ..jv,. ta Z IC lULIflJ earch Labs BS/MS Rahway, NJ; West Point, PA ng Division BS/MS Rahway, NJ; Albany, GA; Danville, PA; Elkton, VA; West Point, PA; Somerset, NJ KELCO Calgon Water Management Division PhD Fields of Special Interest Process changes which address the environ mental aspects of plant operations Support the current technology and contribute toward development of new technology * Process developmentfrom conception through to scaleup and eventual plant start up BS/MS San Diego, CA; Okmulgee, OK BS/MS Pittsburgh, PA Tech Center Locations Merck Manufacturing Division Rahway, NJ; Albany, GA; Elkton, VA; Danville, PA; West Point, PA Merck Research Labs Rahway, NJ; West Point PA ADDITIONAL INFORMATION Merck hires chemical engineers in several divisions to play a critical role in the implementation of our business. In each division we have highly skilled chemical engineers and we will continue to hire highly qualified applicants in the chemical engineering field. Advertisement published in Chemical Engineering Education, Volume 26, No 3 (1992) MILLIKEN & COMPANY College Relations Department, M285 P.O. Box 1926 Spartanburg, SC 29304 GENERAL INFORMATION Milliken is a major manufacturer of textile products for apparel, commercial, home, and industrial markets, as well as specialty chemicals for a wide variety of applica tions. The company was founded in 1865 and has nearly 50 plants and 13,000 associ ates in the US. Milliken is a world leader in quality manufacturing and was the 1989 winner of the Malcolm Baldridge National Quality Award. Careers at Milliken involve challenge, innovation, advanced technology, promotion from within based on indi vidual performance, and extensive education and training opportunities. Entry level opportunities are available in South Carolina and Georgia. CITIZENSHIP REQUIREMENTS: U.S. citizenship or Permanent Resident Visa REGIONS WHERE BS/MS CAMPUS RECRUITING IS CONDUCTED: Southeastern United States HOW TO APPLY IF UNABLE TO SCHEDULE A CAMPUS INTERVIEW: Send cover letter with functional area interests and geographic preference statement, resume, and a copy of your transcript to the above address. ENTRY LEVEL OPPORTUNITIES FOR CHEMICAL ENGINEERS BS / MS Process Engineering: Manufacturing Management: Research: Provides technical support in textile dyeing and finishing operations and in Specialty Chemicals production. Responsibilities include manu facturing compliance with customer product quality specifications and process efficiency/improvement project assignments. Responsible for the production resources of people and machinery. The first line production manager may be promoted to either Ad vanced Production Manager or Process Engineer in the dual career ladder. Develops new products and associated machinery or processes. Prefer PhD, but will consider MS An Equal Opportunity Employer Advertisement published in Chemical Engineering Education, Volume 26, No 3 (1992) PROCTER & GAMBLE R&PD BS/MS Recruiting Winton Hill Technical Center 6090 Center Hill Road, Box A118, Dept. PJ3CE1 Cincinnati, OH 452241793 GENERAL INFORMATION P&G, founded in 1837, having over $28 billion in sales, is the largest consumer goods company in the United States. Of P&G's 100,000 employees, over 3000 are graduate scientists and engineers (including more than 1000 with PhDs), doing research and development in 32 R&D facilities in 19 countries, supported by over $800 million annual R&D spending. More than half of our BS/MS entrylevel hires are chemical engineers. CITIZENSHIP REQUIREMENTS: U.S. citizens and individuals legally authorized for fulltime employment without restrictions. For nonUSA locations, appropriate citizenship/visa. International needs are particularly strong in Latin America (facilities in Venezuela, Mexico, and Brazil). REGIONS WHERE BS/MS CAMPUS RECRUITING IS CONDUCTED: Nationwide at over 50 universities HOW TO APPLY IF UNABLE TO SCHEDULE A CAMPUS INTERVIEW: Send a letter and resume to the above address. Please include both your campus and home addresses and telephone numbers ENTRY LEVEL OPPORTUNITIES FOR CHEMICAL ENGINEERS I I BS / MS I Functional Area Process Development Product Development Products Research Packaging Development PhD Fields of Special Interest Process Development Product Development Applied Research Degree Level BS/MS BS/MS BS/MS BS/MS Degree Level PhD PhD PhD Major Hiring Locations Cincinnati, OH; Hunt Valley, MD; See above locations See above locations See above locations Norwich, NY Tech Center Locations Cincinnati, OH; Hunt Valley, MD; Norwich, NY See above locations See above locations ADDITIONAL INFORMATION P&G's leadership roles have been recognized externally in comparisons published by Fortune, Forbes, Graduating Engineer, NSBE Magazine, Computerworld, Black Enterprise, and Savvy. Internally, five of the twelve Charter Members of the Victor Mills Society, honoring excellence in technology at P&G, are chemical engineers. Advertisement published in Chemical Engineering Education, Volume 26, No 3 (1992) I I UNION CARBIDE CORPORATION 39 OLD RIDGEBURY ROAD DANBURY, CT 06817 GENERAL INFORMATION Union Carbide Corporation is a global leader in the production of basic petrochemi cals and plastics. Founded in 1917, Carbide has 16,000 employees worldwide and generates annual sales of over $5 billion. Key products include: polyethylene, latex and specialty polymeric resins; ethylene oxide/glycol and derivatives; urethane addi tives; alcohols and organic solvents. We also license internationally our Kirkpatrick Awardwinning UNIPOL (polyolefins) and Low Pressure Oxo (alcohols) process technologies. Chemical engineers account for 60% of our entry hires. CITIZENSHIP REQUIREMENTS: U.S. citizens and individuals legally authorized for fulltime employment without restrictions. REGIONS WHERE BS/MS CAMPUS RECRUITING IS CONDUCTED: Gulf coast, northeast, midwest, southeast, southwest, and Rocky Mountain. IF UNABLE TO SCHEDULE A CAMPUS INTERVIEW: We'll be pleased to review your credentials and notify you if interest exists. Send resume, photocopy of transcriptss, and statement of functional/location preference (see below) to the attention of our Professional Employment & Placement Department, Section M4, at the above address. ENTRY LEVEL OPPORTUNITIES FOR CHEMICAL ENGINEERS BS / MS IBS/M I Functional Area Degree Level Major Hiring Locations Design (Process; Control Systems) BS,MS Charleston, WV Environmental/Safety Engineering MS Charleston, WV Manufacturing (Production; Environmental BSMS Bound Brook, NJ; New Orleans, LA; Houston and Protection; Process/Project Engineering) Victoria, TX; Charleston, WV Purchasing and Distribution BS Charleston, WV; Houston, TX R&D (Polymer Applications/Tech Service; MS Bound Brook, NJ; Charleston, WV; Process Development) Tarrytown, NY Technical Sales BS Metropolitan areas, nationwide PhD Fields of Special Interest Tech Center Locations Catalysis, Polymers, Separations Bound Brook, NJ; Charleston, WV ADDITIONAL INFORMATION All major locations operate highly competitive summer intern programs which provide professional track employment for students with sophomore or junior standing. Also, coop programs are sponsored at our technical centers in New Jersey and West Virginia and at our manufacturing complex near Houston. Advertisement published in Chemical Eneineering Education, Volume 26, No 3 (1992) five semesters this project has been prepared and presented by Dow Chemical, U.S.A., Louisiana Divi sion. Dow's personnel present the project to the stu dents and then return two to three weeks later for the students' presentation. The students are asked to deliver an oral presentation and a written report. Once the material on discrete and sequential logic is presented, the lab continues with a oneperiod discussion on control valves and subsequent periods on controller tuning. Through these exercises the students realize the effect of nonlinearities on the controller tunings and loop stability. We have described how APCI has been taught for the past five years. It is quite heavy in laboratory practice and includes topics such as PLC logic which are not usually covered in classical courses. We con tinually question whether such a departure from the norm provides a correct education. Often, we try new things, or modifications, such as a modification in the spring 1992 semester that added a bit more time to the subject of stability. Automatic Process Control II (APCII) This is an elective onesemester course with a twohour lecture and a threehour lab per week. Usually twenty to thirty percent of the undergradu ates take APCII, and it also serves as the first graduate course. We use the APCI textbook with additional notes on digital controls provided. Table 5 shows the material presented in the lec ture. Three modeling and simulation projects are usually also assigned. Table 6 shows the weekly assignment for laboratory practices. The equipment is the same as that used in the undergraduate course. Some exercises take a great deal of time because we show the benefits of the techniques in detail. For example, during the feedforward control, we first ask the student to control a process with simple feedback control. Then steadystate feedforward is implemented and its performance is compared to the performance of feedback control. We then add lead/ TABLE 6 Automatic Process Control II: Laboratory WEEKS) CONTENT 1 Introduction to distributed control systems and standalone controllers 2 Computing relays 3 4 Cascade control 5 7 Feedforward control 8 9 Multivariable control 10 Deadtime compensation 11 Level averaging control 12 "Catchup" 13 14 Discrete and sequential logic Summer 1992 lag and go through a similar comparison. Finally, dead time compensation is added to the feedforward and the results are compared. We ask the students to do the same with two different processes. In the lab, one to one and onehalf periods are used to explain about distributed control systems and standalong controllers. The students actually learn to configure one system. The student also learns in more detail about the available computing power of these controllers, helping them to design and imple ment different control strategies. In the spring 1992 semester a design project was also assigned to each group of two students. Two instrument and control engineers from the local Bad ger Engineers office provided the design projects and acted as "leaders." Each group was asked to design the control strategies for a process. In addi tion, each individual student was asked to completely specify all the instrumentation for a control loop. CONCLUSION This sequence of required courses provides exten sive "handson" laboratory work and lectures that are focused on the practical aspects of understand ing the elements, functions, and properties of a con trol loop. Although process control is used as the unifying theme for the sequence, the students have also been able to relate the material to their reaction engineering course and their capstone project design course. In addition, the department has several ac tive undergraduate research projects that provide an opportunity for the students to use their knowl edge of sensors and computer interfacing in a more traditional university laboratory environment. The course descriptions refer to the courses as they are taught today. As new sensors, computer hardware, and control software become available, the number of interfacing options increases and the implementation of more advanced control strategies becomes possible. It is our intent to make sure such developments are reflected in our course sequence. REFERENCES 1. Smith, C.A., and A.B. Corripio, Principles and Practice of Automatic Process Control, John Wiley & Sons, Inc., New York (1985) 2. Honeywell Inc., Phoenix, AZ 3. Bailey Controls, Wickliffe, OH 4. Yokogawa Corp. of America, Newnan, GA 5. TUTSIM Products, Palo Alto, CA 6. Coughanowr, D.R., Process Systems Analysis and Control, Second Ed., McGrawHill, New York (1991) 7. Gilbert, R.A., and J.A. Llewellyn, Programmable Controllers Practices and Concepts, ITC, Herndon, VA 8. AllenBradley, a Rockwell International Co., Milwaukee, WI 53204 0 laboratory THE EFFECT OF AGITATION ON OXYGEN MASS TRANSFER IN A FERMENTOR RONNIE S. ROBERTS, JAMES R. KASTNER, MAQSOOD AHMAD, D. WILLIAM TEDDER Georgia Institute of Technology Atlanta, GA 303320100 Chemical engineers have traditionally been in volved in a wide variety of industries. Al though specific problems encountered in these diverse industries may appear different, many of them can be solved by using basic chemical engi neering principles. In the classroom and in the labo ratory, basic principles are generally demonstrated by using problems from a number of different indus tries, allowing students to learn about both the in dustries and the application of these basic principles. The increasing importance of industrial opera tions based on biochemical engineering has long been recognized. Biochemical engineering problems have been easily integrated into many courses, including mass transfer, heat transfer, and reactor design. Integrating biochemical engineeringtype experi ments into chemical engineering laboratories, how ever, has been much more difficult. Biochemical engineering experiments must be de veloped which demonstrate basic principles and which can be easily incorporated into undergradu ate chemical engineering laboratories. An experi ment has been developed at Georgia Tech that dem onstrates the effect of agitation on oxygen mass trans fer in a fermentor and that can be conducted by undergraduate students in an ordinary undergradu ate chemical engineering laboratory. BASIC CONCEPTS Fermentation is the production of chemicals from a substrate by using microbes. Many fermentation use microbes which require oxygen to convert the substrates reactantss) into the desired product. Oxy gen is generally supplied by sparging air at one atmosphere pressure directly into the fermentation broth (aqueous mixture of substrates, products, and microbes). The maximum dissolved oxygen par tial pressure in the fermentation broth would be saturated. Dissolved oxygen levels in the fermenta tion broth of 30% of saturation or higher are re quired for many fermentations. A high concentration of microbes in the fermen tor (reactor) is desirable in order to maximize productivity per unit volume of the fermentor. Un fortunately, the high concentration of microbes con sumes oxygen at high rates and thus oxygen mass transfer from the gas phase to the fermentation broth is very important. Ronnie S. Roberts is an associate professor of chemical engineering at the Georgia Institute of Technology. His research interests are in fer mentation and buffered solvent pulping. . James R. Kastner is a PhD candidate in the School of Applied Biology at the Georgia Insti Stute of Technology. His research interest is in the fermentation of five and six carbon sugars. D. William Tedder is an associate professor of chemical engineering at the Georgia Institute of Technology. His research interests are the treat ment of hazardous wastes and solvent extration. Maqsood Ahmad is a visiting scholar in the School of Chemical Engineer ing at the Georgia Institute of Technology. His research interest is in the production of ethanol and other products from lignocellulosic materials. (Photo not available.) Copyright ChE Division ofASEE 1992 Chemical Engineering Education OXYGEN MASS TRANSFER IN A FERMENTOR If the flowrate of air to the fermentor is large, a negligible amount of oxygen will be consumed and the mole fraction of oxygen in the gas phase will be essentially constant. For suspension cultures, the liquid film mass transfer generally controls the overall mass transfer of oxygen from the gas phase to the liquid phase. If we assume that liquid phase mass transfer is controlling in the fermentor, the change in oxygen partial pressure dissolved in the bulk liquid with respect to time can be calculated as shown in Eq. (1): (1) where P02 = dissolved oxygen partial pressure; % of saturation t = time; unit time k, = liquid phase mass transfer coefficient; unit time, unit area' a = surface area; unit area P2 = equilibrium dissolved oxygen partial pressure; 100% saturated r02 = rate of oxygen consumption; % saturation, unit time'1 For a fermentor where the microbial population remains relatively constant, the rate of oxygen con sumption is relatively constant and quasisteady state can be achieved for dissolved oxygen partial pres sure. Thus, kla can be determined as follows: 02 =kla(P0d P02 )(r02)0 (2) (2 *) dt O2 O( ka= r2 (3) P02 PO2 Mass transfer from a gas phase to a liquid phase in a stirred tank is very complex. In particular, agi tation strongly effects bubble size, bubble retention time, and liquid mixing. All of these factors strongly influence k1a. Generally, kla is correlated as shown in Eq. (4):111 kla = a'( (Qair) (4) where a', '= constants which depend on vessel dimensions, agitator dimensions, fluid properties, etc. P/v = agitator power per reactor volume; unit power, unit volume' Biochemical engineering experiments must be developed which demonstrate basic principles and which can be incorporated into undergraduate chemical engineering laboratories. Q. = volumetric flowrate of air; unit volume, unit time For a fermentor which contains a Newtonian fermentation broth, the agitator power is a function of agitator rpm.[2] Thus for a constant sparger air flowrate, kla can be correlated to rpm as shown in Eq. (5): ka = a(rpm) (5) where a,P = constants rpm = agitator rpm; revolutions, min' From Eq. (4), a is equal to zero if the sparger air is stopped. Thus, according to Eq. (1), the change of dissolved oxygen partial pressure in the fermenta tion broth with respect to time is equal to the rate of oxygen consumption. dP (o2 ) (6) The rate of oxygen consumption in the fermentor is equal to the slope of a plot of the dissolved oxygen partial pressure versus time. Experimentally, the effect of agitation on kla can be determined by setting the agitator rpm at various levels and obtaining the corresponding quasisteady state dissolved oxygen partial pressures. Then kla can be determined using the quasisteady state dis solved data and the rate of oxygen consumption. A plot of In(k1a) versus ln(rpm) should give a straight line with a slope of P and an intercept of In a. A good discussion of oxygen mass transfer in an agitated fermentor can be found in Biochemical En gineering Fundamentals. [3 EXPERIMENTAL EQUIPMENT A New Brunswick Scientific Bioflow IIc twoliter fermentor (Edison, NJ) has been used for this ex periment. The fermentor is equipped with a galvanic oxygen probe, temperature control, and a turbine agitator. Compressed air is sparged directly into the fermentation broth and is measured using a rotame ter. A less costly vessel such as a New Brunswick Multigen fermentor or a similarly equipped stirred vessel can be used instead of the Bioflow IIc. Summer 1992 dPdt,=k 22 o) ', Data are collected using an Austin 286 PC and an A/D Board (Mendelson Electronics). Depending on the application, the data are logged, manipu lated, and/or plotted using a PCbased system. PC data collection is not necessary to conduct the experiments. Data can be collected manually with little trouble. A schematic of the experimental equipment at Georgia Tech is shown in Figure 1. The system cost approximately $15,000 in 1990. The equip ment for this experiment is also used to conduct both undergraduate and graduate fermentation research projects. EXPERIMENTAL PROCEDURES One liter of distilled water containing 80 g glucose is added to the fermentor. Temperature control is set at 300C and agitation at 600 rpm. After the system equilibrates, the dissolved oxygen probe is calibrated. (Note: if the probe is being used for the first time, place the probe in distilled water overnight to equilibrate.) Air is sparged into the aqueous solution at the rate of approximately 1.6 L/min until the dissolved oxygen meter indicates quasisteady state dissolved oxygen partial pressure and saturated conditions. The dissolved oxygen meter is then spanned to 100%. Next, nitrogen is sparged at the rate of approximately 1.6 L/min until the dissolved oxygen meter indicates steady state. The dissolved oxygen meter is then zeroed at 0%. The above is repeated by alternately spanning and zeroing until the measurements replicate. Gener ally, twice is sufficient. Next, the agitation is reduced to 500 rpm and approximately 10 g of Fleischmann's dry active yeast (approximately one and onehalf 1/4 oz. packages) is added to the waterglucose solution. Allow three hours for the yeast to become fully active. The dis solved oxygen should be maintained at above 50%. If required, increase the flowrate of sparger air and/or reduce the yeast concentration to maintain the dis solved oxygen at above 50%. After the yeast is fully active, increase agitation to 700 rpm and allow the system to reach quasi steady state dissolved oxygen partial pressure. Next, stop the sparger air and start the flow of nitrogen (1.6 L/min) into the head space in the fermentor. Nitrogen in the head space prevents incidental aera tion. Then obtain a plot of dissolved oxygen partial pressure versus time. The initial part of the plot will not be linear due to probe response. After the initial nonlinear part, a linear plot should be obtained which represents the rate of oxygen consumption in the fermentor. After collecting the transient data the nitrogen flow is stopped and the sparger air is turned on. Repeat the above procedure. Compare the above two runs for similarity and if they are not similar, repeat the measurements. Systematically vary the agitation between approxi mately 500 and 700 rpm. Allow the system to reach quasisteady state each time, and record the dis solved oxygen partial pressure. Generally, ten to twenty minutes are required for the system to reach quasisteady state. After completing the above quasi steady experiments, set the agitation at 700 rpm and allow the system to come to quasisteady state. Repeat the previous procedure again to obtain a final plot of dissolved oxygen partial pressure versus time. If the plot is not similar to previous plots, repeat the procedure. TYPICAL EXPERIMENTAL RESULTS Plots of dissolved oxygen versus time are shown in Figure 2. The nonlinear portion of the plots is due to the response of the dissolved oxygen probe. The slope of the linear portion of the plots represents the rate of oxygen consumption in the fermentor. The similar linear slopes of the plots, 20.4 and 17.0% saturation min1 for the initial and final plots, respectively, indicate that the rate of oxygen consumption is relatively constant during the course of the experiment. Quasisteady state dissolved oxygen partial pressures at various rpm are shown in Table 1. Using the data in Table 1, kla's were calculated using Eq. (3). A plot of k1a versus rpm is shown in Figure 3. We estimated a and 3 to be 2.2*108min1 and 2.75 (agitation in the form of rpm), respectively. NewBrunswick Boflow II C Fermentor Figure 1. Schematic of experimental equipment Chemical Engineering Education The correlation coefficient, r, was 0.986. DISCUSSION A major advantage for this biochemical engineer ing experiment is the simplicity of laboratory proce dures. Yeast (which can be obtained at most super markets) is inoculated at relatively high concentra tion in the fermentor. The high inoculum concentra tion and the relatively short time required for the experiments eliminate the need for aseptic condi tions. Thus, only ordinary cleaning of the fermentor (stirred vessel) and the use of distilled water is re quired to prepare for the experiments. Initial preparation requires less than thirty min utes for the students or laboratory assistant. Little or no additional effort is required during the three M 0 1 2 3 Time, minutes Figure 2. Initial and final dissolved oxygen partial pressures versus time TABLE 1 QuasiSteady State Dissolved Oxygen Partial Pressures at Various RPM Dissolved Oxygen Partial Pressure RPM % of Saturation 600 82 700 87 700 87 500 67 600 82 650 85 550 77 500 68 700 88 hour yeast activation period. The experimental mea surements can be made in approximately three hours. The experimental conditions previously described were developed using a twoliter New Brunswick Scientific Bioflow IIc Fermentor. If other fermentors or stirred vessels are used, these conditions may have to be adjusted. In particular, high shear from the agitator can be detrimental to the yeast. If a microscope and methylene blue stain are avail able, the students can also determine the number of active and inactive yeast cells that are present.[a] The concentrations of active and inactive cells are not required to determine a and p. Inclusion of this step does reinforce the biological nature of this ex periment. The total cell concentration for the typical experiment previously presented was 3.6 x 108 cell/ ml (65% viable and 35% nonviable). The general nature of this experiment should also be emphasized. Although the system investigated in this case involved a fermentation, similar experi ments could be conducted to investigate the effect of agitation on mass transfer from a gas phase to a liquid phase in other unit operations. CONCLUSIONS This biochemical engineering experiment involves basic principles of mass transfer and chemical reac tions. In particular, the effect of agitation on kla is demonstrated using a fermentor. The equipment and material requirements for the experiment are mod est and students and laboratory assistants can gen erally master the experimental techniques with little difficulty. It can be used to demonstrate basic chemi Continued on page 163. 10 r 0.1 L 401 0 600 700 800 Fermentor Agitation, rpm Figure 3. ka versus fermentor agitation Summer 1992 I I classroom "PRODUCT IN THE WAY" PROCESSES NOEL DE NEVER The University of Utah Salt Lake City, UT 84112 any processes in chemical engineering have the same basic physical descriptionthat the thing produced (or something propor tional to it) gets in the way of the process. These diverse processes all lead to the same mathematics and optimization, as will be shown here. A wide variety of chemical engineering processes lead to rate equations of the form 1 2= a(time)+b (1) (production rate) or time g cumulative 2 +h cumulative (2) time=g( product ) n product Here time is measured from the start of production or of the current production cycle, and a, b, g, and h are constants (all symbols are defined in the nomen clature at the end of this paper). Although the pro cesses described by these equations cover the whole range of chemical engineering (including heat trans fer, filtration, condensation, freezing, chemical reac tions, oxidations, etc.), the underlying mechanism is the same for all. This paper is about that mecha nism, its mathematics, and the wide range of places where a chemical engineer can encounter it. In all of these processes, the characteristic physi cal fact is that the product, or something propor tional to it, gets in our way and the more we pro PI Pressure P3 Slurry Flow Filter Clear Filtrate Flow Cake Filter Cloth Figure 1. Flow through a simple filter Noel de Nevers has been a faculty member at the University of Utah since 1963. His prin cipal interests are fluid mechanics, thermody S namics, and air pollution. He has also devel /oped a course and edited a book of readings r on Technology and Society. He recently won local fame by discovering a previously un known major arch in Arches National Park. duce, the more the product is in our way and the slower the process rate becomes. One may easily show that Eqs. (1) and (2) are the same by writing t cumulative product = (production rate) d(time) (3) 0 and then differentiating Eq. (2) and comparing it termbyterm with Eq. (1). The two equations are the same if a = 4g and b = h2. FILTRATION It is easy to see how these equations arise in the classical treatment of constantpressure filtration of a solid from a liquid, to form an incompressible cake.1l] The flow through a filter and its pressure profile are shown schematically in Figure 1. A slurry (a fluid containing suspended solids) flows through a filter medium (most often a cloth, but sometimes paper, porous metal, or a bed of sand). The solid particles in the slurry deposit on the face of the filter medium, forming the "filter cake." The liquid, free from solids, flows through both cake and filter me dium. The flow is laminar in almost all filters, and the changes in potential and kinetic energies are negligible, so that the pressure drop is given by Darcy's equation [(AP/Ax) = gV/k]. Solving that equa tion for the superficial velocity, we find V APk (4) =QA Ax where g is the fluid viscosity and k is the cake permeability. Here there are two resistances in se Copyright ChE Division ofASEE 1992 Chemical Engineering Education Many processes in chemical engineering have the same basic physical description that the thing produced (or something proportional to it) gets in the way of the process. These diverse processes all lead to the same mathematics and optimization... ries with the same flow rate through them. If we let the subscript "f.m." indicate "filter medium," we can write Eq. (4) twice and equate the identical flow rates (see Figure 1) P P 2 k P2 P3(k Ji \AX/cake x \.AXf.m. When we solve for P2, we get P2 = P1 v8 ( cake = P3 VV ( f.m. and then, solving this equation for V., we get (5) (6) = 3 Q (7) ()caLke k)f.m. Afil This equation describes the instantaneous flow rate through a filter; it is analogous to Ohm's law for two resistors in series, so the gAx/k terms are called the cake resistance and the cloth resistance. The resistance of the filter medium is normally assumed to be a constant independent of time, so (Ax/k)f.m is replaced with a constant, a. If the filter cake is uniform, then its instantaneous flow resis tance is proportional to its instantaneous thickness. However, this thickness is related to the volume of filtrate which has passed through the cake by the material balance 'mass volume mass of of of ake cake 1 1 fitrate solids area Pcake) Peake area volume (8) ) Afiltrate ,) Customarily we define mass of w solids _1 volume of cake volume of pake ) volume of filtrate (9) filtrate ) so that cake W (10) Here the cake is assumed to be incompressible, Pcake = constant, which is a good assumption for most filtrations but not for filtration of flocs and gels. The volume of filtrate here is V. (This implies 100% col lection efficiency for the solids in the slurry, which is generally observed.) When we substitute Eq. (10) for the cake thickness in Eq. (7), we find Summer 1992 Time Figure 2. Relation between production rate and cumulative production Q l(dV P1P3 3 =A dt I +a (11) r kA + a) For most industrial filtrations the filter is supplied by a centrifugal pump or blower at practically con stant pressure, so (P1 P3) is a constant, and Eq. (11) may be rearranged and integrated to 2 _Wva =(P31)t (12) 2A k +AWW Eq. (2) and Eq. (12) are identical if product = V/A g = (gW)/2k(P1 P) h = g a/(P P3) Intuitively, we can see what is happening in Fig ure 2. At time zero there is no cumulative product and the production rate equals 1/(A). As soon as we begin to produce filtrate, we also produce filter cake. This increases the resistance, so the production rate declines. The more filtrate we produce, the thicker the filter cake and the higher the resistance. The product (filter cake) gets in the way of producing more filter cake. Often one sees Eq. (2) rewritten as timepr = g(cumulative product)+h (13) cumulative product This allows us to plot (time/cumulative product) vs. cumulative product and to read the values ofg and h as the slope and intercept of the straight line which results. This is shown for filtration (using Eq. 12) by McCabe and Smith.[21 It is equally applicable to other processes described in this paper. That type of plot has less intuitive content than Figure 2, but it makes possible a visual check of whether or not the experi mental data agree with Eqs. (1) or (2) and allow a direct determination of g and h from those data. FREEZING The second example of Eqs. (1) or (2) is the forma tion of ice on a cooled surface, such as occurs in an ice maker.[31 Referring again to Figure 1, we see that in this situation the flow is of heat, not fluid. We would relabel that figure by replacing the slurry with fluid being frozen, replacing the filter cake with the ice which has formed, replacing the filter cloth with the chiller surface (normally a highly conduc tive metal), and at the right the recipient of the heat (normally a chilled brine or evaporating refriger ant). The pressuredistance curve would be replaced by a temperaturedistance curve, with the same shape. If we ignore the heat transfer resistances other than those due to conduction (which is an excellent approximation for this case), then we can rewrite Eq. (7) as q d T _T heat flux = = A (14) A dt r\ Ax A dt L(k)ice k )metal wall ] Where T = temperature k = (which were permeabilities in Eqs. 4 to 12) are now thermal conductivities of ice and metal, re spectively Q/A = total heat transferred per unit area (analogous to the volume, V, of filtrate) q/A = the instantaneous heat flux (analogous to V, = (1/AXdV/dt) for filtration) Here the analogs of Eqs. (8) to (10) are mass of ie _= ice (_1 ice area (Pice J heat mass of IPi. area heat A) (pX)icj transferred where h is the latent heat of fusion. If we substitute this value of Axice into Eq. (14) and perform the integration, we find 2(Q)A 1 ((p Q)({Ak )=(T1T3)t (16) which is identical to Eq. (2) if product = Q/A g = 1/[2(pk)ice(T T3)] h = AXmetal/[kmetal(T T3)] Againthe product (ice) gets in our way. The thermal conductivity of ice is about 1% that of alu minum (the common metal in freezers) so the accu mulated amount of what we are producing (ice) is the determinant of the rate of producing it. This analysis ignores the sensible heat of the ice and any convective heat transfer resistances. KreithE[3 applies the same analysis to the freezing of ice lay ers on ponds in cold weather. That situation is sketched in Figure 3, which is practically the same as Figure 1 if it were rotated by 900. However, in Figure 3 the resistance of the metal wall is re placed by 1/(the icetoair heat transfer coefficient), which is practically a constant. The resulting equa tions are the same as above, with (A eta/kmeta) being replaced by an icetoair heat transfer resis tance, (1/ho). This analysis applies to any solidifica tion process if the solid is a poor heat conductor, but not necessarily to the solidification of metals like aluminum or copper. EVAPORATOR SCALE FORMATION The third example is scale formation in evapora tors.[41 In many such evaporators a scale layer forms on the evaporator surfaces. This scale layer is nor mally a poor heat conductor, so its resistance to heat flow largely determines the overall heat flow rate. The experimental observation is that the thickness of the scale is proportional to the amount of heat which has been transferred to the evaporating solu tion since the last cleaning of the heating surface. Clearly, this is the same as the previous example with the variables renamed, i.e., d T1 T3 heat flux = A = T3(17) A dt [Ax Ad +A k )cae k metal wallJ Normally, the scale thickness is taken as some con stant a times the cumulative amount of heat trans ferred per unit area, and the two fluidfilm heat transfer resistances are added to that of the metal wall, so that the analog of Eq. (11) becomes d = dt X dt and the analog of Eq. (12) becomes (q2( 01 +(sum of other resistances)=(T1T3)t (19) Normally this is seen in the form of Eq. (1).E4,5] The Chemical Engineering Education _ 1 3 (18) a + Ax l h h1J scale )QA \ k metalwall h i transition from Eq. (2) to Eq. (1) may be made by solving Eq. (2) as a quadratic for the cumulative product, finding (cumulative product)= h + 4gt (20) 2 g 2g then differentiating, and noting that the production rate is d(cumulative production)/dt. Thus q = heat transfer rate = 1 (21) A h2+4gt Squaring and taking the reciprocals leads to Eq. (1). GASSOLID CHEMICAL REACTIONS The next application is a gassolid chemical reac tion which forms or leaves a solid residue, e.g. CaO(s)+SO2(g) CaSO3(s) (22) which is important in some gasphase sulfur dioxide capture processes, e.g., fluidized bed combustion, or in catalyst regeneration, e.g. Coked catalyst(s)+ 02(g) Cleaned catalyst(s)+CO2(g) (23) in which carbon is burned off a solid porous petro leum cracking catalyst, leaving behind a cleaned catalyst. In the first of these cases, the product of the reaction is a less porous solid than is the solid reac tant, so that the layer of product forms the principal barrier to diffusion of the gaseous reactant inward to the surface of the unreacted solid. In the second case, the reaction increases the porosity of the solid, but as the regeneration continues the oxygen must diffuse further inward to get to the unregenerated part of the catalyst, and the carbon dioxide must diffuse further to get out, so that the diffusion resis tance of the steadily growing cleaned catalyst layer is the principal resistance in the process. In general, such a reaction is described as[61 A(fluid)+bB(solid) fluid and solid products (24) Tair < 0C Air SConvective resistance laver in air Water Heat Flow Twater = 00C Figure 3. Solid ice formation on a body of water Summer 1992 where b is the stoichiometric coefficient. We may visualize these two reactions on Figure 3 if we visualize the ice as being replaced by the reac tion product (CaSO3 or Cleaned catalyst), visualize the unfrozen liquid being replaced by the unreacted solid reactant (CaO or Coked catalyst), and visualize the airtoice heat transfer coefficient being replaced by a gastosolid mass transfer coefficient. In place of the heat flow we will have gaseous reactant diffus ing to the solid surface and then diffusing through the reacted solid to the surface of the unreacted solid; there will be no flow of anything beyond that surface. For this general reaction, the analog of Eq. (7) is 1 dNA CA Ai (25) A dt (NA/A)b 1 LDolid product kG where (1/AXdNA/dt) = rate at which reactant A is delivered to the surface of unreacted B D = diffusivity of the gas kG = an external mass transfer coefficient, subscripts o, i = bulk gas phase and reaction interface, respectively The analog of Eqs. (8) to (10) is X NAsop b (26) solid product A Psolidproduct NA is normally stated in moles, so that Psolid product must be the molar density of the solid product. Sub stituting Eq. (26) into Eq. (25), we find the analog of Eq. (11) ~~N)2 CAo A1i ) Psolid product (27) 2( A j A CA A o J (27)(k,=Ao In most applications of Eq. (27) it is further as sumed that the concentration of A at the surface of B, (CAi), is negligible, and that (1/kG) is negligible, so that we have 2CA Dbt Axsolidproduct = .o  solid pduct Psolidproduct or, solved for t t Psolid product (29 2 bDC (29) 2bDCAo Most often one sees this equation applied not to a flat surface, but rather to a spherical particle.61 There one sees[6, Chap. 12, Eq. 14] ,_ PeR2 2 3Y r t= PBR2 13(r ) +2( )3 (30) 6bDCA R )( where R is the radius of the particle (assumed con stant, which implies that PB/b = Psolid product), and rc is the radius of the unreacted core of B. This is the "shrinking coreash diffusion controls" kinetic model. In it, the attention is focused on the unreacted B rather than on the reacted material We may see how Eqs. (29) and (30) compare by writing r =RAx (31) substituting in Eq. (30), and simplifying to find t PAX2 2 Ax (32) :2bDCAo [1 (32) A0 If, as assumed above, pB/b = Psolid product, then Eq. (32) is simply Eq. (29) multiplied by 1 (2/3)(Ax/R)], which accounts for the spherical rather than planar shape. For the planar configuration shown in Fig ures 1 and 3, R is infinite and Eqs. (29) and (32) are identical. For spherical particles, Eq. (29) describes only the initial stages of the process during which Ax is small compared to R. This same set of equations applies to the forma tion of oxide films on metals (e.g., rusting, but also solid oxide formation on nonferrous metals) if the film is coherent and does not flake away,[71 and to processes like fluidized bed powder coating and vari ous steps in the production of integrated circuits if the deposited or diffused film is more resistant to the flow of heat or material than the substrate. FILM CONDENSATION Nusselt's classic derivation of the behavior of a vapor condensing on a vertical wall in laminar flow,181 as shown in Figure 4, provides the final example. The derivation normally shown does not make clear that this is one of the class of processes discussed here. If we take the viewpoint of an observer riding with a batch of fluid down the wall (the Lagrangian view), we can see that the situation is exactly the same as the problem of freezing of ice on a solid surface and that Eq. (15) applies, with the proper ties of the condensate replacing those of the ice. In the derivation, the heat transfer resistance of the metal wall is normally ignored, as we do here, so by combining Eqs. (15) and (16) we find 1A2 kATt (33) 2H pl Here the appropriate value of t is the time it has taken this batch of condensate to flow downward from z = 0. At any value of z, for a slice perpendicu lar to the flow we may compute the average velocity from the assumption of laminar flow as (AVag g (34) Vavg = 3g (which ignores the buoyant effect of the vapor and the shear stress between vapor and condensate film). We assume that the piece of fluid we are riding with begins at t = 0, at z = 0, with V = 0. At time t it will be at location z and have the average velocity given by Eq. (34). To find the value of t corresponding to any z, we assume that the average velocity since t = 0 is onehalf of the velocity at t = t, which leads to t = z/ (Vav2). Substituting this value into Eq. (33) and solving for Ax, we find Ax= 12 ATz4 (35) L kRP2g J This is (3)1/4 = 1.3 times the value derived by Nusselt and shown in most heat transfer books. The differ ence results from the approximation made in treat ing a laminar flow as if it were a plug flow. That approximation is not as large a source of error in the derivation as some of the other approximations.[8] OPTIMIZATION One benefit of seeing that all these processes have the same form is that we can then use the optimiza tion equations developed for any one of them for all of them. For any of the processes which require regular shutdown and cleanout (e.g., batch filtra tion, batch freezing, evaporator operation with peri ,Condenser Wall Cooling Fluid Condensing vapor Heat Flow Figure 4. Laminar film condensation Chemical Engineering Education 4 1 odic shutdown and scale removal) if t is the operat ing time and tc is the time required to shut down, clean out and restart, the average production rate is cumulative product average production rate = cumulative product (36) t+t If we substitute Eq. (20) for cumulative product, set [d(cumulative product)/dt] equal to zero, and solve for t, we find (after some algebra) t=t,+h (37) or, if the rate is formulated in terms of Eq. (1), t=t +2 b (38) The latter solution is shown in Peters and Timmerhaus[51 for an evaporator with scale forma tion and regular shutdowns for cleaning, but it is obviously equally applicable to all of the batch pro cesses shown here. CONCLUSION One fundamental kind of processproduct gets in the way of productionappears in many places in chemical engineering. The mathematical presenta tions of these processes vary, but all can be shown to fit a single pattern. By using that pattern, we can use the results and ideas for any one of these pro cesses for all of them. Where Should this Fit in the ChE Curriculum? This material is regularly discussed in our senior year process design class. The students have previ ously taken courses in fluid mechanics, heat trans fer, mass transfer, and chemical reaction kinetics, so the examples should all be review for them. The design course seems a good place for them to see this integration of several diverse topics in their previ ous courses. Professors who use the process design book of Peters and Timmerhaus can introduce the discussion of this topic by assigning the following homework problem: Problems 115 (page 417) and 1416 (page 578) of Peters and Timmerhaus have very different looking rate equations, for processes which are physically similar. Show the choice of symbols which makes the rate equations for these two prob lems the same. NOMENCLATURE a = rate constant in Eq. (1): 1/[(production rate)2 (time)] A = area:m2 Summer 1992 b = rate constant in Eq. (1): 1/(production rate2) b = stoichiometric coefficient: mols/mol CA = concentration of A: mols/m3 D = diffusivity: m2/s g = rate constant in Eq. (2): (cumulative product2Xtime) g = acceleration of gravity: m/s2 h = rate constant in Eq. (2):(cumulative productXtime) h = heat transfer coefficients: J/[(m2)(sXK)] k = permeability: m2 k = thermal conductivity: J/[(m)(sXK)] kG = external mass transfer coefficient: m/s NA = moles of A: mols P = pressure: Pa Q = cumulative heat transferred: J Q = volumetric flowrate: m3/s q = heat flow: J/s R = radius (of spherical particle): m r = radius of unreacted core of spherical particle: m T = temperature: K t = time: s x = distance or thickness: m z = vertical distance: m V = volume of filtrate: m3 V = velocity: m/s V = superficial velocity: m/s W = cake volume/filtrate volume a = cloth resistance/a = (Ax/k)Im.: 1/m a = scale formation constant: m3/J X = latent heat: J/kg g = viscosity: Pa's p = density or molar density: kg/m3 or mols/m3 REFERENCES 1. de Nevers, Noel, Fluid Mechanics for Chemical Engineers, Second Edition, McGrawHill, NY, p. 426 (1991) 2. McCabe, W.L., and J.C. Smith, Unit Operations of Chemical Engineering, Third Edition, McGrawHill, NY, p. 939 (1976) 3. Kreith, Frank, Principles of Heat Transmission, Third Edi tion, IEP, NY, p. 534 (1973) 4. McCabe, W.L., and C.S. Robinson, "Evaporator Scale For mation," IEC 16 478 (1924) 5. Peters, M.S., and KD. Timmerhaus, Plant Design and Eco nomics for Chemical Engineers, Fourth Edition, McGraw Hill, NY, p 357 (1991) 6. Levenspiel, Octave, Chemical Reaction Engineering: An In troduction to the Design of Chemical Reactors, Wiley, NY, p. 338 (1962) 7. Bakhvalov, G.T., and A.V. Turkovskaya, Corrosion and Pro tection of Metals, trans G. Isserlis, Pergamon, NY, p. 9 (1965) 8. Roshenow, W.M., and H.Y. Choi, Heat, Mass and Momen tum Transfer, Prentice Hall, Englewood Cliffs, NJ, p. 240 (1961) classroom A STATISTICAL LOOK AT SIGNIFICANT FIGURES PARK M. REILLY University of Waterloo Waterloo, Ontario, Canada N2L 3G1 Almost all of the numbers used by scientists and engineers are approximations of some t sort, and everyone recognizes the importance of keeping in mind the uncertainty introduced by these approximations. In spite of its severe limita tions, the significant numbers convention is still com monly used by engineers and scientists to express this uncertainty and is widely taught in secondary schools and in the lower levels of universities. The purpose of this paper is to review the convention and its limitations and to present an alternative which is easy to use and is sound statistically. The approach is similar to, but developed independently of, that of Moffat.[1] THE SIGNIFICANT NUMBER CONVENTION Except in trivial cases such as the counting num bers, no decimal number can exactly represent a mathematical or physical reality. The error in a deci mal number is the difference between the "true" value of the reality and its decimal representation. A significant number is defined as one in which the error is less in absolute value than five in the next digit beyond those shown, and under the sig nificant number convention all decimal numbers are assumed to be significant ones. An important con sideration under the convention is the number of significant figures present, which is the number of figures in a significant number not counting preced ing zeros or following ones to which the error crite rion does not apply.[24] There is one important area in which the use of the convention is beyond reproach. That is in com municating purely mathematical numbers such as logarithms, the values of trigonometric functions, etc. The true values of such quantities are perfectly knowable. As an illustration of this, it would be easy in principle to express the base of natural logarithms Copyright ChE Division of ASEE 1992 e correct to 100 significant figures. The significant numbers convention should always be used, and un derstood to be used, in communicating this type of number. There are two basic rules for preserving signifi cance when significant numbers are combined arith metically. The first concerns addition and subtrac tion. Under it, a figure in a sum or difference is significant only if all figures in the same position relative to the decimal point in the numbers being added or subtracted are significant. Thus, in adding five numbers, each of which has one significant fig ure after the decimal point, the sum is said to be significant to one figure after the decimal point. Ob viously this is not true. The maximum error in the sum as indicated by the rule is 0.05, while in fact it is five times thatbecause the maximum error in each number being added is 0.05 by our hypothesis. This is serious because it implies that the rule un derestimates the error. Even more important, the probability of the actual error exceeding the figure given by the rule has the high value of about 44% to a good approximation. If we use the rule, we are in effect guilty of the heinous sin of "showing too many significant figures." Of course, the deficiency of the rule is much more serious when adding, say, 100 numbers. The second rule deals with multiplication and division. Under it the number of significant figures in a product or quotient is the same as in the one of the quantities being multiplied or divided which has the fewest. It is easy to show that this rule also is seriously deficient. Chemical Engineering Education Park M. Reilly is a professor emeritus in the Department of Chemical Engineering at the University of Waterloo. He received his BASc in Chemical Engineering at the University of Toronto in 1943 and his PhD in Statistics at the University of London in 1962.. He has spent about twentyfive years in each of industry and academe. There is one important area in which the use of the convention is beyond reproach. That is in communicating purely mathematical numbers such as logarithms, the values of trigonometric functions, etc. The true values of such quantities are perfectly knowable. The most important fault of the convention and its rules is that they attempt to deal with the maxi mum possible error. While that concept has some limited use, it is much more valuable to know how much error is likely to occur than how much error can occur. MEASURED QUANTITIES Except in trivial cases it is not possible in prin ciple to know the true value of a measurement. Here measurement means the approximate numerical value of a physical quantity obtained by comparison with an accepted scale. The integer results of counting are not considered measurements in this sense. It is not possible to state that the error in a measurement is not greater than five in some deci mal digit because we cannot know where the true value is. Consequently, the significant number con vention cannot correctly be used in connection with measurements. Also, as above it is almost always more important to know how big an error is likely to occur in a measurement than to know its maxi mum possible value. This is another serious limita tion on the value of the convention in dealing with measurements. AN ALTERNATIVE APPROACH As established above, the significant numbers con vention is unsatisfactory. An alternative approach can be developed which deals with which errors are likely. It is based on writing measurements and numbers derived from them in the form x u(x) The symbol x represents the number, and the ex pression u(x) represents the uncertainty in it. A nu merical example is (2.693 0.024) It is useful to enclose the numerical expression in parentheses. The uncertainty is that number which will exceed the magnitude of the error "most of the time." To those who are not familiar with the statistical ap proach it will be the number ordinarily used in stat ing how much error is expected, e.g., "The error is 0.3," or "The error is 2%." Such statements are very commonly made and understood by engineers and scientists. The quantity u(x) is the error magnitude Summer 1992 in this sense. This is also the same sense in which the error is said to be five in some decimal place when using the significant number convention in connection with measurements. This expression of uncertainty is more flexible than the significant num bers convention because it is not restricted to a value of five in some decimal digit. Notice that the uncertainty expressed in this way is additive, not multiplicative, and if the error is taken to be, say, 2%, 0.02 must be multiplied into the magnitude of the measurement to find u(x). From the statistical point of view, u(x) is a con stant times the standard error. Often the constant would be about 2, which would make the statement of the number and its uncertainty consistent with a 95% confidence interval. Ideally, all uncertainties should be halfconfidence intervals or the equiva lent, calculated according to statistical theory. How ever, those who do not use statistical methods for mally may bring some statistical science into their work by using somewhat informally the approach presented here. Some people will tend to be consistently higher or lower than others in estimating the error. As long as they interpret the results of this approach consistently, and other people understand this, no harm is done. In reporting numbers in this form, the number of figures shown in x and u(x) must be large enough that the range is described finely enough. If too few figures are used, the description is too coarse; for example, u(x) = 1.4 represents a 40% wider range than u(x) = 1, which is the same statement of uncer tainty with the last figure rounded. A reasonable suggestion for presentation in most cases would be 2 to 4 arithmetically correct figures in u(x), not count ing leading zeros and the number of figures in x made to correspond, as in the numerical example above. There would be no harm in quoting more except for its awkward appearance. NEW RULES FOR MANIPULATING UNCERTAIN NUMBERS As expressed above, rules for manipulating un certain numbers which are sound statistically may be developed by applying the wellknown formula for the propagation of variance.[51 For its correct application, the uncertain numbers must be inde 153 pendent random variables. This is ordinarily true in the case of measurements. No particular form of probability distribution is implied. As applied to sums and differences, the formula is u(x,X2 ...x)= {[U(x)]2 +[U(,)12+...+[u(x 1)]/2 Thus the uncertainty in a sum or difference of num bers is the square root of the sum of the squares of the uncertainties in the individual numbers. For products and quotients the rule is: let y=x, X x2 x... x Then u(y) = y( ) + uX2 +... + X1 X2 8 or equivalently 2+1/2x y u(y)= y +U( The relative uncertainty in a product or quotient is the square root of the sum of the squares of the relative uncertainties in the individual numbers. If the uncertainties or relative uncertainties, as appropriate, in either of these rules is the same for all quantities being manipulated, the uncertainty or relative uncertainty in the result is equal to that of one of them multiplied by the square root of the number of quantities manipulated. The general rule for the uncertainty in a general function f(xl, x2,... Xn), where the x's are uncertain quantities is given by U(f) = U(xi) i=1 i The formula for the propagation of variance is in general only approximately true. In effect, it de pends on a linearization of the function which is involved. Hence the formula for the uncertainty in a sum or difference is strictly correct, but that for a product or quotient or any nonlinear function is ap proximate. The approximation is good, however, as long as the relative error in the x's is not too large. Davies and Goldsmith[51 suggest a ruleofthumb which implies the generous rule here that u(x) should not exceed 20% ofx. It is interesting to note that at least the first two 154 of the above formulas are equivalent to the old rules for manipulating significant numbers if the error in one of the x's is large enough that it dominates the others. In dealing with mathematical numbers in the above formulas it is possible to use 5 in the first figure not shown, or some multiple of it, as the uncertainty. However it is better, if possible, to ex press them to enough significant figures that their error is negligible compared to those of the other numbers involved. USE OF THE APPROACH IN EDUCATION No background beyond that required for the sig nificant numbers convention is needed to use and understand this approach. It has, however, the di dactic advantage that it is consistent with the use of confidence intervals and their equivalents. It can therefore fill the gap which now exists with those educators who recognize the deficiencies of the sig nificant numbers convention but have little or noth ing to take its place until the student obtains a background in statistics. It also has the considerable psychological advan tage that it requires the explicit statement of the uncertainty in a number. Thus, when a student quotes a number his or her attention is necessarily focused on the uncertainty in it. EXAMPLES The following is a simple example of addition and subtraction: (42.63 0.21) (10 0.05) + (14.0 0.3) = 42.63 L0 + 14.0 (0.212 +0.052 +0.32)1/2 = (55.63 0.37) The second example illustrates the use of significant numbers and perfectlyknown numbers along with multiplication and division. The number 2 is per fectly known and requires no consideration in calcu lating the uncertainty. The number p is a significant number and is carried to seven figures so that its uncertainty has negligible effect. 2n1(10.623 0.500) 129 15 2(3.141593)(10.623) 1+[ 0.500 \2 ( 15 \)21/2 129 [L10.6231 \129 = (0.5174 0.0649) The next example illustrates the use of the for mula for a general function. Chemical Engineering Education (2.31 0.26) exp( 31 0.08) = 2.31 exp(L 31) + {[exp(131) x 0.26]2 + [2.31exp(L31) x 0.08]2 1/2 = (0.6233 0.0861) Summary of Suggested Approach to Calculating and Expressing Uncertainty in Decimal Numbers 1. All mathematical numbers such as e, etc., should be expressed using the significant numbers convention. 2. Enough extra digits should be carried in all arithmetical calculations that the rounded result would not be changed if more were carried. This usually requires two or three extra digits. 3. The significant numbers convention should never be applied for the purpose of expressing uncertainty in measurements or the results of calculations on them. 4. Measurements and the results of calculations on them should be presented by showing the decimal number plus or minus its uncertainty. The best way to deter mine this uncertainty is by statistical treatment of the data. Many people will, however, use their engineering or scientific judgement, formally or informally, to establish it. 5. The uncertainty in the sum or difference of uncertain numbers is found as the square root of the sum of the squares of the uncertainties in the individual numbers. 6. The relative uncertainty in the product or quotient of uncertain numbers is the square root of the sum of the squares of the relative uncertainties in the individual numbers. 7. Providing the numbers used in expressing uncertainty are arithmetically correct, there is no harm except awkwardness in showing too many figures; if too few are shown, however, the uncertainty may not be shown precisely enough. Ordinarily, the uncertainty should be shown with two to four arithmeticallycorrect figures, disregarding leading zeros, and the number itself shown with a total number of figures to conform with the uncertainty. ACKNOWLEDGMENT The author is pleased to acknowledge the helpful criticisms and suggestions of Professor Martin E. Weber of the Department of Chemical Engineering at McGill University. REFERENCES 1. Moffat, R.J., "Describing the Uncertainties in Experimental Results," Exp. Therm. and Fluid Sci., 317 (1988) 2. Anderson, T.W., and S.L. Sclove, The Statistical Analysis of Data, The Scientific Press, 97 (1986) 3. Volk, W., Applied Statistics for Engineers, McGrawHill, 77 (1969) 4. Eshbach, O.W., Handbook of Engineering Fundamentals, John Wiley & Sons, 2 (1936) 5. Davies, O.L., and P.L. Goldsmith, Statistical Methods in Research and Production, Longman, 54 (1977) ri Summer 1992 REVIEW: Plant Design Continued from page 119. Some of these have been developed from previous AIChE Student Contest Design problems. They provide the instructor with a good source of fairly complex class assignments that he or she can then build upon. The student may be able to use these as practice problems in order to prepare for a design contest problem. Although this book is as complete a design text as is possible in a single volume, it is intended to supple ment coursework rather than take its place. This leaves each design instructor many opportunities to embellish the material and give his or her design course its own distinct flavor and personality. What this text does not do (and does not claim to do) is emphasize the importance of choice in design: the alternatives involved in defining the problem, good and bad design choices and how to tell the difference between them, and process integration alternatives and how to evaluate them. Design problems in the less traditional areas of chemical engineering, such as bioseparations, polymers, pharmaceuticals, or food, could be presented in order to give the student a realistic idea of the variety, complexity, and inherent similarity of design issues in chemical engineering practice. The text does not stress the importance of thermodynamic principles in good process design and with the exception of the HAZOP study, neglects the importance of the prin ciples of process control. The instructor must take over where the book stops. Plant Design and Economics, in its fourth edition, ntinues to provide a comprehensive source of .sign principles and information that could be ,!use to both students and professionals. As a text, it includes a wide variety of instructive problems, both solved and unsolved, and many charts, logic flowsheets, and worksheets which aid the student in setting up and solving a design problem. The text is lucid and readable. It serves as an excellent aid to teaching a one or twoterm design se quence as well as a handy reference of uptodate information on regulations and cost. This would be a good text for someone who does not already possess the third edition. We would recommend this text to an owner of the third edition because of the updated and expanded material. We would be happier still if an updated bibliography were in cluded and dual units were incorporated in a future printing. This text remains an excellent buy in terms of value for money. 0 I laboratory ADD SOME FLAVOR TO YOUR AGITATION EXPERIMENT M. ELIZABETH SENSEL, KEVIN J. MYERS University ofDayton Dayton, OH454690246 MaciasMachin, Zhang, and Levenspiel[E1 re cently proposed the unstructured research experiment as an effective means of im proving chemical engineering laboratory courses. This type of experiment has great flexibility from year to year, and it also forces students to be more indepen dent in developing a solution to the problem that is presented to them. We have used this approach in our unit operations laboratory, including the melt ingice heat transfer experiment discussed by Macias Machin and coworkers, and we would encourage other departments to also make use of this type of labora tory assignment. This paper describes one experiment that was developed by students to determine the interphase mass transfer coefficient for a solid dissolving into an agitated liquid. The problem was presented to the students in very general terms, and they were required to search the literature to become familiar with the problem, to develop a realistic mathemati cal model to describe the dissolution process, and to M. Elizabeth Sensel received her bachelor of chemical engineering degree in 1991 and is cur rently studying agitation in gasliquid systems for her master of science in chemical engineering degree (both at the University of Dayton). While she was an undergraduate student, she held a I cooperative education position with the Depart ment of Energy's Mound Laboratory. Kevin J. Myers is an associate professor in the Department of Chemical and Materials Engineer ing at the University of Dayton. He received his Bachelor of chemical engineering degree from the University of Dayton and his doctor of science ' in chemical engineering degree from Washington University. His research interests are in the field of multiphase agitation and chemical reactors. develop a simple experimental technique to deter mine the mass transfer coefficient. Due to the suc cess that we experienced with this experiment, we suggest that it be considered as a means of providing flexibility for agitation experiments. MATHEMATICAL MODEL Badik and Servais[2] have demonstrated the value of a mathematical model for the interpretation of an experiment, and its usefulness is particularly impor tant for an unstructured research experiment. Al though the mathematical model and experimental procedure are developed simultaneously in practice, the mathematical model of solids dissolution will be presented before the experimental procedure and results. Briefly, the experimental method consists of adding a number of solid particles of known mass to the agitated liquid, removing the particles from the liquid after a specified time, and then determining the remaining mass of the particles. Thus, the math ematical model of the dissolution process must re late the mass of solid remaining in the solid phase (the experimental data) to time and the interphase mass transfer coefficient (the unknown parameter). Examination of the agitation literature indicates that the rate of mass transfer between a solid and an agitated liquid is usually described by the following relation (Nienow[31): m=kLA (CSATC (1) The experiment is conducted on a batch system, with the result that a transient mass balance on the dissolving solid takes the form dM = kA (CAT CL) (2) dtSAT while the corresponding mass balance on the liquid phase is O Copyright ChE Division ofASEE 1992 Chemical Engineering Education dC VL T = ln = kLsA (CSAT L) (3) L dt A(CSATC) (3) These model equations are coupled through the liquid concentration term and must be solved simul taneously. The solution procedure can be simplified by noting that the total amount of solid distributed between the solid and liquid phases is constant at its initial value. Mo +VL CLo =M+V CL (4) This equation can be combined with Eq. (2), which can then be solved to yield the model predictions. However, as the solids dissolve, they change their size and shape, and the resulting changes in the interfacial area must be taken into account before the model equations can be solved. Any effect of changing particle size on the interphase mass trans fer coefficient will be ignored in this analysis. The particles studied in the experiment are ini tially spherical and are assumed to retain their spherical shape as they dissolve. The solid particles are also of the same initial size, and it will be as sumed that all of the particles dissolve at the same rate. Under these assumptions, the mass of solid remaining in the solid phase at any time for a sys tem of n particles with radius r is M=4 r3 p n (5) and the corresponding interfacial area is A = 4 r2 n (6) Substitution of Eqs. (4), (5), and (6) into Eq. (2) yields the form of the model equation that can be solved for the mass of solid remaining in the solid phase at any time, k = 2.5E6 m/s k LS = 5.0 E6 mis k15= 7.5 E6  ... [the students] were required to search the literature to become familiar with the problem, to develop a realistic mathematical model to describe the dissolution process, and to develop a simple experimental technique to determine the mass transfer coefficient. 1 dM_ k (36 n nM2 (3()) (7) dP. LSAT L VL This equation can be solved numerically, but an analytical solution is possible if the liquidphase con centration is always much less than the saturation concentration (CL << CSAT) which was the case for the experimental results reported here. Under these conditions, Eq. (7) can be integrated to yield the following relation between time and the fraction of solid remaining in the solid phase M. IC 3 MopS ] Figure 1 presents plots of the fraction of solid remaining in the solid phase as a function of time for typical experimental parameters and the range of interphase mass transfer coefficients observed dur ing this study. Comparison of these model predic tions with experimental data yields the magnitude of the interphase mass transfer coefficient for any experimental run. EXPERIMENTAL PROCEDURE AND RESULTS The experimental procedure is based on the work of BoonLong, et al., [4 who studied the dissolution of benzoic acid particles into water. After initial con sideration of working with benzoic acid, we decided that there must be a better solid for use in an under graduate laboratory. The solid that was selected from a number of possibilities was sourball candy, a mix ture of sugar, citric acid, and color additives. This material is inexpensive and safe, and its high solu bility allows a number of successive experimental runs to be made with a single liquid batch. As described during the model development, the experimental procedure is to add a number of par ticles of known weight to the agitated liquid and to remove, lightly dry, and weigh the particles after they have dissolved for a specified time. This tech nique yields an integral interphase mass transfer Summer 1992 0 a0 Z W" 0 0 s 0 50 100 150 20 250 TIME (S) Figure 1. Model predictions for typical experimental conditions ! coefficient that is representative of the entire experi mental run. Since the particle size changes during the experi ment, the mass transfer coefficient might also change. To check this, experiments were performed to deter mine if a single value of the interphase mass trans fer coefficient describes the entire course of an ex periment. These results are presented in Figure 2, and it can be seen that the model predictions with a constant value of the interphase mass transfer coef ficient accurately describe the experimental data. For the remaining experiments, only a single data point was taken (usually after five minutes of disso lution), and it was assumed that the calculated in terphase mass transfer coefficient was representa tive of the entire experiment. Two impeller types were studied: a fourbladed 450 pitchedblade impeller and a threebladed high efficiency impeller. The dependence of the interphase mass transfer coefficient on the agitation speed for these impellers is shown in Figure 3. All results were obtained using 0.178meter diameter impellers in a 0.445meter diameter tank with a liquid level equal to the tank diameter, an impeller offbottom clearance of onefourth of the tank diameter, and standard baffles. The results of Figure 3 indicate that the inter phase mass transfer coefficient is not strongly af fected by operating conditions as has been discussed by Nienow.[3] The magnitude of the interphase mass transfer coefficients found in this study are some what lower than those reported in the literature, but this can be attributed to the fact that the sourball candy used in this study is considerably larger than the solids used in other investigations (Miller[51). The results presented in Figure 3 appear to indi cate that the pitchedblade impeller performs better than the highefficiency impeller, yielding similar interphase mass transfer coefficients at lower speeds. However, the pitchedblade impeller draws about four times as much power as the highefficiency im peller at the same operating conditions. A proper comparison results when the interphase mass transfer coefficient is considered as a function of the power input per unit liquid volume, as shown in Figure 4. This comparison indicates that the high efficiency impeller yields interphase mass transfer coefficients that are about five percent higher than those of the pitchedblade impeller at equal power inputs. This small difference is near the limit of accuracy of the experiments, and the performance of ~~A MODEL PREDICTION (k LS = 5.5 E6 m/s) \ * EXPERIMENTAL DATA 0 60 120 180 240 300 360 420 TIME (S) Figure 2. Comparison of the model prediction and experimental data over the course of an experiment 0 * HIGH EFFICIENCY 0 PrTCHEDBLADE 0 L_____________Lr I 150 175 200 225 250 275 300 325 350 AGITATION SPEED (RPM) Figure 3. Experimental results for pitchedblade and highefficiency impellers , I'' *HIGH EFFICIENCY O PITCHEDBLADE 0.02 0.04 0.06 0.08 0,10 0.12 POWER INPUT PER UNIT VOLUME (kW/M3) 0.14 0.16 Figure 4. Comparison of impeller performance on an equal powerpervolume basis Chemical Engineering Education . . . . I I I I I II' ' ' l'' l' , L I I I the two impellers is essentially equal. The data of Figure 4 indicates that the interphase mass transfer coefficient increases with the power input to the one fourth power, which is consistent with the data dis cussed by Nienow.131 CONCLUSIONS We have found that the study of sourball candy dissolution can spice up an agitation experiment. The technique is safe, inexpensive, rapid, and is capable of yielding meaningful results. The interpre tation of the experiment requires the students to develop a mathematical model of the dissolution pro cess which adds to the instructional appeal of the experiment. Although the experimental technique was developed as an unstructured research experi ment, it is also possible to supply the students with the technique and instruct them to use it to solve other problems, such as making a sugar solution (make up a good assignment story), comparing the performance of various impellers, determining the effect of vigorous agitation on liquidsolid mass trans fer, and comparing the data with reported values in the literature. ACKNOWLEDGMENTS The assistance of Bonnie Struble and Russ Logue in developing this experimental technique is grate fully acknowledged. NOMENCLATURE A total liquidsolid interfacial area at any time (m2) CL liquidphase concentration of the solute (kg/m3) CSA equilibrium liquidphase concentration of the solute (kg/m3) km liquidsolid interphase mass transfer coefficient (m/s) M total mass of solute remaining in the solid phase at any time (kg) m rate of interphase mass transfer of the solute from the solid phase to the liquid phase (kg/s) n number of solid particles used in an experiment () r radius of the solid particles at any time (m) t time (s) VL liquid volume (m3) p, solid density (kg/m3) o subscript indicating initial conditions REFERENCES 1. MaciasMachin, A., G. Zhang, and 0. Levenspiel, "The Un structured StudentDesigned Type of Laboratory Experi ment," Chem. Eng. Ed., 24(2), p 78 (1990) 2. Badik, C.S., and R.A. Servais, "Experiences with Revamp ing an Introductory Engineering Laboratory Including Data Systems and Modeling," Proc. 1991 ASEE North Cent. Sect. Conf., p 265 (April, 1991) Summer 1992 3. Nienow, A.W., "The Mixer as a Reactor: Liquid/Solid Sys tems," Chapter 18 of Mixing in the Process Industries, ed ited by N. Harnby, M.F. Edwards, and A.W. Nienow, Butterworths, London (1985) 4. BoonLong, S., C. Laguerie, and J.P. Couderc, "Mass Trans fer from Suspended Solids to a Liquid in Agitated Vessels," Chem. Eng. Sci., 33, p. 813 (1978) 5. Miller, D.N., "ScaleUp of Agitated Vessels: Mass Transfer from Suspended Solute Particles," Indus. Eng. Chem. Proc. Des. and Dev., 10(3), p 365 (1971) C EDUCATOR: Wankat Continued from page 123. strong emphasis on counseling. He felt he could put his counseling experience to good use by dealing with students who were at a critical stage in their careers. He feels that the vast majority of students who enter the freshman engineering program have the ability to graduate and become successful engi neers, but that the lack of motivation is a problem for some of them. Phil rejects the "sink or swim" ideathat the best students will rise to the top while the others sink. Rejecting the notion of teach ing only the intellectually elite, he believes that the "purpose of a university is to nurture students' learn ing and to help them get past barriers." That is the goal of the freshman engineering program. PERSONAL Phil has won numerous awards, among them ASEE's Western Electric Award (1984), George Westinghouse Award (1984), and Chester F. Carlson Award (1990). In 1991 he was named a Fellow of ASEE. He has also held several divisional offices, including Chairman of the ChE Division of ASEE. Phil and Dot have two children: Charles (7) and Jennifer (4)both of whom, alone or in tandem, provide him with all of the exercise he needs. When he feels contemplative, or simply in need of quiet moments, he likes to head to a favorite fishing spot; fishing is his Zen meditation. When it is possible, he likes to go canoe camping (especially in the Quetico Superior area) and (hopefully) catch fish every day. A Chicagoarea native, Phil has never outgrown his addiction for the Bulls, the Bears, and the Cubs. For him, 1991 was "Bull Heaven." And finally, what surely labels him as an eternal optimist, he still believes the Cubs will win it all next year. Phil Wankat is a busy man: teacher, researcher, counselor, author, editor, administrator. "Just do it. . but care!" would be a good slogan for him. His career exemplifies what many others strive fora blend of excellence in both research and teaching. O curriculum MOLECULAR ENRICHMENT OF THE CORE CURRICULUM HENRY A. MCGEE, JR. National Science Foundation* Washington, DC 20550 he core courses of chemical engineering are properly taught from the viewpoint of con tinuum physics. Thermodynamics, transport phenomena, and reaction engineering, as they are now presented to students, would be unchanged whether molecules exist or not. This is a strength in that one need never worry about underlying struc ture or mechanism. But it is also a weakness, for so much of practice in chemical engineering, in the thermal fluid sciences part of mechanical engi neering, in materials science and engineering, and in aerothermochemistry rests directly upon molecu lar insights. It is truly molecular engineering. With out a molecular perspective to complement their continuum perspective, our young graduates will be illequipped to be full participants in modern engineering practice. Building on the courses in physical chemistry, we can enrich our continuumbased core courses in chemical engineering by having students study an auxiliary textbook that discusses the same phenom ena and processes, but from a molecular point of view. The molecular discussion can be read at the same time that they read any of our good continuum based textbooks. The needed enrichment must not delve into the exotica of quantum and statistical mechanics that is of interest only to the specialist, however; it must Henry McGee was trained in chemical engi neering and in theoretical chemistry at Georgia STech and the University of Wisconsin, and his subsequent teaching and research reflect this SS *7' continued dual interest. His research has cen tered upon chemical reaction and processing under extreme or unusual conditions. While in Washington, he has been conceded mainly with science policy matters and with priorities in the support of research. * On leave from the Chemical Engineering Department at Vir ginia Polytechnic Institute and State University, Blacksburg, VA 240610211 160 engage students and faculty alike at their existing level of understanding. Molecular understanding and practical examples must be compelling and memo rable to the engineering student rather than elegant to the theoretical chemist.[13 The molecular perspec tive on thermodynamics, on transport, and on chemi cal kinetics may thereby be merged into the four or five semesters now required for the core. DEFINITIONS AND EXAMPLES Molecular engineering encompasses those prob lems wherein a molecular perspective (whether it be computational or merely phenomenological) is an essential part of any optimum design. A number of illustrative examples follow. Chiral synthesis and separation is an essential part of many problems involving pharmaceuticals, manufactured foodstuffs, agrochemicals, flavors, and fragrances. Chirality can be critical in drug manu facture. For example, one isomer of thalidomide (shown below) is a useful sedative while the other is a potent teratogen that caused thousands of birth defects three decades ago. H H o0 0 0 0 Sedative Teratogen How is a stereoselective catalyst designed, or how does one think about separation of chiral molecules? To be sure, Pasteur first separated crystals ofd and 1 tartaric acid by using a pair of tweezers and a micro scope, for the salts of the two isomers have macro scopically recognizable differences in crystal morphology. But more usually one thinks about spe cific completing agents that geometrically fit the one molecule but not the other. Such catalyst and separation designs are exercises in molecular recognition. Designs frequently depend upon either ab initio or semiempirical quantum mechani Copyright ChE Division ofASEE 1992 Chemical Engineering Education cal techniques to calculate structures and energies of candidate hostmolecules. Although only well developed for separating ura nium isotopes, the best procedure for any isotope separation seems to be the atomic vapor laser iso tope separation (AVLIS) scheme. Here one recog nizes small differences in the ionization energy of each isotope and uses a fixed energy input from a properly tuned laser to ionize the one desired iso tope, while producing no effect at all on the other isotope. To produce reactorgrade enriched uranium, one produces a low pressure vapor of natural ura nium and irradiates this vapor with laserphotons selected to have energy sufficient to ionize 235U that is present to 0.3 percent, but the 99.7 percent which is 238U is transparent to these photons. The laser photons function as somewhat of a Maxwell's De mon. A negatively biased plate attracts and collects the ions. This laser process has been pilotplanted, and although still somewhat debatable, data sug gest that both installed and operating costs of the laser plant will be an order of magnitude below such costs for conventional gaseous diffusion technology. This process would not have occurred to chemical engineers unschooled in molecular thinking, and as a matter of fact, this process was largely conceived and developed by physicists and physical chemists. It is disappointing to admit that researchers and designers other than chemical engineers have pro duced attractive solutions to problems in separa tions on a practical scale. In a sense, chemical engi neers have here been beaten at their own business due to a lack of molecular insight. The physical and chemical properties of clusters depend on the size and arrangement of the atoms forming the cluster. Even the color of cadmiumsele nium particles can be red, orange, green, or black due to slightly different particle sizes. Red Cd/Se is 5 nm across and has about 3000 atoms, while orange Cd/Se is 3.5 nm across and has about 1000 atoms. Such clusters are not molecules nor are they bulk metal, but rather they are a new class of materials. They may be effective catalysts, and materials made from clusters may have superior properties. The mo lecular view suggests that a large fraction of the constituent atoms are on or near the surface. Also the conduction electrons are confined to a space only a few atoms across, producing a quantum size effect responsible for the colors of Cd/Se and other proper ties. Ionic Nb19 exists in two forms, jokingly called "chocolate" and "vanilla," wherein the one is highly reactive with hydrogen while the other is unreactive. Presumably, this striking chemical Summer 1992 Building on the courses in physical chemistry, we can enrich our continuumbased core courses ... by having students study an auxiliary textbook that discusses the same phenomena and processes, but from a molecular point of view. difference is due to the arrangement of the nine teen atoms of the cluster. The understanding and application of clusters is impossible outside of the molecular perspective. Suppose we are interested in the specific impulse that might be obtained from an electrothermal arc jet thrustor using H2 as the propellant. At plasma temperatures, we need the thermodynamic proper ties of H, H2, H+, and e, for it is necessary to calcu late the difference in enthalpy of the expanding gas between the combustion chamber and the exit plane of the nozzle if we are to estimate the thrust. There is no way to measure the heat capacity or entropy of atomic hydrogen or of protons. Rather, we calculate all of the properties using the techniques of molecu lar physics. With values for all of the properties, we can calculate the equilibrium extent of reaction of H2 t42H H t4 H+ +e and then the specific enthalpy of the reacting and expanding gas in the nozzle as a function of tem perature and pressure. From this, the expected thrust levels for any particular motor design can be de duced. One believes the calculated values of thermo dynamic properties in regions not experimentally accessible because in all instances where such com parisons between theory and experiment are pos sible, agreement is excellent. Without molecular in sight, such rational rocket motor design would be impossible. One of the best ways to grow diamond films is by chemical vapor deposition (CVD). In the highenergy environment of a lowpressure plasma, a large vari ety of reactive chemical species can exist, and each may play a significant role in the formation and quality of the resulting diamond film. With a feed gas stream ofH2 and CH4, reactive species including CH, C2, C3, C2H2, and C2H are evident. Different electronic and vibrational states are also evident, and these may not be in equilibrium with the trans lation/rotation heat bath. How does the concentra tion of species vary in time and place in the CVD reactor, or with temperature, or with pressure, or with feedgas composition? How do you even think about the temperature of such a reacting gas mixture in the presence of an electric field? Such a CVD gas mixture may not be at equilibrium at all, but rather the concentration of the various species may be kinetically determined. All such questions must be addressed from the point of view of molecu lar engineering, and the optimum design of CVD reactors for diamond deposition depends upon these molecular insights. Such a listing of examples of molecular engin eering could continue, but these few suggest the central importance of molecular insight in engineer ing design. INNOVATION VS. DESIGN The manufacturing ability of the United States is being challenged by worthy competitors, particularly in Germany and Japan. The NSF and the entire federal research and development establishment is developing a major initiative designed to help en sure that American industry maintains its interna tional competitiveness. Terms such as "agile manu facturing," or "21st Century manufacturing," or "en vironmentally benign manufacturing" are seen with regularity. Creative innovation and design are cen tral to success in competitive manufacturing. Whether one is substituting an alternative reaction chemistry or optimizing a separation and heat ex change network, there is creative opportunity. Molecular engineering addresses questions of in novation by stimulating the engineer to think about, say, an alternative separation based on some newly synthesized zeolite with heretofore unavailable pore size. After the innovation, process design allows its optimization. Both are importantbut the senior design class in chemical engineering usually concen trates on process design alone (which has become very logical and analytical). Computerized design methodologies are a triumph of modern chemical engineering. But in contrast, molecular engineering gives the student more opportunity to be imaginative. It gives the chem ical engineer an opportunity not unlike that afforded to an architect who imagines the form of a building and then performs structural design calculations (just as in the chemical engineer's process design) to judge whether that imaginative design is economi cally buildable. Modeling and tools such as ASPEN are important and powerful components of the curriculum. But they will never invent AVLIS or a stereoselective separation or a nanoparticle manufacturing scheme. After the innovation has occurred, conventional teach ing allows the practitioner to pursue the important, but subsequent, tasks of simulation, modeling, opti mization, and control. That initial innovation, how ever, is the point where molecular insight is so im portant. It is not a panacea, and it is not a sufficient condition for innovation. But it does broaden one's scope and opportunities. Some curricula require a year of physical chemis try where some insights into partition functions, energy levels, kinetic theory, and molecular dynam ics is learned. However, ABET requires only one semester of physical chemistry (which is largely clas sical thermodynamics in most courses). Whether the study of molecular physics in physical chemistry is required or elective, it is to be applaudedbut it remains a subject apart from the mainstream (like technical writing, or the German language) and the student never integrates modern molecular physics into his or her engineering Weltanschauung (philosophical worldview). That nowmissing in tegration is the goal of molecular enrichment of the core curriculum. SOLUTION Teaching and learning molecular engineering, like other subjects in the core curriculum, are not diffi cult if the sophisticated researchoriented aspects of the subject are abandoned. For example, starting with a thought experiment with a collection of a halfdozen, labeled molecules, one immediately visu alizes the most probable distribution of molecules among energy levels, with the distribution driven to its most probably state by no other mechanism than simple chance.11] Then, with the same Lagrangian technique of undetermined multipliers learned in calculus or in thermodynamics when calculating re action equilibria, the student immediately derives the Boltzmann distribution; he/she gains immediate insight into why so many macroscopic phenomena (equilibrium constant, rate coefficient, vapor pres sure, etc.) depend on exp(energy/RT). Molecular insight also provides a powerful peda gogical tool in that it enables linkage between other wise (seemingly) disparate macroscopic phenomena and processes. For example, it can be seen that the same intermolecular collision frequency that gov erns the chemical reaction rate also governs thermal conductivity. As a pedagogical tool, molecular engi neering is compelling, provided only that sophisti cated molecular physics is avoided. Similarly, single particle partition functions are easy to understand as compared to ensemble ideas. To be sure, there are troubling consequences when Chemical Engineering Education real gases are studied, but sensible treatment is possible and nothing has to be unlearned by those very few students who later will wish to become expert in statistical thermodynamics. The MaxwellBoltzmann distribution of molecu lar speeds is easily obtained from the most probable distribution of energy, which itself was easily ob tained (as we saw) from simple thoughtexperiments with a few labeled molecules. With the MB distribu tion, all the concepts of kinetic theory of average speed, mean free path, and collision frequency may be immediately shown. Similarly, and of more inter est to students, each of the transport properties can be calculated and collision theories of chemical ki netics can be developed.[11 Compelling comparisons of all such theories with experiment make it real and believable to the students. With this kinetic theory, it is natural to realize that chemical reaction does not occur in one step as we typically write an overall stoichiometric change, e.g., H2 + Br2  2 HBr Rather, reaction occurs by a complex array of usu ally bimolecular encounters which together constitute the reaction mechanism, which for the hydrogen/ bromine flame is Br2 + 2 Br Br+H2  HBr+H H+Br2 HBr+Br S+ H BrH H2 +Br Br + Br Br2 The rate of each of these molecular events of the mechanism depends on its particular reactant colli sion frequency, the relative energy involved in the collision, the energy states of each colliding reac tant, and the relative geometry of the colliding reac tants at the moment of impact. Reaction occurs only in collisions that occur with an abovesomemini mum threshold energy, and even then only in colli sions that occur with certain geometric orientation. Finally, the macroscopic (or observed) rate of the overall stoichiometric change is some sort of a com plex average of these many different microscopic events. And, under a variety of assumptions, this averaging can be calculated, and comparisons with experiment may be made. It is pedagogically essential to present numerous comparisons with experiment and to present many case studies and practice problems to inspire and provide exercise for the students in their develop ment of new skills.[11 Summer 1992 CONCLUSION You may invent a laserbased process for isotope separation, or be concerned with fundamental prob lems in combustion leading to greater fuel efficien cies and less pollution, or be concerned with ion implantation for the development of new alloys of new and unusually doped materials of interest in electronics, or require some property of matter that may be unknown or unmeasurable. From whatever perspective, however, a molecular view is essential, and a purely traditional or classical perspective un acceptably slows invention, hinders creativity, and frustrates original design. This enrichment of the chemical engineering core curriculum will have served its purpose if its atti tudes can be internalized. That is, long after the student has forgotten just exactly how this or that particular argument or calculation goes, he or she will nonetheless instinctively think about any prob lem in terms of what the molecules must be doing. That is the real, bottomline goal of molecular en richment of our core curriculum. ACKNOWLEDGMENTS The author gratefully acknowledges helpful criti cism by the referees of this paper. REFERENCES 1. McGee H.A., Jr., Molecular Engineering, McGrawHill, New York (1991) 0 OXYGEN MASS TRANSFER Continued from page 145. cal engineering principles in undergraduate labora tories. ACKNOWLEDGMENTS This work was partially supported by Grant DTD 910418, U.S. Agency for International Development, Pakistan Participant Training Program, and the Georgia Tech Foundation. The authors are very ap preciative of the assistance of Brenda Chand, Harolyn Ingram, Susan Elliot, and William Ernst. REFERENCES 1. Van't Riet, K, "Mass Transfer in Fermentation," Trends in Biotechnology, 1(4) 113 (1983) 2. Ruston, J.H., E.W. Costich, and H.J. Everett, "Power Char acteristics of Mixing Impellers," Part 2, Chem. Eng. Prog., 46, 467 (1950) 3. Bailey, James, and David Ollis, Biochemical Engineering Fundamentals, 2nd Edition, McGrawHill Book Company 4. Lee, S.S., F.M. Robinson, and H.Y. Yang, "Rapid Determi nation of Yeast Viability," Biotechnol. and Bioeng., Symp. No. 11, 641 (1981) 0 classroom DESIGN OF CSTRs IN TANDEM REVISITED A. A. ADESINA University of New South Wales Kensington, NSW, Australia 2033 since the days ofDenbigh,'1] the optimal design of chemical reactors has held a place of pride in the chemical engineering profession. Aris' unparalleled monograph[2] laid a solid mathema tical foundation for the treatment of this subject, and application of dynamic programming tech niques in the determination of optimal interstate conversions and reactor sizes was painstakingly expounded. Chen[31 also utilized Pontryagin's maxi mum principle to solve this problem, while the twopart serial papers by Chitra and Govind[41 offer considerable insight into situations involving com plex reaction networks. Chemical reaction engineering is usually taught in the penultimate year of a fouryear bachelor (honours) program. Unfortunately, thirdyear stu dents, partly because of their limited exposure to mathematics, are loathe to accept analytical meth ods as a substitute for a graphical procedure involv ing the maximization of rectangles (described in the texts by Levenspiel[5] and Fogler[61). For instance, until recently at the University of New South Wales, introductory concepts in dynamic programming were not encountered until the process optimization course in the last year of the undergraduate curriculum. Additionally, Levenspiel's approach has important pedagogical appeal for the lecturer. Therefore, the analytical techniques must be reserved for a more advanced course in reaction engineering where the students (such as those in the postgraduate pro gram) will then be able to savor the taste of these mathematical treats. Adesoji A. Adesina is a chemical engineering faculty member at the University of New South Wales, Australia. He obtained his BSc from the University of Lagos (Nigeria) and his Masters and PhD from the University of Waterloo S(Canada). His primary research activities are in Z catalysis and reactor design theory. However, there is a compelling drive (especially with the infiltration of computers into chemical engineering) to let the students know that it may sometimes be possible to dispense with a trialand error design procedure and employ an appropriate sequential technique that utilizes only the math ematical methods with which they are familiar (the principle of vertical and lateral organization in a curriculumdevelopment model). The design proce dure suggested here assumes that the undergradu ate student has taken (or is currently taking) courses in elementary calculus, vectors and matrices, and introductory programming techniques. All of these requirements are easily met by the average third year chemical engineering student at the University of New South Wales. DESIGN METHODOLOGY Consider a train of N CSTRs whose inlet and exit conversions (with respect to reactant A) are Xo and XN respectively. The design problem is to find the (Nl) intermediate optimal conversions, X, X2 . XN1 which will minimize the overall reactor size. Following Levenspiel,[51 the optimal selection of the conversion, Xi, (in the ith tank) on the l/(rA) vs X plot is such that the diagonal of the rectangle must possess the same slope as the tan gent to the curve at the point Xi. Thus for the 1st reactor f(X2)f(X) df(X) X1Xo dX (1 where f(X) = 1/(rA) and 1 means "evaluated at X." Rearranging, we have W1 = f(X2) f(XJ)(X1 XO) 0 (X)= O (2a) In general, for the ith reactor, we obtain jit reactor Wi =f(X,+1)f(Xi)(XiX ,_l) (Xi)=o (2b) Copyright ChE Division ofASEE 1992 Chemical Engineering Education (Nl)' reactor wN =f(XN)f(XN)(XN1 XN2) D (XN)= 0 (2c) The simultaneous solution of these equations yields the unknowns X,, X2, . XN, whence the optimal size (V/FA0)i for the ith reactor is (V/FA )i=(XiXi_)f(Xi) i=1,2,...N (3) and D(X) = df(X)/dX. Using the NewtonRaphson method for solving the nonlinear system described by Eqs. (2), we find (m) =x(m X(m1 __J l(x(m1))w((m1)) m=t2,... (4) where Xm) is the column vector of conversions at the mth iteration and is written X(m)=X[m) X(m) ...X(m) (5) m T1 2 N1 ) with X(o) being the guessed initial conversion vector and the Jacobian matrix at the (ml)th iteration, J(X(m1)) has dimension (Nl) x (Nl). Because Eq. (4) has quadratic convergent property, the true solution is obtained after very few iterations. An effective TABLE 1 Computer Program c This program calculates the optimal conversion in each tank c in a train of isothermal cstr's c The program was executed on an IBM compatible 386 machine c using WATFOR77 compiler c ********************SOME HELPFUL HINTS************ ****** C c Reaction example: r kl*CA k2*CR (1st order reversible rxn) c c Declaring program variables c c The dimension of the variables should be at least, ntanks + 2 c where ntanks = number of tanks in the train c c real xinit(22), a(22), c(22), d(22), x(22), w(22) c c c The rate here is, r caO [kl*(lxx)k2*xx] c c Note that this program can handle any form of rate expression c no matter how complex provided it is twice differentiable. c Fortunately, this is always the case even for complex c LangmuirHinshelwood and MichelisMenten type kinetics. c c Defining the statement functions for the rate expression, its c reciprocal and the first two derivatives w.r.t conversion, xx c c *************************************************************** c rate(xx)=ca0*(constl*(l.0xx)const2*xx) rateinv(xx)l=.0/rate(xx) der(xx)=(constl+const2)/(caO*(constl*(1.0xx)const2*xx)**2.0) dder(xx)=2.0*(constl+const2)**2.0/(caO*(constl*(l.0xx)const2*xx *)**3.0) c c open the input and output data files c input data file = cstr.dat c output data file = cstr.res c c open(unit=l, file= 'cstr.dat') open(unit=2. file= 'cstr.res') read (1, *) ntanks, (xinit(i),i=2,ntanks),constl, const2, xO *xf, caO, tol c write(2, 100) ntanks, caO 100 format(//,10x,'RESULTS OF THE REACTOR TRAIN DESIGN',//,15x, *'number of tanks = ',i4,//,15x,'concentration of reactant, A. ir *feed = ',f6.2,lx,'mol./lit') c nntanks1 xinit(l)=x0 xinit(ntanks+l)=xf c kounta is the iteration counter until tolerance limit is met kounta=0 10 continue kounta=kounta+l c calculating elements of the principal diagonal, d(i); c the lower diagonal elements, a(i), c the upper diagonal elements, c(i) and c the vector, w(i) do 20 i=l,n d(i)(xinit(i) xinit(i+l))*dder(xinit(i+l))2.0*der(xinit(i+l)) w(i)=ratinv(xinit(i+2))ratinv(xinit(i+1))(xinit(i+l)xinit(i))* *der(xinit(i+l)) 20 continue nn=n1 do 30 j=l,nn a(j)=der(xinit(j+l)) c(j)=der(xinit(j+2)) 30 continue call soln(n,a,d,c,w,x) sum = 0.0 do 40 i=l,n sum=sum+(abs(x(i)))**2.0 40 continue c c reinitialising the interstate conversions c do 50 j=l,n xinit(j+l)=xinit(j+l)x(3) 50 continue c c determining the average percentage conversion difference, apcd, c in successive iterations apcd=sqrt(sum/float(n)) if(apcd .gt. tol) go to 10 continue write(2, 300) 300 format(//,5x,'final results satisfying tolerance limit',//, *5x,'conversion in reactor train') write(2, 400) (xinit(i), i=2,ntanks), xO, xf, kounta 400 format(//, 5(10x,el2.4/),//,10x,'inlet conversion = ',e12.4 *//,10x,'train exit conversion = ',e12.4,//,10x, *'required number of iterations to achieve tolerance = ',i5,//, */////////,5x,'EXHIBIT 2: COMPUTER PROGRAM OUTPUT FOR 1ST ORDER RE *VERSIBLE KINETICS') close(unit=l) close(unit=2) stop end c c subroutine for calculating the solution, y=JINV*b c where p(i), q(i), and r(i) are the lower, principal and upper c diagonal elements respectively of the matrix J whose inverse c is JINV. The dimension of J is 'norder'. c subroutine soln(norder,p,q,r,b,y) dimension p(norder), q(norder), r(norder), b(norder), y(norder) do 2 i=2,norder ymult=p(il)/q(il) q(i)=q(i)ymult*r(i1) b(i)=b(i) ymult*b(il) 2 continue y(norder)=b(norder)/q(norder) do 3 i=norder 1, 1 y(i)=(b(i)r(i)*y(i+l))/q(i) 3 continue return end Summer 1992 1 i convergence criterion is yX(m)X(mi) 2 i ~ %I x(m) Si < Additionally, the Jacobian matrix is easily shown to be tridiagonal since the elements are given as aw.i X aq q=1,2,...(i2) (7a) W = (Xi 1 Xi) a(X) 2 (Xi) "KiJ axxi(X i) awi Xi1)= (Xi) awi Xi+ =N(Xi+l) s^9 aw. _ ax pX, p=i+2,i+3,...(N1) As such, the Jacobian matrix is C C C C (i 8c c c Vw1 aw )X1 aX2 w2 w2 w2 )x1 ax2 ax3 0 aW 3 3Wa ax2 8Xs 0 0 0 0 aw, aw awi 0 aWi oWi aWi aXi_1 axi axi+1 0 WN1 aWN1 aXN2 8Nl As shown in the text by Cheney and Kincaid,17' it is unnecessary to carry out the inversion required by Eq. (4) at every iteration for the tridiagonal matrix J. In fact, the vector, ym), is easily computed from simple operations between the tridiagonal elements of the Jacobian matrix and the vector, w. This simple algorithm was utilized as a subroutine subroutinee SOLN) in the accompanying program. Clearly, this method is completely independent of the reaction kinetics in question; the only requirement is that the rate expression be differentiable with respect to the conversion. To demonstrate the utility of this method, we provide the following two illustrative examples. Example 1 A Reversible 1st Order Reaction Suppose the reaction A<R occurs in a cascade of six CSTRs isothermally. For 1st order kinetics in both A and R we have rA=kCA k R (9) which rewrites as rA=CA [k,(1X)kX] (10) in terms of conversion X. The results from the com puter program (see Table 1) based on the method described here using k1 = 0.196 min1 and k2 = 2.124 x 103 min1 with 1 mol/lit of pure A in the feed is as shown in Table 2. As expected, the number of itera tions needed (usually less than six) is not affected by the values of initial conversion guesses (if each is less than unity). Changing the number of tanks from TABLE 2 Computer Program Output for 1st Order Reversible Kinetics RESULTS OF THE REACTOR TRAIN DESIGN Number of tanks = 6 Concentration of reactant, A, in feed = 1.00 mol./lit final results satisfying tolerance limit conversion in reactor train 0.2302E+00 0.4069E+00 0.5439E+00 0.6519E+00 0.7342E+00 inlet conversion = 0.0000E+00 train exit conversion = 0.8000E+00 required number of iterations to achieve tolerance = 3 Chemical Engineering Education TABLE 3 Computer Program Output for LH Kinetics RESULTS OF THE REACTOR TRAIN DESIGN Number of tanks = 6 Concentration of reactant, A, in feed = 1.00 mol.lit final results satisfying tolerance limit conversion in reactor train 0.2272E+00 0.4028E+00 0.5399E+00 0.6488E+00 0.7326E+00 inlet conversion = 0.0000E+00 train exit conversion = 0.8000E+00 required number of iterations to achieve tolerance = 3 six to as high as twenty neither influenced the rate of convergence nor significantly the time for pro gram execution on an IBM compatible 386 machine using a WATFOR77 compiler. Example 2 Complex LangmuirHinshelwood Kinetics This example shows that the proposed method is robust with respect to the kinetics of the reaction. The Ptcatalyzed oxidation of CO in an isothermal CSTR is described byt81 kCco rA +KC2 (11) [1+KCo 2 The program was used to compute the optimal interstate conversions for the same number of tanks. The results are shown in Table 3, along with the reactor parameters. Again, in spite of the nature of the rate expression, the conversions were obtained in only three iterations. Considerably more complex rate equations (involving, for example, more than one reactant and product) could be handled since the concentration of each species may be related to the conversion, X, as TABLE 4 Equations for Top, for a Class of Reversible Type of Reaction Topt 1 aA <rR r=k1CA1 k2C01 E2 E1 A 2R A 2E 2A10" (OR VR S 2 2 Ao Rg n A1Ei(1X)a 2. aA + bB < rR r = aB kC1 E2 E1 r = kCCal2 k2 RE (al+a2) A B k2 R CP1(1+2) R in A2 2 Ao Rg AE1(1 X)"a (0 3. aA+bB<>rR+sS kCaC a2 k2CPIC2 E 2 r=kC1 A CB 2 RS A2 2 RgA AE A1E(1X) 4. aA<>rR+sS r=kCc kCCi2 2AE E2El R in 2 2 Ao (OR S e A1E9(1 Summer 1992 Ci= CA (0i +ViX) DESIGN FOR A CASCADE OF NONISOTHERMAL CSTRs In certain situations, especially for highly exo thermic reactions, it may be desirable to let each reactor in the train operate at a different tempera ture. Thus, it is important to find the optimum tem perature, T for each tank which will simulta neously yiel the optimum conversion and hence the overall minimum reactor size. To carry out the de sign for a nonisothermal train of CSTRs, the pro gram can be suitably modified so that the kinetic rate constants which were previously supplied as data to the program can now be computed in situ along the optimum temperature progression (OTP) path. Table 4 contains the equations for the opti mum temperature, Topt, for a class of reversible reac tions frequently encountered in process design (see reference 9 for detailed derivation of equations for T ). The following is a modification of the algo rithm for this situation. 1. Supply an initial vector of interstate conversions, X(O1, X(O) =[X) X() ...X(O) T Reactions 2. Evaluate the optimum tem peratures, Tpt, corresponding to the conversion in each tank using relations from Table 4 for the ap propriate kinetics. 3. Compute the kinetic constants X)oil k, and k, for each reactor at the optimum temperatures Topt1, Topt, . ToptNl) 4. Evaluate the elements of the Jacobian matrix and hence X"(m us ing subroutine SOLN. 1 5. Use the new X(m" to estimate 0R + RX) new optimum temperatures as in B + VX)a2 Step 2. Follow Steps 3 and 4. 6. Continue Step 5 until con vergence criterion for the conver sion is met. The corresponding Top, E are the design optimum tempera R +v X) "(0s +vSX) 2 tures for the train of CSTRs. ) It may be recognized that, in S(0, + ) B practice, the equations in Table 4 are actually quite easy to use since the orders of the reactions a's and p's rarely exceed two, v RX)(0S +v X), while the parameters Oi and vi are usually given by the feed )j specification and the reaction stoichiometry. CONCLUDING REMARKS A method for the design of a cascade of CSTRs isothermall and nonisothermal) has been proffered. The required level of mathematical rigor appears suitable for undergraduate instruction in optimal reactor design. Also, the procedure is particularly amenable, even at that level, for computer coding. NOMENCLATURE A,A, = frequency factors in the Arrhenius relation C &.s = concentrations of species A, B, R, S respec tively, mol/lit E1,E2 = activation energy in the forward and back ward directions respectively, J/mol FAO = feed molar flow rate, mol/min J = Jacobian matrix k1,k2 = rate constants in the forward and backward directions respectively r = rate of reaction, mol/lit. min R = universal gas constant, J/mol K T = temperature, K (subscripts are obvious from text) V = reactor volume, lit X = fractional conversion a. = reaction order w.r.t. reactant i pj = reaction order w.r.t. product 0BAss = feed concentration ratio of species A, B, R, and S to that of A VA.Rs = stoichiometric coefficient ratio of species A, B, R, and S to that of A By convention v is negative for reactants and positive for products. Consequently, vA = 1, vB = b/a, and vR = r/a. Similarly, 8A = 1 and 0B = CB/CAO, etc. REFERENCES 1. Denbigh, K.G., Chemical Reactor Theory: An Introduction, 2nd ed., Cambridge University Press (1971) 2. Aris, R., The Optimal Design of Chemical Reactors: A Study in Dynamic Programming, Academic Press, New York, NY (1961) 3. Chen, N.H., Process Reactor Design, Allyn and Bacon, Inc., Boston, MA (1983) 4. Chitra, S.P., and R. Govind, "Synthesis of Optimal Serial Reactor Structures for Homogeneous Reactions: I & II," AIChE J., 31,177 (1985) 5. Levenspiel, O., Chemical Reactor Omnibook, Oregon State University Press (1984) 6. Fogler, H.S., Elements of Chemical Kinetics and Reactor Calculations, PrenticeHall, Englewood Cliffs, NJ (1974) 7. Cheney, W., and D. Kincaid, Numerical Mathematics and Computing, Brooks and Cole Publishing Company (1985) 8. Carberry, J.J., Chemical and Catalytic Reaction Engineer ing, McGrawHill, New York, NY (1976) 9. Omoleye, J.A., A.A. Adesina, and E.O. Udegbunam, "Opti mal Design of NonIsothermal Reactors: Derivation of Equa tions for the RateTemteratureConversion Profile and the Optimum Temperature Progression for a General Class of Reversible Reactions," Chem. Eng. Comm., 79, 95 (1989) 0 book review AN INTRODUCTION TO NUMERICAL METHODS FOR CHEMICAL ENGINEERS by James B. Riggs Texas Tech University Press, Lubbock, TX 794091037; 460 pages (includes Solutions Manual)(1988) Reviewed by R. Narayanan University of Florida This book is meant primarily for undergraduate students in chemical engineering. It could be used by other engineering students even though the physical connection of the majority of the examples is related to chemical engineering. It is aimed at students who have some background in calculus and some differen tial equationsthe student would typically be in the junior or senior year of chemical engineering. I found the book well structured. It deals with ma trix operations and inversion with clarity and with minimum confusion. The examples from stagewise operations are delightful, and the chapter on single nonlinear equations is well presented. However, some elementary derivations on sufficient conditions for con vergence of the methods and the errors would have been possible but were (unfortunately) omitted. The section on multiple nonlinear equations was adequate, but the geometric interpretation of Newton's method was needlessly confusing. Also, the chapter on polynomial approximations and integration could have been strengthened by inclusion of error bounds, and a section on Richardson's extrapolation method should be included in any future edition. I liked the chapters on ordinary and partial differ ential equations and the subsequent treatment on boundary value problems. The chemical engineering examples were particularly good. One of the most useful chapters for students, I feel, deals with linear and nonlinear regression. In short, I feel that the book is thoughtfully written. Its main weakness is that it lacks some theoretical background (which can be provided by an instructor without much ado). Its strengths are the apt chemical engineering examples that are provided, and in this regard, I find it a suitable alternative to other well established textbooks. I found the level appropriate for our juniorlevel students and feel it can be taught without essential knowledge of the main chemical en gineering courses. It is published by a relatively un known press and does not appear to have received the publicity it deserves. O Chemical Engineering Education 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 Ameri can 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. Specific 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 Manuscripts of less than twelve doublespaced typewritten pages in length will be given priority over longer ones. Consult recent issues for general style. 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. Ifyou 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. 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