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Cornell University ( PDF ) Classical solution thermodynamics: A retrospective view ( PDF ) The nature of adjoint variables and their role in optimal problems ( PDF ) Semiconductor chemical reactor engineering and photovoltaic unit operations ( PDF ) The use of computer graphics to teach thermodynamic phase diagrams ( PDF ) Book reviews ( PDF ) An improved design of a simple tubular reactor experiment ( PDF ) The B. C. (before computers) and A.D. of equilibriumstage operations ( PDF ) 
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cmia eg"i ti on heC ackndwledge4 and d~lk 0.... 3M FOUNDATION CHEMICAL ENGINEERING EDUCATION wito a do alio o/ jaunds. EDITORIAL AND BUSINESS ADDRESS Department of Chemical Engineering University of Florida Gainesville, Florida 32611 Editor: Ray Fahien (904) 3920857 Consulting Editor: Mack Tyner Managing Editor: Carole C. Yocum (904) 3920861 Publications Board and Regional Advertising Representatives: Chairman: Lee C. Eagleton Pennsylvania State University Past Chairman: Klaus D. Timmerhaus University of Colorado SOUTH: Homer F. Johnson University of Tennessee Jack R. Hopper Lamar University James Fair University of Texas Gary Poehlezn Georgia Tech CENTRAL: Robert F. Anderson UOP Process Division Lowell B. Koppel Purdue University WEST: William B. Krantz University of Colorado C. Judson King University of California Berkeley Frederick H. Shair California Institute of Technology NORTHEAST: Angelo J. Perna New Jersey Institute of Technology Stuart W. Churchill University of Pennsylvania Raymond Baddour M.I.T. A. W. Westerberg CarnegieMellon University NORTHWEST: Charles Sleicher University of Washington CANADA: Leslie W. Shemilt McMaster University LIBRARY REPRESENTATIVE Thomas W. Weber State University of New York Chemical VOLUME XIX Engineering NUMBER 2 Education SPRING 1985 4UAa'Sd Iectwte. 72 Semiconductor Chemical Reactor Engineer ing and Photovoltaic Unit Operations, T. W. F. Russell The Educator 54 Joe Hightower of Rice University, Joyce Taber Department of Chemical Engineering 58 Cornell University, Julian C. Smith and Paul H. Steen Classroom 68 The Nature of Adjoint Variables and Their Role in Optimal Problems, O. K. Crosser 78 The Use of Computer Graphics to Teach Thermodynamic Phase Diagrams, C. D. Naik, Paulette Clancy, and Keith Gubbins Laboratory 84 An Improved Design of a Simple Tubular Reactor Experiment, A. A. Asfour Lecture 62 Classical Solution Thermodynamics: A Retrospective View, H. C. Van Ness and M. M. Abbott 88 The B. C. (Before Computers) and A. D. of EquilibriumStage Operations, J. D. Seader 67 Books Received 71 Positions Available 82,83 Book Reviews CHEMICAL ENGINEERING EDUCATION is published quarterly by Chemical Engineering Division, American Society for Engineering Education. The publication is edited at the Chemical Engineering Department, University of Florida. Secondclass postage is paid at Gainesville, Florida, and at DeLeon Springs, Florida. Correspondence regarding editorial matter, circulation and changes of address should be addressed to the Editor at Gainesville, Florida 32611. Advertising rates and information are available from the advertising representatives. Plates and other advertising material may be sent directly to the printer: E. O. Painter Printing Co., P. O. Box 877, DeLeon Springs, Florida 32028. Subscription rate U.S., Canada, and Mexico is $20 per year, $15 per year mailed to members of AIChE and of the ChE Division of ASEE. Bulk subscription rates to ChE faculty on request. Write for prices on individual back copies. Copyright 1985 Chemical Engineering Division of American Society for Engineering Education. The statements and opinions expressed in this periodical are those of the writers and not necessarily those of the ChE Division of the ASEE which body assumes no responsibility for them. Defective copies replaced if notified within 120 days. The International Organization for Standardization has assigned the code US ISSN 00092479 for the identification of this periodical. SPRING 1985 W educator of Rice University JOYCE TABER Rice University Houston, TX 77251 t 'VE BEEN DELIGHTED to be where I am," says Dr. Joe Hightower in regard to his decision 17 years ago to become a chemical engineer and an educator as well. Joe Hightower, of the chemical engineering department at Rice University in Houston, says he started out like many other members of engi neering faculties: "I started as a child by taking things apartbicycles, motorcycles, clocks, every thing!" As early as the fourth grade he proceed ed to take his clarinet apart and to rebuild it shortly after he began taking music lessons. Then throughout high school, in addition to becoming an accomplished musician as a member of the all state band, he made a veritable career out of re pairing the instruments of the other band mem bers. While he was in high school Joe also decided to study chemistry. Later he obtained his masters and PhD in chemistry from Johns Hopkins but couldn't decide if he wanted to do industrial re search or academic work. It was during a three year stint at the Mellon Institute that he decided to teach. "I found that I enjoyed the interaction with the students, the stimulation of the faculty, and the flexibility of the job." He found he had to make another decision, howeverwhether to go into a department of chemistry or chemical engineering. "All my edu cational background was in chemistry, but I had a philosophical desire to work on things that have very practical uses," he says. However, chemistry departments were moving in the direction of quantum mechanics and other more esoteric areas while chemical engineering was moving from unit operations into engineering science. Thus, chemical engineering embraced catalysis, and Joe Hightower embraced engineering. It happened at that time that heterogeneous catalysis, the research area in which Joe was interested, was a field that had been explored primarily in chemistry departments. However, chemistry departments were moving in the di rection of quantum mechanics and other more esoteric areas while chemical engineering was moving from unit operations into engineering science. Thus, chemical engineering embraced catalysis, and Joe Hightower embraced engineer ing. Since then, Joe's research philosophy has been directed toward providing new insights into how existing catalysts work as opposed to discovering new catalysts. "We try to ask the question 'Why?' " he says. Using his chemical training, Joe has worked at gaining information about the chemical nature and concentration of active sites, the in fluence of solid state parameters in determining activity and selectivity, and the mechanisms of re actions that occur over solids that are of interest to the petroleum and petrochemical industries. He and his students have extensively used isotopic Copyright ChE Division, ASEE, 1985 CHEMICAL ENGINEERING EDUCATION tracers (both stable and radioactive) to study the kinetics, reaction networks, rate limiting steps, and incorporation of surface species into product molecules. (Some of his work has been sum marized in an earlier issue of this journal, Vol. XVI, No. 4, p. 148, Fall 1982). A few of the catalyst systems that his group has studied in clude cracking catalysts, auto emission control catalysts, partial oxidation catalysts, and zeolites. But Joe's research does not stop at the gradu ate level. Several years ago he incorporated some research techniques into a sophisticated under graduate experiment. While taking his kinetics and reactor design course, junior level students now investigate all the kinetic parameters for cu mene dealkylation over a silicaalumina cracking catalyst, explore the reaction mechanism with deuterium tracers and a mass spectrometer, and determine the surface area of the material. From their results the students are able to calculate the true surface reaction rate constant, the concentra tion of active sites, the turnover frequency, and the role of intraparticle diffusion on the kinetics (Chem. Eng. Educ., p.118, Summer, 1969). This experiment allows the students to determine ex perimentally many of the parameters that are useful in scaling up such reactions from labora tory to plant size. Joe's research has led him into other situations which he has especially enjoyed. In the early 70's, for example, he was chairman of several National Academy of Science panels which were assigned the task of assessing the feasibility of using catalytic converters to decrease pollutants from automobiles. "No one had ever applied catalysts in this way, and we were assigned the task of de termining if these devices would in fact work," he says. "It was fascinating. There was a lot of secrecy. No company would tell us directly what they were doing, but they would tell us what they thought the other companies were doing, and we had to try to put together a clear picture. Once I was asked to testify before the House of Repre sentatives Committee on Science and Technology which was chaired by Representative George Brown of California. The congressman from Detroit would say things like, 'I want you to know that people from my district are being put out of jobs because of government regulation and control.' Brown would respond, 'People in my district in California are dying because we don't have enough controls, and pollutants are killing people.' How can one give an objective testimony in an atmosphere like that!" Involvement in professional societies has been another rewarding part of Joe's career. He is cur rently chairman of the 24member Petroleum Re search Fund Advisory Board, a group that is re sponsible for a corpus of $150 million. This year the foundation will donate $11 million for uni versity research in petroleumrelated areas. In 1971 he received the National Award in Petroleum Chemistry from the American Chemical Society. Joe finds time to enjoy simple, relaxing activities, such as blowing glass in his lab. As a councilor for the American Chemical So ciety, he represents the southeastern Texas local section. He has been on the national research committee and is on the national awards commit tee of the American Institute of Chemical Engi neers. He has also served as chairman of the petroleum chemistry division of the American Chemical Society. Dr. Hightower has over 50 publications to his credit and is also very much involved in presenting short courses for industry. "Catalysis," he says, "is a field that is not taught in many universities as an area of specialization. Yet, 80 or 90% of all commercially important chemical reactions are catalytic reactions. People are trained as organic chemists, physical chemists, or chemical engi neers and then learn about catalysis on the job. This creates a great demand for the types of short courses that we instituted at Rice years ago and that are being continued in cooperation with other schools such as the University of Houston." It would appear that Joe's day would have to last more than 24 hours in order to accomplish his many activities. Yet, there is still another part of his life that is as important to him as his pro SPRING 1985 . . there is still another part of his life that is as important to him as his professional work. In 1968 he was a leader in establishing the Human Resources Development Foundation . (which) provides free temporary housing for needy families who come for treatment to the Texas Medical Center hospitals. fessional work. In 1968 he was a leader in es tablishing the Human Resources Development Foundation. The foundation provides free tempor ary housing for needy families who come for treatment to the Texas Medical Center hospitals. The foundation started as a project at Dr. High tower's church and has expanded to serve over 700 families from over 38 states and 26 foreign countries in the last 16 years. Joe is president of the foundation whose facilities have grown from an old army barracks into 15 beautiful apart ments. He heads a group of approximately 40 dedi cated volunteers who minister daily to the needs of families who are under enormous stress. "All a person needs to move in are pajamas and a toothbrush!" Joe laughs. But his statement Human Resources Development Foundation apartments. is very accurate. The apartments are furnished down to the pots and pans. A local church even provides meat once a week for the residents. Social workers, ministers, and even former residents refer potential patients. Selection is made on a firstcome, firstserved basis without regard to race, creed, sex, religion, age, or level of disability. Need is the sole criterion used to determine eligi bility. Residents are allowed to stay for up to three months. "The project is valued at over a halfmillion dollars, and most of it has been given because of something Joe has done," states Marge Norman, VicePresident of the foundation. "Joe never hesi tates to go speak to a group if there is some chance that they might have an interest in the founda tion. His work takes him to the far reaches of the U.S. and overseas; on every airplane trip he takes he makes sure his seat partner is very well acquainted with his pet project, and it often leads to very good things for this foundation." Senator Orrin Hatch was one of the latest people to hear about the foundation because of one of Joe's "airplane contacts." Joe sat next to a member of Senator Hatch's staff on one flight, and later he received a letter from the senator, who had been informed by the staff member about the foundation. The letter commended Dr. High tower for his charitable work. Continues Mrs. Norman, "Even though we have a foundation board which is functioning very well, without Joe I don't know if the Board would have been as effective or if this place would have become what it is today. He loves it so much, and he works so hard. A couple of times each year we have a work day when all our volunteers come to clean, repair, and paint. All kinds of people are represented in the workers. Joe is always the first here and the last to leave, working at anything that needs to be done. Even before he comes over, he gets up before dawn and bakes bread so that at 10 a.m. we can have hot bread and butter and coffee. We've been acquiring land to build more apartments next door, so you can be sure he's on the campaign trail again! We have parties for our residents, and again, Joe is always here with bread he has baked. He even brings his mandolin and plays and sings. There's not much Joe can't do!" On any given day the facility may house people from Florida, or from various towns in Texas, Columbia, or Indiana. There are no class dis tinctions. Last winter a brain surgeon from Main land China was allowed to leave his country with only $200 when he came to care for his quadri plegic daughter in the medical center. As a resi dent of the foundation's apartments, he scrubbed floors and took a lot of goodnatured ribbing when he painted an outside door with interior paint. Something all residents share, however, is grati tude to Hightower, who personally greets each newcomer with a loaf of bread, and gratitude to the foundation he helped establish. CHEMICAL ENGINEERING EDUCATION Wrote one resident, "What a tremendous help this facility has been to us. Each of us is faced with an extremely serious medical problem, and the expenses are staggering. To be sure, the financial savings are important, but even more, we have a place to call 'home' and people with whom we can talk as friends. I'm one of the lucky ones. Two weeks ago I had a kidney transplant, and now I am hoping and praying that my body will not reject it. I have been so impressed by the consistent care and visits that I've had from the jogger (Dr. Hightower) and his friends even during my recovery from surgery. I hope in some way I can repay the favors done for me. Right now, though, I'm going to sit back and enjoy an other slice of the hot bread Joe Hightower has brought me before it has time to lose its flavor!" In his professional life also, colleagues have only good things to say about Joe Hightower. Dr. T. W. Leland who was department chairman when Joe came to Rice says, "I was impressed with him right from the start, and I had a great interest in getting him to come to Rice. I think it's been a firstrate choice. He's done a remarkably good job over the years. He's an excellent teacher and has had an active research career. He is well thought of by his students and has perfected a graduate course in kinetics and catalysis to a high degree. He has been effective in giving short courses and he is outstanding in his volunteer public service. Personally, he is outgoing, friendly, and extreme ly wellorganized. He has excellent rapport with people in all walks of life, from the top of the technical ladder in terms of ability to students not doing too well in their courses. He's a remarkable individual who has been a great addition to our department." Joe is modest in describing his daily activi ties. "I just enjoy it all," he says. He gets up at 4:45 every morning to jog and share breakfast and a Bible reading with his wife Ann, a chemical engineer who works for the Exxon Chemical Company. By 6:30 a.m. he is at work, doing all the things he loves to do. "From the very be ginning I couldn't make up my mind about what I wanted to do. I wanted something in both in dustry and the academic world. Now I have both, and I'm grateful for that. I probably border on getting involved in more things than I should .. but they're all so interesting! I guess I just like being where I can interact with people and see them grow, whether it's at the university or whether it's with people who are hurting." E SPRING 1985 TRANSPORT PROCESSES MOMENTUM, HEAT AND MASS Christie J. Geankoplis University of Minnesota 1983 casebound 350pages This text takes a unified approach to basic transport processes: Geankoplis points out the similarities of basic equations and calculation methods, and the differences which occur in the actual physical processes. Each chapter of this classtested text is divided into elementary sections, followed by more sophisticated Selected Topics allowing you to expand or focus your course according to the needs of your students and the time limits of your course. SI Units are used throughout, with important equa tions and tables in dual units. Over 150 example problems and more than 340 homework problems emphasize applications as well as theory. This text offers complete coverage of more essentials than any other book you'll find. Look for these new topics: diffusion in solids and porous solids, bioengi neering transport, nonNewtonian fluids, numerical methods for steady and unsteadystate diffusion and conduction, design and scaleup of agitation systems, mass transport, and an introduction to engineering principles. TRANSPORT PROCESSES AND UNIT OPERATIONS SECOND EDITION Christie J. Geankoplis University of Minnesota 1983 casebound 650 pages This fullyrevised Second Edition includes TRANSPORT PROCESSES: MOMENTUM, HEAT AND MASS as Part One of the text, plus the unit operations (Part Two) so essential to chemical engineers. TRANSPORT PROCESSES AND UNIT OPERATIONS, Second Edition offers an optimal balance of theory and application. Geankoplis emphasizes the diversity of practical applications in chemical, ceramic, mechanical, civil, food process, and bioengineering. Over 220 example problems and 500 homework problems illustrate both theory and applications. The Second Edition features new sections on multi component distillation, unit operations of membrane processes, nonNewtonian fluids, diffusion in solids, porous solids and biosystems, freezing, freeze drying, and sterilization of biomaterials. For examination copies or more information on these two titles, write to Ray Short, Engineering Editor, Allyn and Bacon, Inc., 7 Wells Avenue, Newton, MA 02159. Allyn and Bacon, Inc. College Division Olin Hall from the west. no department  CHEAT CORNELL UNIVERSITY JULIAN C. SMITH AND PAUL H. STEEN Cornell University Ithaca, NY 14853 F OUNDED IN 1868 AND with a long tradition in engineering, Cornell is almost unique in being both private and statesupported; about half the divisions, including engineering, are privately en dowed while the other half are funded by the State of New York. An awkward arrangement, it would seem, but it works surprisingly well. Cornell, a medium sized university with a total enrollment of some 18,000 students, is set on a hill overlooking the city of Ithaca, and the waters of Cayuga Lake, the largest of the Finger Lakes. Ithaca is small but strongly cosmopolitan. The setting is semirural; the scenery is beautiful; the air is clean. Almost the only drawbacks are a modicum of cold gray weather on occasion, and some possible difficulties in travelling in and out. Ithaca has been called "the most centrally iso lated city in the Northeast," but as a graduate student from Greece recently remarked, "If it wasn't for the weather, Ithaca would be Paradise!" The School of Chemical Engineering has 18 faculty members, about 100 undergraduate students (3rd and 4th years only), and over 65 graduate students. During the past twelve years research activity and expenditures have greatly increased, and strong research programs have been established in fluid mechanics, polymers, sur face science and catalysis, thermodynamics, and biochemical engineering. The number and quality of MS and PhD candidates (especially PhD's) have risen rapidly. The growth in research, how ever, has not reduced the traditional concern for undergraduate and professional graduate teach ing. All faculty members are expected to teach undergraduate courses, and many participate in advanced design projects in the professional Master's program. The school occupies its own building (shared with a few other tenants) with CHEMICAL ENGINEERING EDUCATION Copyright ChE Division, ASEE, 1985 a total area of some 90,000 square feet, 54,000 of which is exclusively chemical engineering. A BRIEF HISTORY At Cornell, as at many institutions, chemical engineering began in the Chemistry Department, but its development was somewhat unusual. Very early (before 1900) courses were offered in in dustrial chemistry which had a considerable practical flavor; as taught by Fred H. "Dusty" Rhodes in the 1920's they dealt with the principles and practice of chemical engineering. By 1930 Dusty had established both undergraduate and graduate programs in chemical engineering, but because of rivalries between Chemistry and the Engineering College the undergraduate program had to be a 5yearlong hybrid: four years in Arts and Sciences (leading to the degree Bachelor of Chemistry) followed by one year in engineering (for the degree Chemical Engineer). In 1938 the department with its three faculty members became part of the Engineering College and the 5year program led to the degree Bachelor of Chemical Engineering. In 1942 chemical engi neering moved to Olin Hall, the first building on what was to become engineering's new quadrangle. It was at a considerable distance from chemistry and the old ties quickly weakened. After World War II all the undergraduate engi neering programs at Cornell were lengthened to five years. This lasted until 1965, when the present 4year BS programs, including that in chemical engineering, were established. Dusty Rhodes was director of the school until he retired in 1957 and Charles C. "Chuck" Wind ing took over. Ken Bischoff, now at the University of Delaware, was director from 1970 to 1975; Julian Smith from 1975 to 1983; and Keith Gubbins from 1983 to date. For many years chemical engineering at Cornell was known for its strong undergraduate program. Rhodes felt that good teaching was the most important thing required of a faculty mem ber and while ability to do research should be considered in reviews for promotion or tenure, it should not be a major factor. This is not to say that there was no research or graduate work. Be tween 1932 and 1970 the school awarded 140 MS and 104 PhD degrees, and many of the recipients have had distinguished careers in industry and academia, including John Prausnitz (Berkeley), Ed Lightfoot (Wisconsin), and a string of past or present heads of chemical engineering depart ments: Bob Coughlin (UCONN) ; Howard Greene (Akron) ; Deran Hanesian (NJIT) ; Will Kranich (Worcester Poly) ; Larry McIntire (Rice) ; Steve Rosen (Toledo); Julian Smith (Cornell); Tom Weber (SUNY Buffalo) ; and Jacques Zakin (Ohio State). Bob Finn's pioneering work in biochemi cal engineering was begun in the 1950's. Dusty's policies set a pattern for the school which persisted until the early 1970's. By then it was clear that the research effort had to be greatly The Fred H. Rhodes Student Lounge, redecorated through a gift from Joseph Coors, '40. expanded. Beginning in 1970, new faculty mem bers were added who developed, or brought with them, strong research programs in several areas. This attracted additional researchminded faculty and increasingly stronger graduate students. During an 8year period research expenditures in creased by a factor of six. The number of gradu ate students has risen to 67; more significantly, a majority of them are now PhD candidates. And collaboration with Chemistry and other depart ments of the university is once again close and extensive. RESEARCH GROWTH From 1976 through 1982, the annual research expenditures, in dollars per faculty member, climbed at a rate that was second highest and reached a level that was fourth highest among all chemical engineering departments in the country.* Total sponsored research costs for 198384 were over 1.5 million dollars, for an average of $94,000 *Journal of Engineering Education, March issues, 1976 1983. SPRING 1985 per fulltime faculty member. This is especially striking since only about a third of the faculty was responsible for 75% of the total expenditures. As the current younger faculty develop their pro grams and additional researchminded faculty re place retiring senior professors, the overall re search program should continue its strong ex pansion. Biochemical Engineering Biochemical engineering research has grown from Bob Finn's early studies of microbes and ... many of the PhD recipients have had distinguished careers in industry and academia, including John Prausnitz (Berkeley), Ed Lightfoot (Wisconsin), and a string of past or present heads of chemical engineering departments. microbial populations. The goal was, and is, to develop new and more efficient biochemical con versions. One project seeks to find economical ways of producing ethanol from pentose sugars, a second to develop better treatment methods for wastes containing pentachlorophenol (PCP), and a third to exploit an unusual bacterium which can rapidly ferment arabinose. Mike Shuler shares several specific interests with Bob Finn including the treatment of waste water by specialized microorganisms. Mike's di verse interests are tied together by a view of the living cell as a "catalyst" waiting to be used in chemical reactors. His research embraces studies of plantcell tissue culture, reactors with solid substrates (e.g. mold growth on solid surfaces), photobioreactors, biofilm formation, and the con tinuous protein production from bacteria with re combinant DNA. Particularly noteworthy have been his group's experimental demonstration of the feasibility of hollowfiber membrane units for entrapment of microbial populations (necessary groundwork for the development of hollowfiber reactors) and the construction of a mathematical model of the organism Escherichic coli. Doug Clark, who joined the faculty in 1984, brings the point of view that enzymes rather than the whole cell can be the building blocks for bio chemical reactors. He is studying how the im mobilization, or attachment to a foreign matrix support, affects the structure and function of an enzyme. A related interest is the transport of bio logical macromolecules through porous media; this transport is an essential step in enzyme immobili zation, gel permeation and affinity chromatogra phy, and ultrafiltration. In collaboration with Bill Street, Doug has initiated a study of methane producing bacteria which live at extreme tempera tures and pressures in deepsea hydrothermal vents. Polymers and Materials Science For a million circuit elements to fit on a tiny silicon chip linewidths must be on the order of a tenth of a micron. In one technique for achieving such precisionelectronbeam lithographythe silicon surface is covered with a polymer film polymethyll methacrylate, for example), then ir radiated by an electron beam creating a pattern of soluble polymer. The soluble polymer is washed away leaving a precision mask and the chip is ready for the final step, silicon modification. Ferdinand (Rod) Rodriguez is directing an inter departmental program on polymers for advanced lithography, to improve the performance of the polymer "resist" used in the masking process. This is a good example of Rod's research on polymeric materials which has the broad goal of understand ing the processes of polymerization and gelation (crosslinking) and degradation (chain scission) in order to produce better materials. Claude Cohen uses macromolecular science to interpret the physical properties of polymer systems and to understand the structures that develop during industrial processing. On the fundamental level, predictions from models of macromolecules are used to understand rheo logical and lightscattering behavior, with ex periments to complement the theoretical work and to test the adequacy of the models. On the applied level, the orientation of glass fibers in composite thermoplastics during the molding process is being investigated. This work is in conjunction with an interdepartmental program on injection molding. Surface Chemistry, Catalysis and Reactor Engineering The surface chemistry and physics of hetero geneous systems which have direct technological application is the central concern of Bob Merrill's studies. Examples are CO oxidation on noble metals (automobile exhaust converters), the de composition of hydrazine (rocket monopropel lant), the oxidation of aluminum (catalyst sup CHEMICAL ENGINEERING EDUCATION port technology, corrosion protection, and elec tronic insulators in microcircuitry) and hydrode sulfurization catalysis (sulfur removal from pe troleum). On the one hand, Bob's group answers practical questions; on the other they are develop ing and sharpening several types of analytical tools. These include the use of lasers in surface chemistry, the use of synchrotron radiation (EXAFS) to study the dynamics of gassolid re actions, and the use of spectroscopy in real catalyst systems (highsurfacearea configurations and high pressures). Peter Harriott studies the influence of mass transfer, heat transfer and mixing on the per formance of chemical reactors as well as the kinetics of reactions in heterogeneous systems. One project concerns the regeneration of catalysts used in the pyrolysis and gasification of coal. An other examines the heat and mass transfer and the overall kinetics in limeslurry droplets used in the "dry scrubbing" of SO, from flue gas; the goal is to pin down the ratelimiting step and im prove the design of commercial units. Joe Cocchetto's recent work on catalytic re action kinetics has concentrated on the fuel cell. By controlling the structure of a porous electrode, a better understanding of the interplay between transport and reaction has been gained and tech niques for improving efficiency have emerged. Joe returned to industry in early 1985. Bob Von Berg is interested in the use of gamma radiation in various chemical processes: ammonia synthesis and the reaction of hydrocarbons and liquid nitrogen. Bob has also collaborated with Herb Wiegandt on a longterm project involving the desalination of water by freezing, as described later. Fluid Dynamics and Stability: Rheology Bill Olbricht concentrates on problems in fluid mechanics and rheology with applications in en hanced oil recovery, biomedical fluid mechanics, and the production of semiconductor materials. He is studying the lowReynoldsnumber motion and coalescence of immiscible drops in tubes of various geometries (characteristic of porous media) for critical evaluation of methods for en hanced oil recovery. In the biomedical area, in conjunction with the University of Rochester Medical School, he is modelling the motion of red blood cells in microcapillaries to predict the dis tribution of these cells within tissue. A third area Cornell's Chemical Engineering Faculty, 1984. Back row: Shuler, Finn, Scheele, Steen, Smith, Winding. Middle row: Zollweg (Research Associate), Cocchetto, Harriott, Jolls (Visiting from Iowa State), Von Berg, Rodriguez. Front row: Olbricht, Merrill, Clark, Wie gandt, Thorpe, Clancy, Gubbins, Streett. of research examines the momentum, heat, and mass transfer involved in silicon film growth by chemical vapor deposition with the aim of pre dicting rates of film growth in lowpressure deposition reactors. Paul Steen, who joined the faculty in 1982, studies fluid motions and their stabilities. Buoy ancydriven convection patterns, generated in fluidsaturated porous media, are examined as prototypes of fluid motions susceptible to transi tions in which strong nonlinear effects are domin ant. This work involves the development of tools in applied mathematics. In another area, motions induced at fluid/fluid interfaces due to tempera ture gradients (thermocapillary effects) are being investigated by experiment, with relevance to the floatzone crystalgrowth process and the break up of thick films. George Scheele's study of liquidliquid immis cible systems focuses on the coalescence of drop lets and the breakup of jetsboth at relatively high Reynolds numbers. He also has interests in the computer simulation of chemical processes, particularly in computer graphics. Molecular Thermodynamics and Computer Simulation Keith Gubbins and Bill Streett have coordinat ed their efforts towards understanding, predicting, Continued on page 103. SPRING 1985 P lecture CLASSICAL SOLUTION THERMODYNAMICS A Retrospective View H. C. VAN NESS AND M. M. ABBOTT Rensselaer Polytechnic Institute Troy, NY 12181 T HE PRIMARY VARIABLES of classical thermo dynamics for fluid systems are temperature T, pressure P, and the molar properties volume V, internal energy U, and entropy S. Temperature is a primitive, having no definition in terms of anything simpler. Pressure and molar volume are defined directly by three other primitives: force, mass, and length. These primitivestemperature, force, mass, and lengthare subject to direct sensory perception, and we have little difficulty accepting them as meaningful. Internal energy and entropy, however, are primitives not associated with direct detection by the senses. Nor are they directly measurable; we have no energy meters, no entropy meters. Energy and entropy are mental constructs which have meaning only as mathematical functions. Accepting this, we then need to know what they are functions of. We find by experiment that the molar volume of a homogeneous phase is a function of its temperature, pressure, and composition. Generaliz ing, we postulate that the molar internal energy and entropy of a homogeneous phase are likewise functions of temperature, pressure, and com position. When this is true, the first and second laws lead to a fundamental property relation among the primary thermodynamic variables d(nU) = Td(nS) Pd(nV) + Zidni (1) The ni are mole numbers of the species present, Nor are they directly measurable; we have no energy meters, no entropy meters. Energy and entropy are mental constructs which have meaning only as mathematical functions. Copyright ChE Division, ASEE, 1985 H. C. Van Ness is Distinguished Research Professor of Chemical Engineering at Rensselaer Polytechnic Institute, where he has been a faculty member since 1956. He is coauthor with J. M. Smith of Introduction to Chemical Engineering Thermodynamics and has co authored a number of research papers on thermodynamics with M. M. Abbott, in addition to two books, Schaum's Outline of Theory and Problems of Thermodynamics and (with M. W. Zemansky as a third coauthor) Basic Engineering Thermodynamics. (L) Michael M. Abbott is Associate Professor of Chemical Engineering at Rensselaer Polytechnic Institute, where he has been a faculty member since 1969. Prior to that he was employed by Esso Research and Engineering. His teaching interests are mainly in the thermal sciences and in chemical process design. He is coauthor with H. C. Van Ness of the books Schaum's Outline of Theory and Problems of Thermodynamics and Basic Engineering Thermodynamics (with M. W. Zemansky). (R) n = Ini is the total number of moles, and the [i are chemical potentials. Written for n = 1, Eq. (1) becomes dU = TdS PdV + J.dx. showing that U = U(S,V,x) Thus, in general, the natural independent (canoni cal) variables for U are entropy, volume, and composition. New thermodynamic properties can be defined that are functions of alternative sets of inde pendent variables. In particular, the enthalpy H and the Gibbs function G are defined as CHEMICAL ENGINEERING EDUCATION H E U + PV (2) and G H TS (3) Then nG = nU + P(nV) T(nS) and d(nG) = d(nU) + Pd(nV) + (nV)dP Td(nS) (nS)dT Substitution for d (nU) by Eq. (1) gives d(nG) = (nS)dT + (nV)dP + ZJ.dn. (4) This equation is equivalent to Eq. (1), and repre sents an alternative fundamental property rela tion. Written for one mole of material, it becomes dG = SdT + VdP + lZidxi (5) whence G = G(T,P,x) Because temperature, pressure, and composition are subject to direct measurement and control, the Gibbs function is a defined thermodynamic property of great potential utility. An equation such as Eq. (4) is too general for direct practical application. Its value is in storing much information. Thus, we write by in spection s =  pl (6) V (7) F T,x and i = n (8) ST,P,n. where the subscript nj indicates that all mole numbers are held constant except ni. Application of Eqs. (6) through (8) presumes knowledge of G as a function of T, P, and x; given this, then Eqs. (6) and (7) yield S and V as functions of T, P, and x. Other properties come from defining equations; for example, by Eq. (3) H= G + TS Thus, if we know how G is related to its canonical variables, we can by simple mathematical opera tions evaluate all the other thermodynamic properties; given G = G (T,P,x), we can also find S, V, ,it H, Cp, etc. as functions of temperature, pressure, and composition. All this is the legacy of J. W. Gibbs and in principle nothing more is needed. An expression giving G = G (T,P,x) is an example of a canonical equation of state. Such an equation serves as a generating function for the other thermodynamic properties, and implicitly represents complete property information. For realfluid mixtures, canonical equations of state are unknown. The problem is that such an equation must be based on experimental data. Un fortunately, there are no G meters and no con venient experimental measurements that lead easily to values of G. Without a canonical equation of state, we can make no direct practical use of a fundamental property relation. The slow evolu tion of solution thermodynamics since Gibbs' time has led to new formulations that relate much more directly to experiment. Our purpose here is The slow evolution of solution thermodynamics... has led to new formulations that relate much more directly to experiment. Our purpose is to rationalize the structure of modern solution thermodynamics. to rationalize the structure of modern solution thermodynamics. In the early years of this century, G. N. Lewis introduced several concepts basic to all subse quent developments: the partial property, the fugacity, and the ideal solution. A partial property is defined by the equation =. (nM) (10) i n. (10) 1 T,P,nj where M is the molar value of any extensive property. The simplest interpretation of Eq. (10) is that it apportions a mixture property among the constituent chemical species. Thus, MR has the characteristics of the property of species i in the mixture. Indeed, a mathematical consequence of Eq. (10) is the relation M = iX which shows that the partial properties combine in the simplest rational way to yield the mixture property. We see by comparison of Eq. (8) with Eq. (10) that Ui = G SPRING 1985 Thus, the chemical potential is identified with partial Gibbs function. The fugacity is an auxiliary thermodyna property related to the Gibbs function. Thus, a mixture, the fugacity f is defined by the e tions dG = RT d in f (const T,x) the i,: dGa = RT d in (xiP) (const T) IIIlU The actual properties of a fluid may be com for pared with the properties the fluid would have as lua an ideal gas at the same temperature, pressure, and composition. The comparison by subtraction (13) gives rise to residual properties. Thus, by defini tion f lim = 1 (14) P.0 For the special case of pure species i, these become and dG = RT d in fi (const T) lim 1 P0 For species i as a constituent of a mixture, the fugacity ?. is defined by the equations 1. MR M M' MR E M. Mi (26) Applying this concept to the Gibbs function, we subtract Eq. (19) from Eq. (13) d(G 6') = RT d in P dG = RT d in p (const T,x) (const T,x) (const T) and i lim P 1 (18) P0O i For an idealgas mixture one replaces V in Eq. (5) by RT/P; then dG' = RT d in P (const T,x) where the prime (') denotes an idealgas proper From Gibbs' theorem for such mixtures, we I G' = xiGi + RT xi in xi where 0 is the fugacity coefficient, defined as f Integration of Eq. (27) gives Integration of Eq. (27) gives G = RT in i (19) The integration constant vanishes, because for P = 0, GR = 0 by assumption and In 4) = 0 by rty. Eq. (14). For the special case of pure species i, have this becomes (20) R G. = RT in n 1i By Eqs. (6), (7), and (9), we get For species i as a constituent of a mixture, we subtract Eq. (24) from Eq. (17) S' = xiS R Axi in xi V, = zxiv! f. d(Gi GI) = RT d in i .P R dGi = RT d n i H' = xiH! L 1 (const T) (const T) Each of these is implicit in Eq. (20). Moreover, Eq. (8) yields S= G = G + RT in x whence dG' = dG' + RT d in xi x (const T) By Eq. (19) written for pure species i, this be comes where i. is the fugacity coefficient of species i in the mixture, defined as $  1 X.P Integration of Eq. (31) gives .R = RT n i 3. i CHEMICAL ENGINEERING EDUCATION dG. = RT d in f. 1 3. Unlike a formulation based on a canonical equation of state, the residualproperty formulation cannot provide complete property information. One needs in addition the heat capacities necessary for evaluation of properties for the idealgas state. where again the integration constant vanishes. An alternative form of Eq. (4) derives from the mathematical identity ) 1 nG d nG= d(nG)  dT RJ RT RT 2 Substituting for d (nG) by Eq. (4) and for G by Eq. (3) gives d dT + dP dn (34) R 2I R I T RT2 For the idealgas state, Eq. (34) becomes 'd = dT + W dP + dn. (35) RT RT2 RT RT With ti replaced by Gi in Eq. (34) and p' replaced by G: in Eq. (35), we subtract these two equa tions: SR d nG dT + nV dP + dni (36) where the definitions of Eqs. (25) and (26) have been invoked. This is the fundamental property relation for residual properties. In view of Eq. (33), it may also be written R R VR d T = 2 dT + C dP + in i. dn (37) Working relations for the residual properties can now be written by inspection H ( ( JRT) RT aT p,x  (a Zn 4 I aT )Px VR f (GRT ,x a ki RT P JTx P JT,x in n= ( /RT) n Oi @ani T,P,nj Sa(n an ,, I 3ni T,P,n. an3 where the second form in each case follows from Eq. (29). Equation (39) may be written vR d In =, dP RT (const T,x) where by definition VR = V V' = V RT P Values of VR come directly from experimental PVTx data, and Eq. (41) then allows calculation of In 0; Eqs. (38) through (40) yield other properties of interest. This close link to experi ment is the major reason for a residualproperty formulation of solution thermodynamics. Given a PVT equation of state V = V(T,P,x) we can evaluate all residual properties. Because of its direct relation to experiment, a PVT equa tion of state is far more easily developed than is a canonical equation of state. Furthermore, the principle of corresponding states allows the generalization of PVT data and the development of generalized correlations for the residual proper ties, thus greatly extending the usefulness of available experimental data. Unlike a formulation based on a canonical equation of state, the residual property formulation cannot provide complete property information. One needs in addition the heat capacities necessary for evaluation of properties for the idealgas state. In principle, PVT equations of state apply equally to gases and to liquids. In practice, how ever, the accurate representation of liquid proper ties proves much more difficult. Thus, an alterna tive formulation of solution thermodynamics has developed for liquids. The key idea is that of an ideal solution. By definition fid x.f (42) 1 l 1 where the superscript id denotes an idealsolution property. Expressions for all of the properties of an ideal solution follow from this equation. Integration of Eq. (17) from the state of pure i to the state of i in solution at the same T and P gives SPRING 1985 I fo G Gi = RT n (43) For an ideal solution, this becomes id Gi = G + RT n x (44) and by Eq. (11) Gid = xii + RT Equations (6), (7), and (9) in this case yield Sid = xiSi Rxi n xi (46) Vid= xIVI (47) and Hid = xiHi (48) Just as we may compare the actual properties of a fluid with its idealgasstate properties, so may we compare the actual properties of a fluid mixture with its idealsolution properties at the same temperature, pressure, and composition. Thus, we have definitions of excess properties quite analogous to those for residual properties ME =M Mid E id M. =M. M. 1 1 1 Equation (49) applied in turn to the properties whose idealsolution expressions are given by Eqs. (45) through (48) becomes GE = G Zx.Gi RTx.i An x. (51) SE = S x.iSi + Rjxi in xi (52) VE = V xiVi (53) HE = H xiHi (54) The excess properties are closely related to property changes of mixing AM = M xi.M. (55) These quantities measure the changes that occur when one mole of mixture is formed from the pure constituent species by a mixing process at constant T and P. The definition of Eq. (55) allows Eqs. (51) through (54) to be written GE = AG RTyx. in x. E SE = AS + Rx.i n x. VE V HE = AH Thus, the excess properties are readily calculated from property changes of mixing and vice versa. Interest in property changes of mixing is focused on AV and AH, because these quantities can be experimentally determined by direct measurement. Unfortunately, measurements of AV = VE and of AH = HE for liquid mixtures do not allow calculation of GE. For this, we need vapor/liquid equilibrium data, which are related to GE as follows. Subtraction of Eq. (44) from Eq. (43) gives X. E id i Gi = Gi G = RT nf 1 1 1 x.f. i3 or E = (60) in Yi RT where the activity coefficient yi is defined by f Yi (61) 1ii In accord with Eq. (11) H E GE GE RT= x RT and by Eq. (60) this becomes GE R = x.i n yi (62) Values of yi are calculated from experimental vapor/liquidequilibrium measurements by the equation yiPDi Yi sat (63) ii Here, Ii is a secondary factor of order unity that can be readily evaluated from volumetric data for the equilibrium phases or from correla tions of such data. The fundamental property relation for the excess properties follows from Eq. (34). For an ideal solution, this equation is written CHEMICAL ENGINEERING EDUCATION TABLE 1 Summary of key equations rnc nH nV i d nj = dT d + T dP + dn FRTJ RT2 RT RT InC I II nVT d = dT + nR dP RT j RT2 T + I in i.dni 1 1 (34) (37) d nGrE nHE nVE d dT + __T dP + L n idni (66) RT2 RT R21 G Pi R GE G i E RT ^i RT J in" y, SnGid nlid nVid d (R n I dT + dP RT2 RI id + I dni RT i With ii replaced by G. in Eq. (34) and ~d replaced id by Gi in Eq. (64), we subtract these two equa EiE E nHE E 6 d = n dT + + IdP + dn. (65) RT2 T RT i tions where the definitions of Eqs. (49) and (50) have been invoked. In view of Eq. (60), this equa tion may also be written n _nHE nVE d Ii nH dT + n dP + I n y dn (66) RT2 RT 1 i Equation (66) is analogous to both Eqs. (34) and (37); analogous to Eqs. (38) through (40), we have HE E S= T (/RT)1 (67) RT ^ T JP,x VE (G/RT) (68) RT [ P J T,x n (nGE/RT) T (69) A i ni T,P,n ; SPRING 1985 The formulation of solution thermodynamics through excess properties derives its usefulness from the fact that HE, VE, and yi can all be found by experiment. This relative abundance of experi mental entries provides alternative measurements that yield property data. However, the excess property formulation provides even lesscomplete property information than the residualproperty formulation, because it tells us nothing about the properties of the pure chemical species. In Table 1, we bring together for comparison the parent fundamental property relation for the Gibbs function and the two analogous property relations which follow from it for the residual and excess Gibbs functions. Included as well are the equations which relate the three mixture Gibbs functions to their respective partial properties. These are particular applications of Eq. (11). [ 10 books received I Fundamentals of Chemistry, Second Edition, James E. Brady, John R. Holum; John Wiley & Sons, Inc., New York; $34.95 (1984) Handbook of Powder Science and Technology, Edited by M. E. Fayed and L. Otten; Van Nostrand Reinhold, 135 West 50th Street, New York, 10020; 850 pages, $79.50 (1984) Analytical Pyrolysis: Techniques and Applications, Edited by Kent J. Voorhees; Butterworths, 80 Montvale Ave., Stoneham, MA 02180; 486 pages, $69.95 (1984) Heat and Mass Transfer in Rotating Machinery, Darryl E. Metzger, Naim H. Afgan; Hemisphere Publishing Co., 79 Madison Ave., New York 10016; 713 pages, $74.50 Cheaper, Safer Plants or Wealth and Safety at Work, Trevor A. Kletz; Institution of Chemical Engineers, 165 171 Railway Terrace, Rugby, England; (1984) Engineering Information Resources, Margaret T. Schenk and James K. Webster; Marcel Dekker, Inc., New York 10016; 232 pages, $24.75 (1984) From Technical Professional to Corporate Manager; A Guide to Career Transition, David E. Dougherty; John Wiley & Sons, Somerset, NJ 08873; 279 pages, $19.95 (1984) Natural Product Chemistry: A Mechanistic and Biosyn thetic Approach to Secondary Metabolism, Kurt B. G. Torssell; John Wiley & Sons, Inc., Somerset, NJ 08873; 401 pages, $24.95 (1984) The Wiley Engineer's Desk Reference, Sanford I. Heisler; John Wiley & Sons, New York 10158; 567 pages, $34.95; (1984) Laboratory Manual of Experiments in Process Control, Editor, Ch. Durgaprasada Rao; ChE Education Develop ment Center, Indian Institute of Technology, Madras 600 036 India, $20 (1984) classroom THE NATURE OF ADJOINT VARIABLES AND THEIR ROLE IN OPTIMAL PROBLEMS 0. K. CROSSER University of MissouriRolla Rolla, MO 65401 A DJOINT VARIABLES ARE frequently arbitrarily introduced into the textbook discussion of op timal or extremal theory. For example, Bryson and Ho [1] "adjoin" them to the optimization prob lem, Denn [2] introduces them as a "convenience," and Leitman [11] regards them as a transforma tion to a "useful" vector space basis. Only Jackson [10] has shown that they are desirable as a general transformation from one set of variables which appear naturally during the formulation of the problem to the set of interest in the solution search problems. Adjoint variables are the sensitivity coefficients in optimal search problems. Adjoint variables exist because the coefficient matrix of every system (of describing equations) has a transpose, and there are, therefore, two independent solutions to the homogeneous form of the system. However, it was the late Professor F. M. J. Horn who in 1958 most directly presented the fundamental nature of the adjoint variables and their role in optimal reactors in chemical engi neering [5]. The original papers [6, 7, 8] and the more complete elaboration in his thesis were in German, with results published in English [9] by 1967. Publications about Pontryagin's Principle [8, 10] became the standard literature reference, and the directness of Horn's approach became less available for the beginning student to appreciate. Furthermore, this appreciation or understanding of adjoint variables makes much of Horn's later work in optimal chemical reactorseffect of by passing, cyclical operation of nonlinear process esmuch easier to follow. This demonstration makes use of the example presented in detail in appendix 1.11 of his Thesis [5]. One asks for the optimal temperature profile for a plug flow reactor with several independent chemical reactions. The set of independent chemi cal reactions is any set of the smallest number of time dependent stoichiometric equations sufficient to define all reaction compositions uniquely in time. The proper interpretation of independent is also clearly discussed in this thesis, although a more formidable presentation is now available [12]. We suppose a tubular plug flow reactor with several chemical reactions and arbitrary kinetics (Arrhenius) Xe 0 dx./dz = Vi(xlx2 ..."m, T), 1 1..x ) (i = 1 to m) O. K. Crosser received his PhD from Rice University in 1955 and is currently professor of chemical engineering at the University of MissouriRolla. His primary interests include optimization of pre liminary plant design and fixed bed separations. and presume that the objective function has the form M=M(xle,x ,...xme) = M(xe) (2) M depends only upon the exit composition (ex tents) x. and a straightforward solution to the problem would be to assume a temperature pro file, calculate the exit composition vector to give Copyright ChE Division, ASEE, 1985 CHEMICAL ENGINEERING EDUCATION M, then presume another temperature profile and continue to adapt the profile in some beneficial way until an extreme in M was obtained. Suppose we had two such solutions (we use x for the vector of extents of the independent reactions and V for the vector of reaction rates, and the super dot to imply differentiation with respect to z) x + xl = V(xl,T1) and x2 x2 = V(x2,T2) (3) so that for sufficiently small differences between T, and T, a first difference (perturbation) is sufficient. Then x2 xI = [3Vi/3x.](x2 x1) + (3Vi/ T)(T2 T1) or y = [3Vi/ax.]y + (aVi/aT) T (4a) where y stands for the perturbation in x caused by the perturbation T in T. We will also have the perturbed response m aM M = M(x2) M(Xl) = a Yk(Ze) k=1 ke = (M/axie)' y (Ze) (4b) Since both x, and x, are zero at z equal zero, y(0) is zero. Note that the matrix, [aVi /xj] and vector, (aVi/aT) are functions of z only, so that the system of Eqs. (4a,b) is a set of linear differential equations in which the coefficients are functions only of the independent variable z. y is the re sponse of the system to T What we desire is to solve Eqs. (4a,b) subject to the restriction that M be an extreme, so that it is necessary that dM = (aM/x. e)' dx = 0 (5) (aM/axie)' is the transpose of the vector of partial derivatives of M with respect to xie, that is, x at the end of the reactor. A system like (4) is usually solved by Variation of Parameters (Boyce & DePrima [3] or Hochstadt [4]), finding first the homogeneous (complementary) solutions. The form of these solutions is more conveniently manipulated if we use the solution matrix rather than the solution vector (in contrast with usual forms for systems with constant coefficients). Therefore Y = [B] yC C c where B.. = 3V./ax. ij 1 J has the homogeneous solution matrix [Y] such that [Y] = [B][Y] (7) and we take [Y(ze)] to be [I] the identity matrix. Any other boundary condition on [Y] may be ob tained directly from this one. Using the Variation of Parameters we suppose y = [Y]c and hope to find the vector c to fit the inhomogeneous part, which is the second term of Eq. (4) y = ([Y]c) = [Y]c + [Y]c = [B][Y]c + (aVi/3T)T (8) Substituting from Eq. (6) we have 1 [Y]c = (aVi/aT) T c = [Y] (V.i/aT) T (9) and we see that the vector of the particular solu tions c is directly related to the temperature pro file, T Now, these functions depend only on z, and we intend to keep the same inlet temperature but to alter the shape of the profile. Therefore any one of the particular solutions c must have the property z e Ck() = ck(z)dz 0 and since y = [Y]c and [Y] with ck(O) = 0 = [I] at z = z e y(ze) = c(z ) Then from Eq. 4b, using t as a dummy variable and recalling that (DM/Dxie) does not depend on z f= m M dt = k=l 0 (aM/aXke)ck(e) S(M/axke )ck(t) dt k=l M = (3M/aXke)k(Ze) = (aM/3aie)'c k=1 (Note that M is the derivative, with respect to the independent variable of the response M to the perturbation T .) We now have to solve simultaneously m + 1 linear equations involving c There are m inde pendent chemical reactions, and Eq. (11) for M Since these equations must be linearly de SPRING 1985 Thus, the differential equations for the adjoint variables and their corresponding conditions at the end of the reactor show that the influence of the exit extents upon the value of the objective function can be obtained for any entering conditions to the reactor by integrating their adjoint variable differential equations from the end of the reactor to the entrance. pendent, their determinant must be zero M (av /aT)T (WM/xie)' =0 [Y] i u' or = 0 w [Y] using vectors u and w for notational convenience, then S 0 u' M JYJ + Y = 0 w [Y] M/ax1 aM/ax2 (e 0 1 l(Ze) = o 3M/axi 3M/8x2 2(Z e)= 1 0 Hence 0 u' and M= IYl A(ze) = Ii = u = (aM/xid ) As we expand the numerator determinant of Eq. (12) first about w, the first column (deleting the ith row in Y) and then about u' in the first row (deleting the jth column in Y), we will obtain the cofactors of the elements Yj, in Y, which we label aij and Eq. (12) can be written i uy w'[a ]u w[Adj(Y')]u M Y i = w'A (13) IYI IY1 IYI because [aij] is the adjoint matrix of the transpose of [Y], (a sign change occurs as the i + 1 index in the determinant decreases to the i index for w). The adjoint variables, A, are defined by Eq. (13) and 1 S= [Y']lu [Y']X u Since S= (aM/xie) = (for the extreme in M) {[Y'IX} = 0 = [Y']x + [Y']J = [y'] 1 'l,] But [Y] = [B][Y] + [Y']X = {[B][Y]}' [Y'][B'] then S= [Y']'[Y'][B'] and A = [B']X These are the differential equations for the adjoint variables. The boundary conditions of [Y] = [I] at z = Ze imply (for m = 2 for clarity) because Iij is the unit ij cofactor from the identity matrix. Thus, the differential equations for the adjoint variables and their corresponding conditions at the end of the reactor show that the influence of the exit extents upon the value of the ob jective function can be obtained for any enter ing conditions to the reactor by integrating their adjoint variable differential equations from the end of the reactor to the entrance. These functions, therefore, explain how the optimal re sult is affected by changing the values of the ex tents of reaction at any point along the reactor such as the entrance. Since there is a direct cor respondence between length in a plug flow re actor and time, it is equally clear how the adjoint variables apply to time optimization as well. The adjoint variables are therefore nothing more than the additional homogeneous solution for the linear perturbation. Had the problem been cast in the form of time optimal control, they would have indicated the switching functions; in troduced with an Hamiltonian or Lagrange multi plier problem, they would have been the cor responding multipliers [13]. The thing to see is that all of these structures rely essentially only on a Cramer's rule for solving a dependent set of linear equations and that the adjoint variables appear naturally as the added homogeneous solutions to the transpose of the system coefficient matrix, and they show how temperature changes along the reactor affect the objective function, which depends on the con version at the exit from the reactor. CHEMICAL ENGINEERING EDUCATION ACKNOWLEDGMENTS One of the original reviewers of this article observed that the subject of this paper is contained in modern control theory texts. It is a pleasure to suggest to students that Linear Systems by Thomas Kailath (PrenticeHall 1980) is an ex cellent reference with good examples and exercises. The most directly relevant part is section 9.1 pp 598606 and example 9.13 p. 605, but there are many other items of interest throughout the entire text. The University of MissouriRolla awarded the Faculty Sabbatical during which this note was written. D. W. T. Rippin and his Systems Engi neering Group of ETH Zurich provided the affectionate welcome and gentle scholarly support. Don MacElroy offered a most helpful suggestion toward the end of the work.  POSITIONS AVAILABLE  Use CEE's reasonable rates to advertise. Minimum rate % page $60; each additional column inch $25. MICHIGAN STATE UNIVERSITY Chemical Engineering .. Tenure system faculty positions. Doctorate in Chemical Engineering or closely related field. A strong commitment to teaching and the ability to de velop a quality research program is expected. Preference will be given to candidates with research interests in the areas of Biochemical Engineering, Surface Science, Solid State Phenomena, or Polymeric Materials. However, ap plicants with outstanding credentials and research interests in other fields related to Chemical Engineering are en couraged to apply. Teaching and/or industrial experience desirable but not essential. Michigan State University is an affirmative actionequal opportunity employer and wel comes applications from women and minority groups. Send applications and names of references to Chairperson, Faculty Search Committee, Department of Chemical Engi neering, Michigan State University, East Lansing, Michi gan 488241226. LITERATURE CITED 1. Bryson, A. E., Y. Ho, Applied Optimal Control, Halsted Press, New York (1975) [chapter 2, and pp 4750, pp 149150]. 2. Denn, M., Optimization by Variational Methods, McGrawHill, New York (1969), [pp 102109]. 3. DiPrima, R. C., W. E. Boyce, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, New York (1977). 4. Hochstadt, H., Differential Equations, Dover Press, New York (1975). 5. Horn, R. "Optimalprobleme bei kontinuierlichen chemischen Prozessen," Thesis, Tech. Hochsch. Wien, Ostereich (1958). 6. Horn, F., Discussion of "Optimum Temperature Se quences in Reactors," K. G. Denbigh, Chem. Eng. Sci., Special Supplement, 8, 131 (1958). 7. Horn, F., U. Troltenier, "uber den Optimalen Temperatur verlauf im Reaktionsrohr," Chem. Ing. Tech, 3S, 382 (1960). 8. Horn, G., "Adjungtierte Variable und Maximum prinzip in der Theorie Chemischer Reaktoren," Ostereichische ChemikerZeitung, 6, 186 (1967). 9. Horn, F., "Mathematical Models in the Design and Development of Chemical Reactors," Ber. Buns. Ges., 74, 8189, (1970) (in English). 10. Jackson, R., "Optimization of Chemical Reactors With Respect to Flow Configuration," J. Theo. App. Opt., 2, 240259 (1968). 11. Leitman, G., An Introduction to Optimal Control, McGrawHill, New York (1966), [p 26, Chapter 2]. 12. Smith, W. R., R. W. Missen, Chemical Reaction Equilibrium Analysis, John Wiley & Sons, New York (1982), [Chapter 2also see the reference by these authors in Chem. Eng. Educ. which contains a re view of the literature to 1976]. 13. Report: SEG/R/128(83), Systems Engineering Group, Tech. Chem. Labor, ETH, Zurich. NOMENCLATURE I I Determinant [ ] Matrix (square) ( ) Vector (column) ( )' Transposed vector (row) z Length of plug flow reactor x Extent of reaction V Vector of reaction rates y Perturbation in x T Temperature M Objective function to be optimized m Number of independent chemical reactions B Coefficient matrix from partial derivatives of rates V Y Matrix of homogeneous solutions to Eq. 4a I Identity matrix c Vector of particular solutions for Eq. 4a u' Row vector (aM/axie)' w Column vector (aVi/aT)T X Vector of adjoint variables Subscripts o e i,j,k Entrance to reactor Exit from reactor Row column indexes Superscripts i Index to independent chemical reactions (1 to m) S Differentiation with respect to length 1 Inverse matrix Transpose  Indicates perturbation value SPRING 1985 4waEd .2eceOND SEMICONDUCTOR CHEMICAL REACTOR ENGINEERING The Chemical Engineering Division Award Lecturer for 1984 is T. W. Fraser Russell. The 3M Company provides financial support for this an nual lectureship award. The lecture has been presented at the University of Florida, N the University of Michigan, and Colorado School of Mines. A native of Moose Jaw, Canada, Fraser Russell re ceived his BSc and MSc in chemical engineering from the University of Alberta and his PhD from the University of Delaware. He joined the department at Delaware in 1964 and is presently the Allan P. Colburn Professor of Chemical Engineering. Prior to beginning his academic career, he spent two years with the Research Council of Alberta, where he did early development work with the Athabasca Tar Sands. He later joined Union Carbide Canada as a design engineer, where he completed the reactor and process design for all of Union Carbide's ethylene oxide derived chemicals. His innovative process design for these oxide derivative units became the first multipurpose continuous processing unit built in Canada. In research, Russell's efforts have been directed into two major areas: design of gasliquid systems; and semi conductor chemical reaction engineering and photovoltaic unit operations. His research in gasliquid system design has resulted in over 25 publications which have been widely used by industrial concerns and have led to improved design of gasliquid contractors and reactors and biological waste treatment systems. Russell is recognized as a respected and inspiring teach er and has received the University of Delaware's Excellence in Teaching Award. His efforts in education have resulted in the publication of two texts, Introduction to Chemical Engineering Analysis with Morton M. Denn, and The Structure of the Chemical Process Industries with J. Wei and M. Swartzlander. In the research and development of thinfilm photo voltaic cells, his efforts have centered on the need to apply chemical reaction engineering principles to improve the design and operation of reactors used in making the semi conductor material, and to ensure that solar cells developed at the laboratory scale can be manufactured in commercial quantities. He carries out this research as Director of the Institute of Energy Conversion, a laboratory with a staff of some fifty people devoted to the development of thin film photovoltaic cells. T. W. F. RUSSELL Institute of Energy Conversion University of Delaware Newark, DE 19716 THE QUANTITATIVE ANALYSIS of a reactor pro ducing semiconductor film can be termed semi conductor chemical reactor engineering if the analysis creates procedures which improve the de sign, operation, and product quality of laboratory or larger scale reactors. The creation of a thin film semiconductor, or indeed any thin film, re quires an understanding of both molecular and transport phenomena. The process is analogous to that encountered in a typical catalytic reacting system (Fig. 1). Molecules must reach the surface of a substrate, adsorb on the substrate, diffuse and/or react on the substrate to produce a film possessing specified material and electronic properties. In a catalytic system, the product de sorbs, leaving the substrate for the surrounding fluid phase. A variety of reactors have been employed to move molecules or atoms to the substrate but much can be learned by considering two general types: Physical vapor deposition reactors Chemical vapor deposition reactors In a physical vapor deposition reactor the re quired solid or liquid phase species are placed in a source. Energy is supplied to vaporize these species causing molecular beams to impinge on the sub strate [1]. In a chemical vapor deposition reactor the molecular species are continuously supplied in a vapor phase which flows over the substrate. A SURFACE ADSORPTION SURFACE DIFFUSION SUBSTRATE SURFACE REACTION FILM OR CRYSTAL GROWTH FIGURE 1. Surface molecular phenomena. CHEMICAL ENGINEERING EDUCATION 0 Copyright ChE Division, ASEE, 1985 kND PHOTOVOLTAIC UNIT OPERATIONS quantitative understanding of transfer from the bulk vapor to the surface is required and it may be necessary to contend with complex reactions in the vapor phase [2]. The performance of a reactor which produces a semiconductor film is judged by the quality of the film produced. Much effort is being devoted to ascertaining film quality by measuring optical and Electronic & Optical Properties Material Properties Design & Operation of Reacting System FIGURE 2. Simplified logic diagram. electronic properties but film quality is ultimately determined by the performance of the semicon ductor in some type of electronic device. A success ful semiconductor chemical reaction and reactor analysis should provide experimentally verified models linking the electronic properties of the film to the design and operation of the reactor through a detailed understanding of the material proper ties of the film and the mechanism of film growth (Fig. 2). The logical sequence summarized in Fig. 2 has been followed by the integrated silicon circuit com munity of researchers and industrial practitioners in dealing with the key step in integrated circuit manufacture of dopant diffusion into a film. It has not been a trivial task and well over two decades of effort have gone into the development of models relating device performance to doping concentra tion profile and doping concentration profile to the design and operation of the furnace [3]. Growing a polycrystalline or amorphous film, predictably, with the desired electronic properties is an even more difficult task; one which remains an active integrated circuit research area today. If one is interested in applications which could require millions of square meters per year of semi conductor film, the task of effectively predicting film growth becomes an order of magnitude more complex. Semiconductor films covering an area on the order of a square meter or more are needed for photovoltaic panels for power generation electrophotography electronic devices for thinfilm displays For these largearea applications it is necessary to carry out research on a scale between that used in a typical laboratory and that required for com mercial operations. This unit operations scale re search needs to have both a theoretical and experi mental component which builds upon the labora tory scale research. The position of chemical re actor engineering and unit operations scale experi mentation in the research logic is shown in Fig. 3. I will illustrate the application of semicon ductor chemical reactor engineering with research we have underway in physical vapor deposition of CdS at both the laboratory and unit operations LABORATORYSCALE EXPERIMENTATION MATERIAL DEVICE DESIGN DEVELOPMENT AND ANALYSIS T T CHEMICAL REACTOR ENGINEERING ANALYSIS MATHEMATICAL UNIT OPERATIONS DESCRIPTION EXPERIMENTATION PROCESSING EQUIPMENT DESIGN COMMERCIALSCALE MANUFACTURE FIGURE 3. Role of chemical reactor analysis. SPRING 1985 We originally became interested in the semiconductor research because of a need to design larger scale reacting systems. However, the last five years of research has taught us that the chemical engineering analysis is very useful in the laboratory scale research effort, and indeed essential, if such research is to be done efficiently and with minimum expense (a key issue with today's research costs). bell jar heater box radiant heater i ffoil substrate substrate holder S top heat shield Sheat shield source bottle I 1 "^ tantalum heater current lead and support base plate thermocouple thickness monitor FIGURE 4. Physical vapor deposition reactor. scale and chemical vapor deposition of amorphous silicon at the laboratory scale. PHYSICAL VAPOR DEPOSITION Laboratory Scale Research A typical laboratory scale physical vapor de position unit is shown in Fig. 4. The rate of evapo ration of any material is determined by the surface temperature of the source material. For thermal evaporation this is a function of bottle geometry, the material surface area, and the design of the source heater. To make a semiconductor film, the material of interest is placed in the source bottle, heated to the point at which it evaporates or sub limes, flows out of the source bottle to the sub apv =T, pVCp, tpg AHR FVF (TTI4)A. FIGURE 5. Model equations. state, and then condenses on the substrate, the temperature of which is carefully controlled. The modeling and experimental verification of a model describing the rate of effusion for CdS which dissociates and sublimes has been thorough ly discussed by Rocheleau et al [4]. The mass and energy balance equations written for the material to be evaporated are shown in Fig. 5 (nomencla ture in [4]). These equations can be solved numerically, given the initial dimensions of the material in the source bottle and the appropriate constitutive equations for the flow through the orifice in the source bottle. Fig. 6 gives the required equations in terms of the mass flux, r, related to FLOW REGIME ORIFICE PIPE Free gcR2p 2 Molecular r= (plgc/2 ")12 ( lP) r=  (P2 P2 )[4( ~) xm (l\,/R>l) .16pL R " gR2p1 Viscous r= CY[Zp gc(PlP2)]"2 r=  ( p2P22) (Xm/R<00 ) 16pL FIGURE 6. Constitutive flow equations. pgq through the area available for flow. The solu tion method is somewhat complex and complete de tails are given by Rocheleau et al [4] and Roche leau [6]. Solving the equations yields the rate of effusion versus charge temperature, T1. A com parison of model prediction (solid lines) and ex perimental data (horizontal bars marked with the wall temperature, T2) are shown for two different orifices in Fig. 7. The heat transfer from the source bottle walls to the subliming surface is the key issue in predicting rate of effusion from the source bottle. Another type of experiment in which cadmium and sulfur are used in separate source bottles can be used to obtain information on the surface phe nomena (Fig. 1). An extensive set of data has been obtained by Jackson [5] who also was able to pre dict the impingement rate of the molecular beam at any point on the substrate. The impingement rate of cadmium and sulfur on the substrate was calculated and the corresponding rate of CdS film CHEMICAL ENGINEERING EDUCATION growth measured. About 1000 separate pieces of data were obtained to verify the predicted model behavior. The model equations for each species are shown in Fig. 8. The rate of reaction of cadmium to the CdS comprising the film is assumed to be r(rxt, Cd) = k(CdS) [Cde][Ss] This expression combined with the equations shown in Fig. 8 yields r(rxt,Cd) = K(CdS) [8(Cd) r(i,Cd) r(rxt, Cd) ] [8(S) r (i, S) r (rxt, S)] The parameter, 8, is a condensation coefficient; r (i, Cd or S) is the rate of molecular impingement of Cd or S; r (rxt, Cd or S) is the rate of reaction of Cd or S; and K is a modified specific reaction rate constant whose detailed form is given by Jackson [5]. Comparison of the model behavior with some of the data is shown as Fig. 9 where the rate of deposition of CdS is plotted as a function of the 1.2 1 1334 numbers indicate wall temperature in oK 1.0 / 1309 .5 .8 E "E orifice set 4 1271 .4 12 1 ~orifice set 3 .2 1179 1198 OL 1150 1200 charge temperature, K FIGURE 7. Comparison of model behavior with ex perimental data. rate at which cadmium is delivered to the sub strate. At low values of r (i, Cd) the rate of film growth is proportional to the rate at which cad mium is delivered; as the film growth becomes sur facereaction dependent, the lines curve. The hori zontal line indicates a region in which there is not enough sulfur to react with all the cadmium being delivered to the substrate. Experimental evidence indicates that photo voltaicgrade CdS can only be made when the rate of film growth is controlled by the rate at which Cadmium r(r xt, Cd)=r(i, Cd)r(e, Cd)r(r, Cd) Sulfur r(rxt, S)=r(i, S)r(e, S)r(r, S) Cadmium Sulfide I dM(CdS) I dM(CdS) == r(rxt, CdS) M,(CdS)A, dt FIGURE 8. Component mass balances. cadmium and sulfur react on the surface to form CdS. Furthermore, if sulfur is not present when a cadmium molecule arrives at the surface, the cadmium will reflect and not adhere. We are just beginning to try to relate these observa tions to film properties. This second laboratory scale study of Jackson's is an attempt to learn more about the semiconductor chemical reaction engi neering necessary for the field to progress in an orderly fashion. A much more complicated set of chemical equations will be considered in the section on chemical vapor deposition. Unit Operations Scale Research For large area applications uniform defect free film with the required properties must be de posited over areas on the order of meters in dimension. It may be necessary to deposit on a moving substrate to lower costs to the level re quired to make a large area application like photo voltaics economically feasible. In this section the cooperative research efforts in photovoltaic unit operations between the Department of Chemical Engineering and the Institute of Energy Con E2 4) 0 E 8 0 t." 3.3I I I I I 5 39 67 n 22 19 Model Predictions 8.3 Constant Sulfur Flux Indicated x 108 mi se 220 C Substrate Temperature 0 o I 0 8.3 16.7 25 33.3 41.7 50 58.3 66.7 75 r(i, Cd) x 108 Kgmoles/m2 sec FIGURE 9. Deposition versus incident rate of cadmium in cadmium sulfide. SPRING 1985 I ( ( I i I I I 83.3 91.7 ICO ^ The theoretical and experimental work of Rocheleau, Rocheleau et al, and Jackson provide the verified models of the laboratory scale batch experiments that can be used to design apparatus and experiments at the unit operations scale. version at the University of Delaware will be de scribed. CdS is the wide bandgap window semicon FIGURE 10. ThinFilm polycrystalline solar cell. ductor for the following polycrystalline hetero junction cells (Fig. 10) CdS/CuInSe2 CdS/CdTe CdS/Cu2S All of the above devices have achieved con version efficiencies (percentage of sun energy con verted to electricity) of just over 10%, although in the case of CuS cells some ZnS had to be al loyed with the CdS. At this conversion efficiency, inexpensive electrical power generation begins to become feasible if modules containing the indi vidual cells can be made cheaply. A first step in meeting this goal is to find a way to continually TABLE 1 Approximate Throughput and Size Specifications Unit Ops. Lab. Scale Scale Exps. Expts. Throughput (m2/year) Deposition Chamber (m3) Commercial Scale Production 1100 1,00020,000 100,0001,000,000 0.10.5 13 1030 FIGURE 11. Unit operations scale deposition system. deposit the CdS on a moving substrate. The theoretical and experimental work of Rocheleau [6], Rocheleau et al [4] and Jackson [5] provide the verified models of the laboratory scale batch experiments that can be used to design apparatus and experiments at the unit operations scale. Throughputs and chamber sizes for typical units are shown in Table 1 for the three scales of operation of interest. The laboratory scale ex periments are almost always batch experiments on a static substrate. The unit operations and com mercial scale equipment for photovoltaics need to be designed for continuous deposition on a mov ing substrate. A sketch of the unit operations scale equip ment used at the Institute of Energy Conversion TABLE 2 Deposition Unit Specifications Chamber1.28 m diameter X 1.34 m long VacuumPumpdown to 5 X 106 torr in 2 hours Web Capacity500 mm maximum width 250 mm roll diameter (200 m Cu foil) Web Speed1.2 to 12 cm/min Deposition Zone45 cm X 10 cm SourceA proprietary design (U.S. Patent 4,325,986) providing: Constant rate2 micron/min Uniformity over 20 cm wide zone 80% CdS utilization Web Temperature200 to 2250C Throughput0.6 to 6 m2/shift CHEMICAL ENGINEERING EDUCATION Windup Roll Web or / Substrate. Substrate Temperature Control Payoff Roll Vacuum Chamber SGuide Roll  Evaporation Source is shown in Fig. 11. This piece of equipment was designed using model equations similar to those presented as Fig. 5 and 6 and modified for a multi orifice geometry and the different sourcesubstrate geometry of the unit operations scale equipment. It was also necessary to expand the energy balance analysis to include radiative heat transfer between the source and substrate. The model equations, their behavior and their influence on the design and operation of the unit operations scale reactor are given by Rocheleau [6] and Griffin [7]. The specifications determining the equipment are shown in Table 2 and a photograph of the equip ment is shown as Fig. 12. FIGURE 12. Photograph of unit operations scale deposi tion system. The unit operations experimentation consisted of controlled deposition of CdS on rolls of zinc plated copper foil. Throughput of the foil ranged from 180 to 600 cm2 per hour with film growth rates ranging from 0.5 to 2 Mm/min. Substrate temperatures were varied between 200C and 250C. At throughputs of 400 cm2/hour, up to 3000 cm2 of 25 im thick CdS was prepared in a single run. Fig. 13 compares a crosssection of con tinuously deposited CdS with that of photovoltaic grade batch deposited CdS. Xray diffraction con firms predominantly caxis orientation for the continuously deposited CdS. Resistivities of the continuously deposited CdS films ranged from about 1 to 100 ohmcm. Resistivity of the best laboratory CdS ranges from 1 to 10 ohmcm. 4j FIGURE 13. CrossSection of CdS film. Crosssection of continuouslydeposited CdS on left, crosssection of batchdeposited CdS on right. The principal means of evaluating the CdS was to measure the photovoltaic response of cells fabri cated using the material from the unit operations experiments. Results are summarized in Table 3 which shows the efficiencies of CdS/Cu2S cells made using both laboratory scale and unit opera tions scale CdS. The Cu2S layer can be made using a wet process by dipping CdS into a CuCl solution or it can be made using a dry process in which CuCI is evaporated onto the CdS and then allowed to react with CuS. A quantitative description is given by Brestovansky et al [8]. Cells made by the dry process in the unit operations scale equipment had both layers, CdS and CuS, continuously de posited on the moving substrate. Cells made by the wet process had to have the Cu2S layer made in a batch operation. All cells had an evaporated gold front contact. The efficiency figures show that the unit opera tions scale continuously deposited CdS is virtually of the same photovoltaic quality as the laboratory scale batch deposited CdS. It took some ten years of research to achieve the efficiencies shown for the batch deposited CdS. The continuously deposit ed CdS reached the efficiency shown in well under two years of unit operations experimentation. This could only have been achieved by drawing heavily Continued on page 106. TABLE 3 Cell Efficiencies (CdS/Cu2S) Laboratory Scale (Batch) Wet Process (CdS Only) Dry Process (CdS/CuS) Unit Operations Scale (Continuous) 8% 7% SPRING 1985 Sn classroom THE USE OF COMPUTER GRAPHICS TO TEACH THERMODYNAMIC PHASE DIAGRAMS CHANDRASHEKHAR D. NAIK Singer Corporation Silver Springs, MD 20904 PAULETTE CLANCY AND KEITH E. GUBBINS Cornell University Ithaca, NY 14853 THE TEACHING of thermodynamic phase dia grams poses problems which affect both the in structor and the students. The usual approach in which the threedimensional pressuretempera turecomposition diagrams for binary fluid mix tures are represented on a twodimensional page is difficult for students to visualize. Traditionally, in order to simplify this complex situation, 'cuts' at constant pressure, temperature, or composition are made to show a truly twodimensional diagram de scribing the relationship between two of the three independent variables. However, the interre lationship of all the variables involved is lost with this approach, and the problem of comprehension intensifies as the complexity of the phase behavior increases. Construction of threedimensional Chandrashekhar D. Naik ob tained his B. Tech. degree in chemical engineering from the Indian Institute, India, in 1981. He obtained an MS degree in chemical engineering from Cornell University in 1983 and is currently employed at Singer Link Simulations Systems Di vision in Silver Spring, Mary. land. (L) Paulette Clancy is current ly an assistant professor in chemical engineering and as sociate director of the Manu facturing Engineering Program at Cornell University. She received her BS degree at the University of London and a D.Phil. degree at the University of Oxford. She held fellowships at Cornell University and at London University before joining the faculty at Cornell in 1984. (C) Keith E. Gubbins is currently the Thomas R. Briggs Professor of Engineering and director of chemical engineering at Cornell Uni models offers an alternative solution, but they are difficult and timeconsuming to produce and offer no possibility for student interaction. At Cornell an alternative to traditional ap proaches was sought to improve the quality of teaching and the level of comprehension of the students. Computer graphics offers an innovative solution to these difficulties: presentday graphics hardware can perform rotational transformations of threedimensional images almost instantaneous ly and allows extensive manipulation of the viewed image by the user, making this an extremely powerful tool eminently suited to the task at hand. During the past two years a highly interactive "user friendly" graphics package has been de veloped depicting the threedimensional phase be havior of binary fluid mixtures, and it has been used in both undergraduate and graduate courses with great success. THE GRAPHICS WORKSTATION The Computer Aided Design Instructional Copyright ChE Division, ASEE, 1985 versity. He received his BS and PhD degrees at the University of London, and was on the staff at the University of Florida from 1962 76, when he moved to Cornell. He has held visiting appointments at Imperial College, London, at Oxford University, and at the University of California at Berkeley. He has coauthored two books, Applied Statistical Mechanics (Reed and Gubbins) and Theory of Molecular Liquids (Gray and Gubbins). (R) CHEMICAL ENGINEERING EDUCATION FIGURE 1. An example of a typical workstation show ing the Evans and Sutherland vector refresh graphics monitor with VT100 terminal, electronic tablet and stylus. Facility (CADIF) at Cornell houses "stateofthe art" computer graphics equipment used solely for teaching (and developmental work towards edu cation). The central computers for the facility are Digital Equipment Corporation (DEC) VAX ma chines, an 11/780 and an 11/750, running the VMS operating system, with DEC PDP 11/44 ma chines as postprocessors. Attached to these ma chines are two different types of graphical display equipment for student use: vector refresh stations with dynamic threedimensional capabilities and color raster stations for applications requiring color. In this application, the vector refresh work stations were used exclusively, these being the highly sophisticated Evans and Sutherland Multi picture Systems. Each workstation has a digitiz ing tablet and electronic stylus as the primary in put peripheral for cursor control, with a DEC VT100 terminal for alphanumeric input. A typi cal workstation is shown in Fig. 1. An electrostatic plotter is also available for hardcopy output, a use ful and necessary addition allowing students to submit a record of their progress to the instructor. The software, which is the heart (or perhaps more appropriately, the brains) of this application, was written in FORTRAN making use of system graphics software routines developed at CADIF. The consideration of ergonomic factors to produce a well designed application in terms of its "user friendliness" was considered essential to promote ease and clarity of use of the graphics package as well as increased flexibility. Some of the ways this was achieved include the following points: ex tensive 'help' messages and prompts for required input were made available, clear consistent "menus" for optional choices of interactive re sponse by the program were produced, and the ability to recover from mistakes or unintentional "miskeying" was provided. It was an original tenet of this study that students should not have to read a computer manual before using the pro grams. The emphasis is thus on learning engineer ing principles without requiring prior expertise in computing. REPRESENTATION OF THE PHASE DIAGRAMS The phase equilibrium data for binary mix tures needed for the representation of the phase diagram (i.e. pressures, temperatures and com positions) were generated using a theoretical model. The original RedlichKwong equation of state was employed for this purpose, chiefly be cause of the simplicity of its representation (since only two adjustable parameters are involved) and the reasonably realistic description of binary phase behavior it provides. This approach was also used It has proven to be extremely popular with the students, and has raised the level of comprehension of this potentially difficult subject above that achieved previously by using conventional means. by Willers and Jolls [1] who produced three dimensional phase diagrams on a Cal Comp plotter using the same equation of state. The wellknown RedlichKwong expressions describing the conditions for vaporliquid or liquid liquid equilibrium in terms of the pressures and chemical potentials of both phases were used to generate data points P, T, VL, VG, X1, and y1 cover ing a region from the higher of the pure com ponent triple points to a temperature above both critical temperatures. The nonlinear equations in volved were solved using a multidimensional NewtonRaphson [2] technique. Close to the criti cal region, however, convergence problems were encountered which were due, we believe, to singularities in the Jacobian matrix. These difficul ties were overcome by using the Marquardt [3] method which combines the advantages of New tonRaphson and Steepest Descent algorithms. Here Argonne National Laboratory's 'MINPAK' SPRING 1985 Computer graphics offers an innovative solution . present day graphics hardware can perform rotational transformations of threedimensional images almost instantaneously and allows extensive manipulation of the viewed image by the user . an extremely powerful tool . . software package provided the subroutine for a Marquardt method of solution. Solving for vapor liquid critical lines also provided a challenge. Neither of the previous techniques mentioned was able to reproduce these highly nonlinear equa tions, and a specialized algorithm due to Deiters [4] was employed for their solution. Scott and Van Koynenburg [5, 6] classified the experimentally observed types of fluid phase dia grams into six classes, based on the presence or absence of threephase lines and their connection with the critical lines. So far we have been able to cover the two simplest classes, I and II, although extension of the programs to cover the other classes is well underway. These more complex systems will provide an even more striking visual illustration of the advantage of using computer graphics. In classes I and II both components have similar critical temperatures with the vaporliquid critical line passing continuously between them. In class II, however, the mixture is more nonideal and exhibits liquidliquid immiscibility at low temperatures. For this class, in addition to the vaporliquid region encountered for class I, two other regions exist in the phase diagram, those of liquidliquid equilibrium and a threephase liquid TABLE 1 Examples of Classes I and II Type Behavior Available for Display By the User CLASS I Binary Mixture PentaneNonane Cyclopentane Nonane Pentane Ethylbenzene Acetone Trichloromethane MethaneTetra fluoromethane Perfluoropen tanePentane Azeo Temp Pressure tropet Range K+ Range Barst N 425590(594) 2.3824.30(33.7) N 425590(596) 2.3921.92(45.2) N 425615(617) 4.1836.70(37.4) Y 420530(535) 11.8651.18(55.6) CLASS II N 80224.5(228) Y,Het 240505(506) ,+ Horn 1.25x104 38.76(46.0) 4.18x102 39.28(39.7) tN,Y = no, yes; + = positive or negative deviation from Raoult's law; Het, Hom = heterogeneous, homogeneous azeotrope. +The figures in parentheses are the highest values of T, and Pc occurring along the critical line. FIGURE 2. Threedimensional phase diagram for a typi cal class I system, pentanenonane. The solid and dashed lines show the vapor and liquid boundaries, respectively. FIGURE 3. PTx diagram for the class II system, me thanetetrafluoromethane, showing the coexisting vaporliquid equilibria (solid lines for the vapor, dashed for the liquid) and the region of liquidliquid immisc ibility (shown as solid vertical lines). Superimposed on the diagram (shown in bold) is a Tx cut at a pressure of 0.03 bars. CHEMICAL ENGINEERING EDUCATION liquidgas line. Examples of the binary systems chosen to illustrate the phase behavior of classes I and II are shown in Table 1. Some of the available systems exhibit azeotropic phenomena with either positive or negative deviations from Raoult's Law, and of either a heterogeneous or homogeneous nature. Photographs depicting some of the com FIGURE 4. A 3D view of another class II system, CF,,  CH12. The original display has been rotated by 180 degrees and tilted downward so that the view is from the hightemperature end and somewhat above the phase diagram. The regions of vaporliquid equilibria (showing an azeotrope) and liquidliquid equilibria (solid vertical lines) are clearly visible. SYSTEM: C5F12(1)Pentane(2) I0o mTrArP 9479, F VArni o.o 0.0 X(W) HELP (CUT) ULTAIN EAS[ MUL. CUT FIGURE 5. TwoDimensional xy diagram for the system C5F12C5sH, derived from the threedimensional phase diagram at 247.26K. This diagram shows the character istic behavior of an azeotropic system with liquidliquid immiscibility, as shown by the horizontal portion of the curve. putergenerated phase diagrams are reproduced in Figures 26 illustrating the kind of image dis played for the user to manipulate. USER INTERACTION WITH THE GRAPHICS SOFTWARE PROGRAMS The image of the phase diagram (e.g. as in Figs. 26) can be manipulated by the user by means of an electronic tablet and stylus (pen). As the pen is moved over (and slightly above) the surface of the tablet, a cursor in the form of cross hairs moves over the display. When the pen is pressed down onto the tablet the graphics program is activated and performs an operation appropri ate to the area of screen chosen, given that such an area is one of the several specially designated parts of the screen called "windows" on a socalled "menu" of options. In this application of computer graphics the menu contained the following list of 'entrees' for the user to select a) READ: Allows the user to choose different binary systems to examine by supplying one of a given set of data file names via the terminal. b) ORBIT: This allows the phase diagram to be ro tated about its pressure and composition axes in a continuous fashion as required. c) PAN: Allows horizontal or vertical translation of the phase diagram. d) ZOOM: Provides closer examination of a chosen area of the image by scaling the diagram up or down. SYSTEM: CSF12(1)Pentane(2) L I :i!N. HELP ULN CUT MUL CUT FIGURE 6. A view of the CsF12C5Hi2 phase diagram as it appears in the initial orientation on the screen. Solid and dashed lines have the same meaning as in Figure 2. A Pxy cut is shown superimposed in bold on the diagram at a temperature of 259.9 K. SPRING 1985 _1 1.0 e) STRETCH: Scales either of both of the P, T axes relative to the composition axis for ease of viewing. f) HELP: Summons the HELP text. g) RESET: Voids all previous manipulations and resets the system to the beginning of the program. h) SNAP: Produces a hardcopy image of the screen on a nearby plotter. i) EGRESS: Allows the user to terminate his or her session. j) TX, PX, PT: Each of these windows allows a particular highlighted "cut" of the phase diagram to be chosen by the user as shown in Figure 3 for a Tx cut at 0.03 bars for the system methanecarbon tetrafluoride, and in Figure 6 for a Px cut at 259.9K for CF12pentane. k) CUT: Produces a Px, PT or Tx "cut" displayed alone (i.e. not superimposed on the whole phase diagram) depending on which of these three windows (PX, PT or TX) was last active. Multiple cuts (of Px at different temperatures for example) may be displayed simultaneously. 1) Produces an xy plot at constant temperature, as shown in Fig. 5 for the system CF,,pentane at 247 K. A 16mm movie lasting approximately thirteen minutes has been prepared to illustrate the cap abilities of this graphics package; this was pre sented at the 1983 AIChE annual meeting in Washington, D.C. SUMMARY The interactive graphics package illustrating the phase behavior of binary mixtures which has been described in this paper has been used within the chemical engineering curriculum at Cornell since the fall semester of 1982. It has proven to be extremely popular with the students, and has raised the level of comprehension of this potential ly difficult subject above that achieved previously using conventional means. The major advant age lies in the suitability of computer graphics as a means of visualizing threedimensional objects (here the PTx phase space) ; the capability of the hardware to perform rapid and continuous rota tions of the image; and, perhaps most importantly, the opportunity to interact, manipulate and con trol the image observed on the screen, brought about by flexible "userfriendly" software. All these features combine to contribute to the success of this technique in undergraduate instruction. O ACKNOWLEDGMENTS It is a pleasure to thank the Gas Research Institute for partial support of this work. REFERENCES 1. K. R. Jolls and G. P. Willers, Cryogenics, 329, June 1978. 2. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer Verlag (1980). 3. D. W. Marquardt, J. Soc. Ind. & App. Math, 11, 431 (1963). 4. U. K. Deiters, Diplomarbeit, Univ. of Bochum, West Germany (1976). 5. R. L. Scott, Ber. Buns. Phys. Chem., 76, 296 (1972). 6. R. L. Scott and P. H. Van Koynenburg, Disc. Fara. Soc., 49, 87 (1970). Qs book reviews FOUNDATIONS OF BOUNDARY LAYER THEORY FOR MOMENTUM, HEAT AND MASS TRANSFER by Joseph A. Schetz Prentice Hall, Inc., NY (1983) Reviewed by O. T. Hanna University of California, Santa Barbara This book on Boundary Layer Theory is indi cated by the author to be applicable for students in mechanical, aerospace, chemical, civil, and ocean engineering. Some people would doubt that anyone could succeed in such a broad task. The author's stated goals for this book include (i) pro viding an understandable coverage of advances in turbulence modeling, (ii) presenting application of large digital computers to boundary layer prob lems, and (iii) treating mass transfer in an inte grated manner with momentum and heat transfer. It would appear that the first goal has been met reasonably well; achievement of the second goal is questionable, and the third goal has definitely not been met to the satisfaction of chemical engi neers. The book is generally well written and well organized. The coverage of laminar flows includes chapters on integral and differential equations of flow together with approximate integral solutions and exact similarity solutions. Unfortunately al most all of this material is available in a number of other sources and hardly any of it is more recent than 1960. The meager coverage of mass CHEMICAL ENGINEERING EDUCATION transfer is likely to be of little interest to chemical engineers. Chapters 4 and 5 do contain some use ful discussions of numerical solutions of bound ary layer problems. However, there are no example problems or computer programs. The major contribution of this book would ap pear to lie in Chapters 6 through 9, on turbulence modeling, which constitute more than half the length of the book. This material includes a useful historical perspective and spans the complete range of engineering approaches in this area up to the present time. The chronological discussion of work in turbulence modeling beginning with early mixinglength theory and progressing up to algebraic and various partial differential equa tion models should be of interest and value to chemical engineering. This discussion also inte grates well the contributions to modeling from both experimental and theory. In summary, the present book seems somewhat disappointing in its treatment of laminar bound ary layers, but in contrast it contains material on turbulent momentum transfer which should be of interest to chemical engineers. In this context the book can be recommended as a useful reference. E INDUSTRIAL HYGIENE ASPECTS OF PLANT OPERATIONS by L. J. Cralley, L. V. Cralley, and J. E. Mutchler Macmillan Publishing Company, New York, 1984: $60.00 Reviewed by Klaus D. Timmerhaus University of Colorado This is the second volume of a new three volume series that is being prepared to provide recognition, measurement, and control of potential hazards normally present in various industrial plant operations. The first volume covered process flows while the third volume will treat equipment selection, layout and building de sign. This volume, divided into two major sections of unit operations and product fabrication, en compasses a broad range of industries with au thoritative information contributed by specialists from these industries. In the first section twenty three contributors discuss unit operations as dis tinct entities along an industrywide concept. Some of the unit operations considered include filtration, clarification, mixing, blending, grind ing, and spray, vacuum, freeze and fluidized bed drying. The second section includes thirteen con tributions which cover the operations and pro cedures for assembling parts and materials into final products. The industries considered in this latter survey range from such basic industries as storage battery and tire manufacturing to the high technology industries of semiconductor and liquid scintillation counter manufacturing. One may argue with the manufacturing processes that were selected by the editors; however, the breadth of the selected processes and the hazards associ ated with these processes should provide a good introduction to the hazards associated with those manufacturing processes that were not included. Even though most contributors to this second volume have adequately described each step in the unit operations and product fabrication flow of a specific manufacturing process and have included a discussion of the various health hazards that may be encountered with suggestions for their monitoring and control, many engineering read ers will be disappointed by the qualitative ap proach taken by the contributors to this important subject. Only a few of the chapters in the volume have included quantitative information that would be necessary in the design and construction of process equipment that minimizes or eliminates identified industrial hygiene hazards. Where such quantitative information is included, it is general ly quite sketchy and incomplete forcing the design engineer to consult other literature sources. Un fortunately, no guidance to such quantitative data is included by any of the contributors. Chemical engineers will also be somewhat disappointed in this volume because the "unit operations" portion of the title implies that some of the contributions will examine the conventional unit operations as sociated with heat, mass and momentum transfer. However, many of the key unit operations such as distillation, absorption, extraction, evaporation, heat transmission, etc. found in most typical petroleum and chemical processing plants have not been included. Nevertheless, this volume does manage to bring together a wealth of experience in a broad range of industries and will aid engineers, managers, and industrial hygienists to more fully recognize potential hazards of industrial processes. This, in turn, will permit these professionals to evaluate such hazards and take the necessary steps to effectively control the problem. C SPRING 1985 laboratory AN IMPROVED DESIGN OF A SIMPLE TUBULAR REACTOR EXPERIMENT ABDULFATTAH A. ASFOUR University of Windsor Windsor, Ontario, Canada N9B 3P4 DESIGNING A TUBULAR FLOW reactor experi ment for an undergraduate laboratory is not a simple task. This is because the experiment will have to meet certain criteria, viz It is safe It is simple and cost effective It is instructive Its analytical needs must be simple and easy, to meet the time constraints of an undergraduate laboratory Anderson [1] developed a tubular flow reactor experiment for an undergraduate laboratory at Princeton that utilized the system acetic an hydridewater. This particular experiment re quires relatively elaborate safety precautions. Moreover, since the reaction is exothermic, rather expensive temperature control equipment is re quired. Samples taken at the reactor inlet and outlet are analyzed by the anilinewater method which is relatively lengthy and subject to errors. Hudgins and Cayrol [2] utilized the basic de sign of Anderson in developing a simple and interesting experiment. They utilized the classical reaction system of crystal violet dye neutraliza tion with sodium hydroxide. This system was studied earlier by other investigators, mainly in a batch reactor (Carsaro [3]). The two novel aspects of the HudginsCayrol experiment compared with that of Anderson are A colour change can be seen between the inlet and outlet of the reactor The temperature constraint is removed. This makes the experiment operable at room temperature Also, from the safety standpoint, a relatively dilute sodium hydroxide solution (0.04 N according to Hudgins and Cayrol) is used. However, the experimental setup design given by Hudgins and Cayrol can be significantly Copyright ChE Division, ASEE, 1985 A. A. Asfour received his B.Sc. (Hon) and M.Sc., both in chemical engineering, from Alexandria University, Egypt, and a Ph.D. from the University of Waterloo, Waterloo, Ontario, Canada. He joined the research department of Imperial Oil, Sarnia, Ontario for the period 19791981. In 1981 he joined the Chemical Engineering De partment at the University of Windsor, Windsor, Ontario, Canada. His research interests are in the area of mass transfer in threephase fluidized beds and in membrane processes. improved. The design improvements suggested in this article should make the experiment easier to run and control, significantly improve the repro ducibility of results, and expedite the process of data collection within the time constraints of an undergraduate laboratory. The main objectives of this experiment are To study the effect of residence time on conversion in a tubular flow reactor To compare the experimental conversions with those obtained from plugflow and laminarflow reactor models THEORY As it was established by Corsaro [3], the re action between crystal violet dye and sodium hy droxide is of the first order in the concentration of each of the reactants, i.e., the reaction is of the second order. However, the reaction can be made pseudofirst order if sodium hydroxide is used in great excess with respect to crystal violet dye. In other words rdye = k' [dye] (1) CHEMICAL ENGINEERING EDUCATION The value of the rate constant, k', is needed for this experiment. Students are requested to run a batch experiment to determine the value of k' at the same temperature of the flow experiment.. . (and) to prepare their own calibration curve of the dye concentration versus absorbance. For the purpose of this experiment, 0.02 N sodium hydroxide solution is used with 6.86 x 104 M dye solution, i.e., the sodium hydroxide concentration would be about 282 times that of the dye, if equal volumes of reactants are used. Experimental conversions are calculated, as will be described later, and compared with theo retical conversions predicted from the plugflow model and the laminarflow model. For a first order reaction in a plugflow re actor, the following equation applies assuming constant density of reaction mixture: V 1 7= = In(1x) (2) Vo k = In CA (3) k' CAo If CA is taken as [dye]e, i.e., the dye concentration at reactor exit and CAo as [dye]i, i.e., the dye con centration at reactor inlet, then one can rewrite Eq. (3) as follows: V 1 [dye], T  n (4) vo k [dye]e For a first order reaction in a laminarflow reactor, the following equation applies assuming no change in volume by reaction as well as no mixing in both radial and axial directions (4) x= 1i 2 E ( ) + ( 1)exp(Nn/2) (5) where NR =k'r V Lrr2 V, Vo The function E (y) is defined by: exp (4) E(y) = d y The function E (y) is tabulated in standard tables as Ei(X). EXPERIMENTAL A schematic diagram of the proposed experi mental setup is given in Fig. 1. The experimental apparatus is comprised of the following compo nents. Reservoirs Constant Head Tanks Pumps Mixer Rotameters FIGURE 1. Schematic diagram for the experimental setup. M: mixer, P: pump, R: rotameter, RES: reser voir, S: sampling point, T: constant head tank, TFR: tubular flow reactor. :(RES 1) 200L polyethylene tank for the sodium hydroxide solution (RES 2) 20L polyethylene tank for the dye solution : (T1) 20L polyethylene tank for the sodium hydroxide solution (T2) 4L polyethylene tank for the dye solution :(P1) Magnet drive gear pump; Model P/N 81152 manufactured by Micropump Corp., Conford, Cali fornia. Purchased from Cole Parmer Co. (P2) Centrifugal pump. Cole Parmer catalogue No. K700430 : (M) Little Giant Pump. Model 2E 38NT. Purchased from Can Lab : (R1) Size R615A rotameter. Max flow 450 ml/min with SSfloat. Purchased from Brooks Instru ment Co. (R2) Size R615B rotameter. Max flow 1300 ml/min with SSfloat. SPRING 1985 Purchased from Brooks Instru ment Co. : 40 meters of 3/8 in. I.D. Tygon tubing, wound on spool (made of lexan), 28 cm in diameter and 55 cm in length* : needle valves to adjust flow : Spectronic 20 (Bausch & Lomb) modi fied, as will be described later, to provide continuous measure ments. Two polyethylene tanks (RES 1 and RES 2) of capacity 20 liters and 200 liters serve as reser voirs for the crystal violet dye and sodium hy droxide solutions, respectively. Two pumps (P1 and P2) are employed to pump the reactants to two constant head tanks (T1 and T2). The over flows from the constant head tanks are returned to their respective reservoirs. The underflows from the constant head tanks go via rotameters (R1 and R2) to a small pump that acts as a mixer (M). The reactant streams are mixed in the mixer, M, and are pumped through the reactor. The tubular reactor is in the form of a helical coil wound on support. Connections are provided at the inlet and outlet of the reactor to the flow through curvettes of the spectrometers. The main advantages of the proposed experi mental setup over that suggested by Hudgins and Cayrol are Reservoirs and constant head tanks are used. This arrangement provides more stable rotameter opera tion, especially at low flow rates. A flowthrough accessory which is simpler in design and operation than that suggested by Hudgins and Cayrol has been used. The flowthrough accessory shown in Fig. 2 allows one to use Spectronic 20 for continuous measurements. PROCEDURE Due to the limitation of the headroom in most undergraduate laboratories, the constant head tanks (T1 and T2) are placed about 3 meters above the rotameters' level. This limitation makes it only possible to attain maximum flow rate of 1300 ml/min of NaOH. The maximum flow rate of the dye is set at about 135 ml/min. *One of the reviewers suggested the use of polyethylene instead of Tygon tubing, which discolors to a deep purple making it difficult to observe gradual colourchange along the reactor. It is believed that polyethylene is more resistant to the dye than Tygon tubing. Cap made  of blue glass = FLOW IN F FLOW OUT Parafilm used as a seal Spectronic 20 Cuvette FIGURE 2. Flowthrough cuvette for the Spectronic 20. The flow rates of sodium hydroxide and the dye solution are set such that the ratio is 9:1. One should start at the highest possible flow rate to expel all air bubbles from the reactor. One should wait for slightly longer than the residence time, for a particular flow rate, for steadystate to be reached. The reaction mixture is then allowed to flow through the Spectronic 20 flowthrough cuvettes and the readings are re corded. Usually, one waits for two minutes and takes another reading as a duplicate. Experience has shown that the Spectronic 20 readings are highly reproducible. Other flow rates of NaOH and dye solution are chosen, keeping the flow rates ratio 9:1 as before, and the Spectronic 20 read ings are recorded. The experiment usually lasts for one hour provided that the solutions are pre pared prior to the laboratory period. TABLE 1 Holding Time and Conversion Data Holding Exp. Time Reynolds PFRM LFRM Conversion min Number conv. conv. x 6.42 791 71.7 63.3 71.5 4.3 1181 57.1 50.3 53.3 3.23 1651 47.1 41.6 44.5 2.54 2000 39.4 34.96 36.9 2.13 2385 34.3 30.7 33.85 CHEMICAL ENGINEERING EDUCATION Reactor Valves Spectrometers The value of the rate constant, k', is needed for this experiment. Students are requested to run a batch experiment to determine the value of k' at the same temperature of the flow experiment. This has proven worthwhile, since temperature fluctuations in most undergraduate laboratories do not allow conducting a batch experiment at the beginning of the semester and giving the value of k to the students to perform the required calcu lations. Also, students are required to prepare their own calibration curve of the dye concentra tion versus absorbance. This leads to better results. RESULTS AND DISCUSSION Table 1 reports the residence time and the con versions from the plugflow reactor model (PFRM), laminarflow reactor model (LFRM) and the experimental conversions. Fig. 3, also, depicts the conversions against the residence time. The data reported in Table 1 and Fig. 3 were obtained from an experiment conducted on the setup available in Windsor. It is clear from Fig. 3 that, as expected, the experimental conversions fall between the con 0.7 0.6 + 0.5 S0.4 0.3 II I 2 3 4 5 6 7 HOLDING TIME, 7 (min.) FIGURE 3. Comparison between conversion obtained from experiment and those obtained from LFRM and PFRM. REQUEST FOR FALL ISSUE PAPERS Each year CHEMICAL ENGINEERING EDUCATION publishes a special fall issue devoted to graduate education. This issue consists 1) of articles on graduate courses and research written by professors at various universities, and 2) of announcements placed by ChE departments describing their graduate pro grams. Anyone interested in contributing to the editorial content of the fall 1985 issue should write the editor, indicating the subject of the contribution and the tentative date it can be submitted. Deadline is June 15th. versions obtained from the two theoretical models, viz., the PFRM and LFRM. It is worthwhile to note here that the data re ported by Hudgins and Cayrol indicate that the experimental conversion curve crosses the LFRM curve at short holding times, i.e., experimental conversions are lower than those predicted by LFRM, which is not possible. Such results may be attributed to the obvious design flaws in the set up reported by those authors. The change of colour of the reaction mixture between the inlet and outlet of the reactor is due to the conversion along the reactor. Such a visual effect helps the students to integrate the labora tory experiment with what they learned in the lecture part of the course about conversion in tubular flow reactors. O REFERENCES 1. Anderson, J. B., "A Chemical Reactor Laboratory for Undergraduate Instructions," Princeton University, 1968. 2. Hudgins, R. R., and B. Cayrol, "A Simple Tubular Reactor Experiment," CEE, XV, 1, 26, 1981. 3. Corsaro, G., Chem. Educ., 41, 48, 1964. 4. Holland, C. D. and R. G. Anthony, Fundamentals of Chemical Reaction Engineering, PrenticeHall, Engle wood Cliffs, N.J., 1979. NOTATION o0 i,e CA k' L NR r (r) Vo V x r = subscript symbol for initial = subscript symbols for reactor inlet and exit, respectively = concentration of component A, (mole/L) = pseudofirst order rate constant, (min1) = length of reactor tube, (m) = k' = reaction number for a first order reaction = inside radius of reactor tube, (m) = reaction rate, (mol/L.min) = volumetric flow rate, (L/min) = reactor volume, (m3) = conversion = V/vo = space time (min) SPRING 1985 OnI lecture The B. C. (Before Computers) and A. D. of EQUILIBRIUMSTAGE OPERATIONS* J. D. SEADER University of Utah Salt Lake City, UT 84112 1 T HE ART OF DISTILLATION and other multi component, multistage separation operations has been practiced since antiquity. Although de scribing equations for distillation were formu lated before 1900, flexible, efficient, and robust procedures for solving the equations did not ap pear in the literature until after the availability of digital computers beginning in 1951. This paper is keyed to that date with B.C. referring to "before computers." In 58 B.C., Sorel developed the first theoretical equations for simple, continuous, steadystate distillation, but they did not find wide application until 30 B.C., when they were adapted to a rapid graphical construction technique for binary systems by Ponchon and then Savarit. This was quickly followed in 26 B.C. by a much simpler, but restricted, graphical technique by McCabe and Thiele. Computer methods have largely replaced the rigorous PonchonSavarit Method, but the graphical McCabeThiele Method is so simple and so illustrative, it continues to be popular. A modern version of Sorel's equations (shown in Fig. 1) includes, in the case of a partial con denser, total and component material balances and an energy balance around the top section of the column. Phase equilibrium on each tray is ex The development of a separation process can be an exciting experience when computers and computer programs are available to perform the tedious calculations and allow time for more consideration of synthesis and optimization aspects. *Tutorial Lecture presented at 92nd ASEE Annual Con ference, Salt Lake City, Utah, June 2228, 1984. J. D. Seader has been a faculty member at the University of Utah since 1966. Prior to that, he was employed by Chevron Research and Rocketdyne. His principal technical interests are equilibriumstage operations, process synthesis, process simulation, and synthetic fuels. He is a Director of the AIChE and was the Annual Institute Lecturer in 1983 for AIChE. He prepared the section on distillation for the sixth edition of the Chemical Engineers' Handbook and is a Trustee of CACHE. pressed in terms of component Kvalues and one mole fraction sum per stage for either vapor or liquid is needed. Similar equations are written for the bottom section of the column and for the feed stage. DEGREES OF FREEDOM ANALYSIS A degrees of freedom analysis for the equations was first developed by Gilliland and Reed in 9 B.C. A more thorough treatment for all types of separations and other operations was reported by Kwauk in 5 A.D. If the equations and variables are counted, for a column with N stages (including the con denser and reboiler) to fractionate a feed with C components, it is found that the number of equations is N (2C + 3), while the number of vari ables is N(2C + 4) + C + 7. Variables include stage temperatures, pressures, vapor and liquid flow rates and component mole fractions; feed Copyright ChE Division, ASEE, 1985 CHEMICAL ENGINEERING EDUCATION The nature of the equations of Sorel and the difficulty of their solution for multicomponent systems has long been recognized. The set of equations can be large in number. For example, with 10 components and 30 equilibrium stages, the equations number 690. Sixty percent of the equations are nonlinear, making it impossible to solve them directly. III II flow rate, composition, temperature and pressure; reboiler and condenser duties; and number of theo retical stages above and below the feed. The thermodynamic properties, K and H, are not counted as variables because they can be written explicitly in terms of the other variables just mentioned. The degrees of freedom or number of variables that must be specified equals the differ ence between the number of variables and the number of equations or N + C + 7. A simple set of specifications would include feed flow rate, composition, temperature, and V, TOPDOWN I TVN+ t = LN + VI IN+1 VN+1 Xl,N LN + Y1,1 V] HVN,1 VN+ L HLN Vi V1 0c SKI'N YI,/X , X1, M 1 = YIM V + X+ N LN 1HLM L0 + O; = HVH V, V HLN LN KIM= YI,/X,1M X,N FEED STAGE SF + LF1 + VF+1 = L + V XIF1 Y ZI, F + X,F LF1 YIF+ VF+ F XI,F F ,, YI,, VF HFF + HLF1 LF1 + F+ VF+1 HLF LF + HV VF + I VF KFK = F YI F XIVF XIF YI ,F+ XFF = 1 F FIGURE 1. Modern version of Sorel's equations. pressure; number of trays above the feed and below the feed; the pressure of each stage; and the reflux flow rate, L1. This totals N + C + 6, which is one short of the number of degrees of freedom. From my own practical experience in 5 A.D., failure to supply the one additional specification can result in a calculational pro cedure that will never converge. The additional specification might be the distillate flow rate V1. NATURE OF SOREL'S EQUATIONS AND SPARSITY PATTERNS The nature of the equations of Sorel and the difficulty of their solution for multicomponent systems has long been recognized. The set of equations can be large in number. For example with 10 components and 30 equilibrium stages, the equations number 690. Sixty percent of the equations are nonlinear, making it impossible to solve them directly. The magnitude of the values of the variables can cover an enormous range. For example, the mole fraction of a very volatile component at the bottom of the column might be very small, perhaps 1050. The value of a total flow rate might be 104. Commonly used procedures for solving such sets of equations, as discussed by Henley and Seader, are iterative in nature, requiring start ing guesses for some or all of the variables. Early procedures were complete equationtearing methods, suitable for manual calculations, wherein the equations were solved oneatatime in a se quential manner. With the advent of the digital computer, partial tearing methods appeared, wherein small groups of equations as well as single equations were solved at a time. Most re cently, with the availability of larger and faster digital computers, very flexible simultaneous correction methods appeared wherein all the equa tions were solved simultaneously by a modified Newton's method. An additional characteristic of Sorel's set of equations is sparsity. That is, no one equation contains more than a small percentage of the variables. For example, for the case of 10 com ponents and 30 stages, no equation contains even SPRING 1985 7 percent of the variables. This sparsity is due to the fact that each stage is only directly connected to, at most, two adjacent stages. The nature of this sparsity has been exploited in the development of the abovecited methods by seeking certain sparsity patterns which are best observed by the use of incidence matrices. The rows of the incidence matrix represent the differ ent functions or equations being solved. The columns of the matrix represent the different vari ables contained in the equations. Thus, for N = 30 and C = 10, the matrix is of size 690 by 690. If a certain equation contains a certain variable, some nonzero entry, such as an X, is placed in the matrix at the corresponding location. Other REARRANGEMENT OF A SPARSE MATRIX TO OBTAIN A MORE DESIRABLE PATTERN 12345678 C X X XX X X X XX X X X X X X xx x x x x x UNDESIRABLE PATTERN 5 7 1 3 8 4 2 6 57138426 6 X 8 X X 4 X X 7 X X X 5 X X X X 2 X X X X I X X X X X 3 X X X X X X DESIRABLE PATTERN (LOWER TRIANGULAR) FIGURE 2. Incidence matrices. wise a zero or no entry is made. The sparsity pattern depends upon the order in which the columns and rows of the matrix are arranged. The arrangement shown at the left in Fig. 2 appears to be random without pattern. How ever, by interchanging certain columns and rows, the lower triangular pattern shown at the right is obtained. Such an organized pattern, if it can be achieved, is highly desirable because it indicates that the equations can be solved oneatatime starting with the equation for the first or top row, solving for the only unknown, and then proceed ing down the rows, equationbyequation, solving for one unknown atatime, but where necessary, using values of previously computed variables. Since at least 10 B.C., a number of other de sirable sparse matrix patterns have been recog nized. Shown at the left in Fig. 3 is a block diagonal pattern. The nonzero entries are all contained within the interior boldlined region. Shown in the middle is a banded matrix, where BLOCK DIAGONAL BANDED BLOCKED AND BORDERED FIGURE 3. Desirable sparse matrix patterns. all nonzero entries are contained on the main diagonal and a few adjacent diagonals. Shown at the right is a blockedandbordered matrix. Such organized sparse matrix patterns, when they exist, can be found readily by computer algorithms such as the MA28 subroutine of the Harwell li brary. EQUATIONTEARING STRATEGIES For the four organized patterns just dis cussed, specialized sparsepattern computer al gorithms have been developed to solve, in an efficient manner, linearized forms of the nonlinear equations that describe the system. These sparse matrix methods strive to: (1) eliminate storage of zero coefficients and certain repetitious nonzero elements, (2) reduce arithmetic operations, in par ticular those involving zeros, and (3) maintain sparsity during computations. Lessorganized sparsematrix structures can sometimes utilize organized sparsematrix methods in an iterative manner by employing equation tearing strategies. The structure shown in Fig. 4 is lower triangular, except for an additional non 1 2 3 4 5 EQUATION 6 7 8 9 10 11 VARIABLE 1 2 3 4 5 6 7 8 9 10 11 EQUATIONSOLVING ORDER: 1 10 11 Qy TEAR VARIABLE  BORDER OF INNER CYCLE FIGURE 4. Application of tearing to a sparse matrix. CHEMICAL ENGINEERING EDUCATION I N\ \  zero entry at column 8 in row 3. The linearized equations cannot be solved directly, oneatatime, starting with Equation 1, because when Equation 3 is reached, the value of variable 8 is not known; thus, Equation 3 cannot be solved for variable 3. A tearing strategy can be employed to overcome this difficulty, but an iterative calculational loop or cycle, shown by the dashed border, involving Equations 3 through 8, is necessary. Variable 8 shown as a circled X, is the single tear variable which, when given an estimated value, results in the tearing apart of that subset of equations so they can be solved individually in order. At Equa tion 8 in the cycle, variable 3 is calculated and the value obtained is compared to the value used in Equation 3. If the two values are sufficiently close, the cycle is converged and variable 9 in Equation 9 is computed, followed by solution of Fr = 2x{l + X2 85 = 0 F1 2x12 35 0 F2o201 2 ,* 035=0 TEARING STRATEGY #1: 1 2 S 2 X X TEARING STRATEGY #2t 2 1 2 X GUESSED X1 FROM X2 FROM ITERATION X2 F1 F2 1 8 110 5,9 x 108 2 5.9 x 108 3.0 x 1034 ITERATION 1 2 3 GUESSED 500 4.029936 4.000010 X2 FROM 6.346546 8.999170 9.000000 X1 FROM 4.029936 4.000010 4.000000 BETTER RESULTS WITH THIS STRATEGY BECAUSE F1 IS RELATIVELY SENSITIVE TO X2 BUT NOT TO X0v WHILE THE OPPOSITE IS TRUE FOR F2 FIGURE 5. Sensitivity of the tearing method. Equations 10 and 11 to complete the system. Otherwise, a new approximation for the tear variable must be determined and another iteration of the cycle completed. For the tearing strategy to be successful and efficient, it is necessary that Equation 3 not be sensitive to the assumed value of Variable 8. If too sensitive, it is best, if possible, to reorder the equations and variables to obtain a less sensitive situation. As a simple example of this sensitivity, consider the two equations shown in Fig. 5. If x, is the tear variable and Equation 1 is solved first, SUCCESSIVE SUBSTITUTION BOUNDED WEGSTEIN (16 A.D.) 1 2 3 4 5 6 x x x x[ BEFORE PATITIONI BEFORE PARTITIONING DELAYED WEGSTEIN (28 A.D.) DOMINANT EIGENVALUE (20 A.D.) 1 3 5 2 4 6 AFTER PARTITIONING FIGURE 6. Partitioning when convergence acceleration methods ignore interactions among the variables. convergence is impossible to achieve from any starting guess of the tear variable. For example, if the initial guess for x2 is 8, the sequence ob tained quickly diverges, as shown. After only one iteration, x, has increased in value to 5.9108, which is far from the solution. Alternatively, in tearing strategy #2, where the two columns of the matrix are interchanged to make x, the tear variable, convergence is readily achieved, as shown, from any initial guess, even x, = 500. The solution x, = 4, x2 = 9 is obtained in just three iterations. These two drastically different results are obtained because Equation 1 is very sensitive to the value of x, and almost in sensitive to the value of x,. Thus, in this example, x, should not be the tear variable when Equation 1 is solved before Equation 2. In the example just considered, the guess for x,, at the beginning of each iteration is set equal to the value computed from Equation 2 in the previous iteration. This procedure, called succes sive substitution, can be slow to converge, and, therefore, a number of some simple and some complex procedures have been developed to ac celerate convergence. These procedures are par ticularly useful when more than one tear variable must be used as in Fig. 6. The example at the left has two tear variables. Some methods, including successive substitution, bounded Wegstein, de layed Wegstein, and dominant eigenvalue, ignore interactions among the variables. When inter actions do not exist, it should be possible to inter change columns and rows of the matrix to obtain SPRING 1985 Chemical engineering educators need to closely examine courses on separation processes to make sure that students are being instructed in modern and efficient computational tools. NEWTON BROYDEN (18 A.D.) 1 2 3 4 5 6 1 TI (X) 2 xl 2 X 0 I 5 X XX 6 X X X X FIGURE 7. Convergence acceleration methods that ac count for interactions among the variables. a block diagonal structure, shown at the right of Fig. 6, which allows separate computations of the two individual partitions, each containing just a single tear variable. The more common case is when interactions among the tear variables exist, as shown in Fig. 7, where the two tear variables are 5 and 6. The iterative cycle includes all equations. Then, par titioning is not possible and, if the interactions are strong enough, convergence acceleration by Newton's method or a quasiNewton method, such as that of Broyden, may be desirable. SIMPLE AND COMPLEX SEPARATION OPERATIONS The nature of the sparsematrix pattern ob tained from Sorel's equations and the correspond ing calculational procedure depends on a number of factors, including: (1) selection of the work ing equations, (2) selection of the variables, (3) degree of flexibility in the specifications, (4) order of arrangement of the equations, (5) order of arrangement of the variables, (6) functionality of the physical properties, and (7) method by which any equations are linearized. An additional factor that influences the nature of the sparse matrix pattern of Sorel's equations is the type of separation operation. In simple distillation, a single feed is separated into two products, a distillate and a bottoms; energy re quired to separate the species is added in the form of heat by a reboiler at the bottom of the column where the temperature is highest. Also, heat is removed by a condenser at the top of the column where the temperature is lowest. This frequently results in a large energy input requirement and low overall thermodynamic efficiency, which was of little concern (except for cryogenic and high temperature processes) before 22 A.D. when energy costs were low. With recent dramatic in creases in energy costs, complex distillation opera tions (described by Seader in the 6th edition of Perry's Chemical Engineers' Handbook) and sys tems are being explored that offer higher thermo dynamic efficiency and lower energy input re quirements. Complex columns and systems may involve multiple feeds, sidestreams, intermediate heat transfer, multiple columns that may be inter linked, and in some cases, all or a portion of the energy input as shaft work. COMPLETE TEARING METHOD Simple and complex distillation operations have two things in common: (1) both rectifying and stripping sections are provided such that a separation can be achieved between two compon ents that are adjacent in volatility, and (2) the separation is effected only by the addition and removal of energy and not by the addition of any mass separating agent (MSA) such as in liquid DISTILLATE (YB = 0.75) V1 = 50 (YT 0.25) L "10 FEED BUBBLEPOINT LIQUID F = 100 X( = 0.5 X, = 0.5 (L2 = 10) (T3 = 200'F) ALL PRESSURES = 1 ATM L BOTTOMS FIGURE 8. Example of specifications and tear variables for top down, bottomup, stagebystage method. liquid extraction. Sometimes, other related multiplestage vaporliquid separation operations, such as refluxed rectification, reboiled stripping, absorption, stripping, reboiled absorption, re fluxed stripping, extractive distillation, and azeo tropic distillation, as described by Seader, may be more suitable than distillation for the specified task. All these separation operations can be re ferred to as distillationtype separations because they have much in common with respect to calcula tions of thermodynamics properties, vaporliquid equilibrium stages, and column sizing. For calcu CHEMICAL ENGINEERING EDUCATION lations involving such operations, prior to digital computers, the factors influencing the nature of the sparse matrix pattern from Sorel's equations were chosen so that a complete tearing method could be employed so the equations could be solved sequentially oneatatime. Many techniques were proposed, with the stagebystage methods of Lewis and Matheson in 19 B.C. and Thiele and Geddes in 18 B.C. being the most useful. In 6 A.D., features of these two methods were com bined into a single method, applicable to computa tions with a digital computer. Specifications are the simple set discussed previously and illustrated in Fig. 8 by an example involving two compon ents, benzene and toluene, and five theoretical stages. The tear variables (2C + Nl) in number, and typical initial guesses for them are shown in parentheses. These tear variables include the com ponent mole fractions in the distillate. The initial guesses for the distillate composi tion are conveniently obtained by using a rear rangement of the Fenske equation with the mini mum number of equilibrium stages set equal to onehalf of the total number of specified equilib rium stages. The sum of the component flow rates in the distillate must equal the specified total dis tillate flowrate and, for each component, the sum of the flow rates in the distillate and bottoms must equal the feed rate. The equations used are modifications of Sorel's equations, and include total material balances, component material balances, bubblepoints, dew points, energy balances, an adiabatic flash for the feed stage, and equations for reestimating distil late mole fractions. The incidence matrix, shown in Fig. 9, is lower triangular except for the six tear variables, which are represented as circled X's. They appear in vertical columns in the order L2, Va, V4, Ta, Y1,1, and Y2,1. The latter are the assumed distillate mole fractions. The variables across the top and the equations corresponding to the rows are ordered by stage number, as shown, where the stages are numbered from the top down, with 1 as the partial condenser and 5 as the partial reboiler. The calcu lations begin in the upper left corner and proceed down the diagonal. At the upper left corner, the first two equations, which each contain only a single unknown, are solved directly for the bottoms rate L,, and the toptray vapor rate V2. All remain ing equations are enclosed within the dashedline border, which contains all six tear variables. This large, squared region is the outer loop that con STAGE CALCULATION: j 1 1 2 5 i 4 1 3 1 x x  xxx 42 I X X X _ x xxx x I X X X 2 I xx XI 1I x I1 I xxx x x x 1 Sxx x x x x xx x xL132_ XI X X X X XT / X X XX V5 x X x 1, 1 Xx xix Xx 11 x x x x xx x xxLxXx  3 x l X X I X I xx x x xx x x x x I I xx xx xxxx o Tear Variable  Border of Inner Cycle ! Nonlinear Equatlon FIGURE 9. Incidence matrix for topdown, bottomup, stagebystage method (6 A.D.). tains 40 equations to converge. The matrix is 42 x 42, rather than 35 x 35 (calculated from N[2C + 3]) because the feed stage temperature is computed with three differ ent equations, and V, and the vapor and liquid mole fractions for the feed stage are computed with two different equations. Calculations for the outer loopinner cycle begin with stage 1, the partial condenser. All cal culations involve just linear equations in the case of compositionindependent properties, except for T2, which is computed iteratively from a non linear dewpoint equation. Variables computed from nonlinear equations are boxed. Calculations for stage 2 are completed next. Seven equations are involved, with the last five contained in a dashed inner loop, containing a single tear vari able, L,. At this step in the calculations, the stage above the feed stage has been completed and the calculation steps are now transferred to stage 5, the partial reboiler. Here, two nonlinear bubble point equations are encountered for T, and T4, and another tear variable, V,, is contained in a 5 x 5 matrix. Similar calculations are made next for stage 4. Finally, the feed stage (3) is computed by making an adiabatic flash calculation. The last two equations at the lowerright corner are used to compute a new estimate of distillate composition by comparing the feed flash conditions with those computed from the topdown and bottomup stage SPRING 1985 The method works best for feeds of narrowboilingrang e components. Otherwise, estimates of distillate composition may be too uncertain and cause difficulty in convergence. If feeds are wideboiling, the bubble and dewpoint calculations become sensitive and convergence is jeopardized. calculations, with an adjustment made to main tain the specified distillate rate. In all, four inner loops are contained within one major outer loop. Thirteen equations are in fluenced directly by the guesses for the distillate composition and ten others are influenced in directly by the corresponding bottoms mole fractions. Thus, although the complete tearing method is suitable for manual calculations, the method is relatively inefficient and limited to in sensitive cases of simple distillation of nearly ideal solutions with low reflux ratios. The method works best for feeds of narrow boilingrange components. Otherwise, estimates of distillate composition may be too uncertain and cause difficulty in convergence. If feeds are wide boiling, the bubble and dewpoint calculations be come sensitive and convergence is jeopardized. In any event, convergence may be slow, unless special acceleration techniques are used. However, the calculation by hand of just a few stages for a multicomponent mixture by this method is a very worthwhile learning experience; one not to be missed. EQUILIBRIUM FLASH METHOD Some of the limitations of the previous method were eliminated by McNeil and Motard (23 A.D.) in their development of a stagebystage algorithm that utilizes adiabatic or percent vaporization flash calculations. Their technique requires [(N1) (2C + 2) 2] tear variables, but, as shown in Fig. 10, initial guesses for all but (N2) of these variables can be set to zero. The (N2) vari ables are interior stage temperatures, which are relatively easy to estimate. If the feed is predominantly liquid, the pro cedure starts with an adiabatic flash at the feed stage followed by stagebystage adiabatic flashes in a downward direction until the partial reboiler is reached, where a percent vaporization flash is made. Subsequent adiabatic flashes are made moving up the column until the partial condenser is reached and another percent vaporization flash is made. Additional sequences of flash calcula tions are made moving down, and then up, the column until convergence is achieved. The method is not particularly suitable for manual calculations because adiabaticflash compu tations are tedious. However, flash computer sub routines are readily available, and it is relatively easy to construct an executive routine to apply the method. It is, therefore, another very worth while learning exercise, particularly because column startup is simulated. The method is ap plicable to complex distillation, and initial guesses for the tear variables are easily obtained from vapor pressure data. The flash calculations are usually not sensitive, but convergence, although DOWN AND UP STARTING FROM FEED STAGE FEED BUBBLEPOINT LIQUID F = 100 XB = 0.5 XT = 0.5 DISTILLATE V = 50 *C L = 10 (T2 = 200"F) (L2 = 0) (T3 = 200"F) (T4 = 200*F) (T5 = 200'F) i BOTTOMS FIGURE 10. Example of specifications and tear vari ables for equilibriumflash method of McNeil and Motard (23 A.D.).. almost certain, can be very slow, particularly for large ratios of internal traffictofeed flow rate. MATRIX METHODS Rather than use an equationbyequation com plete tearing technique for Sorel's equations, Amundson and Pontinen in 7 A.D., in a significant development, showed how the equations could be combined in a manner such that they could be solved in the order of type of variable, rather than by stage. However, only partial tearing was achieved and the method involved solving C sets of N x N simultaneous linear algebraic equations. CHEMICAL ENGINEERING EDUCATION To do this, they used full matrix inversion, which often led to computational difficulties. These diffi culties were overcome by taking advantage of the sparse tridiagonal form of the matrices and apply ing Gaussian elimination or LU decomposition in EQUATIONS AND VARIABLES ORDERED BY TYPE FEED BUBBLEPOINT LIQUID F = 100 XB = 0,5 X, = 0.5 DISTILLATE S V1 = 50 (Ti = 170F) = 10 (T2 = 185F) (T3 = 200F) (T4 = 215'F) (T5 = 230F) I BOTTOMS FIGURE 11. Example of specifications and tear variables for bubblepoint method of Wang and Henke (15 A.D.). the manner of Thomas. In 15 A.D. Wang and Henke applied the Thomas algorithm to narrow boiling feeds, while Burningham and Otto re formulated some of the equations in 16 A.D., fol lowing the work of Sujata in 10 A.D., to apply them to wideboiling feeds typical of absorbers and strippers. The need for two such partialtearing methods was shown clearly in 13 A.D. by Friday and Smith, who referred to the two procedures as the bubblepoint and sumrates methods. An N x N tridiagonal equation for each com ponent is formed by combining the component material balance, phase equilibrium, and a total material balance to form an equation in liquid phase mole fractions, stage temperatures, and vapor flow rates. By choosing the temperatures and vapor flow rates as tear variables, the equa tions become linear in the mole fractions, with no more than three mole fractions contained in any one equation, because one stage is connected to no more than two adjacent stages. For each com ponent, the linear equations are ordered by stage. The result is a tridiagonal matrix equation, where the nonzero coefficients are contained only on the three principal diagonals. The solution of the matrix equation is easily achieved by Gaussian elimination, as shown e.g. by Carnahan, Luther, and Wilkes, in no more than 20 lines of FORTRAN code. BUBBLEPOINT METHOD The bubblepoint method of Wang and Henke in 15 A.D. utilized the tridiagonal matrix al gorithm to obtain a computer method for solving distillation problems for relatively narrowboiling feeds. The specifications for the same 5stage, 2 component example used above are shown in Fig. 11. The tear variables are the stage temperatures and vapor flow rates. The distillate rate V1 and the reflux rate L1 are specified. Initial guesses for these tear variables are obtained with a minimum of effort by assuming constant molar overflow, in the manner of the McCabeThiele method. Esti mates of the stage temperatures are obtained by linear interpolation of the distillate and bottoms temperatures, which may be computed by dew point and bubblepoint calculations assuming the most perfect split of the feed components, con sistent with the specified distillate rate. Wang and Henke utilized a form of Sorel's equations that permits the solution by variable type rather than by stage as in the first two methods described. The equations include a total material balance to compute liquid traffic, a com ponent material balance combined with phase TYPE VARIABLE:  Border of inner   Set of Linear Equations cycle i FIGURE 12. Incidence matrix for bubblepoint method. SPRING 1985 Tear Variable O Nonlinear Equation BI x xI I T I VT I I I L TI S  i x x x X XX x  x x xx x x x S x xx I X X I X XX X x x xx x x x x x I x x XX X X X X XX I X X X X X X X X *X X I X X XX X X X X X I X X I x x XX X XX X XX XX equilibrium and total material balance to compute liquidphase mole fractions, bubblepoint equa tions to obtain stage temperatures and vaporphase mole fractions, and energy balances to compute vapor traffic. Although the same variables are computed, starting from Sorel's equations, the incidence matrix, shown in Fig. 12, is quite different from the stagebystage method. The matrix is lower triangular except for the circled tear variables and N x N (5 x 5 in this case) block sub matrices for each component (two in this case). The overall size of the matrix for the example is only 35 x 35 because no variable is computed from more than one equation. At the upper left corner, the first two variables are the same as before and are com puted directly as before. All but the last two of the remaining equations are contained in one large loop. Then the two tridiagonal submatrices are solved separately to obtain the liquidphase mole fractions. All remaining variables are computed oneatatime. Stage temperatures are computed from the nonlinear bubblepoint equation. This is followed by computation of vaporphase mole fractions. Energy balances give the vapor traffic and total material balances give the liquid traffic. The cycle is repeated until the tear variables are converged. Then the duties of the reboiler and con denser are computed. For narrowboiling feeds, the initial estimate of stage temperatures and vapor traffic will be EQUATIONS AND VARIABLES ORDERED BY STAGE FEED BUBBLEPOINT LIQUID FB = 50 FT = 50 DISTILLATE T = 12.5 (T1 = 170'F) (T2 = 185'F) (T3 = 200'F) (T4 = 215"F) (T5 = 230'F) . BOTTOMS BB = 12.5 FIGURE 13. Example of specifications and starting guesses for simultaneouscorrection method of Naphtali and Sandholm (20 A.D.). quite close to the final result and convergence is rapid using successive substitution for the tear variables. For widerboiling feeds, convergence is quite sensitive to the initial estimates of T and V and may not be rapid. In that event, use of a de layed Wegstein [Orbach and Crowe (20 A.D.)] or dominant eigenvalue technique [Rosen (29 A.D.)], rather than successive substitution, can reduce the number of iterations required. The bubblepoint method is not suitable for manual calculations because of the need to solve matrix equations. However, it is easily pro grammed if algorithms are available for solving single nonlinear equations and the tridiagonal matrix equation. The method is applicable to simple and complex distillation columns. Convergence may not be possible if the liquid phase is highly nonideal. The method provides no flexibility in specifications. The user must specify the reflux rate or ratio and the total distillate rate; however, these specifications almost always permit a real, positive solution. An exception can occur where the reflux rate is too small, such that it diminishes to zero at some stage down from the top. The bubblepoint method can be successfully applied to complex distillation e.g. two feeds, two side streams, and one intercooler. Such problems are difficult for stagebystage tearing algorithms, but are relatively easy for partial tearing algorithms like the bubblepoint method, where variables are computed by type. NEWTON'S METHOD More advanced computer methods that can handle a complete range of boilingpoint of feed components and nonideal liquid solutions, as well as offer more flexibility in problem specifications, involve handling the nonlinear equations simul taneously without the use of tear variables. Newton's method, and variants thereof, has long been the popular technique. The use of a simple twoequation manual exercise is sufficient to il lustrate to students the basic concept of Newton's *method, which may then be applied with com puter programs to hundreds of equations when solving a multicomponent, multistage separation problem. Computer methods that solve all of Sorel's dis tillation equations simultaneously may be referred to as simultaneouscorrection methods. Many such methods have been proposed and the Newtonbased NaphtaliSandholm technique of 20 A.D. is repre CHEMICAL ENGINEERING EDUCATION sentative of one of the better ones. The equations and variables are ordered by stage. To reduce the size of the matrix to be handled, component flow rates replace mole fractions and total flow rates. Thus, for the fivestage, twocomponent example, shown in Fig. 13, the number of equations to be solved is 25, rather than the 35 previously. The specifications are different from previous ones in that distillate and bottoms purities replace re flux and distillate rates. Such specifications should be used with caution and the Fenske (19 A.D.) minimumstage equation should be checked to make sure that the minimum number of required stages for the specified purities is less than the specified number of five. Theoretically, initial guesses must be provided for all 25 unknowns, but these guesses can be generated by the program based on guesses of just a few temperatures and vapor rates as shown. These guesses are called tear variables here, but are really not tear vari ables in the strict sense. The initial guesses are generated easily from the T and V guesses by solving the tridiagonal matrix equations of the WangHanke method for the liquidphase mole fractions, from which the initial guesses for the component flow rates are obtained readily from their definition and the component material balances. The NaphtaliSandholm method only involves three types of equations, namely stage component material balances, phase equilibrium in terms of Kvalues, and stage energy balances. The form of the equations is almost identical to the original equations of Sorel. Because bubblepoint, dew point and flash calculations are absent, sensitivity problems are largely avoided. With equations and variables ordered by stage, the incidence matrix, shown in Fig. 14, is block tridiagonal in shape. The blocks are 5 x 5 in this example. The matrix is for the linearized form of the equations, which permits the application of Newton's method. Thus, the matrix is the Jacobian of partial derivatives and an X entry signifies a nonzeroderivative. The entire matrix is iterated to convergence. The solution of the block tridiagonal matrix is obtained readily by modifying the previously mentioned Thomas algorithm for a tridiagonal matrix. The only significant changes are the re placement of matrix multiplication for scalar multiplication and matrix inversion and multipli cation for division. If large numbers of components are present, the (2C + 1) x (2C + 1) submatrices may be large and timeconsuming to invert. The convergence criterion is based on the sum of the squares or socalled square of the Euclidean norm of the three different types of functions. Early iterations are often damped to avoid corrections that are too large. Because of the block nature of the matrices in the NaphtaliSandholm method, it is not at all suitable for manual calculations. Furthermore, the computer program is rather complex. Consequent ly, it is best to obtain the code from one of several STAGE CALCULATION: 1 2 3 5 X XXX X X X X X X x x xx x x x x x x x xxxxxxx x xxx x x xx X X XXXXXX X X X X X X X X X XXXX_ ___ xxxxxxxxXXX X X X X X X X X xxx xx xx xx XXX X x x x x x x xx x X X X X XXXXX x x x x x x x x x x FIGURE 14. Incidence matrix for simultaneouscorrec tion method. sources, e.g. Fredenslund et al (26 A.D.). Versions ranging from PC to Mainframe codes are avail able. The method is applicable to all singlecolumn, complex multistage operations, including those with highly nonideal liquid solutions. Flexibility in specifications is provided at the top and bottom of the column by substituting specification equa tions for the condenser or reboiler energy balances. For example, specification options at the top in clude condenser duty, reflux rate, reflux ratio, dis tillate rate, component molefraction purity and component distillate rate. Convergence is rapid from good initial guesses, but may otherwise be slow and require damped corrections. The method can fail, particularly if initial guesses are very poor. CONTINUATION METHOD AND MULTIPLE STEADYSTATE SOLUTIONS Ideally, especially in practice, one would prefer SPRING 1985 a multistage, multicomponent separation com puter method that would offer complete flexibility in specifications and would always converge to a correct solution. Newton's method and most of its variants are known to be only locally con vergent. That is, the initial guesses must be within a certain region of the variable space or con vergence will not be achieved. This region can be expanded by adjusting the Jacobian in Newton's method or employing a hybrid method such as that of Powell (19 A.D.) or Marquardt (12 A.D.), Equations: 20 x + x2 = 17 I(8X) 3 + x2= 1 H omotopy x2 Path FIGURE 15. Example of regions of convergence for some methods of solving nonlinear equations. which combine the best features of Newton's method with steepest descent. To achieve complete robustness, however, it is necessary to employ a globally convergent technique, such as differential homotopy continuation, for which four algorithms, in FORTRAN, have become available starting in 25 A.D. The one by Kubicek is particularly easy to understand and apply, but is not written for sparse matrices. The regions of convergence for a simple two nonlinearequation example are shown in Fig. 15. Newton's method will converge to X1 = 1 and X, = 4 from an initial guess of X1 = 2, X2 = 5. As seen, another root exists at X, = 4.07 and X2 = 0.65, which can be reached by Newton's method from a nearby initial guess. With Newton's method, the initial guesses must lie within the rather narrow and confined crosshatched regions. With Powell's hybrid method (which is available in the Harwell library, the IMSL library, and MINPACK) the region of convergence is expanded outward to the dashed lines. Both methods will fail badly from a starting guess of 15 and 15. The use of differential homotopy continuation gives convergence from any starting guess, with a typi cal homotopy path to one of the two roots, shown as a dashdot line starting from (15, 15). Many types of homotopy paths have been pro posed, with the linear homotopy being common. The linear homotopy, h, is set equal to the function to be solved, f(X), multiplied by a homotopy pa rameter, t, and added to a function g(X), whose solution is known, multiplied by the function (1 t). The calculations start from the known solution at t = 0, where h = g and X = Xo and move along the path of h vs. t as t is gradually in creased to a value of one, at which point h = f, whose solution X* is to be determined. Choices for the function g(X), with a known solution, Xo, are almost unlimited. For consistency with Newton's method, the Newton homotopy is useful, where g(X) is set equal to f(X) f(Xo), where Xo can be selected arbitrarily. With this homotopy, h equals f(X) minus (1 t) times f(Xo). Alternatively, (1 t) can be replaced by a new homotopy parameter, X, to obtain a slightly more compact form for the homotopy expression. The path will then be from X = 1 to X = 0. If the homotopy path is simple, without turns or rapid changes in X with t, classical continuation can be employed by selecting a sequence of values of t at 0, t,, t2, t, etc., and 1, with X being solved from h at each step by Newton's method using an initial guess equal to the solution from the pre vious step. Thus, Newton's method is embedded into classical continuation. This technique of using continuation is not common though because it is not globally convergent and can not, in general, solve problems that fail with Newton's method alone, which amounts to moving in one step from t = 0 to t = 1. To be robust, one must closely follow the homotopy path and not just continually take steps in t with Newton corrections in Xspace. For example, classical continuation as well as Newton's method will fail on the cubic equation, x3 30x2 + 280x 860 = 0, because of two singular points at about x = 7.418 and 12.582, where the derivative of the function becomes zero. CHEMICAL ENGINEERING EDUCATION C1 40 C2 = 30 C 3= 30 R 3 HP 30 L_. ) Stage 19 FIGURE 16. Example of specifications for an interlinked system of Petlyuk towers. For initial guesses of x less than about 12.6, Newton's method fails to find the single real root at about 15.55. The homotopy path for this cubic function depends on xo the choice of g(x). For g(x) = x x0 or g(x) = f(x) f(xo), the two singularity points become turning points in the path, and it is important that the continuation method follow the path closely around these turn ing points to avoid cycling in the manner of Moses in the Sinai. Consider the application of the homotopycon tinuation method to the Petlyuk system of two interlinked towers shown in Fig. 16. The feed is a ternary mixture, which is to be separated into three products. A sloppy split is made in the pre fractionator, with the final three products being produced in the second tower. Reflux and boilup for the first tower are provided by the second tower. The two towers in the system can be solved by alternating back and forth between separate iterations on the individual towers. But numerous studies have shown that it is more efficient to con verge the two towers simultaneously. The stages for the two towers are ordered as shown starting at the top of the second tower, switching to the first tower after stage 11, and then switching back to the second tower after stage 15. Specifications include interlink flows from stage 4 to stage 12 and from stage 16 to stage 15; the reflux ratio; the middle product flow rate; and the bottoms flow rate. The types of equations solved are as in the NaphtaliSandholm method except that balances include interlink flows and provision, if desired, for entrainment of liquid droplets, occlusion of vapor bubbles, and chemical reaction. Phase equilibrium equations can include a Murphree plate efficiency that can be specified by component and tray location. A FORTRAN computer code for applying differential homotopycontinuation to such a prob lem was reported by Wayburn and Seader in 32 A.D. Considerable flexibility in specifications is provided, including at any stage, total flow rates or ratios, component flow rates or purities, and stage temperatures or heat transfer rates. The equations are linearized in the Newton manner and ordered by stage to a bordered, blockdiagonal structure, which is processed by an efficient and stable blockrowreduction algorithm. An attempt is first made to solve the equations by Newton's method, using a line search on the Euclidean norm of the function residuals to determine the best damping factor. If Newton's method fails, then differential homotopy continuation with a linear f(x) xf(x) = 0 IVP: df(x) dx d ) 1 a x fdx dp )  I dx, + x (x IC: =0 ) = 1, x = 1C: p= 0, Az 1, x = x0 FIGURE 17. tinuation. Equations for differential homotopy con Newton homotopy is employed. The differential form of homotopy continuation was first proposed by Davidenko in 2 A.D. As shown in Fig. 17, the homotopy function, f(X)  Xf(Xo), is differentiated with respect to arc length, p (i.e., distance along the path), to con vert a system of M nonlinear equations to a system SPRING 1985 of M + 1 ordinary differential equations that constitute an initial value problem. Because of the extra variable, p, an additional equation is need ed. This is provided by the Pythagorean theorem applied in (M + 1)dimensional space. Fortunate ly, the set of differential equations is not stiff. Rather than simply integrating the differential equations by, say, a RungeKutta method, it is pre NEARLY BLOCK TRIDIAGONAL AND BORDERED FORM S2 3 4 5 6 7 8 910111213141516171819 1 BC 2 ABC 3 ABC 4 ABC C 5 ABC 6 ABC 7 ABC 8 ABC 9 ABC 10 ABC II A C 12 A BC 13 ABC 14 ABC 15 ABC 16 A ABC 17 ABC 18 ABC 19 AB s4 FIGURE 18a. Example of incidence matrix for homotopy continuation method. ferable and more efficient to follow the homotopy path by alternating between an Euler predictor for the differential equations and two Newtonstep correctors for the nonlinear homotopy equations. The Euler step moves the variables somewhat off the path, but the Newton step corrects the vari ables back sufficiently close to the path. An im portant aspect of such a technique is the size of the Euler step, for which a number of stepsize al gorithms have been proposed, as discussed by Seader and Wayburn. The incidence matrix for the Petlyuk column example is in Fig. 18a, where the organization is by stage. Each letter, A, B, or C actually repre sents, in this example, a nonzero 7 x 7 submatrix, which applies to both the coefficients of the differ ential homotopy equations and the Jacobian partial differentials of the nonlinear homotopy equations. The matrix is almost block tridiagonal with bottom and rightside borders, which contain the nonstandard specifications. The four disperse submatrices of A and C, located above and below the three principal diagonals represent the inter links. By moving the number 4 and 16 rows and columns, which contain the disperse submatrices, to the borders, the block diagonal and bordered matrix form, shown in Fig. 18b, is obtained. Solu tion of the corresponding matrix is achieved block by block, starting at the upper lefthand corner, by a blockrow reduction algorithm, which treats the righthand border as part of the righthand side vector. The differential homotopycontinuation method has been applied to the interlinked system shown in Fig. 19 for a ternary aromatic system, over a range of reflux ratios from 4.55 to 5.75. Purity specifications of between 90 and 95 mole% are made for each product, and a bottoms rate of 380 is specified. The program must compute the re quired interlink flow rates, including L, the liquid interlink recycle from the second column back to the first column. In some cases, Newton's method converged, for this system, while in other cases, the differentialhomotopycontinuation method had BLOCKED AND BORDERED FORM I 2 3 5 6 7 8 91011121314151718194 16 I BC 2 ABC 3 AB C 5 BC A 6 ABC 7 ABC 8 ABC 9 ABC 10 ABC II AB C 12 BC A 13 ABC 14 ABC 15 AB C 17 BC A 18 ABC 19 AB 4 AIC C B 16 A AC FIGURE 18b. Permuted incidence matrix for homotopy continuation method. CHEMICAL ENGINEERING EDUCATION to be applied to obtain a solution. In Fig. 20, a plot of L, the liquid interlink rate versus the reflux ratio, shows unexpected multiple solutions, three in number. For example, at a re flux ratio of 5, the specifications were achieved with three different liquid interlink rates of about 110, 330, and 420 lbmoles/hr. Such multiple solu tions have long been known to exist for certain cases of an adiabatic reaction in a CSTR reactor, but have not been observed previously for distilla tion. When such solutions are close together, as for solutions 1 and 2 at low reflux ratios, possible control problems could arise. BubblePoint Lqulid Fed nzee (B) zo00O Toluen (T) 200 o*Xyler. {X 400 XB = 0.95 5 o 5.75 X = 0.95 B = 380 FIGURE 19. Example that gave multiple solutions. The continuation method is not at all suitable for manual calculations. The computer code is lengthy, but is applicable to all kinds of complex multistage operations, including interlinked columns. Except for tray numbers, complete flexi bility in specifications is permitted. When the homotopy is constructed properly, convergence is always achieved. The method is best suited for cases where the NaphtaliSandholm fails or can't be applied. The method can find multiple solutions if they exist. TRANSPORT MODEL Sorel's equilibriumstagemodel of almost 100 years ago has served us well in the calculation of multicomponent, multistage separation opera tions. However, that model has always been sus pect for applications to systems of known moder S07 . . 4. 5 4. 9 5. 1 5. 3 5. 5 7 REFLUX RATIO FIGURE 20. Multiple solutions to Petlyuk towers. atetolow stage efficiency. For that reason, some programs, such as the SC method of Naphtali and Sandholm and the differentialhomotopycon tinuation method of Wayburn and Seader in corporate a Murphree tray efficiency, which ac counts to some degree for masstransfer effects. However, the plate efficiencies must be specified, and heat transfer effects are ignored. A better approach is to apply a transport model to handle nonequilibrium directly. Such a model has just been developed by Krishnamurthy and Taylor, who account for multicomponent masstransfer interactions and heat transfer. Their modeling equations are written separately for the vapor and liquid phases with coupling by liquid and gas mass transfer rates, and energy transfer rates. These transport rates are estimated from carefully formulated mass and energy transfer coefficients, applicable to multicomponent systems. For non interlinked columns, the resulting equations lead to an incidence matrix that is similar to that of the NaphtaliSandholm method, for which a solu tion technique is well established. Krishnamurthy and Taylor have applied their method, with good success, to several sets of experimental data from the operation of small laboratory columns. Data from commercialsize columns are now being sought to make further comparisons of predicted and measured compositions so as to evaluate the usefulness and applicability of this transport model. CONCLUSIONS AND RECOMMENDATIONS The digital computer has been responsible for sweeping changes in the manner in which multi stage separation operations are synthesized and SPRING 1985 A c P I I f P ~ ~t' TABLE 1 Recommended Additions to Content of Undergraduate Courses 1. Numerical methods for A. Linear algebraic equations B. Sparse matrices C. Systems of nonlinear equations 2. Application of numerical methods to A. Complete tearing, partial tearing, and simul taneous correction methods for multicomponent separation processes 3. Use of computeraided simulation programs to A. Analyze, correlate and predict multicomponent thermodynamic properties B. Solve openended separation process problems in volving energy integration 4. Secondlaw analysis analyzed. Chemical engineering educators need to closely examine courses on separation processes to make sure that students are being instructed in modern and efficient computational tools. Some recommended additions to the content of under graduate courses are listed in Table 1, where many of the items should prove useful in other chemical engineering subjects as well. Numerical methods should be stressed for linear algebraic equations, including efficient handling of sparse matrices, and systems of non linear equations. These methods should then be applied using computers to utilize partial tearing and simultaneous correction methods for multi component separation processes. However, some manual calculations on simple examples should be performed using complete tearing methods to help develop a basic understanding. More complex and openended separation problems should be assigned that stress energy integration. A secondlaw analysis [see Denbigh (5 A.D.), and de Nevers and Seader (28 A.D.)] of a process should be required, and attempts should be made to improve the process by finding eco nomical means to reduce the lost work. The development of a separation process can be an exciting experience when computers and computer programs are available to perform the tedious calculations and allow time for more con sideration of synthesis and optimization as pects. Ol REFERENCES Amundson, N. R., and A. J. Pontinen, Ind. Eng. Chem., 50, 730 (1985). Burningham, D. W., and F. D. Otto, Hydrocarbon Pro cessing, 46 (10), 163170 (1967). Carnahan, B., H. A. Luther and J. O. Wilkes, Applied Numerical Methods, John Wiley, New York (1969). Davidenko, D., Dokl. Akad. Nauk USSR, 88, 601 (1953). Denbigh, K. G., Chem. Eng. Sci., 6, 19 (1956). de Nevers, N., and J. D. Seader, "Mechanical Lost Work, Thermodynamic Lost Work and Thermodynamic Efficiencies of Processes," paper presented at the AIChE 86th National Meeting, Houston, Texas, April 15, 1979. Fenske, M. R., Ind. Eng. Chem., 24, 482485 (1932). Fredenslund, A., J. Gmehling, and P. Rasmussen, "Vapor Liquid Equilibria Using UNIFAC, A Group Contribu tion Method." Elsevier, Amsterdam, (1977). Friday, J. R., and B. D. Smith, AIChE J., 10, 698 (1964). Henley, E. J., and J. D. Seader, EquilibriumStage Separa tion Operations in Chemical Engineering, John Wiley and Sons, New York (1981). Krishnamurthy, R., and A. Taylor, AIChE J., 31, 449465 (1985). Kubicek, M., "Algorithm 502," ACM Trans. on Math. Soft ware, 2, No. 1, 98 (1976). Lewis, W. K., and G. L. Matheson, Ind. Eng. Chem, 24 496498 (1932). Marquardt, D. W., SIAM J., 11, 43141 (1963). McCabe, W. L., and E. W. Thiele, Ind. Eng. Chem., 17, 605611 (1925). McNeil, L. J., and R. L. Motard, "Multistage Equilibrium Systems," Proceedings of GVC/AIChE Meeting at Munich, Vol. II, C5, 3 (1974). Naphtali, L. M. and D. P. Sandholm, AIChE J., 17, 14 (1971). Orbach, 0., and C. M. Crowe, Can. J. Chem. Eng., 49, 509 513 (1971). Ponchon, M., Tech. Moderne, 13, 20, 55 (1921). Powell, M. J. D., "A Hybrid Method for Nonlinear Equa tions," in "Numerical Methods," Ed. P. Rabinowitz, Gordon and Breach, New York (1970). Rosen, E. M., "SteadyState Chemical Process Simulation: A Stateofthe art Review," Computer Applications to Chemical Engineering, R. G. Squires and G. V. Reklaitis, editors, ACS Symp. Ser. No. 124 (1980). Savarit, R., Arts et Metiers, pp. 65, 142, 178, 241, 266, 307 (1922). Seader, J. D., Section 13 of Perry's Chemical Engineers Handbook, 6th ed., McGrawHill, New York (1984). Sujata, A. D., Hydrocarbon Processing, 40, No. 12, 137 (1961). Thiele, E. W., and R. L. Geddes, Ind. Eng. Chem., 25, 289 (1933). Wang, J. C., and G. E. Henke, Hydrocarbon Processing, 45 No. 8, 155 (1966); Hydrocarbon Processing, 45 No. 9, 169 (1966). Wayburn, T. L., and J. D. Seader, "Solutions of Systems of Interlinked Distillation Columns by Differential HomotopyContinuation Methods," Proceedings of the Second International Conference on Foundations of ComputerAided Process Design, June 1924, 1983, Snowmass, Colorado (available from CACHE Corp., P. O. Box 7939, Austin, Texas 787137939). NOMENCLATURE A, B, C, Coefficients in a tridiagonal matrix equa CHEMICAL ENGINEERING EDUCATION B f F HF h H H, H, K L MP p Q R s tion; submatrices of partial deriva tives in a block tridiagonal matrix Bottoms product molar flow rate An arbitrary function Molar feed rate to a stage; mathematical function Molar enthalpy of feed to a stage The homotopy function whose arguments are x and t The homotopy function whose arguments are x and X Molar enthalpy of vapor leaving a stage Molar enthalpy of liquid leaving a stage Vaporliquid equilibrium ratio Molar liquid flow rate leaving a stage Middle product molar flow rate Path length Heat duty (R for reboiler; C for con denser) Reflux ratio Ratio of liquid drawoff to primary liquid (liquid not withdrawn or entrained) DEPARTMENT: Cornell Continued from page 61. and measuring the properties of liquids and liquid mixtures using theory, computer simula tion, and experiment. Cornell is one of very few institutions with strength in all three areas. Keith guides the theory and the computer simulation (with help from Senior Research As sociate Steve Thompson,) making use of recently developed accurate theories for dense fluids of complex molecules as well as improved computer simulation methods and computer hardware. Typically, highly nonideal substances (in the thermodynamic sense) are chosen for study; sub stances for which traditional methods of pre diction fail. Examples include mixtures occurring in coal gasification and liquefaction, hydrogen energy technology, synthetic fuel processing and supercritical fluid extraction. Other research underway or planned includes studies of ad sorption at gasliquid, liquidliquid and solidfluid interfaces, nucleation and droplet phenomena, polarization in polar fluids, and surfactant effects. Bill Streett and Senior Research Associate John Zollweg carry out experimental studies of dense fluids. In progress are (i) experiments in vaporliquid, liquidliquid, and gasgas equilibria at temperatures from 70 to 500 K and pressures to 10,000 atmospheres; (ii) equationofstate (PVT) measurements of pure liquids and mix tures at temperatures from 70 to 500 K and pres sures to 4,000 atmospheres; and (iii) measure ments of enthalpy of mixing in samples of lique fied gases at temperatures from 70 to 300 K and pressures to 20 atmospheres. Bill is currently de veloping new experiments to measure the surface and interfacial tensions and the velocity of sound in fluids under pressure. The researches of Paulette Clancy, who became a member of the faculty in 1984, range from a statistical mechanical study (using perturbation theory) of multicomponent highly polar fluid mixtures to a development of phase diagrams (based on molecular thermodynamics) of semi conductor materials. In addition, she is involved in the application of computers to chemical engi neering. Herb Wiegandt's interest in desalting sea water, using a freezing process based on direct contact with butane, goes back to 1958. Recent efforts, with Bob Von Berg as a partner, have aimed at overcoming the problems associated with washing and separating the ice crystals which are typically very small. Julian Smith, past Director of the School in a period of unprecedented growth, seasoned edu cator and coauthor of Unit Operations of Chemi cal Engineering (now in its fourth edition, with Pete Harriott as coauthor), has expertise in mixing, centrifugal separation, and handling of granular solids. He is teaching fulltime and is active in the guidance of the school. Ray Thorpe, who has advised graduate students in the areas of phase equilibria and separations processes, splits his time between SPRING 1985 S Ratio of vapor drawoff to primary vapor (vapor not withdrawn) t Homotopy parameter. T Temperature; when used as a superscript denotes matrix transpose V Molar Vapor flow rate leaving stage x The vector of independent variables (un knowns) for the distillation equations; liquidphase mole fraction x The starting vector for the nonlinear equation solver x* The solution to the set of nonlinear equa tions X Mole fraction in liquid of a component; variable y Vaporphase mole fraction Y Mole fraction in vapor of a component Z Mole fraction in feed of a component Greek Letters X Homotopy parameter teaching and university administration: he is di rector of the Division of Unclassified Students. Research Interactions Many research projects involve active col laboration with other researchers at Cornell or elsewhere. Some are directly with other depart ments; some are through Cornell's numerous in disciplinary programs, centers, and institutes many industrially supportedthat facilitate inter action among departments and with industry. Examples are the Biotechnology Institute; the In jection Molding Project; COMEPP (Cornell Manufacturing Engineering and Productivity Program); Applied Mathematics Center; Theory and Simulation Center (established by Ken Wilson, Cornell's 1982 Nobel laureate in physics) ; Materials Science Center; National Facility for Submicron Studies. Strong ties have been es tablished with other departments and colleges of Cornell, and with workers at other universities around the world. Paulette Clancy, Associate Director of COM EPP, is joined by Professor Scheele in a study of ways to improve the interface between the user and ASPEN software chemical process syn thesis and design. UNDERGRADUATE PROGRAM Undergraduate chemical engineering enroll ments at Cornell were almost constant during the twenty years before 1975, with about 40 bache lor's degrees awarded annually. Then, although freshman admissions to the engineering college were held constant, the number of students opt ing for chemical engineering roughly doubled, and for nine years the number of BS degrees awarded was between 65 and 75. After 1985, however, the number will return to 40 or so and is expected to stay at that level for the next several years. The subject matter of the undergraduate pro gram is much the same as at other institutions. For the first two years the students are not in chemical engineering but are enrolled in the "com mon curriculum" of the engineering college. Never theless, their curriculum has much that is differ ent from that of other engineering students. In the freshman year chemical engineers take two semesters of chemistry, not one. Sophomores take two semesters of physical chemistry, with labora tory each terma special course taught by Chemis try almost exclusively for chemical engineers and the required introductory course in mass and energy balances. Organic chemistry (two semes ters, one with laboratory) is given in the third year, as are chemical engineering thermody namics, rate processes and separation processes. The fourth year includes required courses in re action kinetics, process evaluation, process con trol, and unit operations laboratory, and a spring term course in process design. Overall, 132 credit hours are required for the BS degree, including two courses in computer programming and ap plications, four engineering distribution courses, and six courses in humanities and social sciences. Ten of the required courses (32 credit hours) are in chemical engineering subjects. The senior laboratory course is considered the most demanding by students and faculty alike. Each student reports on only five experiments during the term, but each report is thoroughly edited for both form and content by the faculty member in charge of that experiment and nearly always must be extensively revised by the student before it is accepted. The emphasis is on technical accuracy, completeness, and clarity of expression. Oral presentations are stressed in the senior design course, in which each team of students makes weekly oral presentations before two faculty members or industrial visitors. In recent years experienced engineers from industry have been hired for fulltime assistance in this course and in the senior laboratory. Their contributions have been supplemented, during shortterm visits, by those of people from Exxon, Union Carbide, and other firms. Despite this, the laboratory and de sign courses demand large contributions of time by senior faculty members, and pose the most difficult problems for future staffing. A Special Cooperative Program For the past ten years the better students in the sophomore year have been encouraged to enroll in an unusual industrial cooperative program which gives them practical experience without lengthening their time at the university. Typical ly 15 to 20 students are accepted into the program after company interviews exactly like those for permanent employment. Coop students take the fallterm thirdyear courses during the summer following their sophomore year; they work in in dustry during the fall and return to Cornell in the spring; work again for the same sponsor the following summer; and complete their senior year CHEMICAL ENGINEERING EDUCATION in the regular sequence. Industrial assignments are carefully monitored to insure appropriateness, and each student is visited at the worksite by a Cornell person at least once during the course of the program. THE PROFESSIONAL MASTER'S PROGRAM This is a twosemester nonthesis master's program leading to the degree Master of Engineer ing (Chemical). It requires 30 credit hours of advanced technical work, including a substantial design project, with emphasis on practical ap plications. Most of the matriculants are not from Cornell or other U.S. schools; instead the program is attractive to foreign students, especially from developing countries such as the Dominican Re public, Guatemala, India, Kuwait, Taiwan and Venezuela. Over the years a chemical company in India has sent, one after another, three of its top technical employees to this program. Required courses for the MEng (Chemical) degree include equipment design and selection, numerical methods, reactor design, the design pro ject, and a chemical engineering elective. The remaining credit hours can be filled by elective courses in science or engineering or in the Gradu ate School of Management. The choice of subjects for MEng design projects is much wider than in the typical undergraduate design course, and more initiative and originality are expected of the students. Some of the projects are done in close collaboration with industrial firms. RELATIONS WITH INDUSTRY The school has always had close relations with industry and an unusually supportive group of alumni. Industry helps us in many ways: in the design courses; in a "Nonresident Lecture Series" (zero credit, but compulsory), given to seniors on the various kinds of professional careers; in un restricted grants; in scholarships, fellowships, and sponsored research. Continuing fellowship sup port has been provided by Amoco, Chevron, Dow, DuPont, Exxon, Shell, Stauffer and Union Car bide, and recent large research projects came from IBM, Kodak and Mobil. In 1981 the Sun Company gave $250,000 over three years to sup port research initiation on ideas too new and ill defined to merit submission of a proposal to NSF or other granting agencies. This unusual grant led to a number of publications and several con tinuing sponsored research programs. ADVISORY COUNCIL An advisory council, largely from industry, was formed a few years ago. It meets in Ithaca twice a year to review progress and help the di rector steer a course for the school. About half the members are alumni. Recently expanded to 15 members, the council now includes four aca demic people: Andy Acrivos (Stanford), Gus Aris (Minnesota), Gary Leal (CalTech), and Bill Schowalter (Princeton). We don't always agree with the council's suggestions, of course, but as a group it has been marvelously effective in pro viding an "outside" viewpoint and keeping us from being too provincial or selfsatisfied. WHAT OF THE FUTURE? Cornell is facing many of the same problems that face other chemical engineering departments around the countryfaculty retirements and fu ture faculty development, staffing of design and laboratory courses, the optimum use of computers for teaching, expansion of research and the gradu ate program, and renovation of aging facilities. The five professors hired right after World War II are nearing retirement, so for the next several years an average of one new faculty member per year will have to be hired to keep the number constant. Because of the loss of professors with industrial experience one or more people with an industrial background will probably be hired on a nontenuretrack basis to teach design and to supervise the laboratory courses. A related problem is in the use of computer software. How much emphasis should be placed on teaching the use of ASPEN, for example? More generally, as personal computers become ubiquitous, what will happen to teaching methods? Will the course in mass and energy balances, for example, become a course in the use of available canned programs? The total number of graduate students in the school, and the fraction going for a PhD rather than an MS, should rise somewhat over the next ten years, depending on the availability of financial support. This will increase the need for equipment and laboratory and office space. Rela tively speaking, the Chemical Engineering School has a lot of space, but much of it is virtually un usable for modern research. A comprehensive building renovation plan, made by a firm of archi tects, proposes a complete reallocation and rear rangement of available space and the conversion SPRING 1985 of the enormous unit operations laboratory into offices and small research labs. New electrical and other services will be provided, along with central air conditioning. The average faculty office will shrink from over 400 to a more modest 200 square feet and the offices will be grouped more closely, to stimulate greater interaction among the oc cupants. The total estimated cost is some fifteen times the original cost of the building. A fund drive for the first stage is being launched. D AWARD LECTURE Continued from page 77. on the batch experience and using verified mathe matical models to both design the equipment and direct the experimentation. CHEMICAL VAPOR DEPOSITION Laboratory Scale Research A low pressure chemical vapor deposition (LPCVD) system for amorphous silicon is shown in Fig. 14 and the simplified process flow diagram as Fig. 15. Reactants, Si2H,, and material for doping the film, PHs and B2He, in a stream of argon are controlled by valves at the inlet to the reactor. The tubular quartz reactor is temperature con trolled inside an electric furnace. System pressure is controlled manually with a valve at the exit. Effluent gas can be analyzed by gas chromatogra phy and unreacted material is decomposed in a furnace before venting. The detailed operation of this system is described by Bogaert [9]. This effort in amorphous silicon research, spon FIGURE 14. Photograph of Low Pressure Chemical Vapor Deposition unit (LPCVD). scored by the Department of Energy through the Solar Energy Research Institute, is ongoing at the present time and is far from being complete. I am discussing it here to allow the reader to con trast and compare with the physical vapor depo sition reacting systems just described. The chemistry is much more complex for amorphous silicon than for CdS and not well FIGURE 15. Simplified process flow diagram of LPCVD. understood. The present stateoftheart is shown below: Gas Phase Si2H, ;SiH4 + SiH2 SisHs Si2He + SiH, SiHloSiiH, + SiH2 SisHijsSiHo + SiH2 SieH,,4SisH,1 + SiH2 SiH1,,SiH,, + SiH, SisHs8SiH,, + SiH2 Film Formation SisH2*3SiHo.os + 2SiH, + 1.88H2 SiH2 *SiHo.os + 0.96H, This is a preliminary set of chemical equations. The gas phase equations are based on the results of Ring [10], John and Purnell [11], and Bowery and Purnell [12]. The film formation equations are based upon our own preliminary research. The component mass balance equations for this tubular reactor system are given below: Gas Phase D4q j dCi= Ir(rxt,i) + kga (CiCs) \v;rD'j dZ CHEMICAL ENGINEERING EDUCATION Surface 0 = kga(Ci Ci,) ka(yiCi) Film 1 dprV' = ka(yiCi) MW, dt Both the gas phase composition and the film growth rate are functions of axial position. Film growth rate (i.e., amount of amorphous silicon deposited) at any axial position can be determined but it has been possible to measure gas composition only at the reactor exit. The gas and solid phase mass balance equations are coupled through the chemistry of film formation and the transfer from the bulk gas to the surface. Solution of the model equations produces the gas phase exit composition versus reactor holding time plots shown as Figs. 16 and 17. The solid lines were obtained using our present "best" estimates of the specific reaction constants. This "best" estimate is now obtained by using the ex perimentally determined growth rate in the solu tion of the set of mass balance equations. The agreement between data and the predicted values is only fair but we expect to improve the model loo 100 I I NT.400*C NSSl2H6 80 P= 24 Torr SIH4 SSi1H,8 S60 S12H * S40 SSIH4 20 /S13H 0 o0 20 30 40 50 60 70 Holding Time. (sec.) FIGURE 16. Normalized molar percentages versus hold ing time: Major silanes. predictions as we learn more about the system. This research on the chemical reactor and re action engineering for amorphous silicon in the LPCVD reactor is closely coupled with studies of the material and electronic properties of the film and much effort has been devoted to finding the best conditions for good photovoltaic amorphous silicon. To date, we have been able to make a 4% solar cell using material from the LPCVD reactor. These efforts are described in the work of Hegedus et al [13]. Holding Time, (sec.) FIGURE 17. Normalized molar percentages versus hold ing time: Minor silanes. CONCLUDING REMARKS Incorporating chemical reactor and reaction engineering analysis into a semiconductor research effort requires the researchers to achieve a quanti tative understanding of both the molecular phe nomena and the transport phenomena associated with the creation of the semiconductor materials. A test of this understanding is the ability to write useful mathematical descriptions of the laboratory scale reacting system. Mathematical descriptions are an essential part of the analysis because they provide the language which allows the profes sionals doing the research to effectively and un ambiguously communicate with each other. Com munication is easier if the models are simple and, of course, the model predictions must be verifiable by experiment. In fact, the model behavior should be used to plan the experimental program because an enlightened use of a chemical reaction and re actor engineering analysis will identify critical molecular and transport phenomena problems and direct experimental attention to them with the proper priorities. We originally became interested in the semi conductor research because of a need to design larger scale reacting systems. However, the last five years of research has taught us that the chemi cal engineering analysis is very useful in the laboratory scale research effort, and indeed es sential, if such research is to be done efficiently and with minimum expense (a key issue with today's research costs). It is not possible, in our view, to effectively de sign and operate larger scale systems without re action and reactor engineering analysis. In photo SPRING 1985 voltaic applications it is also necessary to carry out analysis and experimentation at the unit opera tions scale. Those who have attempted to scale up without following these procedures have wasted time and money building equipment which is in adequate for the commercial scale processing of solar cells. A useful start has been made in applying chemical engineering analysis to the deposition of thinfilm semiconductors but much effort now must be devoted to the task of relating electronic and optical properties to the design and operation of a reacting system. When we have learned to do this properly, we can begin to "tailormake" material with any desired property. ACKNOWLEDGMENTS Semiconductor chemical reaction and reactor research requires a team effort involving a number of professionals. I am particularly indebted to B. N. Baron, R. E. Rocheleau, S. C. Jackson and R. J. Bogaert, my chemical professional colleagues at the Institute of Energy Conversion. Their analysis, their effective experimentation and their discussions with me have been essential to the de velopment of this field. None of the research could have been carried out without the excellent semi conductor material development and analysis and device design and analysis that my other col leagues at the Institute of Energy Conversion do so well. I am also in their debt for their willing ness to educate a chemical engineer in the art and science of applied solid state physics. Science and engineering research today re quires some considerable management talent. The Department of Energy's photovoltaic office and the Solar Energy Research Institute have worked very hard to develop a rational plan for photovoltaic research that both produces results and handles the political pressures that arise in a budget con scious government. The management group within the Institute of Energy Conversion is unique in its capability to protect the director from ad ministrative detail and to allow me to put most of my effort into technical work. I would like to thank S. Barwick and M. Stallings for this gift. Ol REFERENCES 1. Thornton, J. A., Annual Rev. of Material Science, 7, p. 239 (1977). 2. Kern, W. and V. S. Ban, Thin Film Processes, (J. Vossen and W. Kern, editors) Academic Press, New York (1978). 3. Dutton, R. W., "Modeling of the Silicon Integrated Circuit Design and Manufacturing Process," IEEE Trans. Electron Dev., 30, 9, p. 968 (1983). 4. Rocheleau, R. E., B. N. Baron and T. W. F. Russell, "Analysis of Evaporation of Cadmium Sulfide for the Manufacture of Solar Cells," AIChE Journal, 28, 4, p. 656 (1982). 5. Jackson, S. C., PhD Thesis, "Engineering Analysis of the Deposition of CadmiumZinc Sulfide Semicon ductor Film," University of Delaware (1984). 6. Rocheleau, R. E., PhD Thesis, "Design Procedures for a Commercial Scale Thermal Evaporation System for Depositing CdS For Solar Cell Manufacture," University of Delaware (1981). 7. Griffin, A. W., MChE Thesis, "Modeling and Control of a Unit Operations Scale System to Deposit Cad mium Sulfide for Solar Cell Manufacture," Uni versity of Delaware (1982). 8. Brestovansky, D. F., B. N. Baron, R. E. Rocheleau and T. W. F. Russell, "Analysis of the Rate of Vapor ization of CuCl for Solar Cell Fabrication," J. Vac. Sci. Technol. A, 1, 1, p. 28 (1983). 9. Bogaert, R. J., PhD Thesis, "Chemical Vapor Deposi tion of Amorphous Silicon Films," University of Delaware (1985). 10. Ring, M. A., "Homoatomic Rings, Chains and Macro molecules of Main Group Elements," Elsevier, N.Y., 1977, Ch. 10. 11. John, P. and J. H. Purnell, Faraday Trans. I, 69, p. 1455 (1973). 12. Bowery, M. and J. H. Purnell, Proc. Roy. Soc. Lond., A821, p. 341 (1971). 13. Hegedus, S. S., R. E. Rocheleau and B. N. Baron, "CVD Amorphous Silicon Solar Cells," Proceedings of the 17th IEEE Photovoltaic Specialists Confer enceOrlando, p. 239 (1984). NOMENCLATURE a Ci D K k kg MW q r(e) r(i) r(r) r (rxt,i) V Z Greek area concentration of species i diameter of reactor effective reaction rate constant reaction rate constant mass transfer coefficient molecular weight volumetric flow rate rate of evaporation impingement rate, species i rate of reflection net rate of reaction, species i volume axial position in tubular reactor 8 condensation coefficient y stoichiometric coefficient p density Subscripts f film property g gas phase i molecular species s denotes on the surface CHEMICAL ENGINEERING EDUCATION ACKNOWLEDGMENTS Departmental Sponsors: The following 150 departments contributed to the support of CHEMICAL ENGINEERING EDUCATION in 1985 with bulk subscriptions. University of Akron University of Alabama University of Alberta Arizona State University University of Arizona University of Arkansas University of Aston in Birmingham Auburn University University of British Columbia Brown University Bucknell University University of Calgary California State Polytechnic California State University, Long Beach California Institute of Technology University of California (Berkeley) University of California (Davis) University of California (Los Angeles) University of California at San Diego CarnegieMellon University CaseWestern Reserve University University of Cincinnati Clarkson University Clemson University Cleveland State University University of Colorado Colorado School of Mines Colorado State University Columbia University University of Connecticut Cornell University Dartmouth College University of Dayton University of Delaware Drexel University University of Florida Florida State University Florida Institute of Technology Georgia Technical Institute University of Houston Howard University University of Idaho University of Illinois (Urbana) Illinois Institute of Technology Institute of Paper Chemistry University of Iowa Iowa State University Johns Hopkins University University of Kansas Kansas State University University of Kentucky Lafayette College Lamar University Laval University Lehigh University Loughborough University of Technology Louisiana State University Louisiana Tech. 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Ikoku, Pennsylvania State University January 1985 600 pp. A GUIDE TO CHEMICAL ENGINEERING PROCESS DESIGN AND ECONOMICS Gael D. Ulrich, University of New Hampshire Solutions Manual available (0471894834) 1984 517 pp. (0471052767) 1984 480 pp. FUNDAMENTALS OF MOMENTUM, HEAT, AND MASS TRANSFER, 3rd Edition James R. Welty, Charles E. Wicks, and Robert E. Wilson, all of Oregon State University Solutions Manual available (0471874973) 1984 832 pp. NUMERICAL METHODS AND MODELING FOR CHEMICAL ENGINEERS Mark E. Davis, Virginia Polytechnic Institute and State University Solutions Manual available (0471887617) 1984 320 pp. INTRODUCTION TO MATERIAL AND ENERGY BALANCES Gintaras V. Reklaitis, Purdue University Solutions Manual available (0471041319) 1984 683 pp. NATURAL GAS RESERVOIR ENGINEERING Chi U. Ikoku, Pennsylvania State University Solutions Manual available (0471894826) 1984 498 pp. THE ENGINEERING PUBLISHER To be considered for complimentary cop ies, please write to LeRoy Davis, Dept. 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