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Allan P. Colburn remembered by Olaf Hougen ( PDF )
Biographical sketch. ( PDF ) Applied Mathematics, N.R. Amundson ( PDF ) Energy Transport, S.W. Churchill ( PDF ) Momentum Transport, T.J. Hanratty ( PDF ) Particulate Systems, H.M. Hulburt ( PDF ) Mass Transport, E.N. Lightfoot ( PDF ) Book Reviews ( PDF ) Optimal Control, Leon Lapidus ( PDF ) Molecular Thermodynamics, J.M. Prausnitz ( PDF ) Classical Thermodynamics, J.J. Martin ( PDF ) Chemical Reaction Engineering, Doughart, N.A., & Smith, J.M. ( PDF ) Graduate Engineering and Technological Accreditation, L.E. Grinter ( PDF ) The ChemistryChemical Engineering MerryGoRound, R.A. Morgen ( PDF ) 
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CHAIRMAN, CHEM ENGR DEPT UNIV OF FLORIDA GAINESVILLE, FLURIDA 32601 Excellent Text that can be warmly recommended as an introduction to balances, for it takes proper account of the nature of the subject and at the same time leads to the use of the proper tools." Professor Rutherford Aris Department of Chemical Engineering University of Minnesota MATERIAL AND ENERGY BALANCE COMPUTATIONS By ERNEST J. HENLEY, University of Houston; and EDWARD M. ROSEN, Monsanto Company, St. Louis, Missouri. This is the first book to bring material and energy balance into modern focus. An excellent introduction for one or twosemester engineering courses, the book stresses the problem of seeking and applying the physical and mathematical principles of material and energy techniques. The subject is approached from two viewpointscovering the material for the engineer who will do his work with a slide rule, and for the engineer who will work with a computerand the differences between the two approaches are emphasized. A Volume in the Chemical Engineering Outline Series. 1969 577 pages $14.95 Another important text for electrical engineering students FUNDAMENTALS OF MOMENTUM, HEAT, AND MASS TRANSFER By JAMES R. WELTY, CHARLES E. WICKS, and ROBERT E. WILSON, all of Oregon State University. On an introductory level, this textbook presents the traditionally separate fields of momentum transfer (fluid mechanics), heat transfer, and mass transfer (diffusion) all from a unified viewpoint. The similar means of describing the various processes are stressed; and it is shown how information on one area may be extrapolated to provide an understanding of the other types of transfer. 1969 Approx. 672 pages $16.50 The only text focusing on the sociology of engineering THE ENGINEERS AND THE SOCIAL SYSTEM Edited by ROBERT PERRUCCI, Purdue University; and JOEL GERSTL, Temple University. The only book of its kind, this text provides a detailed examination of the engineer ing profession within the social and historical context of American society. The authors study the social origins, values, and career patterns of members of the profession, occasionally contrasting the facts with data from other countries. 1969 344 pages $9.95 JOHN WILEY & SONS, Inc. 605 Third Avenue, New York, N.Y. 10016 In Canada: John Wiley & Sons Canada Ltd. 22 Worcester Roard, Rexdale, Ontario EDITORIAL AND BUSINESS ADDRESS Department of Chemical Engineering University of Florida Gainesville, Florida 32601 Editor: Ray Fahien Associate Editor: Mack Tyner Business Manager: R. B. Bennett Publications Board and Regional Advertising Representatives: CENTRAL: James H. Weber Chairman of Publication Board University of Nebraska Lincoln, Nebraska 68508 WEST: William H. Corcoran California Institute of Technology Pasadena, California 91109 SOUTH: Charles Littlejohn Clemson University Clemson, South Carolina 29631 SOUTHWEST: J. R. Crump University of Houston Houston, Texas 77004 EAST: Robert Matteson College Relations Sun Oil Company Philadelphia, Pennsylvania 19100 E. P. Bartkus Secretary's Department E. I. du Pont de Nemours Wilmington, Delaware 19898 Peter Le'derman Brooklyn Polytechnic Institute Brooklyn, New York 11201 NORTHEAST: George D. Keeffe Newark College of Engineering Newark, New Jersey, 07102 NORTH: J. J. Martin University of Michigan Ann Arbor, Michigan 48104 NORTHWEST: R. W. Moulton University of Washington Seattle, Washington 98105 MIDWEST: Richard S. Mayer Ohio University Athens, Ohio 45701 UNIVERSITY REPRESENTATIVE J. A. Bergantz State University of New York Buffalo, New York 14200 PUBLISHERS REPRESENTATIVE D. R. Coughanowr Drexel University Philadelphia, Pennsylvania Chemical Engineering Education VOLUME 3, NUMBER 4 FALL 1969 Articles on Graduate Courses 174 Applied Mathematics N. R. Amundson 178 Energy Transport S. W. Churchill 184 Momentum Transport T. J. Hanratty 190 Particulate Systems H. M. Hulburt 194 Mass Transport E. N. Lightfoot 200 Optimal Control Leon Lapidus 204 Molecular Thermodynamics J. M. Prausnitz 212 Classical Thermodynamics J. J. Martin 218 Chemical Reaction Engineering Dougharty, N.A., & Smith, J. M. Departments 163 Editorial 165 Division Activities James H. Weber 167 Letters 168 A Founder of the Profession Allan P. Colburn remembered by Olaf Hougen 173 Biographical sketch. 222 Views and Opinions Graduate Engineering and Technological Accreditation, L. E. Grinter 228 The Curriculum The ChemistryChemical Engineering MerryGoRound, R. A. Morgen 199 Book Review Boudart: Kinetics of Chemical Processes by J. R. Butt 216 Problems for Teachers CHEMICAL ENGINEERING EDUCATION is published quarterly by the 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 DeLand, Florida. Correspondence regarding editorial matter, circulation and changes of address should b addressed to the Editor at Gainesville, Florida 32601. 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., 137 E. Wisconsin Ave., DeLand, Florida 32720. Subscription rate U.S., Canada, and Mexico is $10 per year to nonmembers of the ChE division of ASEE, $6 per year mailed to members, and $4 per year to ChE faculty in bulk mailing. Individual copies of Vol. 2 and 3 are $3 each. Copyright () 1969, ChE Division of ASEE, Ray Fahien, Editor. 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. FALL 1969 SEnrigineers grow with the growing world of Sohio We need talented people immediately to work on important projects in: Research on Products and Processes Process and Product Development Economic Analysis Process Design Plant Design, Construction and Operation Market Research Sales and Marketing Management. Sohio is a progressive company whose growth is based on continuous research, development and marketing of products for transportation, industry, agriculture and construction; not boomorbust military or space projects. That means we rely on your technical talent to help us grow, and we offer rapid advancement to those who.are willing to work hard to achieve it. WRITE SOHIO TODAYI Join the Sohio team...where you can get ahead fast because we're moving fast. We assist you with direct aid for education, too... so you can get where you want to go as quickly as your growing interests and abilities will take you. We are the number one marketer of petroleum products in Ohio; our sales equal those of the next four oil companies combined! Our patented singlestep acrylonitrile process accounts for more than 75% of the free world's supply! SaleJhave risen over 50% ir the last six years! Profits over 190%. Write in strict confidence, stating education, experience and salary requirements to: Elwood G. Glass, Jr. Mgr. Technical and Professional Recruitment OH 10986 Midland Building Cleveland, Ohio 44115 THE STANDARD OIL COMPANY (OHIO) An Equal Opportunity Employer (M&F) 4n Cdi A NIal A LETTER TO CHEMICAL ENGINEERING SENIORS Should you go to graduate school? Through this special issue on graduate educa tion, Chemical Engineering Education invites you to consider graduate school as an opportunity to further your professional development. We be lieve that you will find that graduate work is an exciting and intellectually satisfying experience that greatly enhances your ability to obtain re sponsible and challenging positions in industry and teaching. We also feel that graduate study can provide you with insurance against the increasing danger of technical obsolescence. Furthermore, we believe that graduate research work under the guidance of an inspiring and interested faculty member will be important in your growth toward confidence, independence, and maturity. What is taught in graduate school? In order to familiarize you with the content of some of the areas of graduate chemical engineer ing we are including in this issue articles describ ing graduate courses that have been taught by professors who have specialized in these fields. In doing so we wish to make clear the following: 1) that there is some variation in the content of individual graduate courses in the same area as taught at various schools (e.g., many schools teach transport phenomena sequences, while others teach individual courses in fluid mechanics, heat transfer, and mass transfer), 2) that we have not included all of the areas in which grad uate courses are taught (e.g., we have not in cluded a design course, per se), and 3) that the professors who have written articles for us are by no means the only authorities in those fields, nor are their departments the only departments which emphasize that particular area of study. What is chemical engineering research? We are dedicating this graduate education issue to an outstanding chemical engineering researcher and teacher: the late Allan P. Colburn. Although Dr. Colburn's career included work in education, industry, and government, he is best known among educators for his pioneering research in many areas of chemical engineering. This re search, while based on fundamentals, was directed toward the ultimate attainment of an engineer ing answer, usually in the form of the many Col burn equations or correlations that are still used by practicing engineers. As an example of some one to emulate in your own graduate career, we urge you to read the article by Professor Olaf Hougen on Allan Colburn's activities as a grad uate student. (Incidentally Professor Olaf Hougen was himself featured by CEE (Summer 1968) and is also worthy of emulation). Where should you go to graduate school? It is common for a student to broaden himself by doing graduate work at an institution other than the one from which he receives his bachelor's degree. Fortunately there are many very fine chemical engineering departments in the United States, each of which has its own "personality" with special emphases and distinctive strengths. For example, in choosing a graduate school you might first consider which school is most suitable for your own future plans to teach or go into industry. Or if you have a specific research pro ject in mind, you might want to attend a univer sity which emphasizes that area and where a prominent specialist is a member of the faculty. On the other hand if you are unsure of your field of research, you might consider a department that has a large faculty with widely diversified interests so as to ensure for yourself a wide choice of projects. Or you might prefer the atmos phere of a department with a small enrollment of graduate students. In any case, we suggest that you begin by writing the schools that have pro vided information on their graduate programs in the back of this issue. You will probably also wish to seek advice from members of the faculty at your own school. But wherever you decide to go, we hope that you make the decision to continue your education in graduate school. Sincerely, Chemical Engineering Education University of Florida Gainesville, Florida 32601 NOTE TO DEPARTMENT CHAIRMENAdditional cop ies of this Graduate Education Issue are available at no charge (while supply lasts) to your seniors who are in terested in graduate work. Please write the Editor at the above address, stating the number of copies needed. FALL 1969 62c~ ~a&L, would you like to plan a plant in Puerto Rico?   Too late, the plant is planned! In fact construction is already beginning on Sun Oil's new $125 million refinery complex and harbor at Yabucoa. But the project at Yabucoa is sim ply one indication of Sun on the move. We're geared for growth and we need people. Maybe you? Perhaps you'd like to work for the company that also recently boomed into the 2 billion dollar class through the merger of Sun and Sunray DX; that's pioneering a new fertilizer plant on the island of Martinique; that operates a new Computation Center in Philadelphia; that spon sors winning teams and cars in major road racing championships in the United States and Canadato men tion just a few exciting projects We need men and women to grow with us and build the future. We have openings in Exploration, Production, Manufacturing, Research and De velopment, Engineering, Sales, Ac counting, Economics and Computer Operations. LocationsPhiladelphia, Toledo, Tulsa, Dallas and many other areas. Write us for an appointment, write for our book "Sunoco Career Oppor tunities Guide," or contact your Col lege Placement Director to see Sun's representative when on campus. SUN OIL COMPANY, Industrial Rela tions Department, CED, 1608 Walnut St., Phila., Pa. 19103. An Equal Opportunity Employer M/F S.rP 4 CHEMICAL ENGINEERING DIVISION ACTIVITIES Fellow Division Members: Chemical Engineering Education is completing its second year with its new format. Under the able Editorship of Ray Fahienassisted by his colleagues Bob Bennett and Mack TynerCEE has drawn praise from chemical engineers in in dustry, teaching and government. We of the Publication Board believe we have a good thing going and feel a strong obligation to continue to seek the necessary financial support. The financial support for CEE has come chiefly from two sourcesindustrial concerns and the departments of chemical engineering. A grant from NSF (to the Summer School) helped us during our first year. Our financing method ap pears sound; we have balanced our budget. Never the less, we have to abolish free subscriptions to CEE members. Cost is one reason, another is duplicationmost Division members are also faculty members. Furthermore, our present sys tem of sending each department 10 copies regard less of size, has meant that in the large depart ments some faculty members seldom saw the journal, while some small departments received an oversupply. Consequently, the Publication Board recom mended, and your Executive Committee approved the following plans for CEE distribution: 1. Chemical Engineering Departments will be asked to request a definite number of copies at $4/year for each of the four issues in 1970, with a minimum contribution of $25/year. (They may pay for these through departmental funds or faculty contributions or both.) 2. ASEEChemical Engineering Division mem bers may request (on the attached form) indi vidually addressed copies to any address and pay $6/year starting in January 1970. 3. Libraries and other subscribers that are not members of the Chemical Engineering Division of ASEE may subscribe as before at $10/year. As a matter of interest, we are attempting to make arrangements with ASEE to have our sub scription fee collected with the annual dues. This may be in effect in June 1970 for payment of 1971 subscriptions. Again, this applies only to those desiring individual subscriptions. Jim Weber is Regents' Professor and Chairman of the Department at the University of Nebraska. He has been at Nebraska since 1948 when he received the PhD from the University of Pittsburgh. Jim is the author or co author of fifty articles and serves as an industrial con sultant. Besides ASEE, he is a member of AIChE., ACS, and AAAS, and a registered professional engineer. We wish to continue to give you a first rate publication and thank you for past support. We hope for your continued support and that you appreciate the need for this change. Very truly yours, James H. Weber Chairman, Publications Board R. Br. Bennett, Bus. Mgr. CEE Department of Chemical Engineering University of Florida Gainesville, Florida 32601 Please send our department copies of CEE during 1970 at $4/year each (Minimum $25/ year). [ Payment enclosed D Please send invoice School Address Please find my check (made out to CHEMICAL ENGINEERING EDUCATION for my 1970 subscription to CEE. I (am, am not) a member of the ChE Division, ASEE. Name Address FALL 1969 LOCATIONS HAVING CURRENT OPENINGS Olin MAJOR PRODUCTS PRODUCED DISCIPLINE REQUIREMENTS TYPE OF WORK PERFORMED ChlorAlkali Products Ammonia Process Development, Augusta, Ga. Phosphates Design, Maintenance, Brandenburg, Ky. Urea Planning, Scheduling, Charleston, Tenn. Nitrogen ChE Production, Sales, Joliet, Ill. Acids ME Production, Sales, CHEMICALS Lake Charles, La. Hydrazine IE Accounting, Inorganic Little Rock, Ark. Petrochemicals Chemistry Marketing, Organic & McIntosh, Ala. Insecticides Accounting Financial Analysis, Specialty New Haven, Conn. Pesticides Business Adm. Distribution, Agricultural Niagara Falls, N.Y. Polyurethane Transportation Project Engineering Pasadena, Texas Carbon Dioxide Marketing (Plant Startup & Rochester, N.Y. Animal Health Construction), Saltville, Va. Products Research Engineering, Automotive Chemicals Technical Service Other derivatives Alumina ChE Burnside, La. Aluminum IE Manufacturing METALS Chattanooga, Tenn. Aluminum Extrusions ME Production Aluminum Gulfport, Miss. Aluminum Sheet, Plate, Metallurgy Sales Brass Hannibal, Ohio Coils Met. Engineering Maintenance Ormet, Corp. East Alton, III. Brass Fabricated Parts Accounting Mintenance New Haven, Conn. Sheet & Strip Brass Business Adm. nan Sedalia, Mo. Roll Bond Ind Tech. Metals R&D Wire & Cable Ind. Mgmt. Carbonizing Paper Marketing Fine Printing Papers ChE Process Engineering FOREST PRODS, West Monroe, La. Specialty Paper Chemistry Plant Engineering PAPER & FILM Pisgah Forest, N.C. Products Pulp & Paper Research & Dev. Olinkraft, Inc Covington, Indiana Cigarette Paper & Tech. Statistician Ecusta Filters IE Ecusta ellterse ME Systems Engineering Film Cellophane ME Production Kraft Bags Mathematics Management Kraft Paper Business Adm. General IE Kraftboard Cartons Accounting Design and Corrugated Containers Development Olinkraft Lumber Accounting East Alton, III. New Haven, Conn. Marion, III. Kingsbury, Ind. Sporting Arms Ammunition Powder Actuated tools Smokeless Ball Powders Solid Propellants Safety Flares Franchised Clubs Ind. Tech. IE ME Mathematics ChE Accounting Business Adm. Marketing Personnel Mgt. Physics Ind. Mgmt. Production Control Purchasing Manufacturing Plant Engineering Sales Financial Analysis. Personnel Marketing R&D PRODUCT GROUP WINCHESTER WESTERN ACKNOWLEDGMENTS JThe jlo1aWiC companies hiee auppeaded Qehical 74iiee6iwf PfCLucOn $turiwf twmr apCf eaala titc C"4 dmOmtiak.ao in lieAsa ai aeoijUi#q. C. F. BRAUN & CO DOW CHEMICAL COMPANY MALLINCKRODT CHEMICAL COMPANY MONSANTO COMPANY MINNESOTA MINING AND MANUFACTURING COMPANY OLIN MATHIESON CHEMICAL COMPANY THE PROCTER AND GAMBLE COMPANY STANDARD OIL (INDIANA) FOUNDATION THE STAUFFER CHEMICAL COMPANY from our READERS Lynn responds to Fredrickson Sir: Dr. A. G. Fredrickson's essay "The Dilemma of Innovating Societies" (CEE, Summer 1969) points out a number of the problems facing our society today. The effects of increasing pollution, exploiting wilderness areas, and a rapidly expanding population are steadily making the world a less pleasant place to live. It was therefore a disappointment to see Dr. Fredrickson weaken the strength of his message substantially by overstating it in an emotional tirade against straw villains of his own construction. It is, for instance, unconvincing to denounce the effects of man's activities on our environment by proclaiming a higherthanhuman set of values. There is no reason to think that nature prefers alligators to algae, condors to crickets, or any of these to mankind. Such preferences are human value judgments and should be defended as such. The holierthanthou stance only beclouds the issue. If Dr. Fredrickson really questions the relative happi ness of today's farmer astride his airpolluting tractor I would suggest that he try spending a summer of 12hour days plowing behind a mule. A good look at Van Gogh's "The Potato Eaters" might also be instructive. The issue clearly is not one of slowing down technological innova tion but rather of directing innovative efforts to the solu tion of the problems that are now becoming pressing. It may be that society should have foreseen the urgency of these problems one or two generations ago. However, one should remember that 2020 hindsight is a common virtue and also that no amount of castigation will change the events of the past. Emotional polemics directed against oversimplified whipping boys are highly favored today by political extremists of the left and the right, superconservationists, gungho developers, and many others with a Cause. The trouble with such tactics is that they alienate those whose support might be won by a rational approach. If the need to solve the problems arising from the growth of population and technology is real, and I believe that it is, then well reasoned arguments to this effect can surely be found. It is clear that we have or can develop the technology to solve these problems if we can get general agreement within our society that they need to be solved. Attaining such agreement will require persuasive leadership, factual knowledge, and consider able persistence. I submit that very few will be persuaded by being told that they are simpleminded votaries of the Cult of the Product, believers of the Creed of Technology, and preachers of the Gospel of Growth. Scott Lynn University of California, Berkeley Corrections from Lee Sir: Enclosed please find a corrected copy of the short article entitled Transport Phenomena: Equations of Change, which was printed in the summer, 1969 issue of CEE. Please note that equations 7, 9, 11, 15, 16, 17, 19, 20 are corrected, where originally either a small p (for pressure) is missing or is mixed up with p (for density). V. J. Lee University of Missouri Editors Note: CEE regrets that Professor Lee did not correct this error on the galleys he received. Praise from the Veep Sir: I certainly appreciate receiving the copy of Chemi cal Engineering Education and was particularly interested in seeing the articles involving Stu Churchill. A. L. Conn VicePresident, AIChE (Letters Continued on page 207) FALL 1969 '167 N founder In this issue, CEE begins a new depart ment that will feature some of the found ers of our chemical engineering profession. This article deals with the graduate (and earlier) university career of Allan P. Col burn, who has been described by Professor Olaf Hougen of the University of Wiscon sin as "one of the most inspiring friendly and intellectual teachers and leaders in chemical engineering." The article is writ ten by Professor Hougen, who was his Ph.D. advisor. ALLAN P. COLBURN** OLAF A. HOUGEN, Professor Emeritus University of Wisconsin Madison, Wisconsin 53706 ALLAN PHILIP COLBURN was my first stu dent in graduate research directed towards a doctorate degree in chemical engineering. Allan was born in Madison, Wisconsin, on June 8, 1904. His father, Willis P. Colburn*, at that time, after twelve years of high school teaching, was enrolled as a student in philosophy at the University of Wisconsin. Upon graduation in 1905 Willis accepted a position as principal of the high school in Rhinelander, Wisconsin. It was here that Allan spent his childhood and received his ele mentary and high school education. In June 1922 the Colburn family moved to Wauwatosa, a subur ban city adjoining Milwaukee, where Willis was employed as principal of a local high school. In June 1922 Willis Colburn came with his son *The records of Platteville Normal School (now Wis consin State UniversityPlatteville) show that Willis Paul Colburn, a resident of Grant County, Wisconsin, attended Platteville Normal School in three school years 188687 and 188991 receiving a diploma in 1891. After graduation he served as high school principal at Potosi, Cassville and Viroqua, Wisconsin. He married Jennie Grimm of Cassville. In 1903 he attended the University of Wisconsin in Madison as a student in philosophy re ceiving a bachelor of philosophy degree (BPh) in June 1905. In later life he returned for graduate courses in Education in 1914 and in the summer session of 1929. **This sketch was prepared for the dedication cere monies of the Allan Philip Colburn Chemical Engineering Building at the University of Delaware, Sept. 20, 1968. Allan to my office to consider enrolling him in the College of Engineering of Marquette Uni versity in Milwaukee for a period of two years prior to enrollment in chemical engineering at the University of Wisconsin. [The chemical engineer ing building was then located on the south shore of Lake Mendota at the foot of Park Street.] From Allan's superior high school record and his un usual intelligence, I readily agreed that this plan had much merit not only in the economy of living at home but also in the cultural advantages asso ciated with a sectarian school of high repute. At Marquette University Allan received undergrad uate instruction in general chemistry, mathema tics, physics, English, shopwork and surveying. Imagine! Surveying was a required course in many curricula of chemical engineering 44 years ago. On September 12, 1924 Allan enrolled as a Junior at the University of Wisconsin. I served as his adviser in his junior year; and Professor Otto L. Kowalke in his senior year. The period 1920 to 1930 was critical and transi tional in the development of chemical engineering education. The year 1923 marks the beginning of the American system of education in chemical engineering with the publication of the text, "Principles of Chemical Engineering" by William H. Walker, Warren K. Lewis, and William H. Mc Adams. At Wisconsin the curriculum was at that time predominant in conventional engineering courses with instruction in chemical engineering slowly emerging from descriptive courses in in dustrial chemistry supplemented by laboratory experiments largely empirical in nature. CHEMICAL ENGINEERING EDUCATION A humanitarian goal in life was manifest in his selection rounded citizenship. ALLAN SET HIS GOAL EARLY at the highest professional level not only towards advanced studies and research leading to the doctorate degree but also in seeking a career of high pro fessional and civic responsibility. A humanitarian goal in life was manifest in his selection of liberal elective courses essential for a well rounded citi zenship. In this selection Allan was guided by his father whose own major college studies had been in philosophy and by Professor Kowalke. In lib eral courses he was fortunate in choosing five of the most popular and inspirational professors at Wisconsin, namely, William H. Kiekhofer in Eco nomics, Louis Kahlenberg in the History of Chem istry, Max Otto in Philosophy, A. A. Vasiliev in Hellenistic Civilization, and Daniel W. Mead in Contracts and Specifications. Mead's course was essentially one in engineering ethics based upon Mead's world wide experiences in the construction of dams and power plants. Professor Kiekhofer was the campus spark plug of enthusiasm in his animated lectures on conventional principles of economics injecting life into an otherwise dull subject. Professor Max Otto held a similar posi tion in philosophy and logic. The course in Hel lenistic civilization described the Golden Age of Greece, the causes of its origin and decline. With a delightful sense of humor Kahlenberg portrayed the joys and frustrations of scientific discovery in the lives of great chemists. These five profes sors, combined with parental influence and that of Professor Kowalke gave Allan an altruistic outlook on life and in his dedication to highly ethical and benevolent standards. This served him well in his later administrative responsibilities and projects of community welfare. The present day stress on the importance of liberal courses in training of engineers was met by Colburn 40 years ago. In his college years, Allan became a proponent of the Single Tax theory of economist Henry George. This typified student protest forty years ago in contrast to the violence of today. The professional staff of the Chemical Engi neering Department in 1924 consisted of Profes sors Otto L. Kowalke, Oliver P. Watts and myself. Courses in industrial chemistry and unit opera tions were given by Professor Kowalke, applied electrochemistry by Watts and a calculation course, applied thermal chemistry, by myself. Allan was graduated in June 1926 with a of liberal elective courses essential for a well bachelors degree in science (BS) and high honors. An Engineering Fellowship was awarded him for continuation in graduate studies and research. This fellowship was later renewed for two addi tional years. At this time a new dormitory system for men was established at the University of Wisconsin. Allan was one of the first graduate students to be appointed as House Fellow. The responsibilities of this position entailed living with undergraduate students as counsellor, guide and friend. Allan received his MS degree in 1927 and PhD degree in 1929. In his graduate years Allan's two closest friends were Kenneth M. Watson and Louis F. Warrick. The former became a prominent chemical engi neer in his contributions to education and indus trial practice and the latter became the young State Sanitary Engineer of Wisconsin in 1927. A recent letter from Louis Warrick restores an intimate insight into Allan's zest for living and some of the extra curricular activities he enjoyed during student days. Before accepting a position with du Pont, Warrick and Colburn had made enthusiastic plans to form a partnership as consultants in solving problems in the abatement of water pollution and disposal of industrial wastes. Already forty years ago these two young men were aware of the dire consequences of water pollution, the irrevocable evils of which are so strikingly evident today. T HE RESEARCH PROJECT assigned to Allan for his doctorate thesis was to obtain experi mental data on heat and mass transfer coefficients in the condensation of water vapor from saturated air streams in a tubular gas condenser and to formulate correlations based thereon useful for design and operation. This project differed from conventional dehumidification in that it involved air saturated with water at high temperatures with great reductions in volumetric and mass flow rates of the gasvapor stream during cooling and condensation. The Committee on Condensing and Scrubbing of the American Gas Association had collected operating data on tubular gas condensers used for refining crude coal gas with its high initial con tent of water vapor, hydrogen sulfide, ammonia, cyanogen, naphthalene and tar. These data were FALL 1969 gathered from commercial plants scattered widely throughout the United States. Professor Kowalke, as a member of this committee, assigned to me the task of trying to calculate and correlate the overall heat transmission coefficients from these data in terms of operating variables, physical properties and gas composition. I was unsuccess ful in making any meaningful correlations. In deed, correlation in terms of the geographical location of the plants seemed to be better than any rational attempt. The decision was made to establish data from carefully controlled operation of a laboratory scale tubular condenser using saturated airwater vapor mixtures under indus trial conditions of operation. In the period 192629 few graduate students were enrolled in chemical engineering at the Uni versity of Wisconsin. Prior to 1929 only three doctorate degrees had been granted. In guiding research towards a doctorate degree Allan was the only student assigned to me. A preliminary study of the condensation of water vapor from air saturated at high initial temperatures in a tubular condenser revealed many complexities. Three fluid stream resistances were involved, the airvapor stream, the con densate layer and the stream of cooling water, besides the resistance of the metal barrier. The heat transfer coefficients of these three streams were to be established each in terms of its inde pendent variables. Because of large variations along the length of the condenser it was evident that coefficients of the individual streams should be determined at short intervals of condenser length. Calculations of average heat transmission coefficients of the vapor stream from usual log arithmic mean values of temperature drops at terminal conditions were meaningless, indeed, the temperature drop at the midsection of the con denser was usually greater than at either ter minal. In retrospect, considering the primitive status of scientific information and the com plexity of the problem, this investigation would at that time justify three projects with inde pendent approach to each. A vertical tubular condenser, six feet long, was constructed of three concentric pipes, 3, 7 and 10 inches nominal diameters, well insulated on the outer shell. Cooling water flowed through the inner pipe and saturated air flowed downwards through the two annular channels; the outer outer annular space served as a guard ring to gether with external insulation to minimize the Colburn's genius consisted in his extraordinary capacity for intensive concentration with a mind unusually well organized for retention and retrieval. outward flow of heat. With only $100 available for mechanical help and additional apparatus over a span of three years, the equipment and instru mentation had to be assembled from supplies available in the stock room or borrowed from dis tressed laboratories, including piping, pumps, thermocouples, orifice meters and potentiometers. Allan constructed and calibrated all thermo couples and orifice meters. The construction and location of thermocouples were critical for mean ingful measurements. Multijunction couples were constructed for measuring average temperatures of the gas stream at each level of cross section. Single couples were located in isothermal areas to avoid errors by conduction. A traveling thermo couple was constructed and installed for measur ing the temperature of the cooling water. Tem peratures of the three adjoining fluid streams and the central pipe wall were measured simul taneously at successive short intervals of length. Under commercial conditions of operation natural convection predominated in the stream of cooling water; both laminar flow and turbulence occurred in the gas stream; the condensate accelerated from laminar flow to turbulence with rippling at the bottom of the tube. T HE MEASUREMENT AND CORRELATION of heat transmission coefficients of fluid streams was in a primitive stage in 1926 when Colburn began his experimental and theoretical studies. Most published experiments had been conducted within the preceding ten years. The most significant work had been carried out in Germany. This appeared in the German language without published translation in English. In the United States chemical engineers required that formulations of transfer coefficients be expressed in terms of molecular properties and operating variables. Other engineers were still satisfied with specific values of overall coefficients and physicists had virtually abandoned the field upon discover ing the empirical nature involved. Principles of heat conduction in solids had been well known for over a century starting with the mathematical theory of Fourier in 1822 for the unsteady state. These formulations were extended in the texts of Ingersoll and Zobel in 1913 and by Carslaw and Jaeger in 1921. CHEMICAL ENGINEERING EDUCATION Colburn was an ideal student, scientist, and engineer. . he had an unusual capacity for clarity of expression . . (and promoted) self confidence and ambition in others. The analogy between mass, heat and momen tum transfer in flowing fluids was presented by Prandtl in 1910 based in part upon pressure drop formulations of Reynolds in 1874. In England work was reported in 1916 by Pannel for air flow ing through tubes and by Stender in Germany for water flowing through tubes. A theoretical equa tion for the transfer of heat by free convection in fluids was developed by Lorenz in 1881 and greatly improved by Nusselt in 1915. Nusselt in 1910 also pioneered in deriving theoretical equa tions for the transmission of heat through con densate layers flowing over vertical surfaces and horizontal cylinders. At the time of Colburn's studies and just prior thereto five books on general heat transmission appeared in Germany, namely, by Grober (1921 and 1926), by Merkel (1927), Bosch (1927) and Schack (1921). These were then without English translation. The first authoritative book on heat transmission printed in English and applicable to general engineering processes was that of Mc Adams in 1933 but this book did not appear until seven years after Colburn started his research. McAdams' text and researches generated wide attention to research in heat transmission throughout the United States and among other branches of engineering besides chemical. Allan proceeded at once to read intensively the German sources relying on his high school in struction and his preparation for absolving the German language requirement of the doctorate degree. The extraction of complex theoretical principles from lengthy German dissertations re quired exceptional capacity for intensive concen tration. Allan studied the original German sources with intense concentration over long intervals of time to the point of pain and fatigue. Colburn had exceptional capacity for retaining the argu ments and voluminous observations of previous investigators with subsequent instant recall. Col burn's genius consisted in this extraordinary ca pacity for intensive concentration with a mind unusually well organized for retention and re trieval. In his graduate years Allan devoted fully half of his time to theoretical studies and experimenta tion related to his thesis over a period of three years. In his efforts to establish simultaneously the principles of heat and mass transfer in fluid streams, condensate layers and water streams with free convection, Allan suffered many periods of despair and frustration especially in his efforts to reconcile his data with the formulations of others. But he always bounced back with a zest for scientific discovery and to infuse the same spirit in others. FROM HIS BOYHOOD DAYS in the recrea tional area of Northern Wisconsin with its forests and lakes, it was natural for Allan to seek relaxation from his strenuous intellectual pur suits and frustrations in outofdoor sports the year around, in tennis, canoeing, golf, fishing, skating, and iceboating. In connection with recreation his friend Lou Warrick records a vivid and humorous account of Allan's discovery of 'hot ice'. In iceboating to gether on Lake Mendota their speeding boat struck rough ice and Allan was projected there from at high speed over the rough surface on the seat of his pants. Upon recovery Allan feel ing his posterior exclaimed, "Gee, Lou, this is the first time I realized that ice can get hot!" Because of the great tragedy in health which be fell Allan a few years later few people were aware of his early athletic prowess. Colburn's computational facilities were limited to the use of the slide rule, to laborious calcula tions and plotting by hand. In experimental work he received some aid from two seniors working for academic credit, namely Robert E. Zinn and George F. Hrubesky. Today Allan's monumental task would be greatly facilitated by electronic computers with generous financial subsidies for experimental and computational aid. It should be recalled again that Allan had only $100 available for research aid. In his graduate years Allan restricted his ad vanced studies to scientific work related to his thesis. Liberal reading became extra curricular. His advanced studies included a course in heat conduction under Professor Leonard R. Ingersoll, higher mathematics under Professor R. W. Bab cock, and advanced chemistry courses under Pro fessors J. Howard Mathews, Farrington Daniels, and John W. Williams. Allan was also fortunate in taking courses under two visiting professors, with the Russian chemist A. M. Frumkin in col (Continued on page 193.) FALL 1969 CHEMICAL ENGINEERS GET TOTALLY INVOLVED IN A TOTAL ENGINEERING ENVIRONMENT AT ESSO Ci$ ^s. lpI i~ r;" II" i. 1^ ..V.. ^^^^^^^H ^ _.  Esso Research and Engineering Company, the principal technical affiliate of Standard Oil Company (N. J.), provides research and en gineering services to 250 world wide affiliates with assets of over thirteen billion dollars. The Chemical Engineer plays a vital role in helping us meet these vast responsibilities. But most important to him, he functions in an environment as dedicated as that of the university Chemical Engineering department. For our ultimate goal is the same as that of the university; namely the ex tension of knowledge and the bet terment of the human condition through longterm fundamental and applied research, and the accomplishment of immediate ob jectives through the economical design and operation of plants and equipment. Whether he possesses a B.S., an M.S., or a PhD., and whether he works in Product/Process Re search and Development, Appli cations and Technical Services, Process Engineering, Project De sign or Process Selection and Economics, the Chemical Engi neer serves with his professional peers. He learns from them; he teaches them. But he advances as far as his own talents take him, wherever his interests lead him; either in a technical or ad ministrative capacity. Total involvement . in a total chemical engineering environ ment. That's Esso. For full details on the opportunities available, contact: Dr. P. H. Watkins, Employment Coordinator, Dept. CS27 ESSO ESSO RESEARCH AND ENGINEERING COMPANY P.O. BOX 175, Linden, New Jersey 07036 An Equal Opportunity Employer (M/F) A9 /7~ ,p3 J '&, I A. P. COLBURNA DISTINGUISHED CAREER A LLAN PHILIP COLBURN was born in Madi son, Wisconsin, on June 8, 1904. He grad uated from Rhinelander High School and attended Marquette University for two years before com pleting his education at the University of Wiscon sin, where he received his bachelor's degree in chemical engineering in 1926, his master's degree in 1927 and his doctorate in 1929. From 1929 1938, he was engaged in chemical engineering research at the Du Pont Experimental Station in Wilmington. On May 1, 1938, he became associate professor of chemical engineering and acting head of the University of Delaware's Department of Chemical Engineering. As chairman of the department, he was responsible for developing a wide research program in cooperation with industry and govern mental agencies. In 1947 he became Assistant to the President with his chief responsibility that of assisting in the development of research through out the University. Dr. Colburn was Acting Presi dent of the University from April 1, 1950, to No vember 1950 and, upon the arrival of former President John A. Perkins, he was appointed Provost. At the time of his death in 1955 he was Provost and Coordinator of Scientific Research. During World War II, Dr. Colburn was instru mental in directing the use of the chemical engi neering laboratories at the University on solving war research problems for the National Defense Research Committee, the National Advisory Com mittee for Aeronautics, the Office of Rubber Re serve, and for various war industries. He further assisted in the direction of war research else where. With Dr. B. F. Dodge of Yale, he prepared the curriculum on chemical engineering for the Army, which was taught in the A.S.T. Program throughout the war years. In 1948 he was honored as the first recipient of the Professional Progress Award in Chemical Engineering. The award was the first major, gen eral award in the field of chemical engineering. Twelve years earlier the AIChE had given him the Walker Award for outstanding publications. Active in professional societies, Dr. Colburn was director and chairman of the Awards Com mittee and chairman of the Publications Com mittee for the AIChE, on whose council he also served. His memberships included the ASME, for whose Heat Transfer Division he was an advisory associate and former chairman; the ACS, the NEA, ASEE, AAUP, the AAAS, and the Newco men Society of England. He was a member and former chairman of the Committee on Coopera tion with the Military Services for the Engineer ing Colleges Research Council of the ASEE. In the Delaware Section, ACS, he served as both councilor and member of the education committee. He was an alternate member of the Committee on Chemical Warfare of the Research and Develop ment Board of the DOD. Honorary societies of which he was a member were Phi Kappa Phi, Phi Lamba Upsilon, Tau Beta Pi and Sigma Xi. He was director of the Delaware Chapter of the American Red Cross, and was a member of the research committee of the Delaware Branch, American Cancer Society. He also served on the research committee of the Delaware Academy of Medicine and was a mem ber of the Delaware Section of the AntiTuber culosis Society and the Sigma Phi Eplison social fraternity. A nationally recognized authority in chemical engineering, Dr. Colburn was the author of many publications. His papers dealt with heat transfer, fluid flow, distillation, absorption and extracting, and various other technical subjects. He also wrote several textbooks. While his original interest was in engineering and basic science research, Dr. Colburn appre ciated the importance of developing research in the social science area and in broadening the edu cational programs of students so that they might better understand human relations and interna tional affairs. He was active in developing the University's Institute of InterAmerican Study and Research, the Institute for Human Relations, the Marine Laboratories and the evening pro grams of the Division of University Extension. As his longtime friend and associate, Dr. Robert L. Pigford, aptly stated, "He was the 'man of all seasons' at the University of Dela ware, the prime example for some of us of the uses of scholarship in the fullest way, the best administrator I have seen, the truest friend of student and colleague, the man who makes me remember best that teaching engineering can be just plain fun." Reprinted from University of Delaware News, Fall 1968. FALL 1969 A osoie in Applied MAwemaUaS WHY MATHEMATICS? NEAL R. AMUNDSON University of Minnesota Minneapolis, Minn. 55455 COURSES IN APPLIED MATHEMATICS for chemical engineers are relatively recent addi tions to graduate programs, although some go back about twentyfive years. Often such courses were initiated because of a certain dissatisfaction with pure mathematics offerings and the reluct ance of mathematicians to teach topics in applied mathematics. Courses with purely mathematical content should be taught in mathematics depart ments, while those offered in chemical engineer ing departments should contain something else. That something else is usually associated with the name "model building," although if the course is primarily that, it should probably be given as a part of one of the regular engineering science courses. In short, we seem to be speaking here of an offering which neither fits into the regular framework of a mathematics department nor into the regular kinetics, reactor, transport, control, and thermodynamics scheme of the conventional department. In addition to model building, the course must provide instruction in a number of techniques and actually show the student how to solve problems, a feature that is often anathema to the pure mathematician. In this seems to lie the reason for its being. Early courses were primarily exercises in elementary ordinary differ ential equations with applications to chemical kinetics and oversimplified models of the unit operations. The emphasis is still on differential equations but other topics with a more recent origin are now included. Our own course has gone through almost a continuous change in the last twenty years and is taken by almost all graduate students throughout their first year in residence. The purpose of such a course is not to make mathematicians of engi neers but rather to give the student enough ex perience that he can better cope with the other graduate courses in the department. Such a course Neal Amundson is Regents' Professor and Head of the Department at the University of Minnesota. His dis tinguished career as educator includes assignments as Fulbright Scholar and Guggenheim Fellow at Cambridge University and Institute Lecturer for AIChE. He received the Industrial and Engineering Chemistry Award of ACS in 1960 and the William H. Walker Award of AIChE in 1961. His research interests include the application of mathematics to chemical engineering processes. is valuable for the MS student since he may take little other physically motivated mathematics during his one year of course work. For the PhD student it can serve as the first course where significant and complex problems may be solved by advanced techniques and if he has theoretical inclinations frequently urges him on to take more abstruse and rigorous courses from a proper mathematician. As mentioned earlier, our own course has changed considerably through the years and this was forced on us by the fact that new graduate students now enter with a consid erably better background than formerly. The average entering student has now had about three years of undergraduate mathematics, some have had four years, and only a few the minimum required for the BS degree. This creates a prob lem for the instructor, for the class is very hetero geneous not only in terms of quantity of mathe matical experience but also because of the fact that in terms of coverage junior and senior mathematics courses can be much more variable than those of the first two undergraduate years. Because of the former I have attempted to give material which will overlap as little as possible with what I think they may have been exposed to. There is an additional problem since many of them are taking advanced mathematics courses concurrently. A number of theoretical and numer ical problems are assigned and these seem to be a CHEMICAL ENGINEERING EDUCATION It is important that a student understand the engineering g significance of these concepts (linear dependence of solutions, existence and uniqueness, and continuous dependence of the solutions on the data) and what they tell him about a mathematical model . . departure from mathematical experience of most of the students, and I believe may be the most valuable part of the course. These are graded and returned to the student. For the most part the problems are long and an attempt is made to com plement the lectures, bring out points not cov ered, and to illustrate the numerical procedures and difficulties. Over half of the students do the numerical problems on the University Computer (CDC 6600) although no time in the course is spent on programming. Usually a student will do between 25 and 40 problems in each tenweek quarter. The course is run from 8:00 to 10:00 on Tuesday and Thursdays (with a fiveminute break) and largely as a lecture, although because of the small class size (1525 students) there are frequent interruptions for questions. The fall quarter for some years has covered essentially the content of my book on matrices1, although not all of the book is covered in any single offering. Sections of the book may be skipped and assigned as reading. Other sections are omitted entirely and this varies from year to year. Chapters 1, 2, and 3 are covered almost entirely along with Chapter 4, through section 4.8; occasionally section 4.12 is presented. Chapter 5 through section 5.14 is in many respects the most important part of the course. A choice is usually made among the sections in Chapter 6, not all of it being given. Chapter 7 through sec tion 7.13 is almost always presented. On rare occasions a shortened version of sections 8.18.12 is included. The two volumes of Gantmacher2 serve as a reference for the course. All of the material presented in this quarter has a sort of nineteenthcenturyish ring about it and I have thought for some time that it should be modernized, probably in the direction of Shilov" "Theory of Linear Spaces" and with in troduction of material on tensor analysis (covered at Minnesota in the first graduate course in fluid mechanics). This has not come to pass yet, but probably will since the transition to functional analysis would be much easier. The winter and spring quarters are devoted to an organized exposition of ordinary and partial differential equations with related topics. It is assumed that the student understands the gen eration of solutions of simple differential equa tions. Some time is spent on the theory of differ ential equations covering linear dependence of solutions, existence and uniqueness, and continu ous dependence of the solutions on the data. It is important that a student understand the engi neering significance of these concepts and what they tell him about a mathematical model, for in the qualitative theory of differential equations these ideas play a central role. A good bit of time is spent on seeking to extract as much informa tion as possible about the solution from the model without recourse to numbers. It is surprising how much information one can obtain for stirred re actors, tubular reactors4, simple distillation schemes, heat conduction, diffusion, etc., from the equations by using qualitative but rigorous argu ments such as existence and uniqueness and the various maximum principles for both ordinary and partial differential equations. Often all of the intuitively obvious qualitative physical properties of the system can be drawn from the equations and this is the ultimate test of a model. For ex ample, it should not be necessary to compute a solution to prove that a molfraction lies between zero and one in a distillation calculation, that in an adiabatic tubular reactor there can be no tem perature maximum, or that in an absorption column the transient cannot oscillate. After this qualitative theory a brief discussion of numerical methods for ordinary differential equations is given covering predictorcorrector schemes and RungeKutta methods with applica tions. The question of numerical stability is briefly discussed since anyone who does a sig nificant amount of computer work eventually runs into stability problems. At this time a general discussion of the nth order linear differential operator is begun. Most of the interesting problems in ordinary and par tial differential equations are boundary value problems. The concept of the adjoint operator and adjoint boundary conditions is introduced and the general idea of a selfadjoint boundary value problem is presented. For example, given the nth order operator L d"y d"ly d"2y Ly=ao + a, + a2 dxn dx"1 dxn2 . +any the adjoint operator L* operating on z is FALL 1969 Nature operates on inputs to give outputs while mathematical operators, couched in the language of differential equations, operate on outputs to give inputs. d d(1 L*z = (1)n (ao) + (1)1 (a z) + ... + d dx (a._lz) + anz and it may then be shown if the region of interest of x is (a,b) that S(zLy yL*z) dx = 7r(z,y) b where 7r(z,y) is called the bilinear concomitant and contains the functions z and y and their first (n1) derivatives evaluated at a and b. In most physical problems we are given n boundary condi tions on y, n is even, and we have n/2 boundary at x=a and n/2 at x=b which we assume are homogeneous. Suppose these boundary conditions are denoted collectively by Y(y)=0 A fundamental theorem says that there exists a set of boundary conditions on z, called adjoint, unique except for linear combinations such that so that Z(z)=0 H (z, y)=0 The system made up of the operator L and the boundary condition Y is said to be selfadjoint if L=L* and Y=Z Sa S(zLyy*z) dx=0 b L can only be selfadjoint if its order is even. This equation is a form of Green's Theorem and is the key formula in much of that which follows. Our aim is to study linear differential equations on finite domains. In most applications these are second or fourth order operators, the former aris ing in heat conduction and diffusion problems and the latter in elasticity and fluid mechanics. We assume that the students know how to find solu tions of ordinary differential equations either by inspection, expansion in series, or numerically. (A pamphlet on series solutions is handed out to the students but is not discussed.) We consider a selfadjoint eigenvalue problem Lw = Xpw; a where p is a function of x and p(x)>0. W(w) stands collectively for the n boundary conditions. There are a number of theorems on the existence and character of the eigenvalues and eigenfunc tions of such a system. To be brief, however, there exists a discrete sequence of real eigenvalues Xi, X2, ,, ... and a corresponding set of eigenfunctions w, (x), w2 (x), Wa (x),... with an orthogonality prop erty p w jwi dx=0 ; i== j a Provided the set of functions [wn(x)] is complete with respect to a certain class of functions f(x) we can expand f (x) into a series 00 f(x) = cjwj(x) j=1 with c(j) = p (x)f(x)wj(x)dx a provided the eigenfunctions have been normal ized. These two relations play an important role, for if we write c(j) = bpfwj dx f a f(x) = S cj wj(x) j=1 then c (j) is called the finite Fourier transform of f(x) and f(x) is the inverse Fourier transform of c(j). Without laboring the point here this pair of formulae may be used to solve a number of partial differential equations in an almost automatic way once one recognizes the operator L and its asso ciated boundary conditions. If a partial differen tial equation has the form Ly = p(x) M(y) with boundary conditions CHEMICAL ENGINEERING EDUCATION All of our problems are physically motivated and the translation of the problem into mathematical terms is not mathematics. Y(y) = 0 where M is an operator not containing x explicitly and having its own boundary or initial conditions. We can write Wn Ly = pWn M(y) and integrate with respect to x Sb b Wn Ly dx = pwn M (y) dx a a Using the Green's formula we obtain Xn p (x) w (x) y(x) dx=M P pw nydx a a or Xn C = M cn This is a system which is simpler since all refer ence to x has been removed and may be solved (hopefully) to give cn and hence y(x) by using the inverse transform. In the course this idea is exploited to obtain solutions of a wide variety of diffusion, heat transfer, and reactor problems, and, while, in principle, it is no different than separation of variables, it possesses an automatic quality which appeals to the students. At this point we also discuss Duhamel's Theorem and the relationship among solutions for impulse, step function, periodic, random, and gen eral inputs, thereby solving the nonhomogeneous problems which have been avoided up to this time. A qualitative discussion ensues showing the dif ference between mathematical operators and natural operators. Nature operates on inputs to give outputs while mathematical operators, couched in the language of differential equations, operate on outputs to give inputs. For example, a distillation column operates on inputs (feeds) to give outputs (products). The model for a distilla tion column in the steady state is a system of algebraic relations (which must be inverted) among the outputs. Some mention of nonself ad joint problems is also made showing how the bi orthogonal set of eigenfunctions can be used to generate finite Fourier transforms for these problems. However, because of the extreme diffi culty of numerical work the problem is not pur sued in detail. Using solutions to problems on finite domains standard limiting procedures may now be used to find Fourier transforms for a variety of boundary conditions on semiinfinite domains (infinite hol low cylinders, etc.). The bag of the student has thus been equipped with a technique which will produce solutions with ease and his confidence is increased. A discussion of the Laplace transform is also included with applications to partial dif ferential equations. This discussion usually takes until about the sixth week of the spring quarter (a total of approximately fifteen weeks). One of the difficult things about differential equations is that there are no textbooks available intermediate in level between the elementary undergraduate books and books such as Codding ton and Levinson5, Ince6, Hartman7,, etc. The book by Weinbergers is an excellent intermediate book on partial differential equations but there is no corresponding treatment for ordinary differential equations. I have used some parts of Kaplan9 and Ross10 but it is surprising that with the number of books on differential equations and the age of the topic there are none that are really suitable. The remainder of the quarter (5 weeks) is spent in a variety of ways, but for the past two years first order partial differential equations have been presented with applications to chrom atography. This is a topic not wellpresented in the literature (a lacuna which Professor Aris and I hope to fill). At other times topics such as dynamic programming and calculus of variations, stochastic processes, numerical solution of partial differential equations with stability considera tions, continuous models for discrete processes and many others have been presented. The question arises as to how much rigor should be presented in such a course. The writer has a simple answer to this. Rigor is presented when ever the student feels the need for it. The solution of a partial differential equation involves a series of arbitrary operations and the bright student should ask whether what one obtains really is a solution to the problem. Such a proof requires the introduction of some rigor and it is not avoided. Different representations of a solution frequently arise and the student should wonder whether they are the same; a uniqueness proof is in order here and it is given. Expansions of functions into series require completeness of the set of functions (Continued on page 203.) FALL 1969 I eoa4" in Mame4ntu a.d ta"" Tda4^t 4 THEORIES, CORRELATIONS and UNCERTAINTIES for WAVES, GRADIENTS and FLUXES STUART W. CHURCHILL University of Pennsylvania Philadelphia, Pa. 19104 OBJECTIVES AND PROCEDURES The students in the class have done their baccalaureate work in a large number of other schools. Therefore a primary consideration of the course must be the diversity of their preparation, which on each topic ranges from zero through superficial acquaintance to real understanding. The individual students shift between these posi eions as the topics change. Those who are under prepared must be given sufficient encouragement and guidance and time for remedial selfstudy. Those who are overprepared must at the same time be kept challenged. Ideally each lecture begins from a position of security for the least prepared and ends at a level which temporarily distresses even the bestprepared. Problems with progressively more difficult parts are assigned. Special readings and problems are suggested for those "who have the time and inclination." The class is at first surprised that these optional prob lems are discussed as well as the assigned ones, but soon gets the intended message. Students appear to learn most readily by first examining very simple phenomena and models and then considering the effect of added com plexities. Hence the equations, algebraic or differ ential, for onedimensional, limiting cases are first derived. Then terms and dimensions are gradually added. Reduction of unsteady state, threedimen sional partial differential equations to simple cases does not appear to achieve the same rapidity or degree of understanding. I consider it an obligation to demonstrate a convincing and significant application for every model, theory or solution which is presented and Stuart W. Churchill became the first Carl V. S. Patter son Professor of Chemical Engineering at the University of Pennsylvania in 1967. He received the BSE in both chemical engineering and engineering mathematics in 1942, the MSE in 1948 and the PhD in 1952, all from the University of Michigan. From 1942 to 1947 he worked for the Shell Oil Company and the Frontier Chemical Company. He became an Instructor at the University of Michigan in 1950, rose to Professor in 1957 and served as Chairman of the Department of Chemical and Metal lurgical Engineering from 19621967. He is currently directing research in combustion, natural convection, freezing, fluid mechanics and drying. to point out the deficiencies and limitations of each model and the uncertainties in the support ing data. Home problems are assigned corresponding to each class meeting. In addition the students are frequently asked to complete or extend the deriva tions presented in the lectures. Several longer and optional problems, e.g., to be done on the com puter or to be considered as part of the final examination, are assigned during the semester. The students are sometimes asked to prepare an examination question and solution. The problems are a major ingredient of the course and may illuminate a matter barely mentioned.in the lec tures. The aim is therefore for completion and good comprehension by all students. Teamwork is encouraged with the admonition that anyone who fails to participate as a full partner will ultimately injure only himself. Incredibly, a few graduate students still arrive without digital computing experience. Hence it is difficult to make full use of the computer in this first course. Optional and group .problems are assigned and discussed in terms of computational CHEMICAL ENGINEERING EDUCATION S. a textbook . is probably not desirable in any graduate course . . methods when appropriate. The students are en couraged to elect hardcore courses in numerical methods as well as to develop the capability to use the University's computing systems. The entire semester could easily be spent on any one of the general subjects indicated by the headings below. Some departments of chemical and mechanical engineering do indeed offer up to eight courses on this subject matterresulting in more complete but not necessarily deeper cover age. However my general objective is neither to describe the art and equipment nor to cover the literature of heat transfer and fluid mechanics, but rather to present a methodology for inter preting and using data and concepts for the an alysis and prediction of chemical and physical processes. The technology of momentum and heat transfer provides convenient and stimulating ex amples. If a scientific and operational pointof view is the objective, specific subject matter is of secondary importance and can be chosen primarily for illustrative and motivational purposes. It is amazing how much material becomes redundant when the emphasis is shifted to methods of an alysis and solution. My specific objectives are (1) to give an idea of the character and current state of knowledge in fluid mechanics and heat transfer, (2) to show how to construct and test firstorder and better models and (3) to develop capability, motivation and confidence for future selfstudy. A textbook is not used in the course. Indeed a textbook in the classical sense is probably undesir able in any graduate course in engineering. It gives the false impression that the subject is wellknown and completely covered. Books at this level, with the exception of a few monograms, invariably purport to cover a wider range of material than the authors have mastered. The students arrive with the false notion that the equations, data and statements in the standard textbooks are almost sacred. Sometimes I use these familiar books as a foil, pointing out errors, inconsistencies and misinterpretations to encour age them to read critically and skeptically. Readings are assigned in a variety of books and journals. Reference lists are distributed on each topic for optional and future reading to make the students feel that they are scholars rather than mere receptors. I accept and try to implement John W. Gardner's precept that "the ultimate goal of the educational system is to shift to the individual the burden of pursuing his own educa tion." A significant fraction of the lecture mate rial and problems is taken from the very current literature. Notes are distributed in advance for much of the lecture material in order to discour age the frantic transcription of everything written on the blackboard. A course should evolve from semester to semes ter in reflection of new developments and concepts in science, engineering and computation and also in response to the changing attitudes and inter ests of the students. Currently they are concerned about the relevance of their studies. Insofar as possible a response is provided to this demand in the choice of topics and home problems and also by the allocation of some time to the discussion of professional matters. My absences are gener ally used for quizzes. On returning I usually ex plain why I was away. This often provokes a discussion of extracurricular topics. Tom Baron says a lecture should, like a bull fight, combine grace and excitement. An attempt is made to keep the lectures lively, even at the expense of organization. The students are encour aged to flag me down if they are lost and to chal lenge me if they are disbelieving. COURSE TOPICS COVERED Inertial (NonViscous) Flows Inertial flow is chosen as a first topic because of simplicity (onedimensional, algebraic equa tions are adequate for a first approximation) and because much of the subject is new to most of the students. Adiabatic and isothermal, reversible expansions and their application to orifices, venturi meters and nozzles are reviewed rapidly. The equations for weak pressure waves (acoustic waves) are de rived and applied to the problem of maximum flow through nozzles, including overexpansion and underexpansion, then to the behavior of rocket motors and finally to choked flow in pipes (here including viscous losses). The equations for gravity, shock and detona tion waves, each with reflection, are next devel oped and applied to openchannel flow, chemical shock tubes and the failure of process equipment, respectively. Viscous Flows The models and data for the viscosity of gases (including kinetic theory) and of liquids (includ FALL 1969 Right now,Westvaco engineers are revolutionizing paper for printing, reprographics, computers, nothing, packaging, disposables, structures And chemicals for coatings, pharmaceuticals, inks,tires, soaps,and waxes. And adsorbents for environmental control Your job: Our next revolution. See our campus representative. Or write to: Roger Keehn, Westvaco Building, 299 Park Avenue, New York, N.Y. 10017 Westvi CO An equal opportunity employer CHEMICAL ENGINEERING EDUCATION . a lecture should, like a bull fight, combine grace and excitement . (Tom Baron) ing nonNewtonian behavior) are examined. One dimensional material and momentum balances are first derived for laminar flow down inclined planes, inside tubes, etc., then the integral equa tions for the boundary layer, emphasizing the fortunate insensitivity of the drag to the pos tulated velocity distribution. The threedimensional, unsteady state, partial differential equations for the conservation of mass and momentum are next derived with emphasis on the physical significance of the various terms. Although most students have previously been ex posed to this material, they readily admit that they attained no real understanding as under graduates. These equations are simplified to some of the cases of laminar flow previously derived in one dimension, and are simplified and solved for a few more complex problems. The Blasius problem is used to introduce the HellumsChurchill tech nique for the reduction of partial differential equations. This technique is also used to demon strate that the equations do indeed describe tur bulent flow and that some information on tur bulent flows can be obtained by dimensional an alysis alone. Some of the empirical models for the time averaged turbulent shear stress are examined in a historical framework with emphasis on their success in generalizing the data for the velocity distribution and drag. The various ways of plot ting the data for drag in pipes are examined as an illustration of correlation. Other topics in momentum transfer are con sidered depending on the available time. (The schedule and hence some of these extra topics are frequently sacrificed for comprehension or for professional diversions which interest the class). Usually flow through porous media and the drag of particles are retained with applica tions to fluidization, filtration, etc. Conduction Models and data for thermal conduction and thermal diffusivity are first examined, including kinetic theory, heterogeneity and meanfreepath effects. The usefulness of the Van Dusen trans formation for variable conductivity is noted. Conduction between plates, concentric cylinders and concentric spheres, and outside spheres and cylinders is reviewed. The equations for three dimensional, unsteady state conduction are de rived. The comparative usefulness of various exact methods such as conformal mapping, the Laplace transform and separation of variables, and approximate methods, such as mapping, bounding and patching, are discussed. The use of Duhamel's formula both analytically and numeri cally is described. Numerical methods for both steady and un steady state problems are developed in limited depth primarily in the hope of whetting the appe tite for a complete course in numerical analysis. Laminar Convection The integral equations and solutions for the laminar boundary layer and the Graetz solution for fully developed laminar flow in a pipe are examined. (The detailed evaluation of the co efficients and the eigenwerte are deferred to the algebraically eager student as an extracurricular exercise). The analog of the Graetz solution for a powerlaw fluid is an example of an extension of a derivation which is assigned as a home prob lem. Other solutions for laminar flow in a channel under different conditions are examined and com pared. The equations for laminar, free convection are used as an illustration of the power of generalized dimensional analysis to produce the correct form of the solutions and empirical correlations for large Pr, small Pr, uniform heat flux and even fully developed turbulent motion. The success of asymptotic methods in producing firstorder solu tions of great generality is noted. The Nusselt equation for condensation is de rived and the effects of finite heat capacity, curvature and nonNewtonian behavior are con sidered in home problems. Emmon's generalization for condensation, film boiling, free convection and melting is presented as an illustration of the analogous nature of these four gravitational processes and of the great power of firstorder analysis. Turbulent Convection The onedimensional empirical models and analogies for turbulent forced convection are compared with each other, with the twodimen sional solutions and with the data. Again, the insensitivity of the solutions to the several pos FALL 1969 tulates is emphasized as the reason for the sur prising success of all of the models. Radiation The models and data for emission, absorption, transmission and scattering by solids and fluids are first examined. The integral and differential properties are contrasted. Interchange between black, gray, adiabatic and specular surfaces is considered. Analytical methods and results (including differential ex tensions), numerical methods and results, the electric circuit analog and the radiosity formu lation are included. REPRESENTATIVE PROBLEMS The character and scope of a course are per haps most evident from the problems which are assigned. Hence, a few representative problems are given below. Because of space limitations these problems are somewhat atypical in that they are restricted to those with very short state ments and hence are simpler, involve less input data and are less applied than most of those assigned in the course. 1. A shock wave is generated in a tube containing air at an initial pressure of 0.10 atmosphere and 600F. The measured velocity of the wave is 7200 ft/sec. If the wave reflects off the closed end of the tube to what pres sure and temperature will the wall be subjected? Note all assumptions. 2. Will adding to the length of the diverging section of a rocket nozzle increase the thrust? Explain. 3. A planar shock wave propagates indefinitely, pro ducing an increase in temperature, pressure and density. What is the source of energy for this compression and heating? 4. Water and npentane are pumped at equal volumetric rates between two horizontal parallel plates. Determine the location of the interface for fully developed laminar flow. 5. In cylindrical coordinates the Coriolis force is pu u /r and the centrifugal force is pu2 /r. Show r 0 whether or not each of those terms has a zero or finite, timeaveraged value for fully developed turbulent flow in a straight pipe. Do these terms affect the pressure gradients? Explain. 6. On a dry, 700F day with a barometric pressure of 740 mm. Hg, a pitcher is able to throw a baseball with sufficient force so that it arrives at the plate 60 feet away with a velocity of 90 miles per hour. Assuming that he throws with the same force, what would be the maximum and minimum velocities at the plate if all combinations of weather from 400F to 1000F, zero to saturated hu midity and 730 to 760 mm Hg were encountered during the season? Neglect the effect of ambient conditions on the ball itself. 7. A large block of copper at 1000F is brought in contact with a large block of steel at 2000 F along two plane faces of the blocks. Calculate the temperature of the interface as a function of time. 8. The mean monthly air temperature at Detroit varies as follows in OF: J, 29; F, 28; M, 37; A, 48; M, 57; J, 63; J, 65; A, 65; S, 62; 0, 54; N, 43; and D, 32. Assuming that the surface temperature equals the air tem perature and that moisture, curvature and geological ef fects may be neglected, calculate the temperature at a depth of 10 feet as a function of time. The soil may be as sumed to have a specific gravity of 1.75, a thermal conduc tivity of 1.4 x 103 cal/cmsecC and a specific heat of 0.25. 9. Starting with the general integral formulation, de rive a numerical value for the asymptotic Nusselt num ber for fully developed laminar flow and a fully developed temperature field in a circular tube with a uniform heat flux density at the wall. 10. Determine by dimensional analysis alone the mini mum functional relationship between the local heat trans fer coefficient and the other variables for laminar, steady, free convection of a powerlaw fluid to a vertical plate at uniform temperature. The usual assumptions of boun dary layer theory may be employed and the inertial terms may be neglected. 11. Experimental data are to be obtained for con vective heat transfer and pressure drop in smooth pipes. Experience suggests that for the chosen conditions, (hD/k)/(Cp/k)l/3 and (dP/dL)(Dp/G2) can be ex pected to be different functions of DG/u only. Suggest dimensionless coordinates for a graphical correlation of the measured heat transfer coefficient as a function of the measured pressure gradient such that neither coordi nate contains the velocity. 12. A reaction in a very viscous material is to be carried out in a scrapedsurface heat exchanger. The heat of reaction and heat of viscous dissipation may be neglected. It is proposed to double both the length of the exchanger and the flow rate. Will be conversion be greater, equal or less? Why? Will the outlet temperature be greater, equal or less? Why? 13. Calculate the error resulting from the following approximation for the interchange factor between a square surface and a parallel differential surface at a normal distance equal to the side of the square. Subdivide the square into four square elements and sum the inter change factors for these elements using the center points only. 14. A cloud reduces the radiant flux from the sun by 40%. Estimate the reduction to be obtained if the thick ness of the cloud were tripled (a) assuming absorption and negligible scattering (b) assuming scattering and negligible absorption. ACKNOWLEDGMENT Many colleagues, including particularly R. R. White, J. O. Wilkes and D. A. Saville, and many, many students at both the University of Michigan and the Uni versity of Pennsylvania have contributed to the development of the philosophy and details of this course. CHEMICAL ENGINEERING EDUCATION The world of Union Oil salutes the world of chemical engineering We at Union Oil are particularly indebted to the colleges and universities which educate chemical engineers. Because their graduates are the scientists who contribute immeasurably to the position Union enjoys today: The thirtieth largest manufacturing company in the United States, with operations throughout the world. Union today explores for and produces oil and natural gas in such distant places as the Persian Gulf and Alaska's Cook Inlet. We market petroleum products and petro chemicals throughout the free world. Our research scientists are constantly discovering new ways to do things better. In fact, we have been granted more than 2,700 U.S. patents. We and our many subsidiaries are engaged in such diverse projects as developing new refining processes, developing new fertilizers to increase the food yield, and the conservation of air and water. Today, Union Oil's growth is dynamic. Tomorrow will be even more stimulating. Thanks largely to people who join us from leading institutions of learning. If you enjoy working in an atmosphere of imagination and challenge, why not look into the world of Union Oil? Growth...with innovation. Union Oil Company of California. unl@n e Qow"Us in Moamentnm T4a4Mdp04t FLUID DYNAMICS THOMAS J. HANRATTY University of Illinois Urbana, Illinois 61801 Fluid dynamics plays a central role in many problems of interest to chemical engineers. Be cause of this, the semester course in the area pre sented by the ChE Division of the University of Illinois has been one of the most durable offerings in its graduate curriculum. I have taught this course since 1953 and one similar to it had existed many years prior to my involvement. My goal is to present a unified treatment of fluid dynamics to students who have had courses in differential equations and transport phenomena and who have some knowledge of vector notation. The content reflects the philosophy within the Division that the more advanced and more current aspects of any field are covered in our Special Topics courses and in our seminars. (For example, in recent years I have conducted seminars on tur bulence, hydrodynamic stability, continuum me chanics, water waves, numerical solutions of the equations of fluid mechanics, rheology and in modern aspects of boundary layer theory.) As a result, the fluid dynamics course is largely based on material available in a number of textbooks. It is intended that it be a starting point for ad vanced studies of the current literature, which are best done in an informal fashion. At present, the course is in transition because of the introduction by most schools of more ad vanced topics in fluid dynamics at the under graduate level. One of my chief difficulties is to assess properly the background of the students since I find that the courses offered at different universities in transport phenomena vary consid erably in content and depth. Therefore, at the risk of losing the interest of the better prepared students, I give a rapid treatment of key physical notions that should be covered in a basic course in transport phenomena. I also give the students a chance to do some reviewing of their own by assigning at the beginning of the course a num ber of problems from the book "Transport Phe nomena" by Bird, Stewart, and Lightfoot. Thomas J. Hanratty is professor of chemical engineer ing at University of Illinois. He was educated at Villanova, Ohio State, and Princeton University, PhD ('53). His recent professional honors include the Curtis W. McGraw Award (ASEE) and the William H. Walker and the Pro fessional Prograss Awards of AIChE. I have been experimenting with the contents of the course and, as a result, the outline accompany ing the article is meant to represent the types of topics treated and not the total material covered in a semester. For example, this past year I did nothing with topic 10 and only partially covered the notes I have prepared under topic 9. One of the ideas that is not sufficiently developed is that of hydrodynamic stability. I am currently giving some thought to ways of working it into the course. BASIC EQUATIONS The course is initiated by reviewing the con cept of a fluid particle and of a continuum and, in particular, by pointing out circumstances, such as the settling of fine particles, where the continuum model is invalid. It is then indicated that an Eulerian framework is usually more con venient than a Lagrangian framework for solving fluid dynamical problems. The second law of motion and the first law of thermodynamics are reformulated so that they are applicable to an arbitrary fixed volume in space rather than to a fixed mass. Difficulties that sometime can be en countered in applying thermodynamics, which is formulated for an equilibrium system, to flow fields are pointed out. UNIDIRECTIONAL VISCOUS FLOWS The concepts of a shear stress and the sign con vention to be associated with it are introduced by considering fully developed flow in a pipe. New CHEMICAL ENGINEERING EDUCATION ton's law of viscosity is initially presented by assuming that the shear stress is directly pro portional to the velocity gradient. Through the momentum theorem it is pointed out that the shear stress may also be interpreted as a flux of momentum. Through this concept kinetic theory can be used to interpret fluid viscosity. Flow be tween circular cylinders is considered because it is a convenient way to introduce concepts which are used later to extend Newton's law of viscosity to threedimensions. It is pointed out that viscous effects will not depend on that portion of the velocity gradient that gives rise to solid body rotation but will depend on that portion which distorts the shape of fluid area elements. NONNEWTONIAN FLUIDS Experiments are described which illustrate non Newtonian behavior and which yield definite prop erties characterizing the theological behavior of fluids. These include nonlinear dependence on the rate of strain of the fluid, normal stress effects caused by shear flow, and elastic effects. The Rabinowitsch equation is developed either in class or in a homework problem because of its import ance in interpreting nonlinear effects in steady flows. Methods for calculating normal stress co efficients and the use of small amplitude sinusoidal oscillations and relaxation tests to determine elastic properties are discussed. The Maxwell equation for linear viscoelastic fluids is applied to a few simple flow problems. EQUATIONS OF MOTION The differential equations describing the three dimensional flow of a Newtonian fluid are now developed in a cartesian coordinate system. This is not done in general curvilinear coordinates be cause the small number of new physical concepts introduced by this generalization does not seem to warrant the added complexity. The equation of conservation of mass and the momentum theorem are applied to a differential volume. The use of cartesian tensors to simplify the notation is intro duced. The physical interpretation of the sub stantial derivative arising from the momentum terms is presented. It is pointed out that in order to describe the force acting on the surface of a volume the location as well as the orientation of each surface element is needed. The problem of representing these surface forces is greatly sim plifid by showing that the stress on any arbi S. a theory for turbulent flows is still to be developed. My goal ... is to introduce some of the language used in correlating turbulence measurements and in characterizing the turbulent field. trarily oriented surface element can be described in terms of the nine stress components needed to specify the stress vectors acting on three mutually perpendicular planes. Cauchy's equation of motion can then be derived. The properties of the stress components are now explored in order to facilitate the development of constitutive relations. It is shown that the nine stress components are a second order symmetrical cartesian tensor. The concepts of principle axes and principle stresses are introduced. The invariants of the stress tensor are defined. CONSTITUTIVE RELATIONS The constitutive equations relating the stress components to the velocity field are now devel oped for a Newtonian fluid. The problem of doing this for more general fluids is discussed briefly. The velocity gradient is shown to be a second order cartesian tensor which can be represented as the sum of a symmetric and an antisymmetric tensor. The stress components can be related only to the symmetric part of the stress tensor (rate of strain tensor) since this is the part that gives rise to the distortion of fluid volume elements. Newton's law of viscosity is generalized to three dimensions by assuming that the components of the stress tensor are linearly related to the com ponents of the symmetric part of the velocity gradient tensor. This produces eightyone co efficients of viscosity. Only two of these coeffi cients are independent, because the fluid is isotropic and because the stress and rate of strain tensors are symmetric. The assumption of a linear relation between the stress tensor and the rate of strain tensor is now relaxed and the most general relation is developed for an isotropic fluid for which the components of the stress tensor are only functions of the components of the rate of strain tensor. This relation predicts normal stress effects and nonNewtonian behavior for steady flow in a channel. The work of Oldroyd, Rivlin and Erickson, and Colman and Noll aimed at developing constitutive relations which exhibit elastic effects as well as nonlinearity and normal stresses is discussed only qualitatively. The NavierStokes equations are now derived FALL 1969 You won't just get your feet wet. Standard Oil Company of California offers all the experience you can soak up. You'll start out facing practical situations and using your academic knowl edge and skills to solve real problems. You may even have to improvise and develop new approaches to specific questions. We rotate the assign ments of young professionals. You will be able to work with different groups of experienced colleagues and sharpen your skills on a variety of projects. Talk with our representative when he comes to your campus about the opportunities we have for you. Check your placement office for more information or write to: D. C. Reid, Coordi nator, Professional Employ ment, Standard Oil Company of California, 225 Bush Street Room 105, San Francisco, California 94120. Standard Oil Company of California An Equal Opportunity Employer by introducing the constitutive equations for a Newtonian fluid into the Cauchy equation. The solutions of these equations for the case of an incompressible fluid are developed for three very general assumptions: creeping flow, perfect fluids, and boundary layers. CREEPING FLOWS According to the creeping flow approximation the inertia terms of the equations of motion can be neglected at small Reynolds numbers. Creeping flow around a sphere is worked out in much detail since it illustrates the use of a stream function, the method of eliminating the pressure term from the equations of motion, and the calculation of body forces and skin friction on a solid body. The case of flow around a bubble is worked out in a homework assignment. It is shown that the creep ing flow approximation does a good job in predict ing the force on a sphere but a poor job in predict ing the flow field at distances far from the sphere. For flow around a cylinder it doesn't even predict the force on the cylinder. Regular perturbation methods are shown to yield poor higher order approximations to the creeping flow solutions. The method of Oseen is discussed. The singular perturbation method, as outlined by Proudman and Pearson, is then shown to be a proper way of getting higher order approximations. PERFECT FLUIDS A perfect fluid is defined as one for which the viscosity and thermal conductivity are zero and for which the entropy of fluid particles is con stant. The Euler equations then describe the flow field. The conditions under which one might ex pect an irrotational flow field are discussed. The velocity vector is then describable as the gradient of a potential function and the velocity field is given by the equation of conservation of mass. The integration of the Euler equation using the assumption of irrotational flow yields the Ber noulli equation. The Bernoulli equation can also be obtained for rotational flows by applying the momentum theorem to flow of a perfect fluid along a stream tube. The constant of integration then varies from stream tube to stream tube. The solution for the flow of an irrotational perfect fluid around a sphere is obtained. The predicted pressure distribution around the surface is dis cussed. The concept of virtual mass is introduced by considering the unsteady motion of a sphere in a perfect fluid and is applied to some problems of interest. Twodimensional flows of irrotational perfect fluids are then considered. A stream func tion can be defined for a twodimensional flow from the equation of conservation of mass and complex variable theory can be used to solve flow problems. A number of examples are considered including the lift of solid bodies and free stream lines. The treatment of small amplitude, waves at an interface is one of the more successful applications of the theory for irrotational perfect fluids. Some of the problems considered in this area are progressive twodimensional waves, standing waves, group velocity, wave resistance, KelvinHelmholtz instability, and Taylor insta bility. BOUNDARY LAYERS Boundary layer theory attempts to correct per fect fluid theory for viscous effects by assuming the existence of a viscous layer close to a solid surface. If this viscous layer is thin compared to the dimensions of the body a simplified version of the NavierStokes equation applicable to bound ary layers on flat plates and curved surfaces is obtained. The concept of separation is discussed and it is pointed out that boundary layer theory is only applicable up to the point where the bound ary layer separates from the solid surface. The difficulties of applying the theory are discussed, and, in particular, the problems associated with determining the external inviscid flow or the pres sure distribution around the body are emphasized. Dimensional analysis is applied to the boundary layer equations to determine the functional rela tion between the skin friction coefficient and the Reynolds number. Similarity solutions are briefly discussed. The series methods of Blasius and of Gbrtler and the integral methods ofKPohlhausen and of Bohlen and Walz for solvifig the boundary layer equations for some general pressure dis tribution are introduced. I find that the best way to present these methods is to give a homework problem which requires the application of bound ary layer theory to a solid body for which the pressure distribution is known. TURBULENCE This treatment of boundary layers concludes my discussion of laminar flows. It might seem anomolous that even though most flows in nature are turbulent I don't introduce the topic of tur FALL 1969 bulence until this point in the course. The reason for this is simply that a theory for turbulent flows is still to be developed. Therefore my goal in treating turbulent flows is to introduce some of the language used in correlating turbulence measurements and in characterizing the turbulent field. The increased apparent shear stress in tur bulent flows is explained in terms of the momen tum transport caused by the fluctuating velocity field. The concepts of eddy viscosity and mixing length are introduced. They are found not to be as useful for turbulent flows as were molecular viscosity and mean free path for laminar flows because the scale of the turbulent motion respon sible for transport is of the same order as the dimensions of the field. A general discussion is given on the character of fully developed velocity profiles and, in particular, on the roles of fluid viscosity and of the viscous sublayer. The varia tion of the average velocity and the eddy viscosity with location is correlated through dimensional analysis and the "law of the wall", the "velocity defect law", and the "overlap law". It is then shown how the definition of eddy conductivities for heat transfer are useful in explaining meas urements and in particular the effect of Prandtl number on temperature profiles. The interpreta tion of fully developed velocity profiles and Cole's "law of the wake" are used to develop predictive methods for general turbulent boundary layer flows. Taylor's treatment of point source diffusion in homogeneous turbulent fields is presented as one of the few successful theories in turbulence. It is used to explain the gross aspects of turbulent mixing and to interpret the observed variation of eddy conductivities. Statistical methods for de K01 news The following item on CACHE was sub mitted by Professor Warren D. Seider, University of Pennsylvania, Philadelphia, Pennsylvania 19104. The CACHE (Computer Aids for Chemical Engineering Education) Committee has been organized to coordinate the development of com puting systems for use in chemical engineering education. The committee includes twenty educa tors from sixteen universities. The principal goal scribing turbulent flows are discussed and in par ticular the concepts of correlation, scale, and spectrum are introduced. A very brief summary is given of theoretical attempts to deal with tur bulence through the use of the concept of isotropy and the definition of a turbulence structure. COMPRESSIBLE FLOWS The last topic in the outline is a onedimensional treatment of compressible flows. It is usually pre sented after the material on perfect fluid theory but appears here in my outline because in recent offerings of the course I have deleted it. Most of this material with the exception of that on finite amplitude waves and shock tubes are more prop erly treated in undergraduate courses. OTHER APPROACHES? I should conclude this article by saying that the course that I have discussed is only one way, and not necessarily the best way, of introducing graduate students in chemical engineering to basic concepts in fluid dynamics. My own intro duction to and interest in fluid dynamics devel oped from a course in reactor design. FLUID MECHANICS COURSE OUTLINE 1. Basic Equations 2. Unidirectional Viscous Flows 3. NonNewtonian Fluids 4. Equations of Motion for a Viscous Fluid 5. Constitutive Relations 6. Creeping Flow Approximation 7. Perfect Fluid Theory 8. Boundary Layer Theory 9. Turbulence 10. OneDimensional Compressible Flows of the CACHE Committee is to accelerate the in tegration of computing into the chemical engi neering curriculum by sustained interuniversity cooperation in the preparation of curriculum and course outlines and in the specification and crea tion of computing systems. The CACHE Committee's curriculum sub committee has organized a session for the AICHE Annual Meeting in Washington, D.C., entitled "Computers in Chemical Engineering Education." The session will emphasize topics relating to short and long range plans for the integration of computers into the curriculum. Ten members of the CACHE Committee will participate in the panel discussion after brief presentations. CHEMICAL ENGINEERING EDUCATION . A Celanese c IS freedom, satisfaction, reward, I. career a little selfdiscipline. We feel your students should settle for nothing less than the professional opportunity that offers them the most satisfaction and reward. A career in which they can be themselves. Do their own thing. Naturally, that takes a bit of planning on their part. Like a degree in chemistry, chemical, industrial or mechanical engineering. Or accounting. Plus a degree of imagination. Ambition. Responsibility. A little selfdiscipline. With them, they'll find the professional climate excellent at Celanese. For some very good reasons. We're big. But not too big. More important, we're still actively young and growing. In the past ten years, or so, our sales have almost quadrupled. And you can tell your students that we won't tie them up with long, formal training programs. They'll learn as they practice their profession. Reap the rewards of performancenot on age, or how long they've been with us. Of course, they will be working with experienced pros. So, when a hand is needed, they'll get it. Because it's to our mutual advantage to have them grow as fast as they canand go as far as they can. If you, or your students, have any questions about Celanese, we will be pleased to answer them. Promptly. Just write to: John B. Kuhn, Manager of University Relations, Celanese Corporation, 522 Fifth Avenue, New York, N.Y. 10036. CELANESE An equal opportunity employer $4 Semifnaz GCanue: STATISTICAL THEORIES OF PARTICULATE SYSTEMS HUGH M. HULBURT Northwestern University Evanston, Illinois 60201 The theory of transport processes in the past twenty years has been developed into a compre hensive scientific basis for the description of the essential operations of chemical engineering in any arbitrary differential volume element. The theory is essentially a continuum theory which poses differential equations which must be inte grated over the domain pertinent to each particu lar process unit under consideration. Two major difficulties arise in application. The less onerous is the fact that continuum theory does not pro vide the values of the transport properties, equa tions of state and constitutive relationships for specific systems. Molecular statistical mechanics or experimental measurements, or both, must provide this data. The more serious is the ex tremely intricate forms often taken by the solu tions of the transport equations, for example, in turbulent flow. Although turbulence has been studied intensively for nearly 100 years, we still lack a sound theoretical connection with the NavierStokes equations and still less a pre dictively useful one. The principles of fracture under stress are equally important in crushing and grinding operations and are even less well understood in terms of elasticity theory. Aggrega tion, coagulation and agglomeration are likewise processes which cannot as yet be usefully de scribed by fundamental transport theory. All of these processes have in common the cir cumstance that they occur under conditions which permit a large variety of solutions of transport equations and which select no one of this variety uniquely. The flow paths in a stirred vessel, each of which we are confident is a solution of the NavierStokes equations, are so various that we must describe them as a statistical aggregate of paths having some probability distribution. In most applications, the residence times determined by each path are of less interest than the mean residence time, or some other statistical property of the residence time distribution. In some cases, however, the distribution itself is of direct in terest, as in a crystallizer, since here the crystal Hugh M. Hulburt is Professor and Chairman of the Chemical Engineering Department at Northwestern Uni versity. After receiving his PhD in physical chemistry at the University of Wisconsin in 1942, he was National Research Fellow in Chemistry at Princeton University for one year. He has taught at Hunter College, and the Catholic University and he has held positions with Shell Oil Company, Chemical Construction Corporation, and American Cyanamid Companyat the latter firm as Director of Chemical Engineering (195859) and Direc tor of Physical Research (195963) at their Stamford Laboratories. In 1963 he joined the faculty of North western University, and became Chairman of the De partment of Chemical Engineering in 1964. His current research interests include reaction design, chemical kine tics, and electrochemical engineering. size distribution is a direct consequence of the residence time distribution. Thus, there is a class of unit operations for which the basic transport equations are insuffi cient to provide the theory for engineering design. The first stage of improvement over direct em piricism and analysis by dimensionless groups is to pose a theory based on a statistical distribution of possible transport processes. The system in question is defined by a number of properties (or state variables) whose values are subject to stochastic variation. We visualize the possibility of making a simultaneous observation of these properties and believe that, in any sufficiently large set of replicated observations there will be a definite frequency with which any specified set of values will be observed. The function of the state variables which associates this probability with the specified set of values is the probability distribution. CHEMICAL ENGINEERING EDUCATION  The flow paths in a stirred vessel, each of which is a solution of the NavierStokes equations, are so various that we must describe them as a statistical aggregate For example, in an ideal gas the velocity of any particular molecule might (in thought at least) be observed. Repeated observations will not give the same result, but the observations will tend to cluster about a central value. We can define a function P(v) which is the probability that the observed value, v', be at most equal to v, i.e., P(v) = {Probability that v' < v} = F(v) (1) Our fundamental physical hypothesis is that re peated observations define in the limit a definite function F(v). In general, of course, F(v) may be multidimensional, even continuously infinite dimensional, but the conceptually simple cases of one to three dimensions are often of real utility. Our knowledge of the probability P(v) is not limited solely to the results of empirical observa tion. We remain convinced that the transport equations hold for sufficiently local phenomena just as any one molecule in a gas obeys New tonian (or quantum) laws of motion. Hence, the variations in the values of the statevariables must be consistent with the local transport laws and their distribution function, P(v), is not purely random. There must be correlations im posed by the local laws which lead to functional relationships between the statistical properties of the system and which account for the central ten dencies of the distribution of statevariables. Indeed, they should determine the functional form of F(v) itself. The role of statistical theory is to discover and define those classes of distribution function which are consistent with the local transport laws and whatever other conditions are required and can be defined by the physical nature of the process under study. These additional conditions are usually such physically obvious requirements as, for example, that the system is confined to a finite volume, or contains a fixed mass, or has a definite average velocity and/or energy. Since local velocity is a distributed variable, the re quirement that the system have an average energy requires that F(v) have a form that will produce a definite mean square velocity v'dF (v) Thus physical considerations will restrict the class of admissible distribution functions to those of paths having some probability distribution. possessing at least one and usually two moments, or more. In addition, the effect of the local transport laws on the distribution functions can be ex pressed very generally. For the simple case of a single stochastic variable, q, whose local rate of change is q(q, t), a continuity equation holds in the form 3f a (qf) S+ aq = H(f,q,q,t) (3) Here H (f,q,q,t) is a function that expresses the effect of random sudden interactions upon the distribution density f. Two classes of system are met with in applications. In one, the elements of the system interact only rarely, as in kinetics, or not at all, as in collisionless plasmas or non agglomerating crystallizers. Changes in f are then brought about by the influence of external fields on the motion of the noninteracting par ticles, modified by occasion strong interactions (or collisions) between particles. In the second class of system, a particle interacts frequently with others, but weakly, so that each collision produces only an infinitesimal modification of its properties, as in Brownian motion, or in plasma dynamics. In the first case, one estimates the change in q produced by the strong interaction in a collision and sums over the distribution of target particles and impinging particles. The function H then is the net probability that in unit time a particle acquire a value of q = q' + dq by interaction with other particles. It will usually be expressible as a collision frequency multiplied by a crosssection and integrated over the distributed property, q. In the second case, the effect of collisions is to produce an infinitesimal change in f and the H function is expanded in series of derivatives of f with coefficients which depend upon the collision dynamics. This equation or its equivalent is called vari ously the particle balance equation10 the Liou ville equation, the Kolmogoroff equation, the FokkerPlanck equation, the Boltzmann equation, or possibly other names in specific contexts. In recent engineering applications, physical particles have been envisioned and the "particle balance equation" has has been a descriptive term.1,2 In FALL 1969 general probability theory, the ChapmanKolmo goroff equation is studied, usually under restric tions slightly more limiting than desirable for engineering applications.3 Physicists have studied the FokkerPlanck or Boltzmann equations5 (or for aerosols,6 the Smolichowski equation) in theories of molecular statistical mechanics.7 The Liouville equation clearly applies to any stochastic system for which one can define the weakly interacting entities, or generalized par ticles, and the significant statevariables, q, and their local laws of motion, q=Q(q,t). When physical particles are the elements of the system, f can be interpreted directly as the particle num ber distribution in q, which may be taken to be location, size, mass, residence time or other meas urable or significant property. However, in polymers," one can define chain segments and a distribution of chain lengths which obeys the balance equation. For all termination schemes ex cept radical recombination, or branching transfer, the collision term, H, vanishes and the chain length distribution is determined by the rate of growth in a monomer field, modified by catalysts and transfer agents. Chemical kinetic laws thus give us forms for q=Q(q) and where required, for H(q,t). In comminution9 the same equation is obeyed, but now q (size) is altered only by impact collision. The present problem here is discover the appropriate fracture equations for individual par ticles so that an expression can be given for H(q). In aerosol coagulation, the collision mechan isms are well defined but the effects of boundary conditions and external fields have been explored only partially. That there is a pedagogically useful common base for this wide variety of applications was evi dent in a seminar conducted recently at North western University in the Winter Quarter, 196869. Graduate students and faculty inter ested in crystallization, polymerization, aerosol coagulation, hydrosol coagulation and reaction kinetics in dispersed two phase media par ticipated. The ostensible outline of the seminar was afforded by Beran, "Statistical Continuum Theories," but it soon became apparent that some background in basic probability was required and several weeks were spent by the author develop ing the basic probabilistic notions, the Chapman Kolmogoroff equations and the Liouville or FokkerPlanck Equations as applied to simple The role of statistical theory is to discover and define those classes of distribution function which are consistent with local transport laws and . other conditions . and can be defined by the physical nature of the process under study. Markov processes. The books by Feller, "Introduc tion to Probability Theory," Vol. I and especially Vol. II were invaluable in this phase of the semi nar. Blanc LaPierre and Fortet, "Theory of Ran dom Functions," was also very helpful in pro viding physically motivated treatments of proba bility theory. Loeve, "Probability Theory," is a standard reference on matters of rigor and mathe matical detail. The following topic headings suggest the scope of the course. Each occupied three to five lecture periods. 1. Heuristic motivationstirred tank crystallizerprob abilistic model building. 2. Concepts of probability theorydistribution, mean, conditional probabilities, set functions, combination of probabilities. 3. Characteristic functions, moments. 4. Random functions, indicator function. 5. Law of Large NumbersTchebycheff inequality, Poisson distributionCentral Limit Theorem. 6. Standard probabilistic modelsBernoulli trials, com pound events, normal distribution. 7. Random processeswaiting time distributionspro cesses with independent incrementsgamma distribution Poisson processes. 8. Kolmogoroff equationsMarkov processes, convolu tions. 9. The Agglomeration Equationcollision processes. The final eight days were devoted to reports on specific applications to fields of interest to the participants. As the seminar developed, it became increas ingly apparent that much of the engineering lit erature in probabilistic models consists of redis covery or reinterpretation of some basic concepts and models which have been thoroughly studied in general theory, but which need to be made explicit and in some cases extended to cover fields of engineering interest. Most discussions of ran dom processes are based on Markov processes, yet many engineering models are nonMarkov. Some may be considered to be superpositions or con volutions of Markov processes, and this represen tation is often illuminating when one discovers it. Of greatest value, perhaps, was the exercise in the use of probabilistic concepts and the demon stration of a common logical structure in a wide range of seeming independent physical phe nomena. An initial insight was gained into the CHEMICAL ENGINEERING EDUCATION identification of properties of a system which are purely consequences of its being an assembly of weakly interacting parts or particles as distin guished from those which reflect the intrinsic physical nature of the isolated particles. It seems to the author that while such insights depend upon understanding the general theory of stochastic processes, they do not come without deep study and comparison of specific processes. Therefore, the statistical theory of particulate systems is truly an engineering science, approach able in a useful way from the base of the analysis of engineering problems. REFERENCES 1. Randolph, A. D., and M. A. Larson, AIChE Journal 8, 639 (1962). 2. Hulburt, H. M. and S. Katz, Chem. Eng. Sci. 19, 555 (1964). 3. Feller, W., Introduction to Probability Theory, Vol. I and II., John Wiley and Sons Co., New York 1966. 4. BlancLaPierre, A., and R. Fortet, Theory of Random Functions, Vols. I and II, Gordon and Breach, New York. 5. Chandrasekhar, S., Rev. Mod. Phys., 15, (1943). 6. Fuchs, N. A. Mechanics of Aerosols, Macmillan, New York (1964). 7. Cercignani, C., Mathematical Methods in Kinetic Theory, Plenum Press, New York, 1969. 8. Bamford, C. H. and H. Tompa, Trans. Farad. Soc. 50, 1097 (1954). 9. Austin, L. G., "Equations of Grinding", ACS Christ mas Symposium, Cambridge, Mass., 1967. 10. Himmelblau, D. and K. B. Bischoff, Process Analysis and Simulation, John Wiley and Sons, New York, 1968. HOUGEN ON COLBURN (from p.171.) loidal chemistry and Peter Debye in the kinetic theory of gases. AFTER SECURING HIS DOCTORATE degree in 1929, Colburn was employed in the Engi neering Experiment Station of the E. I. Du Pont de Nemours Company under Thomas H. Chilton. The du Pont Company very generously supported Allan in his undertaking to derive independently the mathematical formulations for mass and heat transfer in fluid streams covering a wide variety of industrial devices and conditions. These formu lations appear in Chapter VII of the pioneer bul letin "Studies on Heat Transmission" by Colburn and Hougen [Bulletin 70 of the Engineering Ex periment Station, College of Engineering, Univer sity of Wisconsin, October 1930]. Otherwise this bulletin is based on Colburn's doctorate thesis. The rough draft was improved by the critical review of K. M. Watson. Colburn was an ideal student, scientist and en gineer. In conversation and public lectures he had an unusual capacity for clarity of expression. He met unsound criticisms and arguments of his audience with patience and encouragement, never with disparagement, seeking not prsonal acclaim but rather promoting self confidence and ambition in others. My technical correspondence with Allan continued for fifteen years following his academic career but terminated when I no longer could keep pace with his ever expanding scientific specula tions. My last meeting with Allan occurred in London in September 1951an unexpected meeting at the time of the Festival of Britain. Mrs. Hougen and I met Mrs. Colburn accidentally on a London bus on our way to the Tate Art Gallery. Allan arrived in London one week later while we were awaiting boat reservations for a return trip to the United States. The four of us spent a pleasant summer evening together at the amusement section of the Festival of Britain. We enjoyed watching the teenagers on the roller coasters and whirligigs. In turn the British teenagers enjoyed our strange use of the English language. As an epilogue to this sketch I should add one additional incident. In the fall of 1963 a young coed, Miss Nancy Hall, a graduate student and research assistant in the Department of Oncology at the University of Wisconsin came to my office to inquire about the possibility of a young man, Willis Colburn, entering the University of Wis consin as a Senior in Electrical Engineering. This was easily arranged. When Willis appeared on the campus, his similarity to his father 43 years previous was most striking and brought a surge of nostalgic memories. And thus this account re turns full circle to its beginning, starting with a meeting with Allan and his father Willis in 1922 and ending with the enrollment of his son Willis 42 years later. Nancy and Willis were soon married. Both continued as research assistants and graduate students for two or three years in their respective fields. And thus three generations of Colburns have touched my life over 46 years of time. I understand the fourth generation has recently arrived. I shall close with one sentence from the letter of his college friend Louis Warrick "How proud the parents (of Allan Colburn) would be to know of this latest honor to the memory of an out standing son." This applies to all who knew Allan Colburn. FALL 1969 4 Comase in Mat 't7anapodt DIFFUSIONAL OPERATIONS E. N. Lightfoot University of Wisconsin Madison, Wisconsin 53706 This is an introduction to heat and mass trans fer for graduate chemical engineers and others seriously interested in transport phenomena. It is normally given in the spring semester to provide entering students an opportunity to take inter ,mediate transport phenomena or fluid mechanics by way of preparation. It can be followed to ad vantage by special topics courses offered on an irregular basis and by Professor Curtiss' course, The Transport and Other Properties of Fluids, given in our Chemistry Department. The organization of the course is indicated in the idealized outline of Table 1, but the actual coverage has changed substantially from term to term ever since its original ancestor in 1944 was introduced by Olaf A. Hougen. Emphasis is normally given to those topics in greatest need of reorganization. Thus in 196566 and 196667 emphasis was put on the formulation and solution of multicomponent diffusion problems and the estimation of transport properties of lowdensity gases. During the last two years the highest prior ity has been given to applications of boundary layer theory, with particular emphasis on mass transfer across mobile interfaces. This current bias shows clearly in this paper. It is now our hope to decrease the emphasis on transport prop erties, and, if possible, to put this material in a separate course. If this goal is realized, more attention will be given to applications of current interest, to demonstrate the utility of the tech niques introduced. The direction in which this course changes will, however, be strongly influenced by the instructor in charge since course content and the instructor's research program have always been closely inter related. Until recently the coverage was heavily influenced by Mr. Lightfoot's interests in bio medical mass transfer and fluidfluid contacting devices. With the increasing involvement of Mr. T. W. Chapman more emphasis on electrochemical applications can be expected. Professor Lightfoot was educated at Cornell University, BchE and PhD. In 1953 he joined the Wisconsin depart ment, became active in curriculum revision, and colla borated with Bird and Stewart on the text Transport Phenomena. His major research interests are in mass transfer and separation processes and in the application of transport phenomena to environmental and biomedical problems. There is no one reference on which this course is based. However, the text Transport Phenomena provides a good introduction to much of the ma terial covered, and The Molecular Theory of Gases and Liquids is used very heavily for both the phenomenological aspects of diffusion and the prediction of transport properties. It has also been the custom in recent years to distribute copies of the lecturer's notes, and a substantial portion of these is now available in printed form as Chapter Two: Estimation of Heat and Mass Transfer Rates in Volume 4, Lectures in Trans port Phenomena of the AIChE continuing educa tion series. SUMMARY OF PRESENT COURSE I. The Equations of Change and the Transport Properties The purpose of this first section of the course is to lay a sufficiently firm theoretical foundation for the foreseeable needs of the students. It has been found desirable to spend six to eight weeks on these topics, or eighteen to twentyfour 50 minute class periods, and even then only a partial coverage can be achieved. In the spring of 196869 most of the material on transport properties was omitted to permit greater coverage of boundary layer theory. However, it is felt by the author that this material should not be slighted. Our introduction of the conservation equations CHEMICAL ENGINEERING EDUCATION TABLE 1Topical Outline for Diffusional Operations I. The Equations of Change and the Transport Prop erties A. Conservation Relations for Multicomponent Systems 1. The equations of continuity and energy 2. The equations of motion 3. The equation of entropy conservation B. Relations between Fluxes and Driving Forces: The Transport Properties 1. A thermodynamic basis for the formulation of rate equations 2. Prediction of transport properties on the basis of kinetic theory 3. Semiempirical correlations of the transport properties 4. Experimental determination of transport proper ties C. The Requirements of a Quantitative Description 1. Review of the equations and boundary conditions needed to describe masstransfer systems 2. Dimensional analysis 3. The approximate description of multicomponent nonisothermal systems in terms of binary isothermal solutions II. Estimation of Heat and MassTransfer Rates in Welldefined Systems A. Intraphase Transport in Systems of Fixed Geometry 1. The film model: unidirectional transport 2. Convective heat and mass transfer through laminar boundary layers a. Forced convection b. Free convection 3. The effects of operating conditions on intraphase transfer coefficients a. Net mass transfer across bounding surfaces b. Homogeneous chemical reactions c. End effects B. Intraphase Transport in Systems with Mobile Interfaces 1. The boundarylayer equations of energy and con tinuity near a deformable surface 2. Description of model systems 3. Effects of operating conditions C. Interphase Transport 1. Addition of intraphase resistance: the tworesist ance theory of Whitman and its limitations 2. Effects of heat and mass transfer on hydro dynamic stability: Marangoni effects III. Approximate Descriptions of Complex Systems A. Mass Transfer to Rippling Laminar Films 1. The effect of surface mobility on masstransfer effectiveness 2. The effects of surfacetension gradients on heat and mass transfer B. Mass Transfer across Free Turbulent Surfaces C. Mass Transfer in Dispersed Systems: Performance of a Model Liquid Extractor is a straightforward extension of the treatment in Chapter 18 of Transport Phenomena and hence is largely a review. This discussion is short, and it is a convenient vehicle for reviewing vector tensor operations. The rate equations require a much more exten sive discussion, however, and neither the coverage nor the approach of Transport Phenomena is con sidered adequate for our purposes. We begin with a careful description of the postulates and predic tions of irreversible thermodynamic analyses and put major emphasis on the masstransfer aspects. The most important single result of this discus sion is the generalization of the StefanMaxwell equations which for isothermal transport take the form: 3(1) JtL with i and j = 0 These equations define* the multicomponent dif fusivities. Equation 1 forms a very convenient and flexible basis for discussion of diffusional transport in both free solution and mechanically constrained membranes. It is particularly useful for systems containing electrolytes. Discussion of kinetic theory is largely a conden sation of the pertinent sections of the Molecular Theory of Gases and Liquids, and its purpose is to explain the origins of the lowdensity gas ex pressions summarized in Chapters 1, 8, and 16 of Transport Phenomena. This is difficult material to explain in a short time, particularly in the later highly mathematical stages of the development of expressions for the transport properties. We are still looking for easier ways to "tell the story". Nevertheless this portion of the course has proven useful to the students and has stimu lated a number to take Professor Curtiss' course in transport properties referred to above. Subsection C is rather heterogeneous and in cludes such topics as the specification of boundary conditions at gassolid and fluidfluid interactions, development of scaleup criteria, and solution of multicomponent diffusion problems by matrix techniques. Here too it is important for the lec turer to focus attention on carefully selected areas rather than to attempt encyclopedic cov erage. *Here .j is the molar chemical potential of species j, and except as noted the nomenclature of this article is that of the text Transport Phenomena. FALL 1969 II. Estimation of Heat and Mass Transfer Rates in Well Defined Systems Accurate prediction of heat and mass transfer rates in applications of practical interest is still a formidable problem. This is due in part to the complex geometry and flow conditions encoun tered but also to the large number of parameters affecting system behavior. Fortunately it is often possible to determine the functional dependence, and even the magnitude, of heat and masstrans fer coefficients from highly simplified models of the real system. This section of the course is devoted to discussion of the most widely useful physical models. A. Intraphase Transport in Systems of Fixed Geometry We start by discussion of intraphase transport in systems with fixed boundaries and concentrate on asymptotic boundarylayer models. In each case we begin our discussion with a simple proto type to illustrate characteristic behavior and then proceed to generalize our discussion to the degree possible. After discussion of convective transfer at zero interfacial velocities we consider the com plications introduced by chemical reactions and high net mass transfer rates. We first consider film models of heat and mass transfer and concentrate our attention on tur bulent transport in duct flows. In this discussion we compare available expressions for the turbu lent transport properties and show that the best of these yield simple asymptotic correlations sim ilar to the ChiltonColburn relation for the Rey nolds and Schmidt (or Prandtl) number ranges of most interest. Since adequate empirical corre lations already exist for long ducts this discussion is most useful as a preparation for analysis of transfer in short ducts and mass transfer accom panied by chemical reaction. We next consider forcedconvection transport through laminar boundary layers and use as our prototypes penetration into a stagnant liquid and dissolution of a sparingly soluble duct wall. These are then generalized using the techniques pio neered by Acrivos and Stewart to systems of arbitrary geometry as suggested in Fig. 1. The key results of these analyses are: SF(4/3) (" d.] Sg.r '(2) (H~isLAn; a 1d N,,= (R.A)'4 I o 5 fal e t ) Here* h,, hy, h2 = scale factors for the locally orthogonal S coordinates of Fig. 1. yz the dimensionless shear rate at the system boundary u*o = dimensionless tangential velocity component at the interface. X = Schmidt or Prandtl number. Equation 2 is essentially that given by Stewart in his boundarylayer analysis of heat and mass transfer about threedimensional solid bodies at high Sc or Pr. Equation 3 is a straightforward generalization of Acrivos' development for two dimensional and axisymmetric bodies at very low Sc or Pr. It is, however, actually most useful for describing steady transfer about drops and bub  . ',E ATED Z  SEPARATED ZONE Fig. 1. Heat or mass transfer from a threedimensional surface. The asymptotic analysis applies upstream of the separated and turbulent flow regions. *Notation given by Lightfoot, op cit. CHEMICAL ENGINEERING EDUCATION bles at very high Schmidt number. It is thus a special case of the surfacestretch modification of the penetration model discussed below. After introduction of these rather general re sults the behavior of several real systems is dis cussed briefly to demonstrate both their utility and limitations. We follow this discussion with introduction to freeconvection transport and the challenging problems of transfer accompanied by chemical reaction and high net masstransfer rates. In these latter discussions we place particular em phasis on the development of geometryinsensi tive correlations. B. Intraphase Transport in Systems with Mobile Interfaces This section is devoted largely to extension of the penetration model to systems with stretching and shrinking interfaces. We follow here the approach of Angelo, Lightfoot, and Howard as refined by Stewart, Angelo, and Lightfoot. In doing this we make use of a similarity trans formation first suggested by Ilkovic in 1934 and since independently reintroduced many times by other workers including Acrivos, Lochiel and Calderbank, and Scriven and Pigford. We are able to show as a result of our analysis that the masstransfer behavior of an extremely wide variety of fluidfluid systems is described by the simple equation . iF4 T)/sct)]2d where Nu is the temporal mean Nusselt number for a surface element based on its area so at an arbitrary reference time, and s (7) is the area as a function of the dimensionless time t*. The symbol A is used for Schmidt or Prandtl number. The term in braces in Eq. 4 is simply the penetration theory result for a nondeformable surface ele ment, and the correction factor K is just the tem poral rootmeansquare value of s/So. Equation 4 is useful for drops and bubbles, rippling films, laminar jets, and many other important systems. Some of the most important of these are discussed in detail. C. Interphase Transport This discussion is devoted largely to a critical review of the twofilm theory of Whitman and of Marangoni effects. Surprisingly relatively few vigorous treatments of interphase transport are available, and it is still common practice to estimate overall coeffi cients by addition of separately calculated intro phase resistances. We review the errors inherent in this approach and show that improper area averaging of local coefficients can be a particu larly important source of error. In this discussion we follow much of the original treatment of King. In fluidfluid contractors variations in inter facial tension over the masstransfer surface can have profound effects on both the shape of this surface and the flow conditions near it. These socalled Marangoni effects were first shown to be important in separations processes by Sher wood and Wei, and the first significant attempt to describe them quantitatively for process equip ment was made by Sternling and Scriven by means of a linearized stability analysis. We con sider in our course two simple analyses of Maran goni effects in falling films due to Ludviksson and Lightfoot and Wang, Ludviksson and Lightfoot. In addition to their simplicity these examples offer the advantage of direct application in sim ple process equipment. They can thus be used in our practical examples. III. Approximate Descriptions of More Complex Systems In this section we consider systems too complex for detailed rigorous analysis but which can be usefully analyzed in the light of the preceding sections. This is felt to be a very important part of the course since in practice engineers must usually stick their necks out and deal with a con siderable amount of uncertainty. The purpose here is to show what sorts of approximations and simplifications are likely to be successful and to encourage the development of "engineering judge ment". The specific examples we are presently using are: 1) Semiquantitative descriptions of inertial rippling and Marangoni effects on mass and heat transfer in laminar falling films. We base these discussions on an alyses and measurements performed in our own labora tory by Howard, Irani, Ludviksson, Massot, and Wang. 2) Prediction of reaeration rates of streams using an extension of the analysis of Fortescue and Pearson. 3) Prediction of the performance of a sievetray liquid extractor from fluidmechanic measurements, based on analyses and measurements of Angelo and Howard in our laboratory. FALL 1969 The university where classes neverend. Union Carbide's research centers are in many ways like a university. In any one of them you'd meet a faculty with advanced degrees in practically every science. You'd see them scribbling complicated formulas on blackboards and working with complex scientific instruments. You'd hear them discussing the uses of tremendous pressures, unearthly heat, intense coldas well as problems in oceanography, outer space, atomic energy. In any one of our many research centers and laboratories which we maintain here and abroad, you'd sense the vast and diversified scope of Union Carbide technology. Finding better ways to do things is the aim of this research. And when a better way is found to revolutionize an industrial processor simply develop a new product for your comfort or convenienceour research scientists don't graduate. They move on to new and exciting challenges created by today's advancing technology. M Aoom For additional information on our activities, write to Union Carbide Corporation, Department of University Relations, 270 Park Avenue, New York, N.Y. 10017. An equal opportunity employer. THE DISCOVERY COMPANY We have chosen these for familiarity and because they are (perhaps fortuitously) successful ex amples of the application of fundamentals intro duced earlier in the course. We believe that unit operations should be taught this way when pos sible: from personal experience and in the light of available theory. Plans for Future Development It is obvious from this discussion that our course outline is too long, and each year we must slight some topics very badly to obtain meaning ful coverage of others. We believe we must soon either expand to a oneyear sequence or move much of Section I to a new course on the estima tion of physical properties. We particularly feel the need of more examples, both for consolidating the theory presented and for advertising new fields. We also feel that we are placing too little emphasis on inventiveness and ingenuity. This course is therefore far from satisfactory, even in its philosophy. We hope it will look very different five years from now. BIBLIOGRAPHY Bird, R.B., W.E. Stewart, and E.N. Lightfoot, "Transport Phenomena", Wiley (1960) Bird, R.B., W. E. Stewart, E.N. Lightfoot, and T. W. Chapman, "Lectures in Transport Phenomena", Vol. 4. AIChE Continuing Education Series (1969), Chap. 2, Estimation of Heat and Mass Transfer Rates, By E.N.L. Hirschfelder, J.O., C.E. Curtiss, and R.B. Bird, "Molecular Theory of Gases and Liquids", Wiley (1954) Book reviews Kinetics of Chemical Processes Michel Boudart, PrenticeHall (1968), ix+246 pp. To quote from the jacket of this volume "... Chemical kinetics, once the esoteric domain of the theorist, has become a vital tool in the design, operation and control of reactors . .". With the emergence over the past decade of chemical reaction engineering as an active and vital area in both teaching and research, the need for a book such as that Professor Boudart has provided us with here has become almost painful. To be sure, chemical engineering kinetics (what ever that is) is said to be the topic treated in a number of available texts. Kinetics, however, generally seems to bow out of the picture after quick tours through phenomological rate forms, chain reactions and the steadystate approxima tion, numerous integrated rate equations and the inevitable LangmuirHinshelwood discussion. One is left in these texts with a substantial involve ment in reactor analysis and design, which is cer tainly worthwhile, but kinetics then appear only as something taken for granted. Those who have attempted teaching reaction kinetics as distinct from reactor analysis or reaction engineering, know that the assembly of material for such a course at any level other than the trivial is a tiresome task of searching through a number (in my own case, six) of texts and monographs in the chemical and chemical engineering litera ture. No longer! In this admirable introductory text, Professor Boudart has put all the informa tion together for us, clearly and concisely, while still leaving lots of latitude for us all to incorpo rate our own variations. The approach of the book is straightforwardly put in the Preface: to develop the single chapter on kinetics in a physical chemistry text into a whole course, giving the essence of theory and experiment without indulging in extreme chemi cal detail. The contents include introductory ma terial on rate functions and reactor types, theory of chemical kinetics of elementary steps, steady state treatments for various systems, chain reac tion sequences, the concept of rate determining steps and stoichiometric numbers, irreducible (i.e., those arising from the reaction process it self) transport effects on kinetics, and correla tion methods for both homogeneous and hetero geneous reactions. There are a number of points which demonstrate how well structured and care fully written this text is. For example, transition state theory is developed via the thermodynamic formulation, which very clearly distinguishes between energy and entropy contributions to re action rates. The development given fbr reac tions proceeding via elementary steps involving active centers chain and catalytic sequences  provides a useful generalization for a large por tion of kinetics, and the chapters on correlations in homogeneous and heterogeneous reaction ki netics provide a nice amalgam of information from sources which are diverse and perhaps not the most familiar. There are quite a number of problems included, and they are generally excel lent. In short, this is a book which every chemi cal engineer should have in his library. John B:'Butt Northwestern University FALL 1969 74 CaOwe in anttol and OP&inaKtkOn OPTIMAL CONTROL OF REACTION SYSTEMS LEON LAPIDUS Princeton University Princeton, N. J. 08540 In this article we shall briefly outline some of the major topics covered in a graduate course in Optimal Control of Reaction Systems at Prince ton. This course was originally started in 1955 as a full year of Numerical Analysis and has gradually evolved into its present form. This was achieved by the inclusion in 1959 of selected items (3 and 6 in Table II below) and the subsequent addition of more and more material until the present coverage resulted. TABLE ITEXTS RECOMMENDED FOR COURSE IN OPTIMAL CONTROL OF REACTION SYSTEMS I 1. Athans, M., and Falb, P. L., "Optimal Control", McGrawHill (1966). I 2. Bryson, A. E., and Ho, Y., "Applied Optimal Control", Blaisdell (1969). I 3. Koppel, L. B., "Introduction to Control Theory", PrenticeHall (1968). I 4. Lapidus, L., and Luus, R., 'Optimal Control of Engineering Processes", Blaisdell (1967). I 5. Larson, R. E., "State Increment Dynamic Pro gramming", Elsevier (1968). I 6. Lee, E. S., 'Quasilinearization and Invariant Im bedding", Academic Press (1968). I 7. Luenberger, D. G., "Optimization by Vector Space Methods", Wiley (1969). I 8. Ogata, K., "State Space Analysis of Control Sys tems", PrenticeHall (1967). I 9. Roberts, S. M., "Dynamic Programming in Chemi cal Engineering and Process Control", Academic Press (1964). 110. Sage, A. P., "Optimum Systems Control", Prentice Hall (1968). I11. Wilde, D. J., and Beightler, C. S., "Foundations of Optimnization", PrenticeHall (1967). Table I lists a number of recommended texts which are used through the year and Table II gives some details on the explicit topics covered in the course. The main text references are I2 and 14, but all those shown in Table I are con Leon Lapidus is professor and chairman of the chemi cal engineering department at Princeton University. He was educated at Syracuse University (BS and MS) and the University of Minnesota (PhD, '50). He teaches courses in analysis of transport phenomena and com putational techniques and in numerical techniques in engineering analysis. His research interests lie in the areas of optimization, optimal control and stability of chemical reaction systems. sulted at various points. Recent references have been included in Table II so that the reader or student has a convenient starting point for pub lished material. The course is devoted to the mathematical and numericalcomputational aspects of the state space or timedomain approach as distinguished from the frequency or transform domain. In general it covers deterministic problems although some stochastic control and the effects of noise are briefly treated near the end of the course. Extensive numerical and computer problems are given as exercises to allow the students to try their hand at applying the theory. As an example, the optimal control of a series of CSTR will be discussed in class and the same problem but with control delay will be given to the students to solve by a variety of algorithms. However, em phasis on complex physical reactions systems for complexity sake is kept to a minimum. To give some perspective to the course objec tives we present Figure 1 which is taken from an article of Kalman, Lapidus and Shapiro pub lished in 1959 in the Instn. Chem. Engrs. Journal. Here we show an online adaptive (learning) com puter connected directly to a process or system. Within the computer there are three main pro grams designated A, B and C. The function of program A is to carry out the optimal control of CHEMICAL ENGINEERING EDUCATION TABLE IICOURSE OUTLINE FOR OPTIMAL CONTROL OF REACTION SYSTEMS *1. Numerical Concepts. Vectormatrix manipulation, solution of O.D.E. and P.D.E., iteration methods and accelerating convergence and functional analysis. Refs.: Amundson, "Mathematical Methods in Chem. Eng.", PrenticeHall (1966); Lapidus, "Digital Computation for Chem. Eng.", McGrawHill (1962); I7 and I8 of Table I.; Kantorovich, "Approximate Methods of Higher An alysis", Interscience (1958).* *2. Necessary and Sufficient Conditions for Minimum. Hessian matrix, constraints, Lagrange multipliers and penalty functions. Refs.: I2 and I11 of Table I. 3. Optimal Control Problem. Definitions of systems, constraints, performance index and selection of optimal control. Refs.: I1 and I4 of Table I; Lapidus, Chem. Eng. Prog. 63, No. 12, 64 (1967). 4. Minimum Principle. Continuous and discrete form of Minimum Principle, extensions, simplifications, numeri cal difficulties and advantages. Refs.: 12, I4 and 110 of Table I; Gurel and Lapidus, IEC Fund. 7, 617 (1968). 5. Dynamic Programming and Invariant Imbedding. Continuous and discrete form of dynamic programming, numerical solution of full nonlinear control problem, numerical solution of 2point B.V. problems and numerical questions. Refs.: 14, 15, I6 and I9 of Table I; Seinfeld, IEC Proc. Design and Devel. 7, 475, 479 (1968); Rothen berger, AIChE Jrn. 13, 114 (1967). 6. LinearQuadratic Problem. Solution of special con trol problem via Minimum Principle, dynamic program ming and variational calculus. Continuous and discrete problems, Riccati equation and numerical questions. The ASP computer program. Refs.: I4 and 110 of Table I; Lapidus, Chem. Eng. Prog. 63, No. 12, 64 (1967). 7. Minimum Time Problem. So ution via Minimum Principle, concept of switching times, bangbang control and singular control. Connection to linear and nonlinear programming. Refs.: Lesser, AIChE Jrn. 12, 143 (1966); Flynn,AIChE Jrn. 15, 308 (1969); 12, I3 and I4 of Table I. 8. Optimal Control Algorithms. Numerical algorithms Depending on the background of the students. for iteratively solving the full nonlinear control problem. First (gradient) and second variation methods including quasilinearization, neighborhood extremals, and the use of the linearquadratic procedure. Constraints and penalty functions. Open and closedloop solutions. Refs.: I2 and I4 of Table I. 9. Suboptimal control. Generation of closedloop feed back approximate control of nonlinear systems. Refs.: Internal work only. **10. Sensitivity Analysis. Performance and trajectory sensitivity, adaptive systems, parameter compensation, closed and openloop algorithms. Refs.: Kokotivoc, Int. Jrl. Cont. 9, 111 (1969); Sobral, Proc IEEE 56, 1644 (1968); Seinfeld, Canad. Jrn. Chem. Eng. 47, 212 (1969). 11. Stability. Single and multiple equilibrium points, limit cycles, lumped and distributed systems and Liapunov functions. Refs.: Storey, Brit. Chem. Eng. 13, 1585 (1968) ; Aris, Chern. Eng. Sci. 24, 149 (1969); Luss, Chem. Eng. Sci. 23, 1237 (1968); Gurel, IEC 61, No. 3, 30 (1969); Berger, AIChE Jrn. 14, 558 (1968), 15, 171 (1969). 12. Control and Stability. Linear quadratic problem and Liapunov functions, lumped and distributed systems, timeoptional control and control algorithms. Refs.: I3 and I4 of Table I; Denn, AIChE Jrn. 13, 926 (1967); Chant, Canad. Jrn. Chem. Eng. 46, 376, (1968); Wang, AIChE Jrn. 14, 934, 976 (1968). **13. Control of Distributed Parameter Systems. Mini mum Principle, dynamic programming, finite differencing, method of lines and control algorithms. Refs.: I3 of Table I; Seinfeld, Chem. Eng. Sci. 23, 1461 (1968) ; Denn, IEC Fund. 7, 410 (1968); Seinfeld, AIChE Jrn. 15, 57 (1969). **14. Filtering, Parameter Estimation and Identification. Linear and nonlinear systems determination of para meters, black box representation and computational ques tions. Refs.: Bard, Cat. Reviews 2, 67 (1968); Harris, AIChE Jrn. 13, 291 (1967); Peterson, Chem. Eng. Sci. 21, 655 (1966); I6 of Table I. ** If enough time is available. the process (issue commands to the various in puts) using the latest process measurements and taking into account the control objective and the dynamic model of the process. Program B con tinually analyses the process measurements to identify an accurate model of the process. Pro gram C is used to generate and inject special calibrating signals into the process such that they do not significantly disturb the process; yet they perturb the process in such a way that useful information can be obtained by means of sophis ticated data evaluation techniques. Feedback be tween the programs is also allowed to increase the efficiency of the overall operation. Within this simple appearing arrangement we have all of the features of the publicized learning computer which can build its own mathematical model of a process and then carry out any pre scribed form of control. The present course is directed, as much as feasible, to detailing the mathematics of these different features. Because of the current state of technology, Program A receives the major emphasis although Programs B and C are discussed (14 of Table II). ADAPTIVE CONTROL COMPUTER ConLrol Feedback Informative Feedback Optimal Con ir e l rDeter ilnint Systep m 'Ini ellggenteS turning ... .. H g .L . Figure A possible Adpt Digital Control System. FALL 1969 The course is devoted to the mathematical and numericalcomputational aspects of the statespace or time domain approach as distinguished from the frequency INTRODUCTORY MATERIAL Because of the wide background of students who take this course it is necessary to first pre sent certain introductory topics which are then used throughout the year. These include the com plete vectormatrix notation and its uses such as evaluating the transition matrix and the pseudo inverse, the numerical solution of ordinary and partial differential equations, the fundamental properties of convergence algorithms and some basic material on functional analysis. The solu tion of equations and the numerical stability of these solutions is necessary because they are an integral portion of all control algorithms and must be done correctly. Further, convergence algorithms are used throughout the entire course to actually obtain the optimal control. In addition to these numerical concepts it may be necessary to present some preliminary details on the necessary and sufficient conditions for an unconstrained minimum of a multivariable func tion, the influence and effects of constraints and the use of Lagrange multipliers and penalty functions to handle constraints. These items, which can be developed for a simple twovariable function, can be carried over directly to the most complicated control problem. As such the con cepts are absolutely necessary throughout the course. TABLE III DEFINITION OF OPTIMAL DETERMINISTIC CONTROL PROBLEM 1. General Form Given: 1. x(t) = f(x,u) System Model 2. x(to) Initial State 4. CFinal Time Conditiate Constraints 4. Final Tine Conditions (tf ) to 2. LinearQuadratic Form. 1. x(t) = Ax + Bu Linear System 2, 3, and 4 are same tf 5. I = x(t )'Sx(t) + f/ x'Qx + u'Rudt Quadratic Sto Scalar Index Find u(t), toSttf, such that I is minimized subject to constraints of 14. MAIN TOPICS With these preliminaries in hand the next step in the course is to define the optimal control problem (see 1 of Table III) and to detail the domain. minimum principle and the techniques of dynamic programming and invariant imbedding as general methods of solving the problem. Here it is very important to give numerical examples and to illus trate both the positive and negative features of these solution methods. The linearquadratic problem (see 2 in Table III) is then treated in detail using the minimum principle, dynamic programming and variational calculus. This is important because the techniques developed form the basis for all 2nd order algorithms for solving the full nonlinear control problem. Because of the availability of the ASP computer program (see I4) to numerically solve this linearquadratic problem a number of differ ent exercises are given to the students. The minimumtime problem is then solved via the minimum principle and the connection to linear and nonlinear programming detailed. Com putational considerations and the singular case are stressed. This area is interesting since it con nects to programming methods and is applicable to the analysis of many reactor systems. Next we discuss a wide variety of computa tional algorithms for solving the nonlinear control problem without and with constraints. These methods include the gradient (first variation) approach and the second variation including quasilinearization. Much of this analysis can be connected directly to the special linearquadratic case already treated. In addition, consideration is given to a form of suboptimal control which is easily developed and yields a closedloop type of approximate control. At the same time sensitivity considerations are employed to indicate the influ ence of parameter uncertainties and to generate iterative methods for the optimal control. Since many reaction systems exhibit the fea tures of stability and instability we then turn to a detailed discussion of the concepts of trajec tory paths in the neighborhood (or global) of an equilibrium point. This leads directly into an analysis of multiple equilibrium points, non uniqueness of solutions of nonlinear equations, limit cycles and Liapunov functions. In particular the Liapunov function approach is extended to distributed parameter systems and to provide convenient algorithms for minimumtime and suboptimal control. The extension of many of the above ideas may CHEMICAL ENGINEERING EDUCATION now be used advantageously to analyze the con trol of distributed parameter systems. Here we treat the control problem in its normal form or carry out a partial type of lumping (finite differ encing) to convert the system to sets of ordinary differential equations. In both cases a variety of possible control algorithms following from the minimum principle and dynamic programming are developed. Finally we consider the identification problem either in its full complexity where no apriori in formation about the reaction system is known or where a model is available but the parameters must be adjusted to fit experimental data (para meter estimation). Here we turn to the linear quadratic case treated as a filtering problem, carry out nonlinear lastsquare regression and fit the system data with generalized orthogonal polynomials. Questions such as the noise involved in the inputs and on the measurement are of im portance. AMUNDSON on Math (Cont'd from p. 177) and some comments must be made and analogies are drawn with finite vector systems. The object of such a course should be to pre sent methods for new problems. If a problem has been solved once then the engineer should use it. But with a new problem there is no one to tell him when the problem is properly posed. Has the model been drawn so it makes mathematical sense and how does one test whether it does? Whether a solution fits certain physical and chemical require ments will be determined by comparison with experiment, but this comparison is meaningless if the model is not selfconsistent. There is frequent confusion in the minds of beginning graduate students on what is mathe matics and what is not mathematics, and, if such a course serves no other function, this question should be answered for him. All of our problems as engineers are physically motivated and the translation of a problem into mathematical terms is not mathematics. The generation of the appro priate mathematical model is the job of a good engineer and whether conclusions drawn from the model agree with experiment is the test of how good an engineer he is. If the model does not agree with the experiments, one of two things may be at fault. Either the model was poorly drawn in that it does not describe the physical situation or the model is incomplete or incon sistent. Once the model is put to paper a mathe If the model does not agree with the experiments . . either the model was poorly drawn . or it is incomplete or inconsistent. matical problem must be solved. The engineer must somehow convince himself either by intui tion or rigorous mathematical argument that the mathematical problem is properly posed. The old argument that the problem is a physical one and therefore possesses a unique solution is a useful argument but not infallible, since only nature solves physical problems and she is quite capable of giving a nonunique solution. The argument also betrays an unrealistic confidence in the engi neer for it assumes that he has translated the physical problem into mathematical language exactly, a most unlikely event. This is really a very complicated problem, for in the course of the solution certain changes or approximations in the model, both physical and mathematical, are made and these should be examined in some detail to insure that the structure has not been destroyed. In conclusion, such a graduate course should not only teach techniques but it should also give the student a feel for what he is doing and what is involved. It has been frequently asserted that we teach only mathematics and neglect engineer ing. On the contrary, we are trying to teach the student the proper place and function of mathe matics, showing not only its strengths but also the pitfalls which may befall the unwary and the uninstructed. BIBLIOGRAPHY 1. Amundson, Neal R., "Mathematical Methods in Chemical Engineering," PrenticeHall, Inc., Englewood Cliffs, N. J. (1966). 2. Gantmacher, F. R., "Theory of Matrices," Vols. I and II, Chelsea Publ. Co., New York (1959). 3. Shilov, G. E., "Theory of Linear Spaces," Prentice Hall, Inc., Englewood Cliffs, N. J. (1961). 4. Amundson, Neal R., and Dan Luss, Canad. J. Chem. Eng. 46, 424 (1968). 5. Coddington, E. A., and Norman Levinson, "Theory of Ordinary Differential Equations," McGrawHill, Inc. (1955). 6. Ince, E. L., "Ordinary Differential Equations," Longmans, Green & Co., London (1927). 7. Hartman, Philip, "Ordinary Differential Equa tions," Wiley, New York (1964). 8. Weinberger, Hans, "Partial Differential Equa tions," Blaisdell Co., New York (1965). 9. Kaplan, W., "Ordinary Differential Equations," Addison Wesley, Inc., Reading, Mass. (1968). 10. Ross, S. L., "Differential Equations," GinnBlaisdell, Waltham, Mass. (1964). FALL 1969 74 o4w4ue iW Te',zmemadnatdmci MOLECULAR THERMODYNAMICS OF PHASE EQUILIBRIA J. M. PRAUSNITZ University of California, Berkeley Berkeley, Calif. Chemical thermodynamics started with J. Willard Gibbs nearly 100 years ago but its sig nificant influence on chemists and chemical engi neers in the United States, starting about 50 years ago, is due in large measure to the work of G. N. Lewis, who introduced fugacity and activity. For many years Lewis' strong personal ity dominated the College of Chemistry at the University of California at Berkeley; he served as its dean from 1912 until shortly before his death in 1946. It is therefore not surprising that Berkeley's College of Chemistry (which includes the Department of Chemical Engineering) has retained a tradition of strong interest in the application of thermodynamics to chemical prob lems. While Lewis' early work in thermodynamics was along classical lines, in his middle and later years he devoted much attention to relations be tween molecular physics and thermodynamic properties. Classical thermodynamics establishes broad relations between macroscopic equilibrium properties but, by itself, it cannot generate numerical values of these properties; further, it is limited in its ability to suggest useful tech niques for interpreting and correlating them. For such purposes, we must call on the insights pro vided by statistical mechanics and molecular physics. Lewis' efforts in this area were accom panied and continued by various collaborators, including W. F. Giauque (who won the Nobel prize for his lowtemperature work) and later, Leo Brewer and K. S. Pitzer (now president of Stanford University). But, with respect to chemi cal engineering education and research in thermo dynamics, the most important of Lewis' collabora tors was (and is) Joel H. Hildebrand, who at age 88 is still active in research on the properties of liquid solutions. Hildebrand's numerous publications have shown that when thermodynamics is coupled with J. M. Prausnitz (BChE Cornell, PhD Princeton) has been a member of the Chemical Engineering faculty at the University of California, Berkeley since 1955. His research interests are concerned with the application of thermodynamics and molecular physics to chemical en gineering design. He has published extensively in this area and serves as a consultant to petroleum, petro chemical and cryogenic industries. He has been honored with a Guggenheim fellowship, Miller Research Professor ship, ACS Honor Scroll and with the Colburn and Walker Awards of the AIChE. simple (often semiempirical) molecular models, many practical phaseequilibrium problems can be solved with a minimum of effort, and what is more important, with a minimum of experimental data. It is this feature of Hildebrand's work that has become the major influence on the course "Phase Equilibria" in the Chemical Engineering Department at Berkeley. The course is normally taken by graduate stu dents in their first quarter at Berkeley. It meets twice a week; each meeting is for one and a half hours. Prerequisites for the course are a oneyear course in undergraduate physical chemistry and at least a onequarter course in undergraduate chemical engineering thermodynamics; almost all entering graduate students satisfy these pre requisites. The two main purposes of the course are: (1) to give students the background needed to apply thermodynamics and molecular physics toward the solution of practical phase equilibrium problems as required in typical chemical engi neering design practice, and (2) to provide an introduction to applied molecular science for tackling new problems on the frontier of modern chemical engineering. The student acquires some feeling for the physical (as opposed to the merely mathematical) significance of thermodynamic functions and some insight into the intermolecular forces which determine the magnitude of these CHEMICAL ENGINEERING EDUCATION Table 1. Berkeley's Course in PhaseEquilibrium Thermodynamics 1. The Phase Equilibrium Problem 2. Classical Thermodynamics of Phase Equilibria 3. Thermodynamic Properties from Volumetric Data 4. Intermolecular Forces 5. Fugacities in Gas Mixtures 6. Fugacities in Liquid Mixtures; Excess Functions 7. Fugacities in Liquid Mixtures: Theories of Solutions 8. Solubility of Gases in Liquids 9. Solubility of Solids in Liquids 10. HighPressure Equilibria functions. He gains practice in reduction and in terpretation of experimental data and in devising efficient and physically meaningful methods for data correlation. Most important, he achieves some perspective on what kind of experimental information he requires for a given problem and, lacking that information, he gains experience in quantitatively estimating desired phase equili brium relations from a minimum of experimental data. Finally, he becomes acquainted with some of the literature on phase equilibrium thermo dynamics and learns, often to his great surprise, that he must be critical of what he reads since this literature, unfortunately, is not free of errors. The course is divided into ten somewhat arbi trary sections which are given in Table 1. These sections correspond to the ten chapters of a recent textbook*. The lectures summarize, focus and illustrate the material in the text which the student reads concurrently. The first section introduces the student to the phase equilibrium problem: what are we trying to do and what connection is there between thermodynamics and the distribution of several components between two (or more) phases? The importance of this problem is discussed not only with respect to typical chemical engineering operations (distillation, extraction, etc.) but also with respect to physiology, meteorology and everyday experiences such as brewing coffee or dry cleaning a piece of clothing. The second sec tion reviews the classical work of Gibbs and Lewis: the concept of open systems; definition and utility of the chemical potential; fugacity and activity; a workedout simple example (Raoult's law as a special case of the general equation of equilibrium). The third section deals with the calculation of thermodynamic functions (espe *J. M. Prausnitz, "Molecular Thermodynamics of FluidPhase Equilibria," PrenticeHall, Inc., Englewood Cliffs, N. J., 1969. cially fugacity) from experimental PVT data (or empirical equations of state) for pure gases, for gas mixtures and for pure liquids and solids. Attention is given to the general procedure of calculating vaporliquid equilibria using only an equation of state assumed to be valid for both gas and liquid phases (e.g., the BenedictWebb Rubin equation). After the third section the student begins to understand that for finding successful solutions to phase equilibrium problems, the important bottleneck has little to do with thermodynamics; the real problem is not thermodynamics, which is (essentially) all worked out, but molecular physics: How do we obtain, with a minimum of experimental work, the various constants in the equations? The student recognizes the need for constants which appear because real substances, unlike ideal gases, consist of finitesize molecules which attract and repel one another. He is made aware of the main difficulty in applied thermo dynamics: to use thermodynamics for obtaining numerical results we must add physical under standing to our thermodynam'c formalism. And while the thermodynamic formalism is beautiful, exact and complete, our understanding of molecu lar behavior is still very limited. Section four presents a simplified survey of what is known about intermolecular forces. The nature and origin of intermolecular forces is pre sented along with the concept and significance of potential energy functions and the microscopic law of corresponding states. A brief discussion is given of such "chemical" forces as hydrogen bonding and chargetransfer complex formation. Emphasis is given to qualitative and semiquanti tative relations between intermolecular forces and macroscopic properties. Section five begins with a critical discussion of the virial equation of state: its theoretical sig nificance, its utility and limitations for calculat ing fugacity coefficients in gas mixtures. Atten tion is given to the exact relations between virial coefficients and intermolecular potential functions and to the composition dependence of virial co efficients. The 'chemical" theory of gas imperfec tions is discussed. Semiempirical methods (e.g., the RedlichKwong equation) are presented for finding fugacity coefficients at high densities, and it is shown how these are used to calculate solu bilities of liquids and solids in compressed gases. Excess functions and their application to liquid mixtures are considered in section six. Various FALL 1969 if YOU can answer "YES" to any of these questions, Rohm and Haas can make your life interesting. Do you enjoy being responsible (and the rewards that go with it) for developing a project and carrying it through to completion? Do you like a job that gives free rein to your imagination and tests your skills? Do you like the satisfaction of making a contribution to society by improving a wide range of industrial and consumer products or by protecting natural resources or improving farming efficiency? Would you like to work for a company that is concerned with problems of social responsibility and encourages its employees to be active in urban affairs? We are not altruists. We are a strong and growth oriented chemical company that needs engineers of all typeschemical, mechanical, electrical and industrialwho want to make a practical contribution to improving man's lot and at the same time advance themselves in the world of business. We have doubled our sales in the past 10 years to the $400,000,000 level. We make some 2,500 chemical products including synthetic resins, plastics, fibers, pharmaceuticals and animal health aids. We are on the move and you can move with us into positions of responsibility just as rapidly as you show us you have the ability. There's just one thingyou'll be expected to work hardbut you'll be in good company with the 13,000 other people that make up Rohm and Haas. We have six major manufacturing locations in the United States and producing subsidiaries in 15 foreign countries. We need engineers in research, production and marketing. Write to Manpower & Employment #7069. It could be good for both of us. ROHMf PHILADELPHIARR PENNSYLVANIA 1105 PHILADELPHIA, PENNSYLVANIA 19105 ROHM AND HAAS COMPANY An equal opportunity employer. equations for representing activity coefficients (van Laar, Margules, Wilson, NRTL, etc.) are discussed. Consideration is given to multicom ponent mixtures and to systems with partial mis cibility. The significance and uses of the Gibbs Duhem equation are given special attention. Section seven is an introduction to the theory of liquid solutions. Brief attention is given to the lattice theory of simple mixtures and of polymer solutions; emphasis is placed on the theory of regular solutions, on applications of corresponding states theory, and on the "chemi cal" theory of associated and solvated solutions. All theories are regarded critically; advantages and disadvantages are pointed out. Sections eight and nine are concerned with the solubilities of gases and solids in pure solvents and in solvent mixtures. Physical and chemical effects on solubility are pointed out and special attention is given to the importance of the stan dardstate fugacity of the solute. The course ends with a brief discussion of the uses of thermodynamics to describe systems at high pressure. Special emphasis is given to the important role of the partial molar volume. Vapor liquid, liquidliquid and gasgas equilibria are considered. As outlined above, the course appears to con tain a lot of material. However, experience has shown that wellprepared firstyear graduate students can handle the course without difficulty provided their total course load is not large. LETTERS (Cont'd from p. 167.) AN ADVENTURE IN TEACHING Sir: During a recent conversation one of my graduate students described a course he was taking. The professor wrote everything down on the blackboard. He defined each symbol of every equation. Definitions were given of each technical word. Discussion was limited to students asking for clarification of the professor's handwriting. And then he added, "It wasn't like the Rate Processes course you taught last year". That had been an exciting learning experience for him. It had been the same for me. I was teaching the second quarter of Rate Processes to firstyear graduate students of varied backgrounds. In the first quarter we had covered the Momentum Transport section of "Transport Phenomena" by Bird, Stewart, and Lightfoot (BSL). This quarter I decided to teach Energy and Mass Transfer by the "row" approach. As the authors point out in the preface this alternate approach is suitable for graduate students. It emphasizes the type of transport and the analogies between the transport phenomena. Realistically it also eliminated the possibility that only Typical firstyear graduate students at Berkeley usually take only a total of two lecture courses plus one seminar course per quarter. Careful reading of the text (which was available in mimeographed form until its recent publication) is augmented with reading of "classical" original articles which are kept on reserve in the College of Chemistry library. Eight problem sets give the student practice in applying what he has learned to the solution of practical phase equilibrium problems. All problem sets are corrected by a teaching assistant and the more difficult problems are discussed in class. The course provides a good partial foundation for subsequent graduate courses in separation operations, in process design and in medical en gineering. For those students interested in doing research in molecular science and engineering it provides background and perspective for subse quent courses in statistical mechanics and in advanced chemistry, physics, and materials science. The oftenpraised versatility of the chemical engineer, his ability to tackle a wide variety of new problems, is in large measure due to his knowledge of applied physical chemistry. Berke ley's course in phase equilibrium thermodynamics aims to contribute to that knowledge while at the same time providing the student with some of the skills for the practice of conventional chemical engineering. a week or two at the end of the quarter would be left for Mass Transport. The first topic covered was methods for predicting thermal conductivities and binary diffusivities. Since this touched on the area of my doctoral research, I added current literature methods to the text material and the students evaluated: Variation of thermal conductivity for sulfur dioxide, carbon tetrafluoride, and tungsten hexafluoride over the temperature range of 0 to 10000C. Comparison of thermal conductivities of ammonia, carbon tetrafluoride, and hydrogen at three elevated pressures with experimental values. Variation of the binary diffusivities of tungsten hexafluoride and hydrogen fluoride over the tem perature range of 0 to 10000C. Three lessons were learned from this exercise. First, beware of phase changes when computing transport properties (melting point of tungsten hexafluoride is 20C and critical point is 1780C). Second, use of available sources of or estimation techniques for intermolecular force parameters. Finally, a feel for the accuracy of gen eralized correlations for transport properties. FALL 1969 Shell balances for simple energy and mass transfer problems (Chapters 9 and 17) came next. This was followed by the equations of change for nonisothermal systems and also the equations of change for multi component systems. The latter topic stirred everyone's interest. On3 of the reasons, I guess, was because it was so difficult. Another reason was that we had finally arrived at equations which encompassed all of the fluxes that would normally be encountered in any problem. In other words these equations presented the whole story. Up to this point the course had been interesting but very conventional. All of us had learned some new ideas. But the real breakthrough was to come. Unscheduled. Unexpected. A real example of serendipity in teaching. An AIChE meeting was scheduled for St. Louis in approximately the middle of the quarter. I was planning on attending and was trying to figure out what the students should do in my absence. Sounds familiar, doesn't it? What I hit upon was to assign each student a paper to review. Since mass transfer seemed to be the area of highest interest, most of the papers were in this area. (See list of papers assigned). Some related to the students interests as I was aware of them. None were by BSL because this would have been unfair in the light of the review I requested. Here are the things I wanted them to include in a written review and also in a fifteen minute presentation before the class: Put the important equations and boundary conditions of the paper in BSL notation. Are these equations in BSL or what equations in BSL are they related to? What are the basic assumptions in the starting equations and do the authors clearly state them? What are the unstated assumptions? Prepare a clear diagram showing problem solved. Show concentration, velocity, or temperature pro files in diagrams. State the three most important contributions of the paper. Should BSL include results in revision of the text book and why? What a surprise I had in store when I returned from St. Louis! Fifteen minutes turned out to be woefully inadequate for any of the students to discuss their papers. I extended the time limit to an half hour. Even this proved inadequate. Discussion on some of the papers lasted as long as an hour after the formal presentation. This was exciting, I finally had to limit discussion so that all of the papers could be discussed within the quarter! From this experience I gained new insight into what constitutes effective teaching. First, teaching was not my exclusive domain in the classroom. Students could teach one another and also they could teach me. In other words, teaching can be listening. How foreign that concept is among professors I've known. How foreign it was to me. Also, how threatening! Prior to this I felt that everything depended on my performance in the classroom. If I pre sented a wellprepared lecture, including good examples to illustrate the material, then students could learn. But I always had a gnawing feeling that there must be other ways and probably better ways of teaching. One of these is embodied in the concept of teaching is listening. Secondly, students can evaluate themselves. Each stu dent graded (anonymously) each talk. The average was their letter grade for the presentation. I was so awestruck by the talks I probably would have given them all A's. The students were more objective and gave an equal number of A's and B's. Grades which I gave on the written reports agreed very closely with those of the students. Relevancy is an overworked word in today's student vocabulary. It denotes that classroom learning has mean ing in or can be applied to the real world and its problems. Unexpectedly, this teaching adventure touched on some thing that was relevant to the students. We had all pro gressed to a common ground of understanding transport phenomena. With this each student attacked a paper in the literature and found that what we had learned applied to that paper. And each student could do his own thing within the guidelines laid down. Further each student had the opportunity to experience that greatest satisfac tion of teaching; namely, to teach is to learn. Frankly, we need to share this sense of fulfillment with our stu dents more often. In a future paper I'll tell how I did this with undergraduate students. In closing let me share one of the other things my graduate student told me. He said that one of the students finished his report while I was away. He then cornered each of his classmates individually and went over his report with them soliciting questions and comments. As I recall his lecture was the best. More importantly, I know he had experienced an adventure in teaching. PAPERS ASSIGNED Arnold, K. R. and H. L. Toor, Unsteady Diffusion in Ternary Gas Mixtures, A.I.Ch.E.J. 13, 909 (1967). Evans, E. V. and C. N. Kenney, Gaseous Dispersion in Laminar Flow Through a Circular Tube, Proc. Roy Soc. (London) 284A, 540 (1965). Getzinger, R. W. and C. R. Wilke, An Experimental Study of Nonequimolal Diffusion in Ternary Gas Mixtures, A.I.Ch.E.J. 13, 577 (1967), also Ind. Eng. Chem. 47, 1253 (1955). Grimsrud, L. and A. L. Babb, Velocity and Concentration Profiles for Laminar Flow of a Newtonian Fluid in a Dialyzer, Chem. Eng. Progr. Symp. Series No. 66, 62, 20 (1966). Lever, R. F. and F. P. Jona, Chemical Transport in Non convective Systems A.I.Ch.E.J. 12, 1158 (1966). Lyczkowski, R. W. et al, Simultaneous Convective Dif fusion of Reactants, Products, and Heat with a Surface Reaction, Chem. Eng. Progr. Symp. Series No. 77, 63, 1 (1967). McCall, D. W. and D.C. Douglass, Diffusion in Binary Solutions, J. Phys. Chem. 71, 987 (1967). Satterfield, C. N. and R. C. Yeung, Diffusion and Heterogeneous Reaction in a Tubular Reactor, Ind. Eng. Chem. (Fundamentals) 2, 257 (1963). Toor, H. L. Diffusion in ThreeComponent Gas Mixtures, A.I.Ch.E.J. 3, 198 (1957). Walker, R. E., and Westenberg, A. A., Molecular Diffusion Studies in Gases at High Temperatures, J. Chem. Phys. 29, 1139 (1958). Charles E. Hamrim, Jr. University of Kentucky (Continued on page 233.) CHEMICAL ENGINEERING EDUCATION Shell is a pair of sneakersmade from our thermoplastic rubber. Shell is a milk containerwe were a pioneer in the allplastic ones. Shell is a clear, clean country stream aided by our nonpolluting detergent mate rials. Shell is a space capsule controlener gized by Shell's hydrazine catalyst. Shell is food on the tablemade more plentiful by Shell's fertilizers. Shell is mileage gasolinedeveloped through Shell research. Shell is a good place for Chemical Engineers to build a career. Shell is an integrated research, engineering, marketing and product application methods; exploration and production, manufacturing, and carrying out research and development transportation, marketing organization with to support all of these. Information about diverse technical operations and business openings throughout Shell may be obtained activities throughout the United States. by signing at the Placement Office for an Chemical Engineers are vital to the Com interview with our representative, or by pany, applying their knowledge to recover writing to Recruitment Manager, The Shell ing oil from the ground; designing and Companies, Department C, Box 2099, operating oil, chemical and natural Houston, Texas 77001. Shell is an gas processing plants; developing new equal opportunity employer. THE SHELL COMPANIES Shell Oil Company/Shell Chemical Company/Shell Development Company/Shell Pipe Line Corporation. FALL 1969 '";* :i .2 Pr ;;1 tt .LI~ . I ic .. ; a, :i _~c~T~' .~. :n d I R ~F~Or Venture: Purify water with the fiber that made men whistle. Nylon. Reverse osmosis. A fiber that started making girls' legs more beautiful some 30 years ago. And a process that's been around a lot longer. But when Du Pont scientists and engineers look at them in a new way, they combine into an idea that can change the world. Reverse osmosis is a purification process that requires no phase change. It's potentiallythe cheapest way to desalinate water. Du Pont's innovation? Hollow, semipermeable nylon fibers much finerthan human hair. Symmetrical, with an outer diameter of .002 inch and a wall thickness of .0005 inch, with an accuracy of manufacture maintained at close to 100%. Twentyfive to 30 million of them encased in a precisely engineered unit 14 inches in diameter by 7 feet long. The result: a semipermeable sur face area of about 85,000 square feetthe size of a 2acre lotand up to 10,000 gallons of desalted water per day. So far "Permasep" permeators have been used experimentally to purify brackish and polluted water, and in various industrial separa tions. But the potential to desalt seawater, too, is there. So Du Pont scientists and engi neers are even now working to ward improved fibers, units and plant designs that should make it possible to get fresh water from salt at a price that any town or nation can afford. Innovationapplying the known to discover the unknown, inventing new materials and putting them to work, using research and engi neering to create the ideas and products of the futurethis is the venture Du Pont people are now engaged in. Ventures for better living. 74 Cas'ze in T'slwwo.a4namiaii THE GRADUATE STUDENT VERSUS THERMODYNAMICS JOSEPH J. MARTIN University of Michigan, Ann Arbor, Mich. Thermodynamics has the reputation, enviable or not, of being a worthy adversary for those who dare to engage it in battle. For most young pro teges there are many rounds of rough infighting before mastery of the subject can be claimed. This is not because of its sheer logic, for mathe matics is undoubtedly the queen of the sciences in this respect, and mathematics does not seem to offer the same degree of difficulty. Rather, it is more probably due to the extreme range of application of thermodynamic principles. Although Webster says that "thermodynamics is the science which treats of the mechanical action or relations of heat," it is much more appropriate to say that thermodynamics is the science of energy and entropy which is involved in every equilibrium state of matter and every process or change that occurs in the real ponderable uni verse. As an undergraduate the young protege is ex posed to the socalled "laws of thermodynamics," of which there are just four as follows: Zeroth. If a thermometer shows the same reading when in separate contact with two objects, no change occurs when the two objects are touched to each other. First. Energy and mass are simultaneously conserved (accountable) in all processes and individually con served in most cases. Second. Actual processes occur in one direction and the initial conditions will never be restored without the aid of some outside agent. Third. The entropy of a pure substance in perfect crys talline form vanishes at the absolute zero of tempera ture. The zeroth and second laws are quite acceptable to the undergraduate because he has had some experience with thermometers and because he has learned the irreversible ways of mother nature from the time of his first broken dish or balloon to the time he has burned the last drop of gasoline in his car many miles from a service station. The first law is palatable because energy and mass balances are mathematically neat, but the protege is often shaky on the general concept Joseph J. Martin was educated at Iowa State, Rochester, and CarnegieMellon University (Sc.D. '47). He is pro fessor and associate director of Institute of Science and Technology at University of Michigan. Presently he is a vicepresident of ASEE and a Director of AIChE. of energy. He finds it difficult to explain the nature of internal energy or to tell just where energy is stored when a weight is lifted in a gravitational field. The third law makes sense to him only insofar as he understands the abstract quantity, entropy, and the absolute temperature scale. It is generally not the laws of thermodynamics which present the greatest difficulty to the novice in the field, but the hundreds of equations which have been developed to permit quantification of the elementary principles embodied in the laws themselves. Fortunately, however, there are just four basic equations from which all others are derived by suitable mathematical manipulation. These four questions do not have a one to one correspondence with the four laws, but collectively they incorporate the first three laws within their structures. The fourtequations are: The Fundamental Property Relation of Matter (Gibbs Equation), N dU = TdS PdV + P i dmi Ti= The Energy Balance on a System,** K d(U+mu2/2 + mgz) sys = 6Q 6W + 6 + (H+u2/2 + gZl.d m (2) j =1  *See the Nomenclature at the end for the definition of symbols. **A system is any portion of the universe chosen for analysis, and may have many distinct parts. CHEMICAL ENGINEERING EDUCATION The Entropy Balance on a System, M. K days = T (60/T), + 6LW/To + S sj6 i=1" j=1 Thermodynamics is the science of energy and entropy which is involved in every equilibrium state of matter (3) and every process that occurs in the real universe. The Mass Balance on a System, K m, = T 6m (4) sys j= () Although it may appear to be an oversimplifica tion of the subject, there is little question but what complete understanding of the four basic equations amounts to a mastery of classical thermodynamics. The equations are, therefore, in troduced to the protege at an early stage, even though it is quite unlikely that in a single course he will perceive all of their underlying implications. The graduate student who is ex posed to one or more succeeding courses will grad ually gain a more complete understanding of the extreme power and utility of the four equations to describe the thermodynamic character of all processes and all equilibrium states of matter. The first of the basic equations is probably the most important and the protege learns that a uniform mass of matter has the extensive thermo dynamic properties, internal energy, entropy, volume, and mass of each component, and the intensive properties (potentials), temperature, pressure, and chemical potential of each com ponent.* The point to be emphasized is that these properties are not completely independent, but are interlocked through Eqn. (1). Any change in the condition or state of matter will cause changes in the thermodynamic properties, but the changes cannot occur indiscriminately; they must occur in accordance with this differential equation. In the deduction of the several terms in the equation, the inclusion of entropy is paramount. It is done in one approach by saying that for a simple heat transfer to the exclusion of any other effect the change of entropy is the change of internal en ergy divided by temperature, or dS=dU/T. This makes S extensive in the same manner as U. The desirability of defining entropy this way is best understood by noting that, for example, when a hot object is touched to a cold object in isolated conditions, the energy balance shows nothing is lost since dUBoth=dUH+dUc=0. Yet experience and intuition tell us that something has changed and the ability to do work has been lost. Calculat *Equation (1) is the ordinary version of the funda mental property relation. Additional terms of the form, (intensive) d (extensive) may readily be added to it to account for the more unusual effects such as surface, elongation or tensile, electric, and magnetic. ing dU/T (i.e., entropy change) for both objects shows (dU/T)Both = (dU/T) + (dU/T)o = dSBoth > 0 because dU = dUc and T1 > Tc. The change in entropy, dS, provides a quantita tive measure of the irreversibility of the heat transfer. Its relation to the work lost is given by Eqn. (3). The unusual mathematical character of Eqn. (1) being an exact differential and homogeneous of the first degree permits it to be integrated to N U = TS PV + p.m. (5) and then differentiated to N 0 = SdT VdP + Z midpi (6) i=l which is the GibbsDuhem equation that is worthy of extended contemplation by the protege. Be cause U, TS, and PV often occur together as in (5), it is convenient to define H = U + PV, A = U  TS, and G = H TS, but it is obvious that although very handy and efficient, H, A, and G are not fundamental properties. By rules of partial differentiation and the definition of heat capacity as Cx = T(dS/dT)x (7) it is possible to put Eqn. (1) in such seemingly unrelated forms as dS = O(dT/T (dV/dT) dP (8) AH = TAV(dP/dT) (9) and many others. In fact, the bulk of thermo dynamics involves the application of Eqn. (1) in different ways to a wide variety of situations in volving ponderable matter. In the second basic equation, the energy bal ance, the protege notes how heat and work are introduced with arbitraary signs, and the reason for distinguishing flows of energy to the system by "8", and changes of energy within the system by "d". The most common applications of the equation are in three integrated forms: Closed System, Q W = AUsys (10) SteadyFlow System, Q W = AH + A(u2/2) + gAZ (11) FALL 1969 SingleFlow System, fsHm + Q W = [(Un) (Um)ys (12) Of course, the energy balance may take many other forms by proper specialization of Eqn. (2) for particular cases. The third basic equation, the entropy balance, is useful for calculation of the work lost in real irreversible processes and for analysis of idealized reversible processes. The protege is quite aware that a tank of compressed air can do useful work by connecting it to an expansion engine and that such work can be forever lost by allowing the air to leak out instead of flowing through the engine. He is equally aware that an outside agent must do work on the air that leaked out in order to get it back into the tank. He will readily appre ciate that this work of restoration under certain ideal conditions is the work lost during the irre versible leaking process. For many applications the entropy balance is applied to closed systems as a simple integral, fTdSys = Q + LW (13) In this form it is seen that for a nonflow rever sible process (no lost work) the heat transfer equals the S Tds. If in addition there is no heat transfer (adiabatic as well as reversible), the STds vanishes and the process is isentropic. When Eqn. (3) is applied to a reversible heat engine operating in a steady state (no lost work, no change of entropy of the systemthe engine, no mass flowonly heat and work flow), 1i(8Q/T)i = 0 = 8Qi/TT + 8Qi/T,, This may be combined with the energy balance, 8Qh +68Q 8Wr = 0, to give 6Wr = 6Qh (Th  r h Th S(TTh "> " which is the Carnot relation that is utilized in the analysis of heat engines and heat transfer processes. Here it is desirable that the protege become familiar with heat engine cycles that are used in steam plants, gas turbines, and reciprocat ing internal combustion engines. The concept of available energy should be included also. The energy and entropy balances may be com bined in another manner by eliminating Q be tween Eqn. (11) and Eqn. (13) applied to a unit mass of flowing material, and utilizing the defini tion of G = H TS or AG = AH JTdS /SdT, so that LW W = AG + JSdT + A (u2/2) + gAZ (15) By using the property relation, AG = S VdP  I SdT, which is just a case of Eqn. (1) for the unit mass, another energyentropy balance form is LW W = JVdP + A(u2/2) + gAZ (16) which is Bernoulli's equation. This equation has proven extremely useful in the treatment of a wide variety of fluid flow problems, particularly those in which there is friction lost work. For a reversible process no work is lost and (16) be comes _r I vdP + A(u2/2) + gAZ (17) while (15) becomes r= AG + SdT + A(u2/2) + gZ (18) The last two equations find applications in the flow work concept of equilibrium between two states. By this concept if two states are in equili brium, no work can be obtained from a transfer or flow of mass between them. Direct use of the equations furnishes an analysis of nonisothermal equilibrium, which is equilibrium superimposed on a steadystate irreversible heat transfer. Most applications, however, are to isothermal equilib rium so that Eqn. (18) is written 0 = AGT + (u2/2) + gz = AGT + m(u2/2) + mgAZ (19) If component A in a mixture is free to move be tween two states in equilibrium, (19) may be written 0 = AG + A(u2/2) + gAZ (20) This is the key equation of phase equilibrium and to understand its implications, it is necessary to become familiar with partial extensive properties as derivatives of any extensive property with respect to mass of component A at constant T, P, and masses of all other components. For free energy, as an example, GA = (dG/dmA)T,P m (21) B This may be compared with the property relation (1) to show that GAx =/ (22) Further comparison with (5) shows that G = mA GA + mB GB + ... (23) When one has gained confidence that a partial property of a component of a mixture is essen tially the same as the unit mass property of a pure substance, he sees how most of the equations for pure and mixture components are inter changeable. For example, by proper manipulation of the fundamental property relation (1) he can get both d(GIT) H (24) dT P p 2 CHEMICAL ENGINEERING EDUCATION The entropy balance is useful for calculations of the Fd(GA/T1 HFA LdT (25) which are GibbsHelmholtz equations. The protege's level of sophistication in thermo dynamics is advancing rapidly if he fully compre hends the definition and use of fugacity and activ ity through the equations, dGA = RTdlnTA where A pA as P 0 (26) and G= GA + RT In G + RT In a (27) !A Here integration has been conducted isothermally from a chosen standard state* to any state. When Eqns. (23) and (27) are applied to a reaction, aA + bB= cC + dD, at a given temperature and no kinetic or potential effects, Eqn. (18) yields c d aC aD 0 Wr = AGO + RT n aC g = AGO + RT In J (28) aAa a At equilibrium where W, = 0, this becomes AGo = RT In K a (K = Ja ) (29) Equilibrium Then substituting Eqns. (8) and (25) into the temperature derivative of (29), and integrating between two temperatures, the following equation for situations without appreciable latent heat effects can be obtained by a person who has a good grasp of the subject, aK A 2 f ( C )dT in a2 A 1 1 + o Prod React dT (30) Kal R T1 TR2 At the same time this move is made, it is desir able that the protege learn how to use certain tabular values of thermodynamic functions so that he can evaluate A 0 AHo 0 (A). = E (GOT o G + ) (31) T a T _T Prod R T T React(1 The protege's knowledge should extend to elec trochemical reactions, so that he understands the relations involving electrical work and voltage, such as work lost in real irreversible processes and for analysis of idealized reversible processes. and NSY = Nf 0 RT1nJ r r a (35) By defining the activity coefficient, yi = i./xif i, it is desirable to be able to proceed from Eqn. (6) to get xAdlnyA + xBdlnYB = 0 (36) YA In dx = 0 0 B YB lny =j XB dlYB (38) vBl xA YB=1 XA Rewriting Eqn. (20). for simple phase equilibria. Ua 5 A g (39) and utilizing (27), the extremely useful relation,  a = 7A fA A is obtained. From the definition of the activity coefficient the equation for liquidvaporequili brium is often shown as (YAXAfAo)li = IAYAfAO) vapor (41) For ideal solutions the protege learns that if he appreciates the significance of assuming VA = ,A' he can derive TA = YAfA (42) ideal = RT(nAlnx + nnlnXg) (43) Equally useful relations for enthalpy and entropy may be obtained. For nonideal solutions it is desirable to see how GEx= AGix AGideal= RT(nAlnyA + nBlnYB) (44) and for regular solutions how AGr = XA(1XA)w + RT(xAlnxA + xBlnxB) d (E,/T) AHO dT N T ( ,2 (33) NE or = RT1nK (34) r a *Standard state is a state of the material at the tem perature of interest and usually at a given pressure or other conditions that determine the state. The empirical and semitheoretical techniques for obtaining activity coefficients or required PVT behavior or an empirical constant such as a must be introduced at each step in which real problems are considered. Calculation of actual situations leads to the application, employment, and even development of correlations of the properties of matter. It behooves the protege to master many of these in the course of mastering the funda mentals of thermodynamics. FALL 1969 W = N_3 (32) (45)  (40) Nothing has been said about the fourth basic equation since it is generally used directly as is. Clearly, however, if an application is contem plated to an extremely high energy process, such as a nuclear reaction or high velocity particles, the Einstein relation, E = mc, must be employed. In such a case the individual mass and energy balances must be modified to allow for the equival ence of mass and energy. NOMENCLATRE T Temperature A Helmholtz free energy, UTS a Activity, f/fo C Heat capacity, T(dS/dT) 3 Faraday number f Fugacity G Gibbs free energy, HTS g Acceleration of gravity H Enthalpy J. Ratio of activities aca4/aaB Ka Equilibrium ratio of activities LW Lost work m Mass N,n Number of moles P Pressure p Partial pressure a Quatity of heat U Internal Energy u Velocity V Volume W Work x Mole fraction y Male fraction Z Height above datum plane a,8 Phases y Activity coefficient a Chemical potential u Constant for regular solutions  Below an extensive property makes it per unit mass Above an extensive property makes it a partial property of a mixture. In case of fugacity, shows that component is in a mixture. o Denotes vapor pressure, as P0. S Entropy Several example problems and solutions are presented in the following section. [ N a problems for teachers The following problems with solutions were submitted by Professor J. J. Martin. No. 1. One hundred million standard cubic feet (600F, 1 atm) per day of radioactive waste gas at 10000F must be released at a height of 400 ft. above the ground to avoid contamination of the surrounding area. A circular stack of uniform diameter is to be used. A draft at the ,0 base of the st('ck of 1 in. of water will be required (pres sure inside stack base is 1 in. HzO less than barometric pressure). The barometric pressure at the base of the stack is 740 mm Hg and the ambient temperature 600F. The gas has a molecular weight of 32 and may be con sidered an ideal gas. What diameter will be required? The lost work of gas flowing through the stack may be approximated by the equation: 0.032 lu2 LW = gD LW = lost work in ftlbF/lbM 1 = height in ft D = diameter in ft u = velocity in feet per sec g, = conversion factor from lbM to IbF Solution: For barometric pressure at top consider a stag nant column of air outside the stack. Since it is at equilibrium (no flow) with no kinetic effects, Eqn. (17) applies as top 0 = fVdp + gAZ = dP + gAZ base p RT In Poa = gAZ Base so (10.73) (520)ln PEt = (40 and p = 729.41 mm Hg The pressure inside the base of the stack is p = 740n (25.4) (1) in base = 740 = 78.13 mm Hg The gas velocity inside the stack at an average pressure of 734 nm Hg is S= 108(760) (1460) (4) 4290 ft 24(3600) (520) (734) nD2 D2 sec For flow inside the stack with no kinetic effects and no work, Eqn. (16) applies as LW = VdP + gAZ = fT dP+ gAZ Thus, (0.032) (400) (4290)2 (10.73) (1460)(144) n 729.41 + 400 32.17 D D (32) 738.13 Solving D = 7.2 ft No. 2. A RanqueHilsch vortex cylinder is a device to expand a stream of air in steady flow from high pressure and ambient temperature down to two streams at atmos pheric pressure, one of which is at low temperature and the other at high temperature. Air enters the cylinder through a tangential tube, creating a vortex from which the hot air is withdrawn from the outer periphery and the cold air is withdrawn from the central region, as shown below. In a certain test of the equipment the tem peratures and pressures were measured and reported to be as indicated on the diagram. The mass flow rate of the hot air was stated to be 1.35 times that of the cold air. Show by calculations whether the reported measurements are possible. nlet Air Surroundings P = 20 psia at 80*F and 1 14.7 psia ti = 80o End view ,' Cold air Hot air < outlet outlet o. tlet = l90a P th= 280J t = 1900F Solution: Apply the energy balance Eqn. (2) to the whole device. Assuming negligible kinetic and potential effects, and 6Q = 6W = dE = 0, sys E H.jm = 0 or (H6m)i = (Hm)h + (H6m) Let 6mh = 1.35, 6m = 1.0, 6mi = 2.35 Taking the reference state of air at 800F and 20 psia, and assuming ideal gas behavior with C, = 7.0 Btu/lb moleR, H = 0, h = 7(28080) = 1400, and H = 7(19080) = 1890 Substituting in the energy balance 0 = (1400)(1.35) + (1890)(1)= 0. So energy balance OK. Apply the entropy balance as dS = + ES 6m. = 0 for steady state S 0 j3 :I or 6 = (Sim)i (sim) (6m)c Tc I h (smC CHEMICAL ENGINEERING EDUCATION Integrating Eqn. (8), AS = Cplln R lnPJ Using this to calculate entropies with respect to the refer ence state, Si = 0, Sh = 7 in 4 1.99 in 147 = 2.815 and S = 270540 20 7 In 1.99 in 14.7 4.24 540 20 Putting these into the entropy balance gives S6L = 0 (2.815)(1.35) (1)(4.24) = 0.44 To or 6LW = 0.44 T To is the temperature at which heat may be rejected to the surroundings, 540, so 6LW = (0.44) (540) = 237 This, however, is impossible because the minimum value of lost work in a perfect process (reversible) is zero. There is no such thing as negative lost work (gained work). Thus, the data from the experiment must be in error. In other words, do not invest in this device if suc cessful exploitation is dependent upon the above data. No. 3. A mixture consisting of 331/3% by volume methane and 662/3% oxygen at 25C and 1 atmos pheres total pressure is fed to a combustion chamber where the methane is burned completely at 18000C. The combustion products are then cooled and expanded at constant composition to 250C and 1 atmosphere pressure. Assume ideal gases and surroundings at 250C and no condensation of HO. a) What is the maximum work obtainable from the above process? b) What would be the maximum work obtainable if the combustion! reaction were carried out irrever sibly at 18000C, but all other steps were reversible? DATA: For the reaction at 9 atmospheres pressure, CH4 + 2 02 + CO2 + 2H20. At 25*C: AH = 208,500 cal/gm mole CH4 AS = 1.23 cal/gm mole CH4/K At 1800*C: AH = 211,110 cal/gm mole CH4 AS = 4.07 cal/gm mole CH4/K Solution: (a) At 250C, AG = AHTAS = 208,500 298 (1.23) = 208,140 To expand 3 moles of products from 9 atm to 1 atm AG = /VdP = nflR dP = nRT(lnP) = 3(1.99) (298)ln1 = 3900 cal From Eqn. (18) Wr = AGT = 208,140 3900 = 212,040 cal (b) Irreversible combustion at 18000C means that the work which could have been produced in a reversible engine is converted to heat. This could be put through a reversible heat engine to recover some work. Thus, AG = 211,110 (2073) (4.07) = 202,680 cal Using Eqn. (14) Wr = 202,680( 2073 = 173580 cal Therefore the lost work due to the irreversible combustion is LW = 202,680 173,580 = 29,100 cal Net work by this process is W = 212,040 29,100 = 182,940 cal No. 4. One hundred Ibmole of a solution of 10 mole per cent carbon disulfide and 90 mole percent acetone are to be separated into pure acetone and the azeotrope which forms at 39.250C under atmospheric pressure. The azeotrope which forms 61 mole percent carbon disulfide and 39 mole percent acetone. It is desired to estimate the minimum work to carry out the above separation in a distillation column if the feed solution is all liquid at 39.250C and the products are withdrawn as liquids at the same temperature. At 39.250C the vapor pressure of CS, is 604 mm Hg, while the vapor pressure of (CH,)2 CO at the same temperature is 400 mmHg. At atmospheric pressure the vapor phase may be assumed to behave as an ideal gas. The liquid solution does not behave as an ideal solution, but its activity coefficients may be represented by the Van Laar equation, A B Iny = 2 and InyB = 2 Solution: Let acetone be component A and carbon disulfide be B. For the equilibrium between the liquid and vapor, Eqns. (20)and (27) show that f = "?. For the liquid i = yixif. Taking the standard state as pure liquid under its own vapor pressure and since vapor pressure is low this may be taken as the fugacity f?, f = yixiPi. Also for the gas, assuming ideality, J = Pi " yiP. Thus, ixiPi = yiP P For an azeotrope xi = yi so Yi =  Thus, 760 760 YA = 0 = 1.900 and B = = 1.258 From equations for Van Laar constants x lnyBl 2 A = InyA 1 + XAlny xLAlnY J = 1.558 xlny' 2 B = nyB 1 + x BlA ^L ''~B11 = In 1.90 1g 0.61 In 1.258 2 0.39 In 1.90 0.39 In 1.90 2 = n 1.258 + 0.61 In 1.25 = 1.787 In feed solution, 1.558 Iny or = 1.0202 inyA 1.558(0.9)2 or A 1.0202 + 1.787(0.1) nyB = 1.787 + 1.558(0.9 or yB = 4.08 From AG = AGEx + AGideal and Eqns. (43) and (44), AG = RT(nAlnxAYA + nBlnxBBg) So AGfeed (1.99) (561) [901n(0.9) (1.0202) + 10 In(0.1) (4.08)1 = 18,140 Btu (Continued on page 221.) FALL 1969 4 GaOwsLe in Cenmical Readcoti Cnwaineetyi REACTOR DESIGN N. A. DOUGHARTY and J. M. SMITH University of California Davis, California 95616 At the University of California, Davis (UCD) two quarterlength (3 lectures/week) graduate courses are available in chemical reaction engi neering. The first course, which is required, is a general treatment. The second is an optional offering the contents of which are more special ized and may vary from time to time. The goal of the first course is to complete the outline shown in Table I. However, even though most students have had an undergraduate kinetics course, our experience has been that a semesterlength offer ing would be desirable to cover the subjects listed. Following each major topic in the outline are references to appropriate books and papers. To gether the book references include all the major texts in the field. At UCD several of these (3, 24, 27, 29) have been used as the textbook for the course. On other occasions, the complete list has been assigned as a set of references. We find that even graduate students benefit from thorough familiarity with one reasonably general book. The course starts out with a review of chemical reaction equilibria to ensure that the student understands how to evaluate equilibrium product distributions. At this time it is also convenient to introduce the interrelations between kinetics and thermodynamics. The differential conservation equations (Sec tion II) provide the basis for subsequent design of whatever degree of complexity. The equations of motion are generally omitted, to be picked up in context as needed. The energy equation is ordinarily simplified to neglect kinetic and poten tial energies and shaft work. It is developed care fully in terms of both partial molar enthalpies and temperature, since this seems frequently a source of confusion. The point materialbalance equations also serve to introduce the reaction source term or rate ex pression. At this point the continuity equation Joe M. Smith was educated at Cal Tech (BS) and Massachusetts Institute of Technology (ScD, '43). He has taught chemical engineering at Maryland, Purdue, Northwestern, and California (Davis). His research in terests are in chemical reaction engineering. Presently he is chairman of the department at Davis. Neil A. Dougharty was educated at Lamar Tech (BS) and University of California, Berkeley (PhD, '65). He was a NASNRC Postdoctoral Fellow at Institut de Recherches sur la Catalyse, Villeurbanne, France and a NSF Postdoctoral Fellow at Rice University. His research interests include heterogeneous catalysis, applied chem ical kinetics, and chemical reactor design. Presently he is an assistant professor at the University of California (Davis). for homogeneous reaction is integrated for con stantvolume and variablevolume uniform batch reactors, to stress the nature of the source term in the material balance as a function of instan taneous local properties, independent of the type of constraints on the reacting volume. The discussion of chemical kinetics (Section III of Table I) is necessarily brief. Students will have had an introduction at least to formal kinetics in undergraduate physical chemistry and chemical engineering kinetics courses. We acknowledge that chemical kinetics in its present state of de velopment is only rarely predictive but often permits more reliable interpolation and limited extrapolation. The steadystate approximation is discussed fairly thoroughly (2), and its mathematically very valuable consequencethat it reduces a stoichiometrically complex reaction sequence to a stoichiometrically simple reactionis stressed. With the information presented the student should be able to follow most of the chemical en gineering literature insofar as it attempts to discern reaction mechanisms. On the other hand, CHEMICAL ENGINEERING EDUCATION TABLE I. APPLIED KINETICS AND REACTOR DESIGN SUBJECT OUTLINE References Review of chemical reaction eq. Conservation equations for systems with chemical reaction A. Continuity equations with homogeneous reaction B. Continuity equations with heterogeneous reaction C. Energy equation III. Reaction rate expressions ] A. Material balances with reaction B. Stoichiometrically simple reactions C. Stoichiometrically complex reactions 1. Determination of an inde 7, 26, 29 0 1, 20 3, 27 pendent set 2. Analysis of extents of complex reactions D. Kinetic treatment of reaction mechanisms 6, 17, 23 1. Molecular reactions 2. Steadystate approximation for reactive intermediates a. Open sequences b. Closed sequences (1) Initiationtermination processes (2) Constancy of number of reactive intermediates E. Empirical rate expressions F. Pseudohomogeneous rate expr. 27 IV. Physical transport and reaction in heterogeneous systems A. Pseudohomogeneous rate equa tionsglobal rate B. Intrapellet transport 1. Isothermal effectiveness factors 27, 29, 34 a. Pellet geometry b. Reaction order c. Criteria for absence of diff. retardation of rate 27 2. Nonisothermal effec tiveness factors 27, 28 3. Physical properties of porous catalysts 29 a. Surface area b. Pore volume, porosity c. Pore volume distribution 4. Diffusion in porous media 2729, 34 a. Bulk and Knudsen diffusion b. Surface migration c. Effective diffusivities 5. Heat transfer in porous media 2729, a. Free molecule and normal conduction b. Effective thermal conductivity 6. Effect of poisoning on the global rate 34 1 sensitivity 6. Autothermal operation VI. Transport parameters for packedbed reactors A. Velocity profiles B. Pressure drop C. Radial mass and heat transfer 4, 9, 16, 21 15, 18, 19 5, 2729 parameters 1. Effective diffusivities and thermal conductivities 2. Wall heattransfer coefficients 16 D. Axial mass and heat transfer parameters VII. Design of fluidizedbed reactors 22, 29, 30 A. Mixing phenomena B. Models of reactor behavior VIII. Miscellaneous reactor types A. Gassolid noncatalytic reactions 22, 24 1. Global rate equations 2. Reactor design a. Wellmixed fluid phase b. Moving fluid, stationary solid phase c. Moving fluid and solid phases B. Slurry reactors 28, 29 1. Global rate equations 2. Reactor design IX. Nonideal homogeneous reactors 15, 24, 25 A. Nature of deviations from ideal flow B. Measurement of residencetime distribution functions (RTD) C. Modeling actual reactors with PFR and CSTR assemblies D. Effect on conversion FALL 1969 7. Effect of intrapellet transport on selectivity 34 C. External transport 2729 1. Mass and energy transfer coefficients 2. Effect of external trans port upon global rate a. Single reactions b. Selectivity effects 3. Multiple steady states (stability) 1 V. Reactor design A. Uniform batch reactor B. Continuous stirredtank reactor 1. Steadystate design 2. Multiple steady states and stability 1, 8, 15 C. CSTR sequences D. Tubular reactor 1. Plugflow reactor 2. Tubular reactor with homo geneous reaction 12, 13, 31, 32 3. PFR with axial dispersion 24 4. Threedimensional design for packed beds 5 5. Stability and parametric Teaching, understanding, and applying the principles of reactor design will long remain a major challenge . . students benefit from thorough familiarity with one reasonably general book. a discussion follows in which the wholly empirical nature of much practical rate data is noted, deal ing as it so often does with complex and un analyzed mixtures, empirical parameters such as research octane number, catalyst deactivation, trace poisons, etc. Finally, a brief discussion of rate data for heterogeneous reactors, usually treated alto gether by homogeneous transport models on a global scale, leads naturally into physical trans port questions, that is, Section IV of Table I. In Section IV effectivenessfactor concepts are first developed, and then their numerical evalua tion is investigated. This approach leads con veniently to a discussion of physical properties of porous catalysts, followed by study of diffusion and heat transfer with the effective diffusivity and thermal conductivity the goal. External transport resistances complete this Section. Here care is taken to emphasize the relative import ance of heat and mass transport effects for various types of heterogenous environments such as packedbed, fluidizedbed, and slurry reactors. The integral reactor design equations (Section V) are developed rather quickly and little atten tion is given to their application to isothermal examples. The students are assumed to have had prior experience with this in an undergraduate course. Somewhat more attention is given to non isothermal cases, since time frequently prohibits adequate consideration of these at the under graduate level. Particular emphasis is placed on reactor thermal stability. Stirredtank reactor sequences are discussed briefly, and their use in modeling tubular reactors [e.g., ref. (14)] is noted. Primary emphasis in Section V is placed on the phenomena which occur in packedbed reactors. Again, particular attention is given to thermal effects, which Denbigh (15) has emphasized as "undoubtedly the biggest factor of uncertainty in the design of fixedbed reactors at the present time." Quantitative analysis of these effects re quires numerical values for radial and axial Peclet numbers for heat and mass transfer. Empirical correlations and theories for these quantities are discussed in Section VI. Specific, but practically important, types of heterogeneous reactors are considered in Sections VII and VIII, with fluidizedbed systems singled out for particular emphasis. While gassolid non catalytic and slurry reactors are the only ones listed in Section VIII, it is at this point that other specialized forms can be introduced. The final subject is nonideal flow in homo geneous reactors. The time spent here is depend ent upon the background of the class. Practical situations where nonideal flow has a significant effect upon conversion are stressed; for example, the CSTR with one or more internal cooling coils. RESEARCH IN REACTOR DESIGN A stateoftheart graduate course will natur ally bare the limitations of our present knowledge, as well as a variety of bypassed problems. A discussion of fluidsolid processes reveals many such areas, both for global rates and reactor design. The role of surface migration, inhomo geneity of catalysts, and heat and mass transfer in real solids are examples, for global rates, of research problems of chemical engineering inter est. Work in this area often encounters unusual practical consequences. An interesting recent example is the report of Weisz (33) that catalyst attrition rates in a fluidized catalytic cracking unit can be greatly affected by intraparticle dif fusion limitations in the cyclic formation and burnoff of coke deposits. In the design of non adiabatic packedbed reactors, uncertainties in the calculation of temperature profiles remain a major source of concern. The scaleup of non catalytic fluidsolid reactors such as carbon black reactors and lime kilns is hindered because of lack of research on mixing patterns. Many com putational difficulties remain in the design of integral reactors because of the complexity of boundary conditions in heterogeneous systems. Research on the design and operation of re actors for treating both municipal and industrial waste water is of literally vital significance. Im provement in the operation of biological reactors for treating primary effluents is an urgent need. New schemes are needed for the treatment of secondary effluents, and the chemicalreaction route offers many advantages. The design of photochemical reactors, which offer a new kind of nonuniformity, that of the light intensity, is an area of active research. Though by no means exhaustive, these research areas are illustrative of both the work basic to reactor design yet to be done and the demanding CHEMICAL ENGINEERING EDUCATION practical needs of society which application of our knowledge of reactor design offers hope of meet ing. Teaching, understanding, and applying the principles of reactor design will long remain a major challenge to chemical engineers. REFERENCES 1. N. R. Amundson and R. Aris, Chem. Eng. Sci, 7, 121 (1958). 2. Rutherford Aris, Ind. Eng. Chem., 61, 17 (1969). 3. Rutherford Aris, "Elementary Chemical Reactor Analysis," PrenticeHall, Englewood Cliffs, New Jersey (1969). 4. C. H. Barkelew, Chem. Eng. Progr. Symp. Ser., 55, (25), 37 (1959). 5. John Beek, "Design of Packed Catalytic Reactors," in "Advances in Chemical Engineering," Vol. 3, Academic Press, New York (1962). 6. S. W. Benson, "Foundations of Chemical Kinetics," McGrawHill, New York (1960). 7. S. W. Benson, "Thermochemical Kinetics," John Wiley & Sons, New York (1968). 8. O. Bilous and N. R. Amundson, A.I.Ch.E. Jour., 1, 513 (1955). 9. 0. Bilous and N. R. Amundson, A.I.Ch.E. Jour., 2, 117 (1956). 10. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, "Transport Phenomena," Ch. 18, John Wiley & Sons, New York (1960). 11. Michel Boudart, "Kinetics of Chemical Processes," PrenticeHall, Englewood Cliffs, New Jersey (1968). 12. F. A. Cleland and R. H. Wilhelm, A.I.Ch.E. Jour., 1, 489 (1956). 13. P. V. Danckwerts, Chem. Eng. Sci., 2, 1 (1953). 14. H. A. Deans and L. Lapidus, A.I.Ch.E. Jour., 6, 656 (1960). 15. Kenneth Denbigh, "Chemical Reactor Theory," Cam bridge University Press (1965). 16. G. F. Froment, Ind. Eng. Chem., 59, 18 (1967). 17. A. A. Frost and R. G. Pearson, "Kinetics and Mechan PROBLEMS (Cont'd from p. 217.) AGproducts = AGazeo + AGacetone AGazeo Gazeo = (1.99) (561) 3 In(0.39)(1.9) + 10 In(0.61)(1.258)] = 5,050 Btu Wr AG = AGproducts AGfeed =5050 + 18140 = 13090 Btu/100 moles feed No. 5. Badische Anilin and SodaFabrik AG of Ludwigs hafen am Rhein give this data. "WaterGas Shift ReactionConversion of the carbon monoxide proceeds according to the expression: (1) CO + H20 CO2 + H2; AH = 9,810 cal It is an equilibrium reaction with a temperature dependent equilibrium constant, (2) Kp = (CO)(H20) (CO2) (H2) ism." John Wiley & Sons, New York (1961). 18. E. A. Grens and R. A. McKean, Chem. Eng. Sci., 18, 291 (1963). 19. C. van Heerden, Ind. Eng. Chem., 45, 1242 (1953). 20. J. C. Jungers, et al., "Cin6tique Chimique Appliqude." Editions Technip, Paris (1958). 21. H. Kramers & K. R. Westerterp, "Elements of Chemi cal Reactor Design and Operation," Academic Press, New York (1963). 22. Daizo Kunii and Octave Levenspiel, "Fluidization Engineering," John Wiley & Sons, New York (1968). 23. K. J. Laidler, "Chemical Kinetics," McGrawHill, New York (1965). 24. Octave Levenspiel, "Chemical Reaction Engineering," John Wiley & Sons, New York (1962). 25. Octave Levenspiel and K. B. Bischoff, "Patterns of Flow in Chemical Process Vessels," in "Advances in Chemical Engineering," Vol. 4, Academic Press, New York (1964). 26. G. N. Lewis and Merle Randall, as revised by K. S. Pitzer and Leo Brewer, "Thermodynamics," McGraw Hill, New York (1961). 27. E. E. Petersen, "Chemical Reaction Analysis," PrenticeHall, Englewood Cliffs, New Jersey (1965). 28. C. N. Satterfield and T. K. Sherwood, "The Role of Diffusion in Catalysis," AddisonWesley, Reading, Mass. (1963). 29. J. M. Smith, "Chemical Engineering Kinetics," Second Edition, McGrawHill, New York (publication date 1970). 30. J. M. Thomas and W. J. Thomas, "Introduction to the Principles of Heterogeneous Catalysis," Academic Press, New York (1967). 31. J. P. Vignes and P. J. Trambouze, Chem. Eng. Sci., 17, 73 (1962). 32. J. F. Wehner and R. H. Wilhelm Chem. Eng. Sci. 6, 89 (1956). 33. P. S. Weisz, Ind. Eng. Chem. Fundamentals, 8, 325 (1969). 34. Ahlborn Wheeler, "Reaction Rates and Selectivity in Catalyst Pores," in "Advances in Catalysis," Vol. 3, Academic Press, New York (1951). The location of the equilibrium for a given gas composi tion is independent of the total pressure of the system." Also presented with the above statements is a graph which is reproduced below. Please study this information and demonstrate (a) by numerical calculations whether their data are concordant, and determine (b) the free energy change of the reaction at 5000C. Temperature dependence of the equilibrium constant Ep 0.401 I 0.30 / 0 53 400 450 500 6b Assume Kp = Ka DO0C T (Continued on page 226.) FALL 1969 1 '14 550 mn .views and opinions GRADUATEENGINEERING AND TECHNOLOGICAL ACCREDITATION L. E. GRINTER University of Florida Gainesville, Fla. 32601 Perhaps at no time in its history has engineer ing education been beset by as many problems as exist today. It has been said from many direc tions that those who left engineering teaching as much as fifteen years ago would not recognize much of the course material taught today. Al though this may be somewhat of an exaggera tion we recognize its broad validity, and we must look forward to equally rapid changes in the future. A vicepresident of one of the great electrical concerns mentioned when talking to students that his company based its longterm planning on the assumption that onehalf of its business twenty years from now would be in new products. To transfer this concept to engi neering education we might anticipate that one half of the courses in the engineering college catalogs circa 1985 will be totally new subject matter and the remainder will be considerably altered. Because college catalogs are revised every year faculties are used to the concept of new courses and curricular changes. The picture of a fifty percent change in course material in fifteen years is not startling because it represents a normal evolutionary trend. In fifteen years the volume of published scientific and engineering material will at least have doubled. There are other changes which are just as probable that appear to produce very severe emotional reac tions even when they are merely discussed. Such questions as, what should represent the first pro fessional degree in engineering, and how does technician training articulate with engineering education are highly sensitive areas of interest. These areas require objective analysis which is Linton E. Grinter is vicepresident of the University of Florida. He was educated at the University of Kansas and the University of Illinois (PhD, '26). Dr. Grinter was ASEE Lamme Medalist in 1958. He is a leader in establishing goals and directions for engineering educa tion in this country. The 1955 Grinter report on the "Evaluation of Engineering Education" was instrumental in adding an engineering science base to all engineering curricula. He is active in many professional societies and has served as president of both ASEE and ECPD. difficult to achieve because of emotional reactions based upon the concept of a unified profession that has never truly existed in engineering. We wear blinders if we fail to recognize that techni cians continually, although in small numbers, move upward into the engineering profession, and that scientists move laterally with little resist ance into engineering activities. Both groups achieve the title of engineer in industry. Engi neering is a profession in flux that has still not been defined for the purpose of exclusion either by words or more importantly by actions. An agency involved deeply with the need to define the undefined and perhaps undefinable profession of engineering is the Engineers' Coun cil for Professional Development. Its task of accrediting engineering curricula and therefore degrees has successfully placed a floor under the profession of engineering that has received broad acceptance. However, this floor, based upon the amount of engineering education that can be mastered in four academic years while allowing for required work in mathematics, science, and socialhumanistic studies is not highly restrictive. The great majority of institutions that make an CHEMICAL ENGINEERING EDUCATION The evidence seems to point to the master's degree evolving rather gradually into the main accredited degree whether or not it is called the first professional degree. effort to recruit an engineering faculty of reason able quality, who in turn select students of rea sonable competency, achieve accreditation. Their products become engineers by definition, and those scientists and technicians who retrain or upgrade themselves to compete with the product of the engineering schools are accepted as engineers. Under the procedures just described, we seem to have some 800,000 engineers in this country, of which about onehalf belong to a major en gineering society. A 1967 EJC survey deter mined that 565,000 individuals belonged to 45 technical and professional societies, of an en gineering and applied science nature, of which 438,000 were classified as engineers. The tech nical societies that hold membership in ECPD and EJC do not all restrict their membership to engineers. GRADUATE ENGINEERING ACCREDITATION An old refrain in engineering education has been the fiveyear undergraduate curriculum. It has been proposed, urged and tried over a forty year period without significant success. At times it appeared that nearly a majority of faculty members would favor it. Why then has there been so much emotional resistance to the concept of the master's degree becoming the "first profes sional degree"? It seems doubtful that the re sistance rests upon the concept that future pro fessional engineers can attain an adequate edu cation in four years. Such is patently not true. It may be that the terminology of "first profes sional degree" applied to the master's degree raises the specter that a very large fraction of presently practicing engineers would lose profes sional status. Because only a quarter of practic ing engineers now have master's degrees a long transition period would be inevitable. Terminol ogy can often mask the most desirable objectives. It is becoming evident that the Engineers' Council for Professional Development is gradu ally being drawn into graduate accreditation in a stepbystep fashion. The first step over a decade ago was to accredit the master's programs of the Naval Postgraduate School using undergraduate standards. Then a number, of master's degree programs or curricula developed in engineering departments having no undergraduate curricula. These departments applied for accreditation and were gradually accepted. Now there are requests for accreditation of master's level curricula in colleges that offer bachelor's degrees in engineer ing where the bachelor's program is considered to be preprofessional by the institution con cerned. ECPD certainly cannot insist upon ac crediting a preprofessional curriculum, and it is doubtful that it can logically reject the right of any educational institution to define for itself what it wishes to call its first professional degree in engineering. The evidence seems to point to the master's degree evolving rather gradually into the main accredited degree whether or not it is called the first professional degree. Important influences are the following: (1) It seems doubtful that a fouryear education in engineering can be made sufficiently superior to four years of either science or technology to form the base for a clearly de fined profession. (2) An accredited master's de gree program based upon student desire and aptitude for advanced study, with the opportunity for those whose interests are not highly profes sional to accept employment at the bachelor's level, would aid greatly in defining the profes sion of engineering. Quite independent of emo tional reactions this seems to be the most prob able direction of gradual evolution. This change will be stimulated by changes in the engineering college catalog because the course material added always exceeds the deletions. Additions can be made at the master's level without gross eco nomic waste due to greater motivation of selected and selfselected students. A factor that should not be overlooked in the accreditation of graduate education is its useful ness in upgrading the casual offerings at many offcampus centers. Undergraduate work in the evening was at one time taught mainly by indus trial employees on a parttime basis. Gradually through the accreditation process evening study has been upgraded to achieve as nearly as practi cal an equivalency with day curricula and day procedures. At the graduate level there has been a dissemination of degree work to socalled grad uate centers. These centers operate not only un der the difficulties of evening programs on an overtime basis, but they often use parttime teachers to an excessive degree. Many fail to FALL 1969 A factor that should not be overlooked in the accreditation of graduate education is its usefulness in upgrading the casual offerings at many offcampus centers. provide even the minimum essentials of library or laboratory resources. Until graduate accredi tation in engineering becomes accepted, these graduate centers will lack standards to guide their activities. They need and their students deserve the support that professional accredita tion would provide. Unfortunately, we still cannot provide the upgrading through accreditation that the offcampus graduate programs so clearly need. TECHNOLOGICAL ACCREDITATION At the opposite end of the spectrum from graduateengineering accreditation is found the problem of technology curricula accreditation. Beginning with the Wickenden report in the late nineteen twenties it has been recognized that the productivity of engineers depends upon the num ber and quality of the technicians available as engineering assistants. In World War II the engineering colleges became large scale techni cian training agencies for the Federal Govern ment and made one of their greatest immediate contributions to the war effort through this chan nel. A postwar surplus of technicians may have existed for a time, but if so, this could have been only at the lower levels. The United States has never developed an educational system that has produced highlevel technicians comparable to those produced in most European countries. In stead, our bachelor degree engineers have per formed many technicianlevel activities. This country suffers under the status symbol of the bachelor's degree. Parents make great sacrifices for their children to attain degrees. Any degree often appears acceptable. Hence technician curricula ranging from two to three years have never attracted sufficient numbers of students. The result is that new degreelevel programs in technology have been growing in numbers. They now represent a considerable group of curricula that carry the same descrip tive titles as the branches of engineering, i.e., electrical, mechanical, etc. One technological cur riculum widely adopted is building construction, which found a home in colleges of architecture rather than engineering. It has in part super ceded the curriculum of architectural engineering which was technically too demanding upon the type of student who was interested in this field. The curriculum of building construction provides the degree incentive and the reward of desirable employment in a status position ultimately di rected toward supervision without requiring the rigor of an engineering curriculum. It has grown rapidly in popularity and is entirely outside the control or direction of the engineering profession. The broad field of industrial technology based educationally upon degree curricula now seems to be ripe for a development comparable to the ex ample given of building construction. The engi neering profession can influence this develop ment through its procedures of accreditation or it can stand aside and observe the uncontrolled development of a second channel for the prepara tion of technological personnel. When this prob lem was presented by the establishment of asso ciatedegree technician training curricula in the years immediately following World War II it was decided to lend a hand toward strengthening these technical curricula through ECPD accredi tation. Of course, the question of terminology arose, in particular the use of the adjective en gineering to describe such curricula. Obviously ECPD could serve no function in the field of medical technology or other fields not directly related to engineering. Our interest had to be restricted to the training of engineering techni cians. To make this clear the curricula eligible for ECPD accreditation were classified as "engi neering technology" curricula. However one may feel about the terminology chosen, the logic in volved seems indisputable. Unless a technical curriculum is designed to produce technicians who will work directly with engineers it could hardly fit within the objectives of ECPD. The recent action of the Board of Directors of ECPD to accredit engineering technology curric ula of two, three and four years duration upon the single basis of inspection of about 70 credit hours of technical course work merely fulfills the concept described above. Beyond the required and regulated core of some 70 credit hours the institution may decide to add additional work requirements to justify the award of a bachelor's degree. This additional work may be in liberal arts, business administration, further technical courses in the major, or in a second specialty, or in any combination it may choose. ECPD will restrict its interest to the core program that CHEMICAL ENGINEERING EDUCATION Six Good Reasons To Choose A McGrawHill Text OPTIMIZATION THEORY AND PRACTICE ENGINEERING DIFFERENTIAL SYSTEMS GORDON S. G. BEVERIDGE, HeriotWatt Uni versity, Edinburgh and ROBERT S. SCHECH TER, University of Texas. Available Winter, 1970 This text encompasses techniques from all aspects of mathematical optimization with the objective of introducing these methods to seniors and graduate students. It is organized to illus trate the interrelationships among optimization methods, their ranges of applicability, and their comparative effectiveness. The authors provide fully workedout examples throughout the book to aid the student, and discuss the main tech niques in detail to give the student competence in their applications. OPTIMIZATION BY VARIATIONAL METHODS MORTON M. DENN, University of Delaware. 416 pages, $16.50 In order to present a comprehensive examina tion of optimal process design and control, the author has simultaneously developed both ana lytical and computational considerations and then united them with detailed practical applications. The text utilizes the "variational" approach, in corporating traditional differential calculus pro cedures and associated computational techniques; classical calculus of variations; Pontryagintype "minimum principles" and related computational methods; and dynamic programming. Many of the examples cited are examined at various levels of sophistication and solved by several different procedures. ENGINEERING THERMODYNAMICS WILLIAM C. REYNOLDS, Stanford University and HENRY C. PERKINS, University of Arizona. Available Winter, 1970 The first half of this book develops the funda mentals of thermodynamics using microscopic insight as the basis for macroscopic postulates. Disorder, randomness, and uncertainty notions are used in conjunction with the Gibbs definition of entropy to provide an intuitive basis for the second law postulate. The remainder of the book applies the statistical concepts that have already been developed to actual engineering systems. Material on power systems and chapters on com pressible flow and heat transfer are included. ROBERT D. KERSTEN, Florida Technological University. 224 pages, $13.50 This is the first book to treat both the ana lytical as well as the numerical methods in engi neering. The author's thesis is that a complete solution to a given engineering differential system can be developed by using these approaches to connect four essential parts of the system: (1) properly understood phenomena; (2) a correct mathematical model of the phenomena; (3) a ten tative solution; and (4) a proper application of boundary or initial conditions or both. DESCRIBING CHEMICAL ENGINEERING SYSTEMS WILLIAM E. RANZ, University of Minnesota. Available Winter, 1970 With the intention of demonstrating how phy sical and mathematical models are built, this par ticipation textbook discusses states and actions of physical and chemical systems; shows the detailed development of material and energy balances; and includes interactions of simple connected systems as they are applied to chemical engineering. This workbook is based on the premise that a student learns by doingtherefore, numerous questions and workedout examples dominate the text. MODERN METHODS OF ENGINEERING COMPUTATION ROBERT L. KETTER and SHERWOOD P. PRA WEL, JR., both of the State University of New York at Buffalo. 500 pages, $15.50 This text (1) presents an introduction to the field of modern computational methods in terms intelligible to the second or thirdyear student; (2) develops from these various methods the first principles that are basic and/or in general usage and indicates the interrelationships among them; and (3) views the material specifically but yet generally enough to give the student the background he will need in numerical methods to cope with future engineering courses. Through out, the emphasis is on the methodology of the solution process and the universality of its appli cation to problems in all fields of engineering and the applied sciences. McGRAWHILL BOOK COMPANY 330 West 42nd Street New York, New York 10036 FALL 1969 relates directly to the title of the curriculum. The award of a degree will be primarily the in terest of the regional accrediting agency. Re gional accreditation at the appropriate level (as sociate or bachelor's degree) must precede ECPD inspection. It is believed that this limited ac crediting procedure by ECPD will eliminate, to the maximum degree possible, confusion between engineering education and engineering technician education. RECOGNITION OF CONTINUING EDUCATION tention given to continuing education would doubtless increase. Because of its extensive ex perience with the accreditation process, ECPD seems to be the logical agency to experiment with this concept of formal recognition of achievement in continuing education. It is hoped that an ap propriate channel for such recognition may be devised. It seems to the writer that such recog nition is a serious responsibility of the engineer ing profession that has been neglected merely be cause of its sensitive nature. DnEILmIIfI A BBraeCPIFIL The significance of continuing education for Fr,,,,, rA r"SV0 engineers was recognized a few years ago by a A profession may be defined in part by re comprehensive report sponsored by EJC. ECPD, quired steps of admission and advancement of its ASEE and NSPE that emphasizes its great im members. It can also be defined in part through portance to the engineering profession. Never aiding in the recognition of associated groups, theless, continuing education operates under the who relate clearly to its activities, but by using handicap that the achievement of the individual distinctly different standards for recognition. receives no formal recognition. In contrast, a Such a relationship exists between engineers and reasonable amount of effort directed toward engineering technicians or technologists. There parttime graduate study can result in a master's is reason to hope that these and other actions of degree that receives nationwide acceptance. If engineering societies may aid in defining the pro some type of formal recognition of perhaps an fession of engineering which has resisted inclu equivalent academic year of effort devoted to sive definition by words alone. Nevertheless, the continuing education could be developed, the at writer believes that definitions can be improved. PROBLEMS (Cont'd from p. 221.) Calculate the reversible voltage to electrolyze water at Solution: 400C if the products and reactants are at 5 atm pressure. Assume ideal gas behavior and negligible effect of pres (a) Assuming heat capacity effects negligible, use Eqn. (30) inr e t sure on vapor pressure. For water take the standard the form H/ \ state to be (a) pure liquid under atmospheric pressure in a AH 1 1 T and (b) pure gas under its vapor pressure at 180C. Com l 1 22 pare the two answers. The reaction is HO0 > H, + 1/2 02. From the graph Ka = 0.lat427C and 0.4 at 600 C Thus (1.99)8 (700) n 9750 cal/gm ole .00873 0.1 9750 cal/gm mole This agrees well with the 9810 cal given. The slope of the InKa vs 1/T plot gives the same result. (b) At 5000C, Ka = 0.19 for CO2 + H2 + CO + H20 For the reverse reaction Ka = 0.19 Therefore, from Eqn. (29) AG = RTlnKa = (1.99) (773)1n 1 0.19 =2550 cal/gm mole No. 6. The heat of combustion of hydrogen with oxygen at atmospheric pressure and 180C to form liquid water is 68,300 cal/gm mole HO. The reversible voltage for the electrolysis of water in a very dilute acid solution at 180C is 1.23 volts when all products and reactants are at atmospheric pressure. The latent heat of vaporiza tion of water is 10,500 cal/gm mole, and both this and the heat of combustion vary negligibly with temperature. The vapor pressure of water at 180C is 15.48 mm Hg (neglect effect of small acid content), while at 400C the vapor pressure is 55.31 mm Hg. Solution: (a) Integrating Eqn. (33) assuming little change in AH,  I H 1 IT N I T or , o 1.23(313) 68,300(313) 40 291 2(23,050) 2 = 1.211 volts By Eqn. (35) a1/2 S0 T 02 "2 1.211 .99) (313) n)/2() 1.2438 volts (b) In producing gaseous water AH = 68,300 10,500 57,800 cal/gm mole u, c =1.23(313) + (57,800) (313) [ 1 1 BI Thus, o + 2(23,050) 21 = 1,228 volts Now activity of gaseous water under own vapor pressure is 55.31 al20 Itr = 3.57 So 12 (1.s9s9)(313) (5)1/2(5) So (2( 1.228 3in 1.2434 volts which agrees well with 1.2438 CHEMICAL ENGINEERING EDUCATION CHEMICAL ENGINEERS When you GO with TEXACO you GO with the BEST! ... The BEST Facilities . The BEST Products .. .The BEST People S. The BEST Opportunities Yet, being the best makes it that much harder to get better. With competition constantly at our heels, it takes an aggressive and imaginative R & D team to maintain Texaco's leadership role in the petro leum and petrochemical industry. An important part of this team are Chemical Engineers . .men like yourself .. constantly searching for better ways. It is through their efforts, as well as their professional colleagues and an aggressive management team, that Texaco stays out front. You, too, can be part of this winning combination. For Chemical Engineers with a B.S. or M.S., the professional and economic rewards of a Texaco career in process and product development have never been greater. Texaco's practice of basing pro motions on performance and ability afford you the BEST opportunities for advancement, while enjoying our congenial, shirtsleeve atmosphere. Texaco has immediate openings at its laboratories in Beacon, N. Y., Richmond, Va., and Port Arthur and Bellaire, Texas for qualified Chemical Engi neers who already are, or are in the process of be coming U. S. Citizens. Interested candidates are invited to send their resume to: W. R. Hencke, Texaco, Research & Tech nical Department, P. 0. Box 509, Beacon, N. Y. 12508. Texaco is an equal opportunity employer. kC uvdicuam THE CHEMISTRYCHEMICAL ENGINEERING MERRYGOROUND RALPH A. MORGEN Stevens Institute of Technology Hoboken, N. J. THE MAJOR CHARACTERISTIC WHICH distinguishes the chemical engineer from all other engineers is the foundation in chemistry which is required. There is a generally accepted statement, the origin of which has been lost in antiquity, that says "The day mechanical engi neers dropped physical chemistry from their cur riculum, chemical engineering was born." Since the birth of the AIChE, the question of chemical engineering education has been uppermost in the minds of its members. One of the first acts of the new Society was to establish a committee on Chemical Engineering Education, under the Chairmanship of C. F. McKenna. There was much debate on the curriculum content for chem ical engineering but little consensus until after World War I, which has been known by many people as the "Chemists' War." Prior to that war, there was little chemical industry in this country. As a matter of fact, the United States was so dependent on foreign imports of chemicals that the Germans were encouraged in 1915 to deliver a supply of dyes to the United States by submarine to avoid the Allied blockade. Following the "Chemists' War" there was a major increase in the chemical industry. Along with the growth of that industry, there was a demand for chem ists and chemical engineers to staff these indus tries. The fundamental educational debate at that time was whether or not there really was such a thing as chemical engineering. That chemists were employed in the chemical industry was un questioned. That engineers were employed in the chemical industry was unquestioned. The Ger man concept was to have a team of a chemist and a mechanical engineer perform most of the functions that are today thought of as chemical engineering. It remained for the team of William H. Walker, W. K. Lewis, and W. H. McAdams * Presented at the Annual Meeting of ASEE, June 1922, 1967. Ralph A. Morgen is a graduate of the University of California, Berkeley (PhD '25). He has been active in research, teaching, and administration in engineering education for thirty years. His most recent assignments were President of Rose Polytechic Institute and Dean of Graduate Studies at Stevens. Currently he is an engi neering consultant to Florida Atlantic University. to publish a textbook "Principles of Chemical Engineering" in 1923, to clearly and succinctly delineate, for the first time, the place of chemical engineering. This book made a clear distinction between the industrial chemists on the one hand and all other engineers on the other. The most significant contribution of Walker, Lewis and McAdams was to focus the attention of chemical engineers on the unique place of the unit opera tions. While the term 'unit operation' had been used earlier and is credited to Arthur D. Little, who based a curriculum study on unit operations as early as 19151, the concept did not take hold until after the publication of this book. Dr. Little, while he was Chairman of the Chemical Engi neering Education Committee of AIChE, made the first steps towards the establishment of ac credited chemical engineering curricula. This initial list appeared in 1925, eight years before the rest of the engineering profession established the Engineers' Council for Professional Develop ment. Only fourteen institutions appeared on this list. In 1933, the chemical engineers joined ECPD, but they retained a certain amount of autonomy. They were the only ones who, because of their previous experience, saw the necessity for greater emphasis on the basic sciences in the undergraduate curriculum fol lowed by advanced work at the graduate level. This occurred before World War II, commonly known as the "Physicists' War." 1"Highlights the first 50 years of the American Institute of Chemical Engineers, pg. 56, (1958 published by AIChE). CHEMICAL ENGINEERING EDUCATION IN ORDER TO EVALUATE WHAT HAS happened to the chemistry content of the under graduate curriculum over the past 30 years, the 14 institutions on the original accredited list augmented by those institutions which were deemed to have either distinguished or strong faculties in the American Council on Education study2 have been reviewed. The conclusions in this paper have been confirmed by reviewing what has happened to the chemistry content of these twentyfour institutions. From 1925 until accreditation was tempo rarily abandoned during World War II in 1943, there was general hauling and pulling among the proponents of more chemistry and basic sciences as opposed to those who preferred more applied and more practical engineering emphasis in the curriculum. Dr. Albert B. Newman3 sum marized the situation as it existed in the late 1930's very well. Quoting from that paper' "In modern practice, it seems clear that the chemical engineer must not only have a working quantitative knowledge of the unit operations, but he must have a sound knowledge of chemistry, physics, mathematics, thermodynamics and economics. He must have facility in applying physical chemistry to plant processes, particu larly in relation to reaction velocities and the graphical calculus used in the interpretation of laboratory and pilot plant data on kinetics of chemical reactions." As a result, in order to try to satisfy the two opposing views, more engineering and applied courses versus more chemistry and basic sciences, the credit content of the undergraduate chemical engineering course rose to an almost intolerable level at many of the institutions. In some cases, an average of 20 semester credit hours for a total of eight semesters was required for gradua tion. The four year curriculum was reaching the bursting point. Most of the stronger accredited institutions, in that period, insisted on four years and a summer session, usually between the junior and senior years, to lessen this unrealistic load. The Chemical Engineering Education Committee was far ahead of the education committees of the other engineering societies and of the philosophy of ECPD in two respects in the 1930's. The 2An Assessment of quality in Graduate Education, Allan M. Cartter, American Council on Education, Wash ington, D. C. pg. 70, 1966. 8Development of Chemical Engineering Education in the United States, Supplement to transaction of the American Institute of Chemical Engineers, Volume 34, No. 3A, July 25, 1938. 4Ibid. Pg. 12 chemical engineers did not consider the four year undergraduate course as terminal education, but rather "that education5 has just begun at the end of the four year course. No student should prepare for chemical engineering unless he is enthusiastic about the idea of a lifetime of study." The Committee further believed that re search activity by the chemical engineering staff and graduate students is important and was usually found in those institutions which quali fied for accrediting. IN FACT, THE TRADITION OF GRADUATE WORK in chemical engineering was one of the close ties between chemists and chemical engineers which fostered graduate education in both disciplines following (World War I). The recommended content of an undergraduate chemical engineering curriculum in 1938 is shown in Table I. The percent figures are those taken from New man's publication6. There was little dissension regarding TABLE I CHE CURRICULUM RECOMMENDED IN 1938 (Normal Credits in) Percent Semester Hours Chemistry 2530 3644 Chemical Engineering 2015 3022 Other Engineering 12 18 Mathematics 12 18 Physics 8 12 Mechanics 6 9 Other Sciences 2 3 Cultural Subjects 15 22 Total 100 148 Development of Chemical Engineering Education in the United States, Albert B. Newman (Trans. AIChE 34, 3a (1938). the percentages, but the difficulty arose when such a curriculum had to be translated into a reasonable number of credit hours. For convenience, in Table I, the column of normal credits is given for comparison with present day curricula, but many of the actual curricula contained total credits up to 160. It appears, therefore, that in the 30's, most of the accredited curricula included four whole year courses and at least one additional one semester course in chemistry. The year courses were usually general chemistry, quantitative analysis, organic chemistry and physical chemistry. The additional courses varied widely depending on the interests, competence and backgrounds of the faculty at the various institutions. At this time, it was generally agreed that the undergraduate load was too heavy and further that graduate work was to be encouraged. Quoting again from Newman's paper7 "The Committee is of the opinion that the tendency to extend chemical engineering study into graduate years, especially on the part of those students whose special aptitude in theoretical divisions, is one that should be 5Ibid. pg. 16 6Ibid. pg. 23. 'Ibid. pg. 23. FALL 1969 229 encouraged, because of the widely recognized difficulty of giving adequate instruction within a period of four years, especially if any attempt is made to teach methods of research in science or engineering." Thus, it appears that the Chemical Engineering Education Committee in 1938 reached the conclusion which apparently the rest of the engineering profession is tentatively approaching in 19678. There was general agreement in the thirties that if the amount of chemistry in the chemical engineering curriculum is reduced drastically, then the need for chemical engineering as a sepa rate entity becomes academic. This was about the situation when World War II intervened. A temporary cessation of the accrediting function took place between 1943 and 1946. When the ECPD Education and Accreditation Committee reconvened after World War II, the effect on the undergraduate engineering curricula was obvious to many. The engineering education of the 30's was found to be insufficient in its content of mathematics and the basic sciences. The need for adding large doses of the engineering sciences (which the chemical engineers had called unit operations in their area for many years) became obvious. The physicists became enamored with subatomic phenomena and tended to abandon classical physics. Thus, the engineering sciences and much of classical physics tended to merge. The resulting effect on the chemistry content of the chemical engineering curricula was serious and in some cases drastic. It became obvious that a thorough restudy of the needs of undergraduate engineering edu cation was in order. At the request of ECPD, ASEE undertook a study which has come to be known as the Grinter Report9. The report of this committee reads strikingly similar to the recom mendations of the Committee on Education and Accreditation of AIChE as annunciated by its chairman in 193810 The obvious difference, how ever, is that when most of the members of the Grinter Committee talked about the basic sci ences and the engineering sciences, they were almost uniformly talking about physics and al most uniformly neglecting chemistry. It was only through the valiant effort of the few chemi cal engineering members of the ASEE Commit sInterim Report of the Committee on Goals of Engi neering Education, E. A. Walker, Chairman, American Society for Engineering Eduction, April, 1967. 9Report of the Committee on Evaluation of Engineer ing Education, L. E. Grinter, Chairman, ASEE Pamphlet, June 15, 1955. i"Ibid. It appears that the ChE Education Committee in 1938 reached the conclusion which apparently the rest of the engineering profession is tentatively approaching in 1967. teen that some of the normal chemical engineer ing unit operations were included as some of the engineering sciences. The publication of the Grinter Report was a signal for rather drastic revisions of engineering curricula throughout the country. The kind of dichotomy which was men tioned previously as occurring among the chemi cal engineers in the 30's, now infected all the branches of engineering, i.e., one group advocat ing more mathematics and basic science as op posed to those who recommend more applied courses and practical training. The course con tent recommended by the ASEE Committee"1 illustrates the dilemma. (See Table II) The whole four year curriculum allows less time for mathematics and all basic sciences than the chem ical engineers thought was necessary for chem istry along in the 1930's. As a result, the chem istry content in 1966 of all of the curricula studied contains less chemistry than those same institutions had in the 30's or when they were first accredited by ECPD. TABLE II COURSE CONTENT RECOMMENDED BY THE COMMITTEE ON EVALUATION OF ENGINEERING EDUCATION11 19521955 Proportion of Curriculum (1) Humanistic Social Studies About 20% (2) Mathematics and Basic Sciences (About equal weight) About 25% (3) Engineering Sciences About 25% (4) Sequenceof Engineering Analysis, Design and Engineering Systems, in cluding the Technological Background About 25% (5) Options or Electives About 10% Total Four years TWO OPPOSING FACTORS FURTHER aggravate the current situation, the explosion of scientific knowledge since World War II, argues for the inclusion of more subject matter while the ASEE Committee recommends decreas ing the total number of credit hours to lighten the burden on the student. This places engineer ing education squarely on the horns of two dilem mas: How to increase the science content of the "Report of the Committee on Evaluation of Engineer ing Education, L. E. Grinter, Chairman (ASEE Pamphlet 1955). CHEMICAL ENGINEERING EDUCATION curriculum on one side, decrease the total number of contact hours on the other side and still make an engineering curriculum without going beyond four years. The conclusion is obvious. Sooner or later it must be recognized that an adequate four years curriculum in chemical engineering is a misnomer. The Goals of Engineering Education Committee realize that there are many ways to reach the desired objective of a well educated professional engineer. However, in each case the inescapable conclusion must be reached that an engineer has an insufficient background at the end of the Bachelor's degree program to fit him for a productive technical career in engineering. The report further contends that formal educa tion to the Master's level followed by continuing education throughout his professional life is a must for the engineer of the future. Equating the Goals report to chemical engineering it ap pears that there is room for various kinds of chemical engineering curricula, all the way from a very "light" chemistry content at the under graduate level followed by more chemistry at the graduate level to a "strong" chemistry content at the undergraduate level followed later by more engineering at the advanced level. (See Table III) TABLE III FOUR YEAR COMPROMISE BChE CURRICULA Light Chem. Semester cr. Mathematics 21 Chemistry 24 Other Science 12 Chem Eng Science 34 Other Eng Science 10 Chem Eng Design 9 Other Eng Design (Electives) 6 HumanisticSocial 28 Total 144 Strong Chem. Semester cr. 16 36 16 28 10 4 6 28 144 F A FOUR YEAR CURRICULUM IN CHEMICAL engineering is to continue to be the norm for first accreditation and if the student is not to be given an intolerable overload, then some compromises must be accepted. A reasonable compromise can be achieved among the relative amounts of basic science (in this case the amount of chemistry), the engineering sciences and the analysis, synthesis and design sequences. This com promise is coupled with the assumption that a course load greater than 18 credits per semester or 144 semester credit hours for four years is undesirable. The "light chemistry" curriculum provides for a year course each in general, organic and physical chemistry. This is agreed as the irreducible minimum for a chemical engineer. The "strong chemistry" program allows for about three semesters of additional chemistry, but in so doing some mathematics, chemical engineering science and chemical engineering analysis, design and systems must be sacrificed. The twentythree institutions in Table II with accredited undergraduate chemical engineering curricula in 1967 all come within these limits. Once the young Bachelor's degree recipient from either of these curricula becomes engaged in technical work in industry, he will feel his inadequacy in one direction or the other depending on his needs. He will be encouraged by his employer to fill the gaps by pro ceeding to the Master's degree. A typical program (See Table IV) illustrates how either man can reach the same general Master's degree plateau by selecting the appro priate courses. There is considerable question in this writer's mind whether the graduate from the "strong chemistry" curriculum (which contains the mini mum amount of chemistry recommended by AIChE in the 30's) has sufficient engineering content to justify a designated degree (or an accreditable degree) in chemical engineering. The "light chemistry" curriculum has a reason able engineering content but is shy in chemistry. The problem has now come full circle. When the AIChE Committee on Accreditation published its first accredited list in 1925, the concern was to TABLE IV THREE POSSIBLE ROUTES TO THE MChE DEGREE Mathematics Chemistry Other Science Ch.E. Science Other Eng. Science Ch. Eng. Analysis, Design and Systems Other Eng. Analysis, Design and Systems HumanisticSocial Mathematics Chemistry Chem Eng Science Chem Eng Analysis, Design and Systems (includes thesis) P4 22 21 16 28 24 36 16 12 16 22 34 28 18 10 10 4 9 6 28 144 w 44 41 0 8 6 16 30 6 28 144 3' 12 3 12 30 4 6 28 144 0 6 0 6 18 30 FALL 1969 No student should prepare from ChE unless he is prepared for a lifetime of study. distinguish the chemical engineer as an engineer distinct from the industrial chemist. Now the problem appears to be to provide the chemical engineer with enough chemistry to distinguish him from other engineers. AS EARLY AS THE 1900'S, NEWMAN12 AND his Committee recognized that education be yond the Bachelor's level was required if the chemical engineer were to have both sufficient chemistry and engineering. The intervention of the Physicists' War showed everyone the need for basic physics for all engineers. Thus, at the time of the Grinter Report in 1955, the amount of basic science in all engineering curricula was raised for the first time to the level required by the chemical engineers as early as 1933. The net result has been that chemical engineering, in order to increase the physics content, had to decrease the chemistry content. The Grinter Re port recommended that an engineering curricu lum include in the more general engineering science courses much of the material that in the 1930's the chemical engineer covered (less thor oughly to be sure) under the unit operations label. The result is a squeeze in the chemical engineering sequence in favor of engineering science courses in other departments, i.e. fluid dynamics in place of the unit operations fluid flow. Conversely, some very fine courses in high temperature chemistry are being conducted by departments of astronautics and aeronautical en gineering and some courses in radiation chemis try are being taught by departments of physics and nuclear engineering rather than depart ments of chemistry. Thus, many of the old labels are being confused. At this point, it does not seem desirable to debate the virtues and vices of these changes, but merely to report them as facts. The result is similar to the meeting of the immovable body and the irresistible force. Somebody has to give or the result is chaos. This writer favors a com promise solution in which all engineers will be given a Bachelor's degree in engineering un designated. Each student in the general engi neering curriculum (See Table IV) would be given rlIbid. pg. 23 ... it must be recognized that an adequate four year curriculum in ChE is a misnomer. ... This writer favors a compromise solution in which all engineers will be given a Bachelors degree in engineering undesignated. . the first designated degree would be at the Master's level. sufficient latitude in electives so that he can choose the basic science and the engineering science that will give him a sufficient flavor of his proposed major. At the same time, the concen tration in his major would be limited so as to permit his getting a broader engineering educa tion than would be the case if there were a des ignated degree at the Bachelor's level. With this type of broad engineering degree, the first desig nated degree would be at the Master's level. It should be a stronger degree with a broader back ground than would be the case with a Master's degree built on either the "light chemistry" BChE degree or the "strong chemistry" BChE degree. (See Table IV for comparison). Nevertheless, it seems perfectly clear that there are at least three routes toward the Master's level in chemical engineering, any one of which will produce a satisfactory product. It is also evident that more chemistry is needed by the chemical engineer than he is now getting in many of the "light chemistry" BChE curricula in institutions listed in Table III. It is further assumed, however, that the better students are wise enough to get that chemistry either by taking additional courses after they graduate or are being exposed to this material by taking courses otherwise labeled in other departments. The inevitable conclusion is that the explosion of knowledge since World War II has emphasized the im portance of giving to the present day chemical engineer at least as much chemistry as he had before World War II. In addition, his curriculum must include more from the other basic sciences plus more mathematics as well as new and expanded engineering sciences. If he is to be an engineer, he must have his share of courses in analysis, and design. All this material cannot fit in the old standardized package. There will be ample jobs for anyone who wishes to terminate his formal education at the traditional Bachelor's level. All three routes, the "light chemistry" BChE, the "strong chemistry" BChE and the general engineering with chemical electives BE, will find many opportunities for productive careers. But in 1967, as in 1938 the chemical engineer has just begun at the end of four years of formal study. No student should prepare for chemical engineering unless he is prepared for a lifetime of study with a maxi mum of chemistry. CHEMICAL ENGINEERING EDUCATION LETTERS (Cont'd from p. 208.) Wills surveys publication frequencies Sir: Publication of the results of their continuing re search is a major responsibility of those holding academic positions in the field of Chemical Engineering. Frequently, and at least once a year during salary review, questions arise concerning these scholarly publications. Presumably both individuals and departments as a whole are eval uated. While it is possible to determine an average per formance for an institution, for its component schools and departments, it is not ordinarily possible to compare individuals and departments with their peers (i.e., similar departments and disciplines at other institutions) even though this would be highly desirable. The deficiency in the use of peercomparison is due to the lack of suitably detailed statistics for each discipline. The purpose here is to furnish the data necessary for peercomparison in the field of Chemical Engineering. Detailed reporting of publications by departments and by individuals is available for ChE for the two year period July, 1965 to July, 1967. The source of this in formation is the 'Directory of Graduate Research," pub lished by the ACS. The information contained in this publication was solicited directly from all of the ChE departments in the United States offering graduate de grees. While detailed information concerning publication records is available in the ACS "Directory of Graduate Research", there is no statistical correlation of these data. Given here is a correlation of these data. The pub lication records by professional rank are given in Figures 50 MEDIAN NO. PUB./MANYEAR 40 0 30 20 UPPER 10% J a 10 0M R 0 0 0.51 L52 2.53 3.54 455 556 >6 PUBLICATIONS/ MANYEAR FIGURE (1) PUBLICATION RECORDS OF ALL ASSISTANT PROFESSORS OF CHEMICAL ENGINEERING 1, 2 and 3. Figure 4 gives overall departmental records. Table 1 gives additional information concerning the pub lication data. It should be pointed out that the estimates of publica tions should be considered as slightly inflated due to the reporting of items that ordinarily would not be considered publications. However, some editing has been done in this regard and the distributions and averages shown should be substantially correct. Also, the data correlated reflect the period 196567. The decreasing graduate enrollments of the past two years may well result in a reduction in the current rates of publication. George B. Wills Virginia Polytechnic Institute Table 1. Publication Rates in 78 ChE Departments Pub. per Professors Number manyr. Remarks Assistant 229 0.73 11.3% published 2 or more papers/yr. Associate 216 1.09 10.2% published 3 or more papers/yr. Full 308 1.95 10.1% published more than 4 papers/yr. Avg. 9.0% of all depts. pub. 2.25 (78 depts.) 1.27 or more papers/yr. 50 0 n 40 C MEDIAN NO. PUB./MANYEAR 0 a <20 SUPPER 10% o I '. I 1 0 1, 0 051 152 253 354 4.55 5.56 >6 PUBLICATIONS/MANYEAR FIGURE (2) PUBLICATION RECORDS OF ALL ASSOCIATE PROFESSORS OF CHEMICAL ENGINEERING 0 0.51 152 253 3.54 4.55 556 >6 PUBLICATIONS/ MANYEAR FIGURE (3) PUBLICATION RECORDS OF ALL FULL PROFESSORS OF CHEMICAL ENGINEERING 50 S40 S30 i MEDIAN NO. PUB./MANYEAR < 20 I UPPER 10% ae 0 0.5 51 11.5 152 2:25 >2.5 PUBLICATIONS/MANYEAR FIGURE (4) PUBLICATION RECORDS OF 78 CHEMICAL ENGINEERING DEPARTMENTS FOR 6/656/67 PERIOD FALL 1969 BERKELEY ACROSS 2.Berkeley overlooks beautiful San Francisco__ . 5.The Athens of the West. 9.City on Monterey Peninsula, 3 hrs. south of Berkeley. 11.California State University 12.Telegraph Avenue Is a veritable__ . 14.Recent Basketball star for southern branch of U.C. 15.People over 30 are likely to be _ 2O.Princlple item developed during thesis research. 23.Noted early Californian. 27.French deity. 28.Cal occasionally scores one in Memorial Stadium. 30.Goes halfway to the stars. 32.Initials of American Conser vatory Theater, noted San Franclso repertory group. 33.Graduate students do spend some time here. 34.In Berkeley,75 across makes the weather never . 36.Prominent resident of Carmel Valley,south of Berkeley. 37.0pen up that Golden . 38.Tall tree on Berkeley campus. 41.San Francisco airport. 43.The Queen of England. 45.The_ and fauna of the Sierra Nevada are famous. 47.At Berkeley, a graduate stu dent never feels like a . 49.Location of notorious steps oft seen on TV. S2.Swiss Sierra Nevada. 56.Berkeley Inltials of 1964. 58.Where the action s. 59.In oral exams.graduate students hope not to __ 61..ame for ancient Troy. 63.Waterfalls, lakes and granite domes,4 hrs. east of Berkeley 66.First word represented by initials of 56 Across. 68.The entering Berkeley grad student Is a11 . 69.Component of cyclotron. 70.Adversary of Bond,Dr. . 71.Prominent California politico 73._ Valleylowest spot in US DOWN BRAIN BENDER Sequoia Sempervr,. s the largest. 2 Common verb. Island in San Francisco Ba. 75:Cause of pleasant Berkeley 4.See 16 Down. summers. 51Thlng to do In Berkeley 17.Ernest O.Lawrence first built Chemistry library. one In Berkeley. 6.Just write EHH heree. O0.If you're not looking at the couldn't think of any bow, you are looking .._ thing: 83.Object of Berkeley chemits 7.Dutch air line. research. B. River, Source of gold 64.Opposite of u.v. In 1849. 85.Berkeley electrochemical eng 10.Honest  . Liners are concerned with this. 13.lnollsh suffix. 88.Principal product of valley 14.Same as 33 Across. Just north of Bay Area. 16.Source of ego. 89.Another West Coast state (abbr) 17.River In Arizona. 90.Where Berkeley students go to 18.Bush with purple flower. see Willie Mays. common In Sierra Nevada. 93.Et ._ Brutus? 19.Potato. 94.Hippie home. 21.Type of current. g7.Whltney is the highest one; 22.A lake, gem of the Sierra. It's In California, too. 24.What this is all about 98.Theme of many a rally. (initials). 99. __ alley, site of 1960 25.Industrial recruiters Winter Olympics easy drive pick up the from Berkeley. 26.Mass_.Heat__.Momentum. 101.Noted San Francisco hill. 28.Scholarly area stressed 103.At Trader Vic's In San Fran. at Berkeley. or Oakland you may have a 29.SIltcone queen of North __ Ta Beach. 105.We endeavor not to make our 31.Type of record. oral exams a__ 35.First letter of acronym 107.Shot In tennis, played year of recent Berkeley move round by faculty and students, meant. O1g. Free ___ 37. t was found at Sutter's 110.A graduate student is never Mill. __ for very long after he comes 38.After you come to Berk to Berkeley. eley, your future is 112.Noted wineproducing area near 39.Modern art form. Berkeley. 40.Gol of grad studies at 114.French plane of World War I. Berkeley (degree). 115.French king. 41.Type of food, music.etc. S116. ___ Cobb. 42.erkeleyc Is . 117. Galahad. 44.It Intersects Ashbury. COURTESY OF: DEPARTMENT OF CHEMICAL ENGINEERING UNIVERSITY OF CALIFORNIA, BERKELEY ...WHERE WE USUALLY PURSUE MORE SERIOUS GOALS. INQUIRE DIRECTLY FOR ADMISSIONS (PUZZLE ANSWERS, ALSO!) 46.Last two letters of acronym begun in 35 Doen. 48.Scottish no. 50.11_ only In wnter In Berkeley (French). 51 Initials of see water conversion process,.patents held bv U.C. 33.All but "he" would greet you warmly. 54.oted Berkeley faculty member. 5S.Stat University (abbr) 57.Mhen you go fishing In the Sierra, you bring your 60.What old professors become. 62.Contalner for beer and ale at LaVal's near campus. 64.Reciprocal of cosine. 65.Nature of Berkeley faculty. 66.Mr. Manchu. 67.Registered nurse. 68.University of California Berkeley. 72.Length of typical mid term quiz. 73.Another result of 75 Across. 74.Type of flower. 76.Poem. 77.Berkeley's computer isn't IBM or HAL. 78.Substance of concern to chemical engineers. 79.Computer sts sometimes count In 82.Attending Berkeley starts you on the to success. 85.Berkeley Is always free of snow and__. 86.Desired verdict from Oualifying E amination. 87.Element No. 93. discovered It Berkeley. 91. ype of water under research by Berkeley Chem. E's. 92. F lllar name for Headless Horseman. 9S.hever regard the faculty as your . 96.Most classes are held in the 100.The California sun keeps one from looking . 101O.Greek letter. 102.Baghdad the Bay (SF). 104.Conjunction 105.Component of distillation column. 106.Curse you._ Baroni 10S.Berkeley chemists found CH. and NH on __ lO9.Popular musical, now playing In San. Fran. Ill.Our graduate students are ll scholars. 113.Hawailan food. CHEMICAL ENGINEERING EDUCATION . . . UNIVERSITY OF CALIFORNIA, SANTA BARBARA The Department of Chemical and Nuclear Engineering offers a full program of graduate courses and research projects leading to the M.S. and Ph.D. degrees in chemical engineering. Eight fulltime faculty members direct research over a wide variety of chemical engineering and related nuclear engineering problems. Modern, well equipped research laboratories and computer facilities (IBM 360/75) back up all research programs. FACULTY . John E. Myers, Ph.D., Univ. of Michigan 1952. Professor of chemical engineering and chair man of Department. Research program: Two phase flow in porous media, mechanisms of boiling heat transfer. Henri J. Fenech, Sc.D., Massachusetts Institute of Technology 1959. Professor of nuclear engineering. Research program: Reactor engineering and reactor analysis, heat transfer. Owen T. Hanna, Ph.D., Purdue Univ. 1961. Asso ciate professor of chemical engineering. Research pro gram: Applications of mathematics in chemical engi neering. A. Edward Profio, Ph.D., Massachusetts institute of Technology 1963. Associate Professor of Nuclear Engineering. Research program: Reactor experimental physics, neutron shielding, nuclear interaction with matter. Robert G. Rinker, Ph.D., California Institute of Tech nology 1959. Associate professor of chemical engi neering. Research program: Kinetics and reactor de sign, energy conversion, air pollution control. Duncan A. Mellichamp, Ph.D., Purdue Univ. 1964. Assistant professor of chemical engineering. Research program: Dynamics of chemical processes, hybrid computer applications to adaptive and predictive con trol problems. Paul G. Mikolaj, Ph.D., California Institute of Tech nology 1965. Assistant professor of chemical engi neering. Research program: Thermodynamics and phase equilibria, structure of liquids and dense gases, oil pollution control. Orville C. Sandall, Ph.D., Univ. of California, Berkeley 1966. Assistant professor of chemical engi neering. Research program: NonNewtonian heat trans fer, interphase mass transfer, fluid mechanics of film flow. CAMPUS . Santa Barbara is located on the Pacific coast one hundred miles north of Los Angeles. The campus occupies a 630acre scenic promontory with the Santa Ynez mountains immediately behind. Twelve thousand students are enrolled in programs in diverse fields of engineering, science, humanities and the arts. Attractive housing of all kinds is available within walking distance of the campus. FINANCIAL ASSISTANCE AND ADMISSION PROCED URES . Teaching assistantships are available to qualified students; the stipend begins at $3,402 for the academic year with merit increases as progress is made towards a degree. A number of University Fellowships, Research Assistantships and various Train eeships are also available for qualified students. In formation concerning departmental procedures can be obtained by writing Professor J. E. Myers, Department of Chemical and Nuclear Engineering, University of California, Santa Barbara 93106. Application forms for admission and financial assistance should be re quested from the Dean of the Graduate Division, Uni versity of California, Santa Barbara 93106. FALL 1969 GRADUATED. SUD IN ~ CHMIA ENGINERIN PROGRAM OF STUDY Distinctive features of study in chemical engineering at the California Institute of Tech nology are the creative research atmosphere in which the student finds himself and the strong emphasis on basic chemical, physical and mathematical disciplines in his program of study. In this way a student can properly pre pare himself for a productive career of research, develop ment, or teaching in a rapidly changing and expanding technological society. A course of study is selected in consultation with one or more of the faculty listed below. Required courses are minimal. The Master of Science degree is normally com pleted in one academic year and a thesis is not required. The Ph.D. degree requires a minimum of three years subsequent to the B.S. degree, consisting of thesis re search and further advanced study. FINANCIAL ASSISTANCE Graduate students are sup ported by fellowship, research assistantship, or teaching assistantship appointments during both the academic year and the summer months. A student may carry a full load of graduate study and research in addition to any assigned assistantship duties. APPLICATIONS Further information and an application form may be obtained by writing Prof. C. J. Pings Executive Officer for Chemical Engineering California Institute of Technology Pasadena, California 91109 It is advisable to submit applications before February 15, 1970. FACULTY IN CHEMICAL ENGINEERING WILLIAM H. CORCORAN, Professor and Vice President for Institute Relations Ph.D. (1948), California Institute of Technology Kinetics and catalysis; gas chromatography; plasma chemistry. SHELDON K. FRIEDLANDER, Professor Ph.D. (1954), University of Illinois Aerosol physics; particlesurface interactions; interfacial transfer; diffusion and membrane transport.. GEORGE R. GAVALAS, Associate Professor Ph.D. (1964), University of Minnesota Mathematical methods applied to problems of chemical reactions and transport, process dy namics and control. CORNELIUS J. PINGS, Professor and Executive Officer Ph.D. (1955), California Institute of Technology Liquid state physics and chemistry; statistical mechanics. BRUCE H. SAGE, Research Associate Ph.D. (1934), California Institute of Technology Eng.D (1953), New Mexico State College. JOHN H. SEINFELD, Assistant Professor Ph.D. (1967), Princeton University Optimization and systems studies in chemical process control. FRED H. SHAIR, Associate Professor Ph.D. (1963), University of California, Berkeley Phenomena associated with magnetohydrody namic power generation; chemical reactions and diffusion in electrical discharges. NICHOLAS W. TSCHOEGL, Professor Ph.D. (1958), University of New South Wales Mechanical properties of polymeric materials and dilute polymer solutions. ROBERT W. VAUGHAN, Assistant Professor Ph.D. (1967), University of Illinois Solid state chemistry and physics, particularly effects of high pressure. Professor W. N. Gill, Chairman Chemical Engineering Department Clarkson College of Technology Potsdam, N. Y. 13676 Please send further information on your graduate program to Name Number and Street Undergraduate School State Zip Code DREXEL IS NOT AN AVERAGE SCHOOL For example, Drexel awards more engineering degrees than any other private university. PROGRAMS: M.S. and Ph.D. in Chemical Engineering. RESEARCH: Drying Dynamics Process Dynamics and Control Atomization Phenomena Fluid Mechanics of Films Environmental Problems Optimization of Drying Processes Biomedical Engineering Catalysis of Reverse Flow Reactors SUPPORT: Fellowships, Research Assistantships, and Teaching Assistantships are awarded to qualified students. The minimum stipend is $275/month plus remission of tuition and fees. LOCATION: In Philadelphia, the hub of the industrialized Delaware Valley. Contact with many of the chemical and petroleum companies of the area is convenient and frequent. For further details, please send this form to: NAME: Dr. John A. Tallmadge, Graduate Advisor; or ADDRESS: Dr. Donald R. Coughanowr, Chairman Department of Chemical Engineering SCH SCHOOL: I AM INTERESTED IN: M.S. __ Ph.D.. FullTime__ PartTime_ DREXEL INSTITUTE OF TECHNOLOGY j PHILADELPHIA, PENNSYLVANIA 19104 oQ9 CHEMICAL ENGINEERING EDUCATION THE UNIVERSITY OF FLORIDA * Remote IBM 360 Terminals * Computer Controlled Laboratory * Individual Student Attention * A Dynamically Developing Department * Modern Airconditioned $1,500,000 Building * Balanced Department Faculty of 17: diversified interests Wide course selection Four degree programs * Participation in NSF "Center of Excellence" Grant GRADUATE PROGRAMS IN SCIENCE AND SYSTEMS Since many of you are interested in industrial careers in development and design, while others intend to teach and do basic research our gradu ate program is divided into two main areas and several interdisciplinary activities. CHEMICAL ENGINEERING SCIENCE Transport phenomena Fluid dynamics Thermodynamics Kinetics Materials science Applied Math CHEMICAL ENGINEERING SYSTEMS Chemical reaction engineering Process dynamics Separations processes Process control Computer aided design Optimization INTERDISCIPLINARY Energy conversion Polymer science Biomedical Process economics Microelectronics Bioengineering DIVERSIFIED DEGREE PROGRAMS Master of Engineering with project on de sign, cost analysis, experimental investiga tion, or computer study. Master of Science with thesis. Master of Engineering PrePh.D. Doctor of Philosophy. New Chemical Engineering Building located at center of new Engineering Building Complex. BASIC GRADUATE COURSES Models and Methods Multidimensional and Discrete Systems Thermodynamics of Reac tion and Phase Equilibria Fundamental Transport Phenomena Process Dynamics 1 or Process Dynamics 2 Reactor Design and Op timization (Systems Program) or Chemical Kinetics (Science Program) TYPICAL ADDITIONAL COURSES Mathematical Methods in Chemical Engineering * Applied Field Theory Computer Control of Processes Optimization Techniques Trans port Properties and Irreversible Thermody namics Applied Statistical Mechanics Sta tistical Thermodynamics Interfacial Trans port Phenomena Turbulent Transport Phe nomena Advanced Transport Phenomena * Rheology NonNewtonian Fluid Dynamics * Chemical Energy Conversion Particulate Sys tems Applied Fluid Dynamics Process Engi neering Process Equipment Design Process and Plant Design Process Economy Analysis * Tensor Fields and Fluid Dynamics Biochem ical Engineering Chairman, Chemical Engineering Department University of Florida Gainesville, Florida 32601 Please send information on your graduate program to: FALL 1969 Iowa State University in Ames, Iowa, the first school to be established under the 1862 Land Grant Act, has a long tradition of lead ership in Engineering and Applied Science. Today it ranks seventh in the nation in Ph.D. degrees granted in Engineering and ninth in degrees in Chemical Engineering. Its College of Engineering is the largest west of the Mississippi River. To those interested in Chemical Engineer ing, Iowa State offers a variety of courses and research areas leading to the M.E., M.S. and Ph.D. degrees. The Department of Chemical Engineering is one of the oldest in the United States and enjoys a rich heritage of excellence in teaching and research. The staff numbers 22 and the enrollment consists of 300 under graduate and 70 graduate students. In addition to facilities available in a new Chemical Engineering building, research is conducted in the Ames Laboratory, a Nation al Laboratory of the US Atomic Energy Com mission, located on the Iowa State campus. A staff of nearly 1,000 at the Laboratory con ducts basic research of longrange interest to the nuclear industry. Ames lies amid the gently rolling hills of central Iowa. Typical of the picturesque yet modern campus is the new cultural center shown above, now half complete. This fall the Festival of Concerts at the center auditorium was opened by the New York Philharmonic. The 14,000seat coliseum will host many Big Eight Conference athletic events. A large variety of assistantships and fellow ships are filled each year by new graduate stu dents in Chemical Engineering. Living accom odations are available for single students in a new eightstory graduate dormitory, and for married students in more than 1300 apart ments operated by the University. IGeorge Burnet, Head IChemical Engineering Department SIowa State University Ames, Iowa 50010 Please send application forms and further information on your graduate program. IName Undergrac luate School Number and Street City State Zip Code__ UNIVERSITY OF KENTUCKY M.S. and Ph.D. Study in Chemical Engineering including A Unique Program in AIR POLLUTION CONTROL Kinetics and equilibria of atmospheric reactions Micrometeorology Diffusion in the atmosphere: modelling of urban areas Air sampling and analysis Process and system control; air cleaning Effects of pollutants on man, materials, and environs Excellent, U.S.P.H.S. Traineeships available At U.K.a nineman faculty, new laboratory and class room facilities, a complete graduate curriculum, a variety of research topics . . Contact: Robert B. Grieves Dep't of Chemical Engineering University of Kentucky Lexington, Kentucky 40506 FALL 1969 LOUISIANA STATE UNIVERSITY Department of Chemical Engineering Program of Study Research Facilities This department offers work leading to the Master of Science degree in chemical engineering and the Doctor of Philosophy degree in chemical engineering. The Master of Science degree may be earned under either thesis option or a course work option. Where practical, the thesis option is encouraged for students planning a terminal Master of Science. A Master of Science in sugar engineering is also available through the department. Each of twenty graduate courses is taught at least once each academic year. Undergraduate preparation should normally be the equivalent of that established as the minimum requirement for accreditation by the Engineers' Council for Professional Development. Special cases will be considered by the Head of the Department. The Master of Science degree requires a thesis plus 24 course work credit hours of which a minimum of 12 must be taken in chemical engineering. For nonthesis Master of Science students, 36 credit hours are required. For the Ph.D. degree a minimum of 60 credit hours beyond the baccalaureate are required. These must include 18 to 27 credit hours in chemical engineering, 12 to 15 in one or two minor subjects, 3 to 6 in cultural electives, and 12 to 15 in technical electives. A maximum of 9 credit hours is allowed for dissertation research. In addition each candidate must demonstrate a reading knowledge in at least one foreign language. Within the Chemical Engineering Department are a number of special purpose research facilities which include a reacting fluids laboratory, thermal fluids laboratory, a high polymers laboratory and a modern computing laboratory which includes analog, digital and hybrid computers. The department is also serviced by such University facilities as the Nuclear Science Center, the Computer Research Center, and one of the most modern libraries in the South with holdings of more than 1,300,000 volumes. Financial Aid A number of fellowships and assistantships are available for graduate students during the academic year. Fellowship and research assistantship support is provided by the NSF, NDEA, HEW, NASA, DOD, the University, and private industry. Typical academic year stipends for halftime graduate assistantships or fellowships range between $2250 and $2700 (taxfree) plus an additional tuition and fee exemption (not including an activity fee of $55). Graduate students have no difficulty in obtaining technical employment during the summer because of the local concentration of chemical and allied industry in the area. Alternatively, fulltime summer research support is also possible. Cost of Study For each regular semester (up to 12 credit hours) graduate student tuition includes a general fee of $50, a University fee of $55, an activity fee of $55, and a nonresident fee (if applicable) of $100. Cost of Living For single students, dormitory rooms vary from $81 to $225 for men and $119 to $227 for women per semester. Unfurrished apartments for married students rent for $65 to $90 monthly. Many reasonably priced offcampus apartments and residences also are available for students within the University environs. Student Body Undergraduate enrollment on the Baton Rouge campus averages 14,800, 40 per cent of whom are women. Graduate and professional enrollment is over 3,500. Students are drawn from every state in the Union and more than 60 foreign countries. About 730 international students are registered each academic year in both the undergraduate and graduate programs. Graduate enrollment within the Department of Chemical Engineering is approximately 90, including parttime and fulltime students. There are 40 fulltime graduate students, of whom 19 are doctoral candidates and 21 are master's or predoctoral candidates. Currently, 38 fulltime graduate students in the department are receiving financial aid. The Community LSU, situated within the city of Baton Rouge, has the unique advantages of a growing metropolis of over a quarter of a million people as well as those of the outlying countryside, a paradise for fishing, boating, and hunting enthusiasts. In addition the University supports a vast cultural and recreational program in music, drama, art, and literature as well as fine programs in athletics. Just eighty miles southeast of the main campus, at the entrance to the Gulf of Mexico, is New Orleans, internationally known for its southern hospitality, charm and recreational activities. The University Louisiana State University and Agricultural and Mechanical College is a multicampus, multipurpose system of higher education, exerting a major influence on the economic, social, and cultural life of all its citizens. Founded as a landgrant institution in 1860, LSU has grown to become one of the leading universities in the South. The main campus consisting of 15 colleges and specialized schools is located on a beautifully landscaped 300acre plateau just east of the Mississippi River. Although physically retaining the beauty of its southern heritage, the University is nevertheless a modern facility reflecting the scholarship and culture of the present. Applying Applications for admission, although considered throughout the year, should be made as early as possible. For those seeking financial assistance it should be made preferably by March 1 for a fall appointment and November 1 for the spring semester. The aptitude portion of the Graduate Record Examination is required for all applicants. Correspondence and Information Department of Chemical Engineering Louisiana State University University Station Baton Rouge, Louisiana 70803 A CAREER IN THE PAPER INDUSTRY? Manufacture of Pulp and Paper is one of the largest and fastest growing industries in the United States and the world. A research ren aissance in product and process diversification provides exceptional growth opportunities for men and women in all disciplines of engi neering and science. Such talents are in demand for research, indus trial engineering, business management, marketing, systems planning you name it, this industry offers it. Train for it at the University of Maine The Department of Chemical Engineering at the University of Maine, Orono, pioneered the first paper studies program in the United States, and continues to lead in teaching multidisciplinary application of engi neering sciences to the varied and complex operational decisions of this forest resources industry. The modern and rapidly expanding paper industry of this state provides an exceptional opportunity for cooperative interaction of University based programs with real life problems of industrial development. Students with a B.S. degree in most scientific or engineering disci plines can program a fifth year extension of their undergraduate cur riculum to fulfill requirements for a Certificate of Advanced Study in Pulp and Paper Management. One half of the fifth year covers basic fiber science and the technology of pulp and paper production. The other half can be an elective sequence to develop special interests in: COURSES Systems Engineering Process Control Environmental Engineering Plant Design Applied Computer Sciences Operations Economy Polymer Science Engineering Management ... and others Students who apply and qualify for admission to graduate school can / fit a substantial part of their fifth year Certificate Program to graduate school requirements for a Master of Science degree in Pulp and Paper Technology, in Systems Engineering, or in Chemical Engineering. GRANTS The University of Maine Pulp and Paper Foundation offers grants to qualified students who undertake the fifth year program. Such grants of full tuition plus $1100 cover all essential academic costs. Fellowships and Assistantships are available also for a limited number of students beyond the fifth year who aim for a Ph.D. in Chemical Engineering. FOR DETAILED INFORMATION For more detailed information about the pulp and paper programs at the University of Maine and the financial assistance available write: Dr. Edward G. Bobalek Chemical Engineering Department 255 Aubert Hall University of Maine Orono, Maine 04473 FALL 1969 THE UNIVERSITY OF MICHIGAN OFFERS EXPERIENCE What are YOU looking for in a GRADUATE PROGRAM? The University of Michigan, Department of Chemical and Metallurgical Engineering, has operated gradu ate degree programs for over 50 years. We have awarded over 300 doctorates and 1000 master's degrees. VARIED RESEARCH The 35 faculty members work in all the traditional areas of research and also such fields as plasma reactions, process dynamics, catalyst structure, bio chemical processes, electrochemistry, multiphase systems, computerassisted design, nonNewtonian fluids, and reservoir engineering. CULTURAL ENVIRONMENT Besides the usual campus activities the University and the Ann Arbor community offers the students scores of concerts by famous artists, lectures held throughout the year, plus the three drama series all handy to campus. Ann Arbor is located in a river valley and is ideal for both winter and summer sports. FINANCIAL ASSISTANCE Most of our American and Canadian students receive financial assistance. Also, the University has excellent employment opportunities for student wives. Write for information and a special book to: Robert H. Kadlec, Chairman of the Graduate Committee Department of Chemical and Metallurgical Engineering The University of Michigan Ann Arbor, Michigan 48104 CHEMICAL ENGINEERING EDUCATION 1811 *) H DEPARTMENT OF CHEMICAL ENGINEERING UNIVERSITY OF MARYLAND COLLEGE PARK, MARYLAND 20740 The Department offers graduate work in chemical, materials, and nuclear engineering leading to the M.S. and Ph.D. degrees. Some of the fields of specialization of the faculty are: Chemical Engineering Process Control Systems Heat and Mass Transfer Turbulent Transport Solvent Extraction Design and Cost Studies Reaction Kinetics Catalysis Multiphase Flow Process Dynamics Computer Simulation Biological and Environmental Engineering Aerosol Mechanics Membrane Separations Artificial Organs Bioengineering Environmental Health Air Pollution Control Nuclear Engineering Nuclear Reactor Physics Nuclear Reactor Design Nuclear Reactor Operation Radiation Induced Reactions System Dynamics Radiation Shielding Radiation Engineering Thermionics Engineering Materials Reaction of Solid Surfaces Solid State Behavior Composite Materials Statistical Thermodynamics Structure of Metallic Solutions Applied Polymer Science Polymer Physics Graft Polymerization Polymerization Kinetics NonNewtonian Flow The general requirements are set forth in the Graduate Catalog. The chemical engineering program is designed for qualified bachelors chemical engineering students. The materials and nuclear en gineering programs are open to qualified students holding bachelors degrees in engineering, the physical sciences, and mathematics. Address inquiries to Dean, Graduate School or Chairman Department of Chemical Engineering FALL 1969 Department of Chemical Engineering UNIVERSITY OF MISSOURI ROLLA ROLLA, MISSOURI 65401 Contact Dr. M. R. Strunk, Chairman Day Programs M.S. and Ph.D. Degrees Established fields of specialization in which re search programs are in progress are: (1) Fluid Turbulence and Drag Reduction Studies Drs. J. L. Zakin and G. K. Patterson (2) Electrochemistry and Fuel CellsDr. J. W. Johnson (3) Heat Transfer (Cryogenics) Dr. E. L. Park, Jr. (4) Mass Transfer StudiesDr. R. M. Wellek (5) Structure and Properties of PolymersDr. K. G. Mayhan In addition, research projects are being carried out in the following areas: (a) Optimization of Chemical SystemsDr. J. L. Gaddy (b) Evaporation through nonWettable Porous MembranesDr. M. E. Findley (c) Multicomponent Distillation EfficienciesDr. R. C. Waggoner (d) Gas Permeability StudiesDr. R. A. Prim rose (e) Separations by Electrodialysis Techniques Dr. H. H. Grice (f) Process Dynamics and ControlDrs. M. E. Findley, and R. C. Waggoner (g) Transport Properties and KineticsDr. O. K. Crosser Tuition: Out of state tuition waived for Graduate Stu dents. Fees are approximately $200 per semester for 10 credit hours or more. Financial aid is obtainable in the form of Graduate and Research Assistantships, Industrial Fellowships and Fed eral Sponsored Programs. Aid is also obtainable through the Materials Research Center. CHEMICAL ENGINEERING EDUCATION GRADUATE STUDY IN CHEMICAL ENGINEERING AT THE UNIVERSITY OF NEBRASKA I PROGRAMS LEADING TO THE M.S. AND PH.D. DEGREES WITH RESEARCH IN Biochemical Engineering Computer Applications Crystallization Desalination Food Processing Heat Transfer Kinetics Laser Applications Mass Transfer Mixing Polymerization Thermodynamics Ultrasonics and other areas FOR APPLICATIONS AND INFORMATION ON AVAILABLE FINANCIAL ASSISTANCE WRITE TO Prof. J. H. Weber, Chairman Department of Chemical Engineering University of Nebraska Lincoln, Nebraska 68508 THE CITY COLLEGE OF THE CITY UNIVERSITY OF NEW YORK ANNOUNCES A SPECIAL PROGRAM IN ENGINEERING APPLICATIONS OF PROBABILITY THEORY Engineers working in the areas of control, communications and reliability have long been familiar with stochastic processes and have contributed to the development of the theory. The powerful mathematical tools developed in these contexts, such as the Theory of Markov processes, Queueing and Renewal Theory, etc., have till recently found only limited application in the chemical engineering profession and many research engineers are still unfamiliar with its use. In the last few years these methods have gained wider acceptance and their usefulness has been demonstrated by different workers in a number of fields. Our department is conducting one of the largest concentrated research efforts in this field. The research group is headed by Professors Stanley Katz and Reuel Shinnar and is active in a wide variety of problem areas endeavoring to demonstrate the usefulness and power of probability methods when applied to contemporary chemical engineering problems. Emphasis is given both to extensions of the theory and the development of new methods, as well as to new applications of the methods developed in other fields. Areas of investigation include: 1) Mixing and turbulence in chemical reactors. 2) Control of process plants. 3) Theory of tracer experiments. 4) Applications of tracer experiments to physiological prob lems. 5) Behavior of particulate systems, with emphasis on crys tallization, polymerization and more recently, meteo rological problems. The program is specially suited for students with a mathe matical inclination, providing them an opportunity to enter a new and growing field in chemical engineering. Research assis tantships for students intending to study for the doctoral degree as well as a few postdoctoral fellowships are available. Inquiries should be directed to Professor A. X. Schmidt Department of Chemical Engineering The City College of the City University of New York New York, N. Y. 10031 CHEMICAL ENGINEERING EDUCATION W7o * Oklahoma, a vigorous state with space to dream and grow, with a unique heritage and a promising future... * a friendly city with a tradition of fine arts excellence and exceptional recreation opportunities. . * a University where human beings share values and con cerns, where innovation and relevance are more than words .. * a balanced department oriented to its missions . . quality teaching for graduate and undergraduate stu dents . .providing basic knowledge through research S.. translating that knowledge to practical use through public service . . * bright, young faculty members with energy and dedi cation, highlymotivated in teaching, research and public service, beginning to achieve deserved recogni tion. .. * a curriculum that has produced outstanding doctor of philosophy recipients, 68 in the past 7 years, 17 of whom are in professorial positions... * academic opportunity, intellectual challenge, and an atmosphere of creativity. The School of Chemical Engineering and Materials Science The University of Oklahoma Norman, Oklahoma 0 IL 4l GRADUATE STUDY IN CHEMICAL AND PETROLEUM ENGINEERING University of Pittsburgh :. M.S. and Ph.D. Degrees ~.iV Pr. 3 < Dr. Dr. Dr. Dr. Dr. . Dr. Dr.  '* i /  ** .,.  I, ,, ~ (l*u /* ^ ^  h'^^' /':.^. 4 .,  'II FACULTY AND FIELDS OF RESEARCH :, IN CHEMICAL ENGINEERING  Charles S. Befoes.  pas Dynamics, Process Desigrtn& Oplimiza ,Unsteady State Heal Transmissi Alan'J. Brainard...... ..... .Thermodynamics. Mass Trans George D. Byrne. Applied Manre rati ShiaoHung Chiang .... .Mass Transfer, Inleriacial pinS .Morton Corn .. .. ......... A f James Coull.. .......... Chemical Kinetics. Catalysis =bLe Thermogravilalional Separali Benjamin GalOr ... ......... . Transport Phenomer Relativistic Therrnodynami Harold E. Hoelscher .......... . ... Reaction Kinetic Interfacial Phenome George E. Klinzing .. .......Fluid Dynamics, Transport Phenome ChungChiun Liu . : ............ Electrochemical Engineeri Yatish T. Shah.. ........ .........Transport Phenome Edward B. Stuart ... .......... .. .Thermodynamics, Adsorpti John W. Tierney . .. Process Dynami Equilibrium Stage Calculatio Lemuel B. Wingard : .. . Biomedical Engineerir Enzyme Catalys IN PETROLEUM ENGINEERING Dr. Paul F. Fulton. . ... ......... .... Multiphase Flow in Poro Media, Wettabil Prof. James H. Hartsock. ... . ... Computer Applicatio to Unsteady State Fl Dr. Joseph J. Taber ................. Interfacial and Surface Phenomer Miscible Displaceme PROGRAM Chemical and Petroleum Engi neering is one of six School of Engineeri4n departments which offer graduate degrees. Interdisciplinary programs 4.. with other engineering depart S mnts and with other PITT . schools and divisions such as i. Public Health, Natural Sci enc and Medicine are en couraged. SCourses begin in Septem ber, January and April; gradu ate students may enter in any term. FINANCIAL ASSISTANCE Graduate assistantships, re S search assistantships, fellow fer ships and tuition scholarships S~ available to qualified stu inancal support is pro r," vded by Ihe University, indus oit try, and various government ia, agencies. Among sponsors of cs current research programs are cs, Petroleum Research Fund, Na na tional Science Foundation, na U.S. Department of Agricul ng ture, National Aeronautics and na Space Administration, and on United States Steel Corpora cs, tion. ns For application forms and 9g, detailed information on FEL sis LOWSHIPS, ASSISTANT SHIPS, and ACADEMIC AND RESEARCH PROGRAMS, write us to: ity Graduate Coordinator ns Chemical and Petroleum w Engineering Department 601 Engineering Hall la, University of Pittsburgh nt Pittsburgh, Pennsylvania 15213 *t^ia. .^ SSYRACUSE UNIVERSITY SJUOS SYRACUSE, NEW YORK 13210 ULTORES iCIENTIA , :ORONAT Syracuse University is a private university situated among S the hills of Central New York State. A broad cultural climate t which stimulates interest in engineering, science, the social ED sciences, and the humanities exists at the university. DEPARTMENT OF CHEMICAL ENGINEERING AND METALLURGY Programs leading to Master's and Ph.D. Degrees in Chemical Engi neering, Master's Degree in Metallurgy, and Master's and Ph.D. Degrees in Solid State Science. GRADUATE CURRICULUM EMPHASES: Computer Science Mathematical Modeling Solid Mechanics XRay and Electron Diffraction Separation Processes Solid State Physics INDEPENDENT STUDY AND RESEARCH PROBLEMS: Water Renovation Optimization of Multistage Processes Rheology and Viscoelastic Fluid Phenomena Membrane Processes Mechanical Behavior of Solids Surface Science Metal Physics Biomedical Applications Thermodynamics and Kinetics Electron Microscopy FINANCIAL ASSISTANCE: Graduate Fellowships and Assistantships. Stipends range from $2,000 to $5,000 with most students receiving $4,000 per annum in addition to remitted tuition privileges. For Information Contact: Dr. James A. Luker, Chairman Department of Chemical Engineering and Metallurgy Syracuse University Syracuse, New York 13210 Telephone: Area 3154765541, extension 2559 FALL 1969 AT THE UNIVERSITY OF TENNESSEE GRADUATE STUDY IN CHEMICAL & METALLURGICAL ENGINEERING PROGRAMS for the degrees of Master of Science and Doctor of Philosophy are offered in both chemical z and metallurgical engineering. The Master's program may be tailored as a terminal one with emphasis on .. systems and design, or it may serve as preparation for more advanced work leading to the Doctorate. Interests of the staff include thermodynamics, physical metallurgy, diffusional operations, heat transfer, fluid mechanics, polymer science, reaction kinetics, informationoperations, and systems analysis and A design as applied to both chemical and metallurgical engineering. FACULTY AND RESEARCH INTERESTSWilliam T. Becker, Ph.D., Illinois, Mechanical Properties and Deformation; Donald C. Bogue, Ph.D., Delaware, Rheology; Charlie R. Brooks, Ph.D., Tennessee, Electron Microscopy, Thermodynamics; Oran L. Culberson, Ph.D., Texas, Operations Research, Process Design; George C. Frazier, Jr., D. Eng., Johns Hopkins, Kinetics and Combustion, Transfer with Reaction; HsienWen Hsu, Ph.D., Wisconsin, Thermodynamics, Transport Phenom ena, Optimization; Homer F. Johnson, D. Eng., Yale, (Department Head), Mass Transfer, Interface Phenomena; Stanley H. Jury, Ph.D., Cincinnati, Sorption Kinetics, Hygrometry, Information Operations; William J. Kooyman, Ph.D., Johns Hopkins, Reaction Kinetics in Flow Systems; Carl D. Lundin, Ph.D., Rensselaer, Physical Metallurgy, Welding; Charles F. Moore, Ph.D., L.S. U., Process Control and Dynamics; Ben F. Oliver, Ph.D., Pennsylvania State University, Solidification, High Purity Metals; Joseph J. Perona, Ph.D., Northwestern, Mass Transfer and Kinetics, Heat Transfer; Joseph E. Spruiell, Ph.D., Tennessee, Xray Diffraction, Electron Microscopy; E. Eugene Stansbury, Ph.D., Cincinnati, Thermodynamics Kinetics of Phase Transformation, Corrosion; James L. White, Ph.D., Delaware, Rheology, Polymer Chemistry. REGULAR PART TIMELloyd G. Alexander, Ph.D., Purdue, Fluid Flow, Heat Transfer; Bernard S. Borie, Ph.D., M.I.T., Xray Diffraction; Albert H. Cooper, Ph.D., Michigan State, Process Design, Economics; Kenneth H. McCorkle; Ph.D., Tennessee, Colloidal Systems; Carl J. McHargue, Ph.D., Kentucky, Physical Metallurgy; Jack S. Watson, Ph.D. Tennessee, Fluid Mechanics; Monroe S. Wechsler, Ph.D., Columbia, Physical Metallurgy, Effect of Radiation on Metals. LABORATORIES AND SHOPSAnalog computer (Expanded EAI,'PACE 221R) and digital computer (DEC, PDP 15/20 with analog interface), Highspeed automatic frost point hygrometer, Mass and heat transfer in porous media, Polymer rheology (Weisenburg rheogoniometer, Instron theological tester, roll mill, extruder), Polymer characterization (gel per meation chromatograph, osmometer), Mass spectograph, Continuous zone centrifuge, Process dynamics, Xray diffraction (including single crystal diffuse scattering analysis), Electron microscopes (Phillips EM75 and EM300), Calorimetry (25 10000C), Electrical resistivity measurements for studies of structural and phase changes, Single crystal preparation facilities, Mechanical fabrication and testing, (metallograph, optical microscopes and melting, etc.), High purity materials preparation, Electronic and mechanical shops staffed by thirteen fulltime technicians and craftsmen. FINANCIAL ASSISTANCE from a number of sources is available, including graduate assistantships, graduate teaching assistantships, research assistantships, industrial fellowships, industrial grantsinaid, NSF Traineeships, NASA Traineeships, NDEA (Title IV) Fellowships, and University NonService Fellowships. COST OF STUDYFulltime students who are Tennessee residents pay $105 per quarter maintenance fee; outofstate students pay an additional tuition of $205 per quarter. Holders of fellowships, graduate assistantships, and certain teaching appointments pay no fees or tuition. COST OF LIVINGDormitory rooms costs for single students range from $75 to $100 per quarter; combined roomand board arrangements are available at $305 per quarter. Attractive one and twobedroom apartments for married students rent from $60 to $110 per month unfurnished, approximately $15 higher furnished. Privately operated apartments are available to single or married graduate students at equivalent and higher rates. Food and other living expenses are below national averages. STUDENT BODYAbout 16,000 undergraduate and 4,000 graduate students are enrolled at the Knoxville campus of The University of Tennessee. In the College of Engineering there are approximately 2200 undergraduate and 300 resident graduate students. KNOXVILLE AND SURROUNDINGS Knoxville, with a population near 200,000, is the trade and industrial center of East Tennessee; convenient transportation is available to all parts of the country. The University is located about five blocks from the downtown business area. In the nearby AuditoriumColiseum, Broadway plays, musical and dramatic artists, and other entertainment events are regularly scheduled. Knoxville has a number of points of historical interest, a theaterintheround, an excellent symphony orchestra, two art galleries, and a number of museums. Within an hour's drive are many TVA lakes and mountain streams for fishing, boating, and water sports; the Great Smoky Mountains National Park with the Gatlinburg tourist area; two state parks; and the atomic energy installations at Oak Ridge, including the Museum of Atomic Energy. Avastnumberof cultural, recreational, and social activities are available on the University campus. A WORD ABOUT U.T.Founded in 1794 as Blount College, The University of Tennessee has grown to a large multi campus, multipurpose system of higher education covering the entire state. Graduate programs in science and engineering centered at the Knoxville campus have developed to major size and strength over the past 25 years. A major stimulus to the growth of these programs has been the proximity of the atomic energy facilities at Oak Ridge and the close cooperation that has developed between these facilities and the University. CHEMICAL ENGINEERING EDUCATION DEPARTMENT OF CHEMICAL ENGINEERING BUCKNELL UNIVERSITY LEWISBURG, PENNSYLVANIA 17837 For admission, address Dr. David S. Ray, Coordinator of Graduate Studies Graduate degrees granted: Master of Science in Chemical Engineering Courses for graduate credit are available in the evenings. Typical research interests of the faculty include the areas of: mass transfer, particularly distillation and liquidliquid extraction; thermodynamics; mathematical applications in chemical systems; reaction kinetics; process dynamics and control; metallurgy and the science of materials; nuclear engineering. Assistantships and scholarships are available. For the usual candidate, with a B.S. in Chemical Engineering, the equivalent of thirty semesterhours of graduate credit including a thesis is the requirement for graduation. COMPLIMENTS OF THE DEPARTMENT OF CHEMICAL ENGINEERING CarnegieNMellon University PITTSBURGH, PENNSYLVANIA Howard Brenner Duane Condiff Edward Cussler Kun Li Clarence Miller Carl Monrad Matthew Reilly Stephen Rosen Robert Rothfus Herbert Toor Raymond Zahradnik FALL 1969 25c CHEMICAL ENGINEERING EDUCATION @.*. CLEMSON UNIVERSITY SChemical Engineering Department *ooo....... M.S. and Doctoral Programs THE FACULTY AND THEIR INTERESTS Alley, F. C., Ph.D., U. North CarolinaAir Pollution, Unit Operations Barlage, W. B., Ph.D., N. C. StateTransfer Processes in NonNewtonian Fluids Beckwith, W. F., Ph.D., Iowa StateTransport Phenomena Bruley, D. F., Ph.D., U. TennesseeProcess Dynamics, Biomedical Engineering Hall, J. W., Ph.D., U. TexasChemical Kinetics, Catalysis, Design Harshman, R. C., Ph.D., Ohio StateChemical and Biological Kinetics, Design Littlejohn, C. E., Ph.D., V.P.I.Mass Transfer Melsheimer, S S., Ph.D. TulaneProcess Dynamics, Applied Mathematics Mullins, J. C., Ph.D., Georgia TechThermodynamics, Adsorption FINANCIAL ASSISTANCEFellowships, Assistantships, Traineeships Contact: C. E. Littlejohn, Head Department of Chemical Engineering Clemson University Clemson, S. C. 29631 CASE WESTERN RESERVE UNIVERSITY Case Institute of Technology, a privately en dowed institution with a tradition of excellence in Engineering and Applied Science has long offered a variety of courses and research areas leading to the M.S. and Ph.D. degrees in Chemi cal Engineering. In 1967 Case Institute and Western Reserve University joined together. The enrollment and endowment make Case Western Reserve University one of the largest private schools in the country. FOR FURTHER INFORMATION YOU ARE INVITED TO WRITE: ROBERT J. ADLER, Head Chemical Engineering Science Division Case Western Reserve University University Circle Cleveland, Ohio 44106 UNIVERSITY OF COLORADO CHEMICAL ENGINEERING GRADUATE STUDY The Department of Chemical Engineering at the University of Colorado offers excellent oppor tunities for graduate study and research leading to the Master of Science and Doctor of Philosophy degrees in Chemical Engineering. Research interests of the faculty include cryo genics, process control, polymer science, catalysis, fluid mechanics, heat transfer, mass transfer, air and water pollution, biomedical engineering, and ecological engineering. For application and information, write to: Chairman, Graduate Committee Chemical Engineering Department University of Colorado, Boulder GRADUATE STUDY IN CHEMICAL ENGINEERING UNIVERSITY OF HOUSTON IMPORTANT FEATURES * Research in most areas of current chemical engineering activity Recipient of a $420,000 NSF "Center of Excellence" departmental development grant New 4.5 million dollar engineering building Graduate studies through Ph.D. with a faculty of 16 and over 90 graduate students Located in the petroleum and petrochemical capitol of the world Yearround recreational activities Qualified applicants are encouraged to apply. Fellowships and assistantships are available in amounts of $3600$5000 in annual stipends. For details and applications contact: Chairman, Graduate Admissions Committee Department of Chemical Engineering University of Houston Houston, Texas 77004. FALL 1969 LEHIGH UNIVERSITY ESTABLISHED & ACTIVE FLUID MECHANICS KINETICS & CATALYSIS ADSORPTION RHEOLOGY ENVIRONMENTAL PROCESS CONTROL SIMULATION HEAT TRANSFER THERMODYNAMICS POLYMERS CHEMICAL METALLURGY WRITE: Chairman Department of Chemical Engineering Lehigh University, Bethlehem, Penna. 18015 UNIVERSITY, HAMILTON, McMASTER CANADA. INTERDISCIPLINARY BALANCE 4 INNOVATION DEPTH Simulation, Optimization and ComputerAided Analysis Water & Waste Water Treatment Chemical Reaction Engineering Transport Phenomena Contact: Dr. T. W. Hoffman, Chairman Dept. of Chemical Engineering GRADUATE OPPORTUNITIES IN ChE AT NEWARK COLLEGE OF ENGINEERING Students seeking a commitment to excellence in careers in Chemical Engineering will find a wealth of opportunity at Newark College of En gineering. The ChE Department at NCE has a well de veloped graduate program leading to the degrees of Master of Science in Chemical Engineering or Master of Science with major in such interdisci plinary areas as Polymer Engineering or Polymer Science. Beyond the Master's degree, NCE offers the degrees of Engineer and of Doctor of Engi neering Science. Over sixty ongoing projects in Chemical En gineering and Chemistry provide exceptional re search opportunities for Master's and Doctoral candidates. Research topics include the follow ing areas: * Fluid Mechanics Heat Transfer * Thermodynamics Process Dynamics * Kinetics and Catalysis Transport Phenomena 0 Mathematical Methods NCE is located on a modern, twentyacre cam pus in Newark, within 30 minutes of Manhattan. Tuition for New Jersey residents is $24 per credit; for nonresidents, the cost is $35 per credit. Fellowships and financial assistance are available to qualified applicants. FOR FURTHER INFORMATION ADDRESS: Mr. Alex Bedrosian, Assistant Dean Graduate Division Newark College of Engineering 323 High Street, Newark, N. J. 07102 CHEMICAL ENGINEERING EDUCATION MICHIGAN STATE UNIVERSITY The Department of Chemical Engineering of Michigan State University has assistantships and fellowships available for the academic year 197071. CURRENT RESEARCH AREAS Transport Phenomena Radiation Engineering Novel Separations Biomedical Engineering Hybrid Computation Chemical Process Systems Theory Process Dynamics and Control Flow Through Porous FOUNDED Media 155 Kinetics and Reaction Engineering VER Diffusion in Liquids Applied Chemical Engi neering Mathematics 256 NORTHEASTERN UNIVERSITY Graduate Program CHEMICAL ENGINEERING In The Areas Of Heat Transfer and Fluid Mechanics Kinetics Mathematical Applications in Chemical Engineering NonNewtonian Phenomena Optimization of Chemical Processes Process Dynamics Thermodynamics LEADING TO THE DEGREES OF M.S. AND Ph.D. For application forms and further information address: Department of Chemical Engineering NORTHEASTERN UNIVERSITY 360 Huntington Avenue, Boston, Massachusetts 02115 graduate study in ,CHEIALI ENGINEERING AT OKLAHOMA STATE UNIVERSITY offering... Master of Science in Chemical Engineering Master of Science in Nuclear Engineering Doctor of Philosophy in Chemical Engineering programs designed to S. develop and expand your scientific and engineering background . equip you to play an effective role in R&D and in production and design S. prepare you to undertake major responsibilities for scientific and technical aspects of chemical engineering plus: A diversified faculty with wideranging research interests . . excellent laboratory facilities especially equipped for graduate research . modern computing facilities, including a "hands on" facility exclusively for engineering students and available 24 hours daily . financial support available Your inquiries are invited. Address: Dr. Robert N. Maddox, P.E. Professor and Head School of Chemical Engineering Oklahoma State University Stillwater, Oklahoma 74074 FALL 1969 CHEMICAL ENGINEERING EDUCATION THE UNIVERSITY OF SOUTH CAROLINA AT COLUMBIA Offers the M.S., the M.E. and the Ph.D. in Engineering. Em phasis on transport processes and thermodynamics. Strong in terdisciplinary support in chemistry, physics, mathematics, ma terials and computer science. Small classes taught by professors with industrial experience. Research and teaching assistantships, fellowships, and traineeships are available. For particulars and application forms write to: Dr. M. W. Davis, Jr., Chairman Graduate Studies Committee College of Engineering University of South Carolina Columbia, S. C. 29208 New Mexico State University M. S. Program in Chemical Reaction Engineering P. O. Box 3805 Las Cruces, N. M. 88001 THE UNIVERSITY OF NEW MEXICO ALBUQUERQUE, NEW MEXICO GRADUATE STUDY TOWARD THE M.S. AND Ph.D. DEGREES IN CHEMICAL ENGINEERING Graduate Assistantships, Teaching Assistantships and Fellowships Available For Further information and applications for graduate study in the Land of Enchantment, contact: Dr. T. T. Castonguay, Chairman Department of Chemical Engineering University of New Mexico Albuquerque, New Mexico 87106 
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