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Jennifer Sinclair Curtis of the University of Florida, Nicholas Peppas, Gintaras Reklaitis, and Erik Ydstie ( PDF )
Chemical Engineering at Tufts University, Nakho Sung, and Daniel Ryder ( PDF ) Chemical Kinetics, Heat Transfer, and Sensor Dynamics Revisited in a Simple Experiment, Maria E. Sad, Mario R. Sad, Alberto A. Castro, and Teresita F. Garetto ( PDF ) Active Problem Solving and Applied Research Methods in a Graduate Course on Numerical Methods, Eric L. Maase and Karen A. High ( PDF ) Student Ratings of Teaching: Myths, Facts, and Good Practices, Richard M. Felder, and Rebecca Brent ( PDF ) Introducing Stochastic Simulation of Chemical Reactions Using the Gillespie Algorithm and MATLAB: Revisited and Augmented, A. Argotti, L.T. Fan, J.Cruz, and S.T. Chou ( PDF ) Applications of the PengRobinson Equation of State Using Mathematica, Housam Binous ( PDF ) Celebrating ChE In Song, Alan M. Lane ( PDF ) Can I Trust This Software Package? An Exercise in Validation of Computational Results, Mordechai Shacham, Neima Brauner, W. Robert Ashurst, and Michael B. Cutlip ( PDF ) ( PDF ) 
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chemical engineering education Visit us on the Web at Jennifer Sinclair Curtis ... of the University of Florida Introducing Stochastic Simulation of Chemical Reactions Using the Gillespie Algorithm and MATLAB: Revisited and Augmented (p. 35) Argoti, Fan, Cruz, Chou Applications of the PengRobinson Equation of State Using Mathematica (p. 47) Binous Random Thoughts: Student Ratings of Teaching: Myths, Facts, and Good Practices (p. 33) Felder, Brent Editorial: Engineers Deserve A Liberal Education (inside front cover) Kiilg Celebrating ChE In Song (p. 52) Lane Active Problem Solving and Applied Research Methods in a Graduate Course on Numerical Methods (p. 231 Maase, High Chemical Kinetics. HeatTransfer, and Sensor Dynamics Revisited in a Simple Experiment (p. 17 1 Sad, Sad, Castro, Garetto Can I TrustThis Software Package? An Exercise in Validation of Computational Results (p. 53) Shacham. Brauner, Ashurst. Cutlip V Tufts University AIL 40 w editorial Engineers Deserve a A s much as or more than for any other career, en gineers need the benefits of a liberal bachelor's education. Other professions have recognized this need for many years. It is time for engineering to change to the master's as the professional degree. Bachelor's education should then be modeled after pre medical educationa set of generally required courses within thecontextofa liberal arts bachelor's degree, with a wide choice of majors. I have presented this proposal and the rationale for it in more detail elsewhere. Some of the drivers for this change are Engineers need the flexibility to move among careersto management, public service, or out of the engineering profession altogether. SEngineers must interact effectively with people of different backgrounds and with the public on issues such as environment, energy, food sup ply, water supply, and national security. SWith instant broadband communication and access to information, these days "the world is flat." Straightforward engineering jobs are going overseas, leaving U.S. engineers with more complex, interactive jobs and the need to understand and work closely with other cultures. There is simply more to be had from life if one has a broader background and interests. All these factors call for a broader engineer, with more understanding of society and the human nature and condition. Engineering is the only major profession with the bachelor's degree as the first professional degree and accreditation primarily at that level. Other professions, such as medicine, law, business, architecture, planning, journalism, and public health, build upon a liberal arts bachelor's degree. To its credit, ABET has recently sought more breadth inundergraduateengineeringeducation.Buttheproblem is that a nominally fouryear bachelors degree is just too small a boxin which to package both a professional engi neering education and an adequate liberal education. Some will say that fewer students would enter engi neering becauseofthe prospectsoflongertimeand more Liberal Education C. Judson King is Provost and Senior Vice President, Emeritus of the University of California. He is also Professor Emeritus of Chemi cal Engineering on the Berkeley campus, where he was formerly \ Provost Professional Schools and Colleges and Dean of the College of Chemistry. He presently directs the Center for Studies in Higher Educa tion at Berkeley. cost. But the change will also enable students to make a much more informed choice as the end of the bachelor's degreeapproaches. Breadth and theopportunityfora de layedcommitmentshould bring morestudentstoconsider engineering seriously, especially women and minorities. The community college transfer route will be much more attractive and workable. The added time and cost are excellent investments for a future career, and the time is offset bythefactthat mostengineering students nowtake well overfouryears to completetheir degree.The change to model premedicaleducation mayactuallyincreasethe number and diversity of engineering students. Some preengineering bachelor's graduates will not proceed onward to the professional master's.They will still have good options, including other professional de grees and jobs linking engineering with otherfunctions. They will contribute toward the goal of a technologically literate population. Some will observe that there is a strong industrial de mand for bachelor's engineers. But here the interests of corporations and individuals diverge. Corporations will gladly hire engineers for entrylevel functions when the openingsarethere. Individual engineers need to enhance theirfiexibility and opportunities for moving to different functions astheircareerdevelops and/orastheeconomic cycle turns downward. Still others will observe that it will be difficult to make this change. But medicine and law did it in the early parts ofthelastcentury.Surelyengineeringthe problemsolv ing professioncan find the way to make the change. My class at Yale had its 50th reunion this year. In the reflective 50yearclass book, oneofour numberobserved and regretted that he had"missed hisYaleeducation"be causeof having majored in engineering.With thechange that I am urging, he would have had that education. p Author Guidelines for the LABORATORY Feature The laboratory experience in chemical engineering education has long been an integral part of our curricula. CEE encourages the submission of manuscripts describing innovations in the laboratory ranging from largescale unitoperations experimentsto demonstrationsappropriate for the classroom. The following guidelines are offered to assist authors in the preparation of manuscripts that are informative to our readership. These are only suggestions, based on the comments of previous reviewers; authors should use their own judgment in presenting their experiences. A set of general guidelines and advice to the author can be found at ourWeb site: c Manuscripts should describe the results of original and laboratorytested ideas. The ideas should be broadly applicable and described in sufficient detail to allow and motivate others to adapt the ideas to their own curricula. It is noted that the readership of CEE is largely faculty and instructors. Manuscripts must contain an abstract and often include an Introduction, Laboratory Description, Data Analysis, Summary of Experiences, Conclusions, and References. An Introduction should establish the context of the laboratory experi ence (e.g., relation to curriculum, review of literature), state the learning objectives, and describe the rationale and approach. The Laboratory Description section should describe the experiment in sufficient detail to allow the reader to judge the scope of effort required to implement a similar experiment on his or her campus. Schematic dia grams or photos, cost information, and references to previous publica tions and Web sites, etc., are usually of benefit. Issues related to safety should be addressed as well as any special operating procedures. SIf appropriate, a Data Analysis section should be included that concisely describes the method of data analysis. Recognizing that the audience is primarily faculty, the description of the underlying theory should be referenced or brief. The purpose of this section is to communicate to the reader specific studentlearning opportunities (e.g., treatment of reac tionrate data in a temperature range that includes two mechanisms). *The purpose of the Summary of Experiences section is to convey the results of laboratory or classroom testing. The section can enumerate, for example, best practices, pitfalls, student survey results, or anecdotal material. A concise statement of the Conclusions (as opposed to a summary) of your experiences should be the last section of the paper prior to listing References. EDITORIAL AND BUSINESS ADDRESS: Chemical Engineering Education Department of Chemical Engineering University of Florida Gainesville, FL 32611 PHONE and FAX: 3523920861 email: cee@che.ufl.edu EDITOR Tim Anderson ASSOCIATE EDITOR Phillip C. Wankat MANAGING EDITOR Lynn Heasley PROBLEM EDITOR James O. Wilkes, U. Michigan LEARNING IN INDUSTRY EDITOR William J. Koros, Georgia Institute of Technology PUBLICATIONS BOARD CHAIRMAN John P. O'Connell University of Virginia VICE CHAIRMAN * C. Stewart Slater Rowan University MEMBERS KristiAnseth University of Colorado Jennifer Curtis University of Florida Rob Davis University of Colorado Pablo Debenedetti Princeton University Diane Dorland Rowan Thomas F. Edgar University of Texas at Austin Richard M. Felder North Carohna State University H. Scott Fogler University of Michgan Carol K. Hall North Carolina State University Jim Henry University of Tennessee, Chattanooga Steve LeBlanc University of Toledo Ron Miller Colorado School of Mines Susan Montgomery University of Michgan Ronald W. Rousseau Georgia Institute of Technology Stan Sandler University of Delaware Donald R. Woods McMaster University Chemical Engineering Education Volume 42 Number 1 Winter 2008 > DEPARTMENT 10 Chemical Engineering at Tufts University Nakho Sung, and Daniel Ryder > EDUCATOR 2 Jennifer Sinclair Curtis of the University of Florida Nicholas Peppas, Gintaras Reklaitis, and Erik Ydstie > RANDOM THOUGHTS 33 Student Ratings of Teaching: Myths, Facts, and Good Practices Richard M. Felder and Rebecca Brent > CLASSROOM 23 Active Problem Solving and Applied Research Methods in a Graduate Course on Numerical Methods Eric L. Maase and Karen A. High > CURRICULUM 35 Introducing Stochastic Simulation of Chemical Reactions Using the Gillespie Algorithm and MATLAB: Revisited and Augmented A. Argoti, L.T Fan, J. Cruz, and S.T Chou > LABORATORY 17 Chemical Kinetics, Heat Transfer, and Sensor Dynamics Revisited in a Simple Experiment Maria. E. Sad, Mario R. Sad, Alberto A. Castro, and Teresita F Garetto > TEACHING TIPS 52 Celebrating ChE In Song Alan M. Lane > CLASS AND HOME PROBLEMS 47 Applications of the PengRobinson Equation of State Using Mathematica Housam Binous 53 Can I Trust This Software Package? An Exercise in Validation Of Com putational Results Mordechai Shacham, Neima Brauner, W Robert Ashurst, and Michael B. Cutlip inside front cover Editorial: Engineers Deserve A Liberal Education C. Judson King CHEMICAL ENGINEERING EDUCATION (ISSN 00092479) is published quarterly by the Chemical Engineering Division, American Societyfor Engineering Education, and is edited at the University ofFlorida. Correspondence regarding editorial matter, circulation, and changes of address should be sent to CEE, Chemical Engineering Department, University of Florida, Gainesville, FL 326116005. Copyright @ 2005 by the Chemical Engineering Division, American Societyfor Engineering Education. The statements and opinions expressed in this periodical are those of the writers and not necessarily those of the ChE Division,ASEE, which body assumes no responsibility for them. Defective copies replaced if notified within 120 days of publication. Writefor information on subscription costs and for back copy costs and availability. POSTMASTER: Send address changes to Chemical Engineering Education, Chemical Engineering Department., University of Florida, Gainesville, FL 326116005. Periodicals Postage Paid at Gainesville, Florida, and additionalpost offices (USPS 101900). Vol. 42, No. 1, Winter 2008 rij] educator  U s_____________________________________ Jennifer Sinclair Curtis of the University of Florida NICHOLAS PEPPAS The University of Texas GINTARAS REKLAITIS Purdue University ERIK YDSTIE Carnegie Mellon University P professor Jennifer Sinclair Curtis of the University of Florida cel ebrates 20 years of an academic career that has made her not only an internationally known researcher and educator, but also a great role model for younger generations of chemical engineers. During her life, she has faced family adversities and almost insurmountable crises. But through her perseverance and convictions, as well as a deep Christian faith, she has triumphed. She has touched many others with her concern, enthusiasm, and dedication. She is now recognized as a leading figure in the chemical engineering profession. Jennifer is a wonderful educator recognized with numerous teaching awards, a student counselor par excellence, and a leading administrator (with two chairmanships one in Fresh man Engineering at Purdue University and another now in Chemical Engineering at the University of Floridaas well as service as an associate dean of Engineering at Purdue). She is a national leader in numerous societies, including AIChE where she has been a major motivating force of the Particle Technology Forum, and as a recently elected AIChE director. But Jennifer Curtis is also the internationally known leader in particle technology, an innovative engineer, and a scholar. Along with all these achievements she enjoys a wonderful fam l ily life with her husband, Barry Curtis, and her children, Jennette and Derek. THE EARLY YEARS Jennifer is a product of the rich Mid western tradition of hard work and ap preciation for education. She was born Jennifer Lynn Doloresco, the daughter of Jerry Doloresco and Carolyn Stortz Doloresco, on Christmas Day 1960 in Cincinnati, OH. Her father enlisted in the Navy immediately after high school to acquire enough money to go to Miami University of Ohio. He Received a B.S. in accounting in three years a necessity because he didn't have enough money to go for a lon ger period of time. Nicholas Peppas remembers, "Jerry Doloresco was especially happy when Jennifer went on to graduate school and received her Ph.D. from Princeton University. He was sad that he could not afford more education himself but he was so happy his children (Jennifer and her brother, Bryan Doloresco) could. I guess this is a pretty common story from the generation before ours that grew up in the postDepression period. Jerry was a loving father, always proud of Jennifer's achievements. We kept frequent correspondence, and he was particularly proud when Jennifer started her academic career at Carnegie Mellon University." Jerry passed away from cancer in 1995, when Jennifer was 35. Jennifer's mother, Carolyn, is of German descent and still lives in Cincinnati. This marvelous mixture of Italian and German background has given Jennifer her wonderful enthusiastic spirit and optimism that everything will go well, but also her dedicationbordering on stubborn nessto achieve her goals Copyright ChE Division of ASEE 2008 Chemical Engineering Education Jennifer graduated from Northwest High School in Cincin nati in May 1979 as valedictorian of her class. Having worked in a summer job at Procter & Gamble, she was sure she wanted to become a chemical engineer and enrolled in the School of Chemical Engineering at Purdue in the fall of 1979. While taking Honors Chemistry during her freshman year she met fellow student Gavin Sinclair. They started dating in their sophomore year and were married in 1981. At that time, the School of Chemical Engineering at Pur due had instituted a new advising program for all students. Each professor was in charge of 30 undergraduates with full responsibility for advising and registration. Nicholas Peppas remembers, "I was a young associate professor, just in my third year at Purdue, and I had been as signed my first advising group of 30 incoming sophomores. My first advisee was Jennifer. And to the door of my office at CMET 210A came this shy 19yearold with golden locks, holding an alreadyfilledout departmental class registration form. Without even stepping into the room she extended her right hand and announced 'Here is my form, please sign it.' I asked her to come in with the explanation that I'd like to learn more about her family and educational background, as well as her future goals. We ended up talking for an hour. I was truly impressed with her intellect and enthusiasm. My intrigue was similar to that of, say, a collector who finds a rare art object. There and then we decided that Jennifer would start doing research with me during her sophomore year." Jennifer's performance at Purdue was stellar. She was se lected a top sophomore woman in engineering in 1981, a top junior woman in engineering in 1982, and a top senior woman in engineering in 1983. She received an Olin Fellowship for undergraduate research, the first prize of the 1983 University of Akron Polymer Engineering competition, and a best paper award at the MidWestern regional AIChE competition at Dayton, OH, in 1983. In addition, she received the Best ChE Senior awardin 1983. She graduated in May 1983 with "high est distinction," the Purdue equivalent of a summa cum laude. Next to her was another graduateher husband of two years Gavin Sinclair, who had achieved the impossible, i.e., to be awarded two difficult degrees (management and ChE) in four years. While at Purdue, Gavin published his first paper, on the analysis of drug release mechanisms from swellable polymers. The paper was published in 1983 in the Journal of Membrane Science and has been cited more than 200 times. Not unexpectedly, the question of graduate education came up very early in Jennifer's undergraduate studies, and the pos sibility of an academic career had appeal as early as 1980. So, that fall Jennifer started working with Nicholas Peppas on a project that led to a twoyear collaboration, her first two pub lications, and her first exposure to fluid mechanics and mass transport. The project involved the development of the first advanced models of nonFickian and anomalous transport in glassy polymers with consideration of the relaxational effects Vol. 42, No. 1, Winter 2008 Jennifer in third grade, left, and with her dad, Jerry Dolo resco, when she was finishing high school in 1979, below. Opposite page: Jen nifer with fellow researchers in her lab at the University of Florida. S 7  S AN  I' 'I of the macromolecular chains and the associated anomalous transport. The problem had been addressed by others with the addition of a pseudoconvective term (resulting from the relaxational effects) until Jennifer's model set the physics correctly, recognizing the importance of the moving bound ary value problem (glassy/rubbery transition and erosion fronts). The paper, "Anomalous Transport of Penetrants in Glassy Polymers," published in Colloid and Polymer Science almost 25 years ago [261,404409 (1983)] has received 180 citations and is one of the most cited articles in the history of this journal. Another paper was published in the Proceedings of the International Congress ofRheology of1984, where the work had been presented. Planning for graduate school included some long evening discussions between Jennifer and Gavin. Jennifer had received an NSF Fellowship and was clearly interested in a university with a scholarly reputation where she could continue working on transport phenomena. The choice came down to Princeton University or the University of Pennsylvaniawhere Doug Lauffenburger had put together a great academic and re search program and was particularly successful in attracting a wonderful group of exceptional women chemical engineers. Finally, Jennifer picked Princeton, but Doug has remained a great supporter and mentor throughout her life. GRADUATE SCHOOL AND FAMILY Jennifer started at Princeton University in 1983, while Gavin was employed a state awayat Air Products in Al lentown, PA! They decided to live in Flemington, NJ, which meant that every day Jennifer had to drive 50 minutes each way to school at Princeton. In the demanding classes there, she met a number of other successful chemical engineers who have remained good friends, including Paul Johnson (now dean at Arizona State University), Harry Ploehn (now at the University of South Carolina), and Spyros Anastasiadis (now at the University of Thessaloniki, Greece). At Princeton, she had many inspiring teachers such as Carol Hall, Bill Russell, and Roy Jackson. Her rapport with the latter was exceptional; he became the main advisor of her Ph.D. thesis, entitled "Ver tical Transport of Gas and Solids with Radial Solid Density Variations." It was publishedamong othersin a seminal issue of the AIChE Journal in 1989 ["GasParticle Flow in a Vertical Pipe with ParticleParticle Interactions," AIChE J. (1989) 35, 14731486]. It's one of the 50 most cited papers in the history of the journal. While she was in graduate school, Jennifer and Gavin decided to start a family. Nicholas Peppas remembers, "In October 1985 Jennifer called me with the good news she was expecting her first child. And it was just two months later that she called me back to share the terrible news that Gavin had been diagnosed with cancer, probably in the lungs, although that was not clear yet. We spent several days and many hours talking about doctors and possibilities and we ended up with the solution of SloanKettering, where Gavin was indeed saved after more than six months of treatment. I will never forget that in all these months, with her graduate research, her commuting almost two hours a day, and her being pregnant, Jennifer did not sound weak even for one minute. I never heard or saw her cry, complain, or give up. This was an incredible lesson for me and for all of us around her." Indeed, Jennifer was able to face these family crises because of her deep faith and Christian beliefs that have given her strength all her life. Fortunately, 1986 was a much better year! A daughter, Jen nette, was born in April and the couple was reunited after the completion of Gavin's surgery and many treatments. Gavin went on to write a book, All Things Workfor Good (based on Romans 8:28), about his cancer experiences, focusing on the many lessons he learned and how his Christian faith was strengthened. As for Jennifer, these were the days when she began to master the art of juggling family, career, and life. Her major professor in those days, Roy Jackson, a member of the Royal Society, is now retired and lives in North Caro lina. He remembers, "As you may know Jennifer was working for her Ph.D. in my group at Princeton at the time her husband, Gavin, was diagnosed with cancer and a very grim prognosis. At the same time she was pregnant with their first child, so the pressures on her 4 Winning the 1989 Lafayette College Teaching Award (pictured with students Paul Verderber and Dave McVeigh). were enormous. In addition she was working on the problem of predicting, in detail, the behavior of gas particle flows in vertical tubes. The behavior of these flows turned out to have an unexpectedly complex (and interesting) structure, whose details were revealed only by meticulously careful computation and interpretation. In this she succeeded brilliantly, despite everything else weighing on her mind at the time. Her resilience and cheerfulness were simply amazing." He continued, "After graduation the demands of her career still had to be balanced against difficulties associated with Gavin's health. But, in spite of this, her reputation grew steadi ly and I was proud to follow her progress. She is an amazing woman who combines academic ability and administrative competence with a warm and friendly personality that has made her so many friends. Of my many academic 'children' she is, perhaps, the one in whom I take the greatest pride." AN ACADEMIC CAREER IN THE SERVICE OF STUDENTS In August 1987, while still a graduate student at Princeton, Jennifer answered the calling to teach. As the family could not leave the area because of Gavin's medical insurance (the dreaded "preexisting condition"), she accepted an offer from Lafayette College and began teaching as an assistant profes sor in that school's Department of Chemical Engineering. She stayed there until December 1989, one semester after her formal graduation from Princeton. During her years at Lafayette she became the master teacher and educator that she is, winning numerous teaching awards. As an educator, she taught a wide range of courses such as Separations, Transport, Transport Lab, Fluid Mechanics, Heat and Mass Transfer, Control, Fortran, and Differential Equa tions. To these courses she later added Kinetics, Advanced Transport (grad level), Advanced Math (grad level), Fluid Mechanics, and Heat and Mass Transfer at Carnegie Mellon; Chemical Engineering Education Jennette reads Daddy's book, All Things Work for Good, to Derek (early 1996). Kinetics, Fluid Mechanics, and Control at the University of Arizona; and Materials and Energy Balances, Particulate Systems, and Transport Phenomena at Purdue University. By 1989, Gavin's medical condition had improved and he was able to move to a new locationPittsburghwhere he was employed by PPG Industries. Jennifer interviewed and was offered a position as an assistant professor at Carnegie Mellon University. Nicholas Peppas remembers again: "She was truly excited with the opportunities at CMU. In December 1989 she flew to Indianapolis and we spent three wonderful days in our homeJennifer, Lisa, and I, planning the future, defining new problems, and talking about possible collabora tions on biomedical applications of micro and nanoparticles. I could see the future leader of particle technology in all her intellectual glory." AT CARNEGIE MELLON At CMU, Jennifer found a truly exceptional faculty with colleagues who provided the scholarly environment for her professional development. During the interview process she was particularly affected by then Head of the Department John Anderson (now president of IIT in Chicago) as well as by colleague Ignacio Grossmann. Anderson remembers, "When I first met Jennifer during her interview at Carnegie Mellon, I knew she was not only an excellent scholar and teacher but also a future academic leader. She has proven me correct. She leads by example, and her enthusiasm is infectious. The way Jennifer has handled her personal and professional lives provides a role model for all of us." In a short amount of time Jennifer developed new courses and attracted firstrate research students, such as Eduardo Bolio, who is currently working for McKinsey in Siberia, Vol. 42, No. 1, Winter 2008 and Christine Hrenya, who is an associate professor at the University of Colorado in Boulder. The students helped her develop a strong research program supported by industry and government. Ignacio Grossman remembers that "Jennifer was the strongest and most popular teacher in the department, and received the highest scores on student evaluations. This is especially noteworthy because she was teaching core courses like Mathematics for Chemical Engineers and Heat and Mass Transfer." She also started to give invited seminars and re search presentations outside the department while engaging herself in university and professional service. While chair of the undergraduate committee, she conceived a plan and raised funds for a new computer cluster for undergraduates. The computer cluster is still a very popular and well used place for students to work and meet. While the computers have been upgraded many times since, credit for the initial concept goes back to Jennifer's ingenuity. "Jennifer managed to do all of this, not only extremely well, but always with a big smile, and while placing the ma jority of her attention on her husband and children," writes Grossmann. Erik and Genevieve Ydstie moved to Pittsburgh in 1992 when he joined the CMU faculty, and they made a point of moving into the same neighborhood as Jennifer and Gavin. "We have a daughter in the same age group as Jennette and we wanted to spend time together," Erik Ydstie remembers. "We knew we could not go wrong as far as location was concernedGavin and Jennifer had done extensive research and had even written a computer program to take as many considerations into account as possible to help in their hous ingdecision process! We have many fond memories of good family times spent together after church and late evenings discussing politics and all manner of other things." Yet in the mid1990s, after Jennifer received promotion to associate professor at CMU, happy times gave way to worries once again. There was some deterioration in Gavin's health, especially due to the cold winters in Pittsburgh. Grossmann notes, "I still recall the day that Jennifer walked into my office, closed the door, and let me know she was leaving for Arizona because Gavin's health required that he live in an environ ment with dry air. Despite the fact that her departure was a big setback to our department, I admired how she was able to put her family's priorities above her professional ambitions. This to me speaks volumes of Jennifer's integrity, and her great dedication to her family and her profession." In 1995, the family moved to Tucson, where Gavin con tinued to work for PPG remotely and Jennifer became an associate professor at the University of Arizonawhose ChE department was under the expert leadership of Tom Peterson, now dean of Engineering. The relocation rounded out happily with the birth of Jennifer's and Gavin's son Derek in early 1996. But while Tucson was a great place for Gavin's health, and Jennifer's career continued flourishing, the remote work with PPG could not continue and medical insurance problems re surfaced. It was extremely difficult for Gavin to get insurance from a new company due to his preexisting condition. And that's when Gavin and Jennifer remembered their roots. THE PURDUE YEARS In summer 1997, Jennifer and Gavin returned to Purdue to launch the next phase of their careers she as a tenured associate professor and he as an assistant professor in organizational leader ship and supervision as well as in agricul tural economics. Al though he was physi cally limited due to the many health problems he faced, Gavin had w  completed a master's degree in economics at Lehigh University while working full time at Air Products, and Above, the Sinclair family (Jenn then completed a Ph.D. 1998 Caribbean cruise. An annu in social and decision Jennifer and her family. Initially sciences at Carnegie then they became Alaskan and r Mellon while working as well. Below, Gavin Sinclair's full time at PPr which was featured in Newswe' full time at PPG. front page of The Wall S This dualacademic career couple immedi ately set about having an impact on their respect ti i departments and students. In spring 1998, Jennil . launched a survey course in Particle Technolo,_' (described in a CEE article in fall 1999). Begin ning in 2000, this course was video broadcast iti include industrial audiences, and it was cotaughi t with Carl Wassgren (M.E.) from 2002 on. The , initiative has continued to flourish, with Mike 0~i Harris taking up the course when Jennifer no longer could. In 1999, Jennifer was elected trustee of CACHE (Computer Aids for Chemical Engineel  ing), where she championed the introduction of compuiduioial fluid dynamics (CFD) into chemical engineering teaching (see her CEE article in spring 1998). It was through her initia tive that Rodney Fox and Dick LaRoche were elected to the CACHE board to further promote this domain in chemical engineering research and education. In fall 2000, while Gavin launched his book, Life, Love and Economics, based on his very successful introductory econom ics course, Jennifer launched the administrative component of her rapidly accelerating career: She accepted the position of head of Freshman Engineering, leading a department that was home to some 2,200 beginning engineering students. She took on this major challenge with her usual optimism and cando style, reenergizing the staff, recruiting new fac ulty, and rallying the firstyear students to overcome that big transition to college. She was remarkably adept at soothing . . . _ ette, Jennifer, Gavin, and Derek) on a al cruise has remained a tradition for / they were all Caribbean cruises, but nore recently Mediterranean journeys textbook, Life, Love and Economics, ek magazine in May 2000 and on the street Journal in October 2000. the concerns of parents while encouraging their sons and daughters to work hard but also find time for campus activi ties. She was also very effective at both attract ing new resources and capitalizing on those the department had. She worked closely with an Industrial Advi sory Council for Fresh man Engineering that helped to refocus that department's strategic directions to its central role of preparing stu dents for success in the professional schools. The chair of that coun cil, Bob Davis (Air Products & Chemicals) comments on her effec tive leadership style: "Jennifer added energy to everything she touched. Her real gift was empowering all the members of her department so that the resources of the department were seem ingly multiplied. The impact on results and morale was remarkable!" Sadly, into this whirlwind of positive pur 'ntits came a devastating downturn: Gavin's h .alth deteriorated again, and he was moved Iick to Tucson because of the excellent im.dical facilities there. He passed away during S'linstmas 2000. ` FORGING ON Jennifer's optimistic outlook prevailed. In spite of her many administrative duties and the neverending meetings, she continued growing her research group to 10 solely orjointly advised graduate students. In Fall 2001 she was promoted to full professor and shortly thereafter (March 2002) she was named the associate dean of the College of Engineering with Chemical Engineering Education responsibility for Undergraduate Education, while continuing her duties as head of Freshman Engineering. And, of course, the particle technology course had to go on as well! In her associate dean role, Jennifer was instrumental in setting the stage for the reorganization of the Freshman Engineering Department and the subsequent launch of a new Department of Engineering Education, the first in the United States. Other challenges included restructuring and restaffing the Minority Engineering Program, the Women in Engineering Program, and the Cooperative Education Program. Her highly effective leadership in engineering education was recognized through the 2003 Sharon Keillor ASEE Award for Women in Engi neering Education. By fall 2003, it was clear that Jennifer was ready to move on to further challengesto the good fortune of the University of Florida. Without question, her Purdue years spanned a very productive phase of her profes sional life as teacher, researcher, and academic leader, but also involved very low as well as high points in her personal life. She remains a valued colleague and has certainly left her impact on Purdue Engineering. AN INTERNATIONALLY KNOWN RESEARCHER Throughout her academic career, the impetus for Jennifer's energy and creativity has been her attraction to, and thus seminal contributions toward, key research issues. In this area, she has contributed on manifold fronts both on a national and international scale. Jennifer has an internationally recognized research program. Her work in the development and validation of numerical models for the prediction of gassolid flow phenomena is in an interdisciplinary research area of both great technologi cal and commercial importance. Particle flow processes are encountered in the pharmaceutical, chemical, mining, agri cultural, food processing, and petroleum industries. Jennifer has received various awards for her pioneering work in this area. One of her most notable successes is the adoption of her multiphase flow models by the two key commercial CFD softwarepackage vendors. Both companiesFluent and CFXhave adapted her models for dilute and densephase gassolid flow in software that is used internationally by both academic and industrial experts. Jennifer received the NSF Presidential Young Investigator Award and an engineering college research award when she was a professor at Carnegie Mellon University. Also early in her career, she was awarded NSF equipment grants that funded the purchase of important equipment such as a laser Doppler velocimetera purchase that helped her propel her early research program further toward the leading edge of science. GRADUATE STUDENTS WHO DID THEIR THESES WITH JENNIFER CURTIS 1. John Doney ChE, M.S., Carnegie Mellon Uni versity, "Heat Transfer in Dilute ParticleLaden Turbulent Flows," 1993 2. Eduardo Bolio, ChE, Ph.D., Carnegie Mel lon University, "Dilute Turbulent GasSolid Flows with ParticleParticle Interactions," 1994 3. Christopher Alexander, ChE, M.S., Carnegie Mellon University, "Pneumatic Transport with a Bimodal Particle Size Distribution," 1995 4. John Doney, ChE, Ph.D., Carnegie Mel lon University, "Electrostatic Behavior of Charged Particle Mixtures," 1996 5. Christine Hrenya, ChE, Ph.D., Carnegie Mel lon University, "Dense GasSolid Suspension Flow,"1996; 1997 recipient of Best Ph.D. in Particle Technology by the International Particle Technology Forum (Dr. Hrenya is now associate professor of chemical and biomolecular engineering at the University of Colorado) 6. Pawan Agarwal, ChE, M.S., The University of Arizona, "Modeling GasSolid Flow with a Bimodal PSD," 1997 7. Michael Schabel, Materials Science, M.S., The University ofArizona, "Characterization of Particle Traps in Plasmas," 1997 8. Tim Mallo, ChE, Ph.D., Carnegie Mellon University, "Modeling Heat Transfer in Dilute and Dense Phase GasSolid Fow," 1997 9. BerendvanWachem, ChE, Ph.D., DelftUniver sity, "Flow in a Fluidized Bed with a Bimodal PSD," 2000 (Dr. van Wachem is now associate professor at Chalmers University, Gothenburg, Sweden, Department of Applied Mechanics) 10. Cynthia BlakePowell, ChE, M.S., Purdue University, "Comparison ofAdHoc Theory Predictions and CFD Predictions for Con fined, Recirculating Flows," 2000 11. Agus Sumantri, ChE, M.S., Purdue Universi ty, "Comparative Analysis of Computational Fluid Dynamic Models for GasSolid Flow in Risers," 2000 12. Edward (Nick) Jones, ChE, Ph.D., Purdue University, "The Effect of PSD on GasSolid Fows," 2001 13. Chie Min Chung, ChE, M.S., Purdue Univer sity, "CFD Modeling of a Fluidized Bed with a Wide Particle Size Distribution," 2002 14. Stephanus Budilarto, ChE, Ph.D., Purdue University, "Experimental Study of Velocity Ratio, Particle Size and Size Distribution Effects in ParticleLaden Jet Fow," 2003 15. Kim Hayden Henthorn, ChE, Ph.D., Purdue University, "The Effect of Particle Rough ness on the Fow Behavior of Fine Powders," NSF Graduate Fellow, 2004 (Dr. Henthorn is now assistant professor in chemical and biological engineering at University of Mis souriRolla) 16. Kunn Hadinoto, ChE, Ph.D., Purdue Uni versity, "DensePhase Fow Measurements using LDV," 2004 (Dr. Hadinoto is now assistant professor in chemical and biomo lecular engineering at Nanyang Technologi cal University) 17. Michael Lasinski, ChE, Ph.D., Purdue University, "Particle Clustering in Granular Flows," 2004 (coadvised student) 18. Bill Ketterhagen, ChE, Ph.D., Purdue University, "DEM Simulations of Granular Discharge from Hoppers," 2006 (coadvised student) 19. Robert Hamilton, ChE, Ph.D., Purdue Uni versity, "Evolving Particle Size Distribution in the Flow of GasSolid Mixtures," (1999 ) (coadvised student) 20. Benjamin James, ChE, Ph.D., University of Florida, "Effect of Particle Shape on Particle Phase Stress," (2004 ) 21. Mark Pepple, ChE, Ph.D., University of Florida, "LDV Measurements of LiquidSolid Flow," (2004 ) 22. Anshu Anand, ChE, Ph.D., University of Florida, "Effects of Particle Cohesion and Shape on Particle Segregation in Hopper Flows," (2005 ) 23. Julio Cesar Castro Vazquez, ChE, Ph.D., Uni versity of Florida, "Effect of Wall Roughness on Wall Friction Angle," (2006 ) 24. Juan Pedro Marval, Departamento de Ener getica, Universidad Nacional Experimental Francisco de Miranda, Punto Fijo, Edo. Falc6n, Venezuela, "Numerical Simulation of Saltating Particles using Granular Kinetic Theory," (2006 ) (coadvised student) 25. Byung Chu, ChE, M.S., University of Florida, "Experimental Measurements of Sifting Segregation in Hoppers," (2005 ) Vol. 42, No. 1, Winter 2008 Right: At Carnegie Mellon University with John Anderson and Ignacio Grossman in 1995. Below: In Australia with a friend in 2005. Facing page: Jennifer, Derek, and Barry celebrating Jennette's graduation from her parents' alma mater, Purdue University, in May 2007. The importance of her research has been validated through multiyear funding from major corporations such as Chevron, DuPont, Alcoa, Dow, and Pfizer. Jennifer consults for a wide range of organizations. She has also served as one of five aca demic participants in a U.S. Department of EnergyOffice of Industrial Technology Consortium on Multiphase Phase Flow. This consortium involved collaboration between industry, academia, and national labs. It received national recognition, and as part of it Jennifer gave a panel presentation before Congress in 1999. Jennifer is a popular speaker not only for scientific but also inspirational talks. In her academic career, she has given more than 180 invited lectures. Over the years, she has served the scientific community in various positions. For example, she served on the Board of Directors for the American Chemical SocietyPetroleum Research Fund (ACSPRF), as a member of an NSForga nized delegation to South Africa promoting collaboration in particle technology, as a member of an external review panel to review the entire Chemical and Transport Division of the NSF, as a member of the editorial advisory board of the Journal of Powder Technology and the Journal of Phar maceutical Development and Technology, and since 2007 as associate editor of the AIChE Journal. Within AIChE, she served as the national meeting cochair, along with Gintaras Reklaitis, for the 2002 Annual AIChE Meeting that was held in Indianapolis. She was elected to the executive committee of the Particle Technology Forum for eight years and to the AIChE Fluid Mechanics Programming Committee for four years, and was chair of the Transport and Energy Processes Division in 2006. A unique contribution to the annual chemi cal engineering meetings has been her continuous leadership and support (with Rosanne Ford of the University of Virginia) of the Christian ChE Fellowship meetings that are held on Tuesday or Wednesday mornings of the AIChE meetings. These meetings were initiated in 1989. AN EDUCATOR, ADVISOR, AND MENTOR Jennifer is a highly effective teacher and mentor. She re ceived departmental and university teaching awards while at Lafayette College and the University of Arizona. At Purdue, she received the departmental Kimberly Clark Mentoring Award and was named to the universitywide Teaching for Tomorrow Program. In 2001, Jennifer served, along with then Purdue Engineering Dean Linda Katehi, on a 10mem ber invited NSF panel on the Future of Engineering Educa tion. From 20032006, she also served on the NAE's Engineering Education Committee. She is very active in training new chemical engineering professors. She has participated as an instructor for new chemical engineering educators at the Ameri can Society for Engi neering Education's Chemical Engineer ing Summer School, held every five years. In 1997, she served as a lecturer at this work shop. In 2002 and 2007, she organized and conducted a very well attended workshop on CFD. In August 2000, when she was named head of Purdue's Freshman Engineering Department, she embarked upon attending a series of workshops that better prepared her for administration. For example, in 2001 she attended the twoweek Management Development Program at Harvard University and in 2002 the American Council on Education's Chairs and Deans Workshop. As a Ph.D.student advisor, Jennifer has excelled by trans ferring to her students the same style and the same warm concern that Roy Jackson, her Ph.D. advisor at Princeton, had transferred to her. Her former Ph.D. student Kimberly Hayden Henthornnow an assistant professor in chemical engineering at the University of MissouriRollaremarks, "I think the thing that impressed me most about Jennifer was her leadership styleshe led by example. She kept her graduate students motivated and working hard by showing us that she was working just as hard as (and sometimes harder than) we were." Chemical Engineering Education 1 Naturally, Jennifer has not only been an exceptional engineer, researcher, and educator, but also a wonder ful role model for the young generation of women chemical engineers. More than 20 years ago, she had the fortune to find a great role model in distinguished ChE and NAE member Carol Hall, then at Princeton. And many years later, Jennifer is now recognized as an example for others. Again, Kim Hayden Henthorn notes: "Having a strong, female role model like Jen nifer has made a tremendous impact on me. She has been through so much, but has kept an upbeat, positive attitude throughout. When I feel overwhelmed trying to balance everything on my plate, I always remind myself that Jennifer has been through all of this before, and I give her a call. She is still willing to listen and give good, thoughtful advice about everything from family life to being a woman in academia." Christine Hrenya, another former Ph.D. student of Jennifer's and now an associate professor at the Uni versity of Colorado, admires Jennifer's ability to focus and be successful. "Jennifer is incredibly productive, but she has an unfair advantage over the rest of us she only needs four to five hours of sleep per night, and this is her longtime average! Along the same lines, she is an expert at multitasking. Once when I was visiting her house for dinner, she promptly pulled out the iron ing board when dinner had finished (though perhaps I was overstaying my welcome!)." This is the love and admiration shown by her students. And Hrenya concludes: "On a more serious note, Jennifer was an outstanding advisor, and continues to be a role model, friend, and sounding board for all topics under the sun. I am extremely fortunate to have spent so many of my formative years under her guidance." A NEW LIFE AT THE UNIVERSITY OF FLORIDA In 2001 Jennifer met Barry Curtis, another Purdue chemical engineer, who had graduated two years before her (in 1981), hav ing done the fiveyear coop program. Jennifer comments, "I met Barry for the first time at a Purdue Chemical Engineering Industrial Advisory Board dinner in Houston. Jeff Hemmer then hooked us up on a strategic planning project for Freshman Engineering, when I was head of that department." The rest is history. They were married in 2002 and moved to Gainesville in 2005. Barry has been a wonderful father to Jennette and Derek and a most supportive and admiring husband and partner to Jennifer. Just before the move to Florida though, the family took yet another tripto Australia! Jennifer was a visiting professor at the Chemi cal Engineering Department of Monash University in Melbourne in 2005, received the Eminent Overseas Lectureship Award from the Institution of Engineers in Australia, and visited a number of Australian universities for seminars. In 2005, Jennifer Curtis assumed a new responsibility as the chair of the Department of Chemical Engineering at the University of Florida. In a very short time, she has taken an already nationally recognized program (achieved under the continuous and imaginative leadership of the editor of this journal, Tim Anderson) and further improved its national ranking. Here is how Anuj Chauhan, associate professor in this depart ment, sums up the thoughts of the younger generation of professors there: "Jennifer runs the department in a very fair, transparent, and efficient manner. She is helpful to the entire faculty, particularly the juniors, and she deserves a lot of credit for the rapid growth of the department and the improvement in rankings." At the University of Florida Jennifer keeps an active research program and collaborates with the nationally known ERC program on particle technology. Meanwhile, the Sinclair Curtis family is thriving. Derek is now 11 years old and in sixth grade. Jennette attended Purdue in the School of Managementthe same school from which her father graduated 24 years earlierand received her B.S. in management in May 2007 with highest distinction (a perfect 4.00). She is now a graduate student at Michigan State University, specializing in organizational behavior. Like her parents, she wants to become a professor. Looking back at Jennifer Curtis's life and accomplishments, one cannot help but admire her concern for others, her high standards, and her dedication to education and advising. But one cannot forget the adversities she has faced in her life. As Nicholas Peppas likes to point out: "At some time or another all of us have had crises or adversities in our lives. How Jennifer was able to cope with them has been an inspiration for all of us. I think Jennifer, Barry, and the children have found the golden rule of a good, inspiring, and productive life. They are a wonderful example that all ;lii,,o work for good." 0 Vol. 42, No. 1, Winter 2008 rj[ =department  U s_____________________________________ ChE at Tufts University The atrium of the Tufts Science and Technology Center, home to Chemical and Biological Engineering. BY NAKHO SUNG AND DANIEL RYDER Tufts College was chartered in 1852 as the 163rd institution of higher learning in United States. Two years later, its first class matriculated. Today, Tufts University is a diverse ensemble of undergraduate, graduate, and professional schools that includes the Schools of Arts and Sciences, Engineering, the Cummings School of Veterinary Medicine, the Schools of Dental Science and Medicine, the Sackler School of Graduate Biomedical Sciences, the Friedman School of Nutrition Science and Policy, the Tissue Engineering Resource Center, and the Fletcher School of International Law and Diplomacy. Fulltime students total 8,830, of which 4,950 are undergraduates. Copyright ChE Division ofASEE 2008 Chemical Engineering Education The School of Engineering includes the Department of Chemical and Biological Engineering as well as the Depart ments of Biomedical Engineering, Civil and Environmental Engineering, Electrical Engineering and Computer Science, and Mechanical Engineering. Chemical Engineering debuted at Tufts in 1898 to become only the fifth such program in this country, and accepted its first students in 1900. The department was officially part of the Chemistry Department until February 1949; its undergraduate program was initially accredited three years later. An M.S. program was added in 1962 and a Ph.D. program in 1965. The first M.S. degrees were granted in 1964 to Dewey Ryu (now a professor in the department of Chemical Engineer ing and Materials Science at the University of California, Davis) and Wayne Sanborn (who worked for Matheson Gas Products). The first Ph.D. degrees were awarded in 1971 to Donald Merchant, Joseph DelPico (who worked for Polaroid Corporation), and Edward Denk, Jr. (now a consultant at Evergreen Solar). This year the department will award some 15 graduate diplomas, including five Ph.D.s Though it will award only 20 B.S.ChE degrees, enrollment in each of the continuing undergraduate classes exceeds 30 students. A perennial hightech incubation zone, the greater Boston area has long been one of the nation's hubs for biotechnology R&D. Within this community, numerous professionals are Tufts Chemical Engineering alumni. Many remain actively involved with the department, which offers teaching and research programs in biological engineering and has contrib uted nationally to development of the relevant undergraduate curricula. For the past 15 years, the department has been assembling a core group of biological engineering faculty. Graduate research is divided within three main areasbiological en gineering, energy and environment, and advanced materials. The current plan is to focus on expanding clean energy and environmental sustainability along with systems engineering research and teaching. TUFTS SCIENCE AND TECHNOLOGY CENTER The department is currently housed in the Tufts Science and Technology (SciTech) Center, which was constructed in 1989 with major financial support from the U.S. Depart ment of Energy. At that time, the Department of Chemical Engineering was the only academic department housed at the facility. The Tufts High Energy Physics Laboratory, Biotech nology Engineering Center, ElectroOptic Research Center, and Laboratory for Materials and Interfaces (LMI) were also housed at the site. Both the Biotechnology Center and LMI were associated research facilities of the ChBE Department. Later, through the generous support of Raytheon Corporation, the Pollution Prevention Projects Laboratory was established within the department's facilities. In addition to the Depart ment of Chemical and Biological Engineering, the complex currently houses the recently established Department of Biomedical Engineering and the Gordon Institute for Engi neering Management. CHEMICAL AND BIOLOGICAL ENGINEERING The department has a longstanding history relating to the development of educational programs in bioengineering. In 1986, the then Chemical Engineering Department initiated a Massachusettssponsored continuing education program to train technical staff for biotechnology business. The Bio technology Engineering Certificate program, which began as a series of handson, introductory, technical courses and remains a vital link to local industry, has led to the develop ment of a graduatedegree program. Today, the department offers graduate degrees in both chemical engineering and biotechnology engineering, including courseonly M. Eng. degrees attractive to working professionals, as well as tradi tional thesisbased M.S. and Ph.D. programs. In 2004 the department hosted a series of NSFsponsored workshops focused on the development of chemical and bio logical engineering curricula. Participants from academia and local bioindustries helped to shape our current undergraduate program (see Table 1), which has evolved to include intro ductory courses on biology, statistical thermodynamics and molecular structure (Physical Chemistry II), and biochemistry. Simultaneously, the chemical engineering science core has incorporated biological systems and processes, while the upperlevel course offerings have been supplemented with applied biologybased electives (see Table 2). We are currently redesigning the senior laboratory courses to comprise indus trysponsored research projects in biotechnology, metabolic engineering, energy sustainability, systems engineering, and advanced materials. FACULTY Current Faculty The current department faculty includes nine tenuretracked members and two jointly appointed members: Professor Linda M. Abriola, currently dean of Engineering, does research in the mathematical modeling of the transport and fate of organic chemical contaminants in porous media. She developed one of the first mathematical models to de scribe the interphase mass partitioning and nonaqueous phase migration of organic liquid contaminants in the subsurface. Her recent research involves the use of models and laboratory experiments to examine abiotic and biotic processes influenc ing the persistence of organic and controlling the effective ness of aquifer remediation technologies. She is also a member of the Department of Civil and Environmental Engineering, and a member of the National Academy of Engineering. Vol. 42, No. 1, Winter 2008 Professor Maria FlytzaniStepha nopoulos spear 4 heads clean energy technology and catalysis research in the School of Engineering. Her work focuses on Faculty member catalytic fuel pro Martin Sussman. cessing, including hydrogen generation, for fuel cells; and new materials for air pollution abatement and fuel gas desulfuriza tion. Her group's fundamental research on catalyst structure and function has led to the identification of a family of very active oxidation catalysts based Some of the on Au, Pt, and Cu on cerium oxide. Kyongt An exciting new development in the Bottom hot gas cleanup area is the group's recent identification of rare earth oxides that remove hydrogen sulfide to subppm levels via reversible adsorption. FlytzaniStephanopoulos's research group presently comprises one postdoctoral fellow, seven Ph.D. and four M.S. research assistants, and several undergraduates. Her research is supported by the NSF, the DOE, the Army Research Laboratory, and industry. Since 2002, she has served as the editor of Applied Catalysis B: Environmental. Professor Christos Georgakis established the Systems Research Institute for Chemical and Biological Processes in 2005. Modeled on the Chemical Process Modeling and Con trol Research Center at Lehigh University, the Institute fosters interdisciplinary research in modeling, optimization, and operation of chemical and biological systems and processes in collaboration with leading pharmaceutical and chemical companies. The institute's first initiativethe Industrial Consortium on Batch Manufacturing of Pharmaceuticals and Chemicals (BMPC)began operation in 2006. The consor tium is a cooperative research program that will focus on improving the development, design, and operation of batch pharmaceutical and chemical processes, with the ultimate goal of halfing the time it takes to design new processes. Initial research areas encompass designing batch reactors for multistep synthesis; enantiomeric and polymorphic crys tallizations; plantwide simulations, scaleup, and process sensitivities; and statistical process monitoring and process control that will effectively contribute to the implementa tion of the FDA's Process Analytical Technology Initiative. The consortium officially launched its research program on Oct. 25, 2006. Professor David L. Kaplan's group applies biopolymer engineering to elucidate structurefunction relationships, including those of fibrous proteins (collagens, silks) and polysaccharides. They apply physiological and genetic engi 12 :hBE faculty. Top, LR: Christos Georgakis, Blaine Pfeifer, bum Lee, Jerry Meldon, Daniel Ryder, Hyunmin Yi. LR: Ramin Haghgooie, Nakho Sung, Eric Anderson. neering approaches to control biopolymer chemistry in order to then explore selfassembly and function from a materials science and engineering perspective. The biopolymer sys tems are studied for potential biomaterials applications, such as matrices or scaffolds to support cell and tissue growth, for controlled release of pharmaceuticals, and as biosen sor platforms. Selective processing, surface chemistry, or genetic engineering redesigns are employed to functionalize biopolymer material surfaces or control morphology and structure, in order to enhance biomaterial interactions with inorganic and biologic interfaces. In combination with human stem cells and complex bioreactor systems, the biopolymer biomaterial systems are employed in the generation of func tional human tissues including bone, cartilage, and adipose systems. These systems are studied both in vitro and in vivo for tissue regeneration, as well as to serve as human disease models with which to understand disease progression, and subsequently for pharmaceutical discovery. Transport issues are addressed via vascularization strategies in vitro and in vivo to promote improved functional tissue outcomes. Kaplan's lab collaborates with many colleagues within and outside of Tufts. He also serves as the chair of the Department of Biomedical Engineering. Professor Kyongbum Lee's research in systems biology, metabolic engineering research in systems biology, and meta bolic engineering and tissue engineering is aimed at better understanding the chemical design of living systems through the development and application of analytical platforms, mod eling tools, and other enabling technologies. He is especially interested in characterizing and manipulating cellular metabo lism for a variety of basic and applied studies. His research is also motivated by the growing need to find and evaluate new therapies for obesity and type 2 diabetes. Current work focuses on the metabolism of liver and white fat cells. Chemical Engineering Education Left, one of several students in the senior projects lab who got handson experience when the department's distillation column needed repair. Below, Danny Pierre, a graduate student in Pro fessor FlytzaniStephanopoulos's rntnlxi. lanh There is increasing evidence that the functions and struc tural features of a biochemical network are intertwined. Thus, a useful approach to studying the design of these complex systems has been to characterize their layout, or "wiring." Lee's research aims to develop a modeling framework to integrate the analysis of the structural and functional layout of biochemical networks. For example, he has recently de veloped and applied a series of algorithms for the rational decomposition of metabolic networks based on experimen tally derived reaction activity data. In the context of metabolic engineering, Lee's group uses mathematical models of metabolic pathways in conjunction with experimental biochemistry techniques to quantitatively study the metabolic basis of adipose tissue development and growth. In addition to fundamental insights into adipocyte metabolic regulation, they seek to obtain: 1) novel drug tar gets, for example enzymes that catalyze the key controlling steps in adipocyte lipid accumulation; and 2) robust, sensitive, and easily measured metabolite biomarkers for diagnosis of obesityrelated diseases and efficacy assessment of thera peutics. To these ends his group has performed in silico and in vitro studies characterizing the metabolic flux profiles of de novo adipocyte formation, identified target pathways for metabolic intervention, and demonstrated the feasibility of achieving a global shift in lipid accumulation through the forced expression of a single protein. Lee also uses tissue engineering to design, build, and optimize model systems whereby important cellular func tions may be studied under wellcontrolled conditions while Vol. 42, No. 1, Winter 2008 Above, Gregory Botsaris gives a presentation at the 2005 sympo sium in his honor. maintaining a high degree of physi ological relevance. The primary focus here is on engineering the cellular component and its medium (as opposed to the materials or reactor housing). Current projects leverage developments in microfluidics and stem cell biology to generate flowthrough incubators that afford spatial control over the chemical en vironment of the cultured cells. This incubator is used for a number of ongoing studies involving heterogeneous cell components (e.g., adipocytepreadipocyte coculture) and solution gradients of exogenously added chemicals (e.g., drug transformation by hepatocytes). Professor Jerry H. Meldon's research interests are in mass transfer and separation processes, particularly when revers ible chemical reactions control the rate and selectivity of a diffusional separation. His primary focus is on membrane processes, gas scrubbing, and mathematical modeling of mass transfer with chemical reaction. Recent work includes theoreti cal and experimental investigations of hydrogen transport in palladium membranes and its role in catalytic membrane reac tors for the production of pure hydrogen. One Ph.D. student is building and modeling a novel spinning disk system for reaction and separation; another is modeling NOx scrubbing. Professor Blaine A. Pfeifer's research group seeks to influence genetic, metabolic, cellular, and process events in the synthesis of therapeutic products. For example, they ap ply molecular and process engineering in conjunction with molecular biology, microbiology, analytical chemistry, and bacterial genetics to develop microbial bioprocesses and products that can potentially target cancer, bacterial infections, diabetes, and other diseases. Pfeifer's group also seeks more efficient and economical ways to generate biological products. One approach is to Left, Zheng Zhou, a graduate student in Professor FlytzaniStephanopoulos's catalysis lab. Right, graduate student Lisa Schupmann works with a multireactor system in Professor Georgakis's Systems Research Institute. transplant genetic material responsible for an important thera peutic product into a convenient and processfriendly bacterial microorganism (such as Escherichia coli) for eventual product scaleup and development. Current projects include: 1) cellu lar and metabolic optimization for production of the antibiotic erythromycin; 2) production of new and established anticancer agents; 3) development of new biological vaccine systems; and 4) modeling and experiments to identify and elucidate cellular bottlenecks in natural product biosynthesis. Professor Daniel F. Ryder's primary research lies in the processing and characterization of ceramic and organically modified ceramic materials for electronic and optical appli cations. Specific areas of study include: Solgel processing of PLZT ceramic thin films for applications in ferroelectric memories; chemical processing of hexagonal ferrite thin films for application in high frequency devices; chemical process ing of superconducting oxide materials; the development of organically modified ceramic materials for abrasion resistant and antireflective coatings on optical polymeric substrates; and chemical vapor deposition methods for IIIV nitride semiconductors. Professor and Chair Nakho Sung's research in the area of structurepropertyprocessing relationships of polymers and composites includes development of in situ monitoring techniques for characterization of reactions in polymers and composites via fiberoptic fluorescence and UV reflection. Both extrinsic and intrinsic reactive fluorophores are devel oped for various matrix polymers for advanced composites to follow cure reactions as well as water uptake using distalend and evanescent fiber optic probes. UV reflection spectroscopy, coupled with bifurcated fiberoptic probes, is also developed to characterize reactions in composites that are fluoresce insensitive, such as those based on polyimides, epoxies, and unsaturated polyesters; this technique complements fluores cence monitoring. Another area of research has been to develop a methodol ogy for tailor designing composite interphases. Using a low temperature plasma of various monomers and inert gases, interfacial structure is modified to enhance such properties as adhesion strength and longterm stability against tempera ture and moistureinduced degradation. Sorption/diffusion behavior of organic molecules in multiphase polymers is investigated using a model composite consisting of glassy and rubbery domains where morphology is systematically varied. These model composites facilitate elucidation of complex diffusion behavior and development of a predictive diffusion model. Recently, Sung's group has been developing lowcost membranes for fuel cell application and hydrogen purification. Current focus is on anionic membranes suitable for lowtemperature direct methanol fuel cells. Professor Hyunmin Yi's research interests lie in biochemi cally driven nanometer scale fabrication (nanobiofabrication) of high throughput biosensors, biophotonic devices, and nanocatalysts for biomedical, environmental, and energy ap plications using smart biopolymers and viral nanotemplates. His primary research areas are nanobiofabrication with geneti cally modified viral nanotemplates, and biophotonic device fabrication with smart biopolymers. Yi's group has developed nucleic acid hybridizationbased surface assembly strategies of genetically modified tobacco mosaic viruses through their own genomic mRNA. These biologically derived nanotubes with the precise dimensions of 300 nanometer (nm) length,18 nm outer diameter and 4 nm inner channelcan serve as nanotemplates for covalent coupling of functional nanoparticles. Building upon these facile assembly strategies of these potent nanotemplates, his group is working toward the development of highthroughput biosensors and BioMEMS devices for environmental and biological threat detection. In closely related collaboration with FlytzaniStephanopoulos, they are also developing na Chemical Engineering Education A closeup of the Tufts Chemical and Biological Engine Xray diffractometer. noscale gold particlebased catalysts for energy applications using the viruses as nanotemplates with high capacity and precise nanoscale spacing. In his second research area, Yi seeks to develop biocom patible highthroughput biosensing platforms and implant able biophotonic devices for biomedical and environmental applications. They exploit the stimuliresponsive properties of smart biopolymers to fabricate nanoscale patterns and waveguides with high spatial, temporal, and orientational control. Example biopolymers include structural proteins such as gelatin and silk as well as polysaccharides such as agarose and chitosan. Recently his research group has shown that the thermoresponsive morphology transition of common biopolymers such as gelatin and agarose can lead to efficient means for consistent manufacturing of nanometerscale sur face diffraction gratings under mild processing conditions. They are currently working on building allbiopolymeric implantable waveguides, photonic bandgap crystal fiberbased biosensors and nanoimprinted biopolymeric gratings. Research Professor Aur6lie Edwards's research falls broadly in the domain of biological transport phenomena, developing theoretical models of transport in the renal medullary micro circulation, both at the macroscale and the cellular scale. The microcirculation in the renal medulla plays an essential role in the regulation of fluid and electrolyte excretion and in the long term maintenance of arterial blood pressure. Understanding how medullary blood flow is regulated and how it influences in turn arterial blood pressure is the main objective. Specifically, a detailed math ematical model of transport in the tubular and microcirculatory systems of the renal medulla is in development in order to ex amine how interactions between oxygen, nitric oxide (a potent vasodilator), and superoxide affect tubular sodium reabsorp tion, medullary blood flow, and blood pressure. Mathematical simulations of ion concentration changes Within the bulk cytoplasm and microdomains are under devel opment to elucidate contractil ity of resistance vessels, such as medullary descending vasa recta, regulated by variation of intracellular Ca2+ concentration ([Ca2+]cyt) in both endothelium ring Department's and smooth muscle. This model accounts for the characteristics of the channels and transporters that exchange ions between the plasma membrane, SR stores, and cytosol. Edwards's group is investigating which factors are most likely to serve as principal determinants of [Ca2+]cyt and delineate the mechanisms through which the bulk cytosol, microdomains, and SR communicate. Their goal is to ultimately relate local medullary NO concentrations to [Ca2+]cyt changes and to blood vessel diameter variations. Emeritus Faculty The early transition years of any department present a series of challenges that require the selfless dedication, vision, and determination of its faculty. The accomplishments of these early efforts are often taken for granted with the passage of time, but whatever level of success that this department presently enjoys is due in large part to a small group of faculty that has served Tufts throughout their professional career. Martin V. Sussman joined the department in 1961 and served on the faculty for 37 years before retiring due to ill ness in 1998. His popular "Technology as Culture" course, which offered liberal art students insights into the world of technology, made him a household name across the Tufts community. He earned his doctorate in chemical engineer ing from Columbia University and was a Fellow of both the American Institute of Chemical Engineers and the American Chemical Society. He was the author of two books on ther modynamics as well as numerous technical articles. Martin's Vol. 42, No. 1, Winter 2008 research interest spanned the fields of materials science and biotechnol ogy. He was author to some 20 pat ents that included the "Incremental Draw Process" for the processing of synthetic fibers and the "FibraCel" tissue culture matrix. The department was saddened by his passing in 2005, but his legacy and connection with Tufts survive through the generous funding of the Jeanne and Martin Sussman Endowed Fellowship. Gregory D. Botsaris joined the department in 1965 immediately after earning his Ph.D. in chemical engi neering at M.I.T. He officially retired in 2004 after 39 years of service, but remains on campus to continue his research in crystallization processes. He is the author of numerous technical articles in that field and other sepa ration processes. His contributions to crystallization technology were recognized by his peers through the establishment of the Gregory Botsaris Lecture series. Ken Van Wormer is the new est emeritus faculty member. Van Wormer joined the department as an instructor in 1954 while completing his doctoral studies at M.I.T. His continuous service to the department for an impressive 53 yearsincluding 10 years as chairis a testament to his dedication to the profession. TABLE 1 Chemical & Biological Engineering Curriculum at Tufts University First Year Fall Term Spring Term EN 1 (Intro. to Computers in Engineering) EN 2 (Intro. to Engineering Communication) Math 11 (Calculus 1) Math 12 (Calculus 2) Chem 1 (Chem. Fundamentals w/ Lab) Chem 2 (Chem. Principles w/ Lab) English 1 (Expository Writing) Humanities/Social Sciences Elective EN Elective (Intro. to Engineering) Physics 11 (General Physics w/ Lab) Second Year Fall Term Spring Term ChBE 10 (Thermo & Process Calc I) ChBE 11 (Thermo & Process Calc II) Math 13 (Calculus 3) Math 38 (Differential Equations) ES 11 (Intro. to Biology) ChBE 39 (Appl. Math and Numerical Methods) ES 10 (Intro. to Materials Science) Chem 32 (Physical Chemistry II) Chem 33 (Physical Chemistry Lab I) Humanities/Social Sciences Elective Third Year Fall Term Spring Term ChBE 21 (Fluid Dynamics & Heat Transfer) ChBE 22 (Mass Transfer) Chem 51 (Organic Chem I) ChBE 102 (Reactor Design) Chem 53 (Organic Chem Lab I) Bio 152 (Biochemistry & Cellular Metabolism) ES 3 (Intro to Electrical Engineering) Advanced Chemistry Elective Humanities/Social Sciences Elective ChBE Concentration Elective Free Elective Fourth Year Fall Term Spring Term ChBE 45 (Separation Processes) ChBE 60 (Chemical Process Design) ChBE 51 (ChBE Unit Ops Lab) ChBE 52 (ChBE Projects Lab) ChBE 109 (Process Dynamics & Control) ChBE Concentration Elective ChBE Concentration Elective Advanced Chemistry Elective Humanities/Social Sciences Elective Humanities/Social Sciences Elective ACKNOWLEDGMENTS Sources for this feature article came primarily from the ChBE Department archives, the Tufts Office of Institutional Research, and the Tufts Office of Alumni Relations. Professor Kenneth Van Wormer's "Speech on the Occasion of the 85th Anniversary of Chemical Engineering at Tufts, April 1986" was an especially valuable source for the department's early history. His speech contained references to Russell F. Miller's Light on the Hill: A History of Tufts College (Boston: Beacon Press, 1966). Authors acknowledge the help from Joanna Adele Huckins in drafting this article. 7 TABLE 2 Biological Engineering Electives ChBE 160 Biochemical Engineering ChBE 161 Protein Purification ChBE 162 Introduction to Biotechnology ChBE 163 Recombinant DNA Techniques Laboratory ChBE 164 Biomaterials & Tissue Engineering ChBE 166 Principles of Cell and Microbial Cultures ChBE 167 Metabolic and Cellular Engineering ChBE 168 Biotechnology Processing Projects Laboratory Chemical Engineering Education M] =1laboratory CHEMICAL KINETICS, HEAT TRANSFER, AND SENSOR DYNAMICS REVISITED in a Simple Experiment MARIA. E. SAD, MARIO R. SAD, ALBERTO A. CASTRO, TERESITA F. GARETTO Universidad Nacional del Litoral Santa Fe, Argentina Since its beginnings, chemical engineering education has incorporated practical activities into the curricula (known as laboratory practices) in varying degrees (number of teaching hours, number of credits) according to the type of course or level of advancement in a given program. The historical evolution of this type of activity in chemical engineering programs has been recently discussed in literature.[1,2] It seems quite clear that, with the exception of certain courses and subjects directly involved with standard practical experiments (e.g., general and instrumental analytic chemistry), there is ample choice in their design and imple mentation. On the other hand, new developments both in the processing of experimental data and in process control and automation techniques make it possible to design and imple ment laboratory practices in practically every area at any level of sophistication and scientific accuracy. This requires important investments in time and money that, unfortunately, many schools are not willing to undertake. A valid alternative, then, is to redesign these practical activi ties to maximize the use of resources. This work describes a simple, inexpensive lab experiment that was originally de signed for qualitative demonstration on the existence of "hot spots" in discontinuous reactors. It has been deeply modified from the viewpoint of the processing and interpretation of the data collected through the use of a relatively sophisti cated mathematical model and the independent validation Copyright ChE Division of ASEE 2008 Vol. 42, No. 1, Winter 2008 of certain parameters. With these modifications, the field of knowledge has been broadened (development and validation of mathematical models, heat transfer and sensor dynamics) at practically null costs (considering time and economic resources). In what follows, both the experimental procedure and the development and validation of the model are described. Re sults illustrate this type of activities performed by students. The subject to which this activity belongs is Chemical Reac Maria Eugenia Sad is a Ph.D. student in chemical engineering. She received her bachelors degree from Universidad Nacional del Litoral in chemical and food engineering. Her research interests are in chemical reaction engineering and heterogeneous catalysis by zeolites. Mario Ricardo Sad is an assistant professor of chemical reaction engineering. He received his bachelors degree from Universidad Na cional del Litoral in chemical engineering. His research interests are in chemical reaction engineering and catalytic reactions of hydrocarbons on metallic catalysts. Alberto Antonio Castro is a full professor in chemical reaction engineer ing. He is dean of chemical engineering faculty and director of Institute of Research in Catalysis and Petrochemistry (INCAPE). His research interests are in chemical reaction engineering and catalytic reaction of hydrocarbons by metallic catalysts. Teresita Francisca Garetto is an associate professor of chemical reac tion engineering. She received her bachelors degree from Universidad National del Literal and her Ph.D. degree from University of Zaragoza, both in chemical engineering. Her research interest are in environmental catalysis and catalytic reactions of hydrocarbons. tion Engineering I, which also has other practical activities of different design that range from pilot scale experiments with semiindustrial equipment to computational simulations of real processes. CONTENTS: THEIR RELEVANCE AND DESCRIPTION Several authors have reported studies with adiabatic reac tors using the strongly exothermic reaction between sodium thiosulfate and hydrogen peroxide in aqueous media. At first, this reaction was used to experimentally assess theoretical predictions about hot spots in discontinuous or plug flow reac tors and steady state multiplicity and hysteresis or oscillations in continuous stirred tank or recycle plug flow reactors.[37] More recently, this reacting system was used to evaluate some new techniques in control and optimization engineering and in risk management.[818] Heat transfer in chemical reactors is an important topic both from theoretical and practical points of view. A common practice in chemical reaction engineering courses (which may be taken in parallel with energy transfer courses by the students) is to assume very simple models of this phenom enon to avoid further complications in the classical chemical reactor models. Another aspect added to thermal kinetic data processing is the incorporation of the "time lag" concept in the temperature measurement. This concept naturally arises in the course of process control, which students take normally afterward, but may be easily modeled as a simple, first order heat transfer lag to help introduce students to the idea of noninstanta neous measurement. In this case, chemical reaction is very rapid, and we have chosen a sensor (digital thermometer) with a deliberately high thermal mass so the lag effect is more noticeable. THEORETICAL FUNDAMENTALS AND MATHEMATICAL MODEL The reaction between sodium thiosulphate and hydrogen peroxide in aqueous solutions is highly exothermic (AHR ~ 147 kcal/mol Na2S203) and practically irreversible. The cur rently accepted stoichiometry for the reaction is: 2Na2S203 +4H202 Na2S30 + Na2SO4 + 4H20 (1) 2A+4B C+ D +4E The kinetic behavior of this system has been well studied by several authors[3 6] and is first order with respect to each of the reagents: r = k.CA.CB [r]= gmolNa2S203 / cm3.s (2) The dependence of k with T will be taken as the usual Ar rhenius form: k =A.eE/RT (3) even though some slightly different form was reported in earlier studies. For a batch reactor the mathematical model is made up of two differential equations resulting from mass and energy balances. Concentration of reagents sodium thiosulphate (C,), hydrogen peroxide (CB), and temperature (T) can be described by using these balances, the stoichiometry ratio between both reagents, and the initial conditions: dCA / dO = k(T.CA .CB (4) V.Cp,.(dT / dO) = k(T).CA.CB.(AHR) A,.U.(T t) (5) CB = CB0 2.(CA CA) (6) T(0) = 0 CA(O) =AO CB(O) = CB (7) The thermal mass of the beaker used as a reactor has been neglected in the energy balance (Eq. (5)), due to its low mass (less than 4g), to avoid further complications in the proposed model. This set of differential equations can be solved using an appropriate numerical method (Euler, RungeKutta, etc.) in order to obtain the evolution of products concentration and temperature with time. It must be noted that this system may exhibit a "stiff'"17I18] behavior, so the numerical integration method should be a robust one.[191 For exothermic reactions such as this one, with heat ex change resulting from an external cooling media, the tem perature vs. time profile has a maximum value ("hot spot"). At this point: k(T).CA.CB.(AHR) = A.U.(T t) = dT / dO = 0 (8) After this point the temperature will decrease with a di minishing slope, as the reaction rate tends asymptotically to zero. At infinite time, the temperature within the reactor will be the same as that of the surrounding media: [k(T).C.CB .(AHR)1 = 0 (9) [dT /d '.. =(A,.U/V.Cp,).(Tt)= Toe = t (10) This behavior at sufficiently high times is useful to "inde pendently" estimate the value of U (from At.U/V.Cpv) using values of temperature/time in this time domain. For the vessel used as a reactor, values of V and At were 100 cm3 and ca. 120 cm2. The value of Cpv was 1 cal/g.K.81 These values were used to estimate the heat transfer coefficient. The "time lag" of the sensor (digital thermometer) was taken into account through the simple firstorder model:[22 24] dTR/ d = (1/ T).(T TR) (11) which may result from the finite heat transfer rate between the fluid, which is at the "real" temperature T, and some internal point in the sensor probe, which is at the read temperature TR. Both the high chemical reaction rate and the deliberate choice of a sensor with high thermal mass justifies the incorporation of the new dependent variable TR in the model with its ad ditional differential equation and a new parameter (T). Chemical Engineering Education EXPERIMENTAL The equipment used is simple, inexpensive and is comprised of the following elements: a 100 mL disposable, lightweight polystyrene beaker, a magnetic stirrer, a digital thermometer of the type used for frozen meat control at coldstorage plants (273 K/423 K range, 0.1K resolution) and a 500 mL crystal lizer to be used as an ice bath. A few millimeters of clearance was left between the bottom of the beaker and the stirrer plate to achieve a more homogeneous external heat transfer envi ronment. The chemical reactants employed are of analytical grade, but commercial hydrogen peroxide (1020 vol, with stabilizer) or commercial thiosulfate have been used with no substantial result alteration. A series of experiments is synthetically described below, for typical work performed by students that can be easily modified from operating conditions or sequencing. C EXPERIMENT #1 Chemical reaction involving heat exchange with the environment (kinetic experiment) This experiment consists of loading the reactor with 50 mL of a Na, S, ~ 0.8M solution, placing the thermometer in an adequate fixed position, and stabilizing the stirring rate to finally add 50 mL of HO, ~1.2M as rapidly as pos sible. Then, temperature readings should immediately start at appropriate time intervals. Figure 1 shows typical results obtained by students conducting this experiment at 303 K (room temperature). 370 360 350 2 . 340 3  330 I 320 310 300 0 10 20 30 40 50 60 1 Time (sec) Figure 1. Reaction between Na2S20, and H202 in a batch reactor with heat exchange by natural convection (room temperature: 300 K). Vol. 42, No. 1, Winter 2008 It was observed that the temperature increased rapidly at the beginning, went through a wellmarked maximum for about one minute, and then decreased increasingly at a slower pace. This results were obtained because when the limiting reactant is practically exhausted, the reaction rate becomes negligible and the system behaves as a simple vessel with fluid at a temperature close to room temperature, where it exchanges heat by convection (internally forced, externally natural) with low heat transfer coefficients.[2021] ( EXPERIMENT #2 Natural cooling of the vessel with no chemical reaction This experiment consists of following the liquid cooling (preferably using the same solution used in Experiment #1 after its completion) under the same conditions previously analyzed, from a temperature of about 353363 K. The final liquid from the previous experiment, preheated up to that temperature level, is left to evolve spontaneously under the same stirring conditions. Many time/temperature values are then registered (10 15 values) to estimate a value of the U global exchange coefficient under conditions similar to those of the kinetic experiment. To do so, it suffices to use the asymptotic integrated form of the energy balance differential Eq. (5): ln[(T0 t) (T t)]= (A,.U V.Cpv). = ca.0 (12) The slope of a straight line in the graph of ln[(T t)/(T t)] vs.6 is the value of a. Figure 2 (next page) shows the results of an experiment of this type together with statistical analysis of the data. ( EXPERIMENT #3 Determination of the thermometer time lag (time constant) This experiment should also be performed with the liquid sample of the first experiment, preheated to a convenient temperature level (343353 K) and existing under the same conditions (stirring, etc.). In this case, the thermometer (previously in thermal equilibrium at room temperature) is introduced and fixed as quickly and as smoothly as possible and the registered temperature evolution is read as a function of time (given the ordinary response times of tools such as the thermometer employed, the readings should be carried out at a frequency of about 25 s). An essential condition for this experiment is that the temperature of the liquid mass be constant during the experiment. Since the time consumed is short, this condition is practically accomplished. The integrated form of differential Eq. (11) yields the classical step input time response for a first order system: In[(TR T,) /(T R T) = ( ] ( r).O from which the value of time constant, i by regression of the experimental data. Th "final" value shown in the display when (T, = TRI >). Figure 3 shows typical results as well as the estimation of T. Analysis of the data: Models and interpretation of results The results obtained with a typical kinetic experiment may be correlated using a simple model (Model I) inte grated by the mass balance (Eq. (4)) and energy (Eq. (5)) differential equations with proper boundary conditions and parameters A, E and U (or a). They can also be correlated by a more so phisticated model (Model II) including the differential equation accounting for thermometer time response with the ad ditional T parameter. On the other hand, the number of parameters (and hence the error associated with its determina tion) are reduced to 2 in both models if values of a and T, obtained from the analysis of the results of Experiments # 2 and 3, are taken as known. Figure 1 shows the adjustment of both types of models (with its associated floating parameters), and Table 1 summarizes the results obtained as well as the compari son between the parameters obtained and those from independent experi ments or reported in the literature. No tice the better adjustment of the model in Figure 1, including the time lag of the thermometer readings. Likewise, in Table 1, a good correspondence can be observed between the parameters de termined from the kinetic experiment, those from independent experiments, and those reported in the literature. This correspondence is more marked in the case of the model including the thermometer time lag. The interpreta tion and analysis of the results is usually performed using any available software capable of performing the regression of the temperaturetime experimental values, by way of numerical integra tion of the differential equations (with methods adequate for "stiff' systems) using any parameter search engine (13) , can be estimated ie value ofT is the it varies no longer such as simplex, steepest descent, Monte Carlo, etc. There is a lot of general software available for these ends (Matlab, Matematica, MathCAD, to mention just a few) and some directoriented free applications (such as Scientist from Micromath) to solve the problem of statistical determination 370 2.5 In [(TTmo)/(TTm)] = 0.001009.0 + 0.0034 360 R2 0.9987 A.U/V.Cpv= 0.001009 2 350  1.5 F7 340 E " 330 E 1 l = 320 * 0.5 310 * 300 0 0 400 800 1200 1600 2000 Time (s) Figure 2. Determination of parameter a defined in Eq. (12). 360 8 .0 ..... 0 ..... 4 ... .... . 350 "" . ..  6 340  LJ 2I 330 . S 4 ^ 320 E ) : c  I : In[(TR TF)/(TR TF)]= 0.1259.08 310 R2 = 0.9973 0 1/ = 0.9973s, 2 300 290 q 0 0 10 20 30 40 50 60 Time (s) Figure 3. Determination of the thermometer time lag. Chemical Engineering Education of the parameters and/or to simulate the system. It is not the aim of this work to approach the wide field of mathemati cal and statistical methods employed in the correlation and modeling of experimental data, so only the results are shown as long as alumni are not compelled to use any particular software. The analysis of data presented in this paper was done using software with Episode (stiff) integrator, which includes the analytical calculus of the Jacobian matrix and a least square method (minimizing relative error) for the parameter adjustments. Testing the predictive power of the model: Ice bath experience An additional experiment to assess the usefulness of the model to predict the behavior of the system can be done by modifying the heat transfer scenario (i.e., using an ice bath as external transfer medium). In this case the values of kinetics parameters (A and E) and the sensor time constant are taken as the previously determined ones, and the heat transfer param eter is separately determined in an individual cooling experi ence as described before (as heat exchange coefficient value is expected to change for the ice bath surrounding media) or estimated from temperaturetime values for higher times. Typical results of such experiences are not shown here, but in Fig. 4 a good concordance can be seen between predicted and 0 20 40 60 80 100 Time (min) Figure 4. Testing the predictive model in ice bath experience. measured temperature vs. time profiles in this case, confirming the usefulness of the model for predictive purposes. CONCLUSIONS Some very simple laboratory experiments can be inexpen sively modified to obtain higher pedagogical output through a more open analysis and correlation of obtained data and some minor operative modifications, including concepts and methods that strictly pertain to another (but closely related) area of knowledge. Modeling and simulation, two fundamen tal tools in chemical engineering, should be used despite the modest quality of equipment and obtained data to illustrate important aspects of the formulation and validation of mod els, design of complementary experiences for estimation of parameters, and the utilization of current numerical analysis and statistical procedures and software. Rather surprisingly in this case, as long as only one variable is measured, the results of parameter estimation and simulation are good and constitute an additional stimulus for students. NOMENCLATURE A Arrhenius preexponential factor [cm3 mol /s 1] A, Effective heat transfer area [cm2] C Molar concentration [mol/cm3] Cpv Volumetric heat capacity [cal/cm3/K 1] E Activation energy [cal/mol 1] k Kinetic constant [cm3 mol /s 1] r Reaction rate [mol/cm 3s 1] R Gas constant [cal/mol 1/K 1] T Temperature of system, real [K] t Temperature of surrounding media [K] T Initial temperature [K] T Constant temperature of medium in Experiment #3 [K] TR Digital thermometer temperature reading [K] TRO Initial digital thermometer temperature reading [K] U Global heat transfer coefficient [cal/s 1/cm 2/K 1] V Reactor volume [cm3] 0 Time [s] T Thermometer time constant [s] a Parameter defined as At.U/V.Cp [s 1] AH, Enthalpy change of reaction [cal.mol 1] REFERENCES 1. Feisel, L.D., and G.D. Peterson, "The Challenge of the Laboratory in Engineering Education," J. Eng. Ed., 91(4), 367 (2002) TABLE 1 Comparison of Parameter Values PARAMETER VALUE OBTAINED VALUE OBTAINED VALUE OBTAINED VALUE REPORTED IN MODEL I MODEL II INDEPENDENT THE LITERATURE EXPERIMENT A 2.1010 2.1010 2.1010 6.85.10" E (Cal/mol) 8120 8238 82389200 U (KCal/s m2 K) 0.0031 0.0035 0.0036 0.0020.02 T (s 1) 0.1259 0.1259 Vol. 42, No. 1, Winter 2008 292 288 284 a. E 280 I  Simulation o Experimental 2. Feisel, L.D., andA.J. Rosa, "The Role of the Laboratory in Undergradu ate Engineering Education," J. Eng. Ed., 94(1), 121 (2005) 3. Root, R.B., and R.A. Schmitz, "An Experimental Study of Steady State Multiplicity in a Loop Reactor," AIChE J., 15(5), 670 (1969) 4. Vetjasa, S.A., and R.A. Schmitz, "An Experimental Study of Steady State Multiplicity and Stability in an Adiabatic Stirred Reactor,"AIChE J., 16(3), 410 (1970) 5. Chang, M., and R.A. Schmitz, "An Experimental Study of Oscillatory States in a Stirred Reactor," Chem. Eng. Sci., 30, 21 (1975) 6. Guha, B.K., G. Narsimhan, and J.B. Agnew, "An Experimental Study of Transient Behaviour of anAdiabatic ContinuousFlow Stirred Tank Reactor," Ind. Eng. Chem. Process Des. Dev., 14(2), 146 (1975) 7. Schmitz, R.A., R.R. Bautz, WH. Ray, and A. Uppal, "The Dynamic Behavior of a CSTR: Some Comparisons of Theory and Experiment," AIChE Journal, 25(2), 289 (1979) 8. Grau, M.D., J.M. Nougues, and L. Puigjaner, "Batch and Semibatch Reactor Performance for an Exothermic Reaction," Chem. Eng. and Proc., 39(2), 141 (2000) 9. Ferrouillat, S., P Tochon, D. Della Valle, and H. Peerhossaini, "Open Loop Thermal Control of Exothermal Chemical Reactions in Multi functional Heat Exchangers," Int. J. of Heat and Mass Transfer, 49 (1516), 2479 (2006) 10. Bhanot, S., S. Kumar, and M.K. Vasantha, "Online Instantaneous and Delayed Control of a Fast Tubular Reactor," TENCON '98. 1998 IEEE Region 10 International Conference on Global Connectivity in Energy, Computer, Communication and Control, Vol. 2, New Delhi, India, 448 (1998) 11. Palanki, S., and J. Vemuri, "EndPoint Optimization of Batch Chemi cal Processes," Proceedings 42nd IEEE Conference on Decision and Control, 5, 4759 (2003) 12. Kulkarni, A.J., S.V. Patil, and V.K. Jayaraman, k!..... Based Local Learning: Application to Process Engineering Problems," Int. Journal of Chem. Reactor Eng., 1, A23 (2003) 13. Prat, L., A. Devatine, P Cognet, M. Cabassud, C. Gourdon, S. Elgue, and E Chopard, "Performance Evaluation of a Novel Concept Open Plate Reactor Applied to Highly Exothermic Reactions," Chem. Eng. & Tech., 28(9), 1028 (2005) 14. Ausikaitis, J., andA.J. Engel, "The Behavior of the Iron(III)Catalyzed Oxidation of Ethanol by Hydrogen Peroxide in a FedBatch Reactor," AIChE J., 20, 256 (1974) 15. Ruiz, D., "Fault Diagnosis in Chemical Plants Integrated to the Infor mation System," Thesis, Universitat Politecnica de Catalunya (2001) 16. Shukla, PK., and S. Pushpavanam, "Parametric Sensitivity, Runaway, and Safety in Batch Reactors Experiments and Models," Ind. & Eng. ( I ..... Research, 33(12), 3202 (1994) 17. Gao, Q.Y., Y.L. An, and J.C. Wang, "A Transition from Propagating Fronts to Target Patterns in the Hydrogen PeroxideSulfiteThiosulfate Medium," Phys. Chem., 6(23), 5389 (2004) 18. Karer G., G. Musi, and B. Zupan, "Predictive Control of Temperature in a Batch Reactor with Discrete Inputs," Proceedings of the 13th Mediterranean Conference on Control and Automation Limassol, Cyprus, 855, June 2729 (2005) 19. Solodov A., and V. Ochkov, "Stiff Differential Equations" in Differen tial Models. An Introduction with Mathcad, Springer Berlin Heidelberg (2005) 20. 21. Perry R.H., and C.H. Chilton (Eds.), Chemical Engineers'Handbook, 5th Ed., McGrawHill Book Company, Chapt. 10 (1973) 22. Sutton, C.M., "In Situ Measurement of Resistance Thermometer Self Heating and Response Time," Meas. Sci. Technol., 5, 896 (1994) 23. Dally, J.W., WE Riley, and K.G. McConnell (Eds.), "Instrumentation for Engineering Measurements," 2nd Ed., Iowa State Univ. (1993), 11.6 Dynamic Response of Temperature Sensors. 24. Chemical Engineering Education classroom  ^ K.___________________________ ACTIVE PROBLEM SOLVING AND APPLIED RESEARCH METHODS In a Graduate Course on Numerical Methods ERIC L. MAASE AND KAREN A. HIGH Oklahoma State University Stillwater, OK 74078 For the past 20 years, educators in the engineering field have implemented ways of better engaging students, including active and cooperative learning, learning communities, service learning, cooperative education, inquiry and problembased learning (PBL), and team projects.[1 Peda gogies of these types have been shown to clearly be effective in allowing students to improve their learning. It is no longer important to consider simply the content covered in a course, but how the material is going to be taught. Researchers at McMaster University[21 found that to solve problems successfully students need both comprehension of chemical engineering and what are called general prob lemsolving skills. The researchers developed workshops to explicitly develop skills deemed appropriate for improving students' problemsolving skill and confidence. Prince,31] in his article on active learning, provides additional motivation for incorporating problembased learning in the classroom. One benefit that he shows is that PBL typically involves significant amounts of selfdirected learning on the part of the students. The professor is no longer solely respon sible for delivering course content. Prince provides extensive and credible evidence suggesting that faculty should consider nontraditional models for promoting academic achievement and positive student attitudes. Based on the above evidence of the value of PBL and active problem solving, the instructors reconstructed their graduate course in Chemical Engineering Modeling. In Fall 2006, the instructors introduced components to the course exposing Eric L. Maase is a visiting assistant profes sor at Lafayette College and a research associate in the Laboratory of Molecular Bioengineering at Oklahoma State Univer sity. He received his B.S. degree from the University of Maryland, his M.S. degree from the Colorado School of Mines, and his Ph.D. from Oklahoma State University. His research interests include biochemical/biologicalmod eling and simulation, polymer science and tissue engineering, thermodynamic modeling of electrolytes, computational modeling and design, and engineering education. Karen A. High earned her B.S. and M.S. degrees from the University of Michigan and her Ph.D. from Pennsylvania State University. She is an associate professor in the School of Chemical Engineering at Oklahoma State University. Her research interests are Sus tainable Process Design, Industrial Catalysis, and Multicriteria Decision Making. She is a trainer for Project Lead the Way preEn gineering curriculum. She is also involved with the development of an entrepreneurship minor at Oklahoma State University. Copyright ChE Division of ASEE 2008 Vol. 42, No. 1, Winter 2008 students to case studies, active problem solving, teamwork, and experimentation with an the aim of promoting creative and critical thinking as students identify and practice the art of modeling. MODELING COURSE The Chemical Engineering Modeling course has been structured to give the students experiences to meet the fol lowing objectives. At the end of the semester, the students should be able to: (a) Develop mathematical models describing chemical engineering phenomena. (b) Evaluate the assumptions, limitations, and restrictions necessary to solve practical problems by mathematics. (c) Use classical numerical techniques to solve the equa tions that resultfrom model formulation (ordinary and partial differential equations, and linear and nonlinear simultaneous algebraic equations). (d) Become familiar with available computational tools that incorporate these numerical techniques (specifi cally in MATLAB and VBA/Excel). The course topics include: (a) Classification of Equations (b) Matrices (c) Model Classifications/Concepts (d) Nondimensionalization (e) Taylor's Series (f) Ordinary Differential Equations (g) Linearization (h) Linear/Nonlinear Equations (i) Finite Differences (j) Initial Value Problems (IVPODEs) (k) Boundary Value Problems (BVPODEs) (1) Partial Differential Equations (PDEs) (m) Applications of .1. i,,,, .' i, I / Analysis All students in the class are graduate status and, more im portantly, are also in their first semester of graduate school. The course is a requirement for all chemical engineering graduate students at Oklahoma State University. Students are concurrently enrolled in graduatelevel thermodynamics and diffusional operations courses during this semester as well. The diffusional course covers the equations of continuity and focuses on analytical solutions to equations. Activities and instructional materials teaching program ming and problemsolving principles using Visual Basic for Applications (VBA) linked through Microsoft Excel and MATLAB have developed during previous semesters as 24 course materials. These computer tools are employed by the students in solving algebraic, ordinary differential equations (initial value and boundary value) and partial differential equations (parabolic and elliptic). The instructors predominantly use two references during the semester that help students develop and set up mathematical models of physical phenomena. The Smith, Pike, and Murrill (SPM) 41 methodology employs an "IOGA" (input, output, generation, and accumulation) approach of building models using a control volume. The Himmelblau and Bischoff (HB))51 methodology relies on beginning with equations of continuity and "crossing" off terms that do not apply for any specific situation. The course includes homework assignments in mathematical theory (paper solutions), computer exercises in numerical methodsboth programming with VBA and MATLAB and exams evaluating student's understanding. In past years students have worked on projects that have them evaluate the effectiveness of the ODE boundary value problem (BVP) strategies of matrix method, shooting method, and the successive overrelaxation method (see Hornbeck&61 and Ri._' ~ I. The students are given the opportunity to pick chemical phenomena that give rise to ODE BVPs. Prior students have stated that they learned a great deal from such projects but also have wondered if there were anyway to be given a project that allowed them to model actual experimental phenomena. AN ACTIVE LEARNING ENVIRONMENT In modification to the course the instructors introduced a semesterlong group project where the students perform experimental work in a lab, report and discuss their efforts via memoranda to the instructors, and apply mathematical theory, numerical methods, and developing modeling skills to reach a specific goal. The students were presented with the following problem statement at the beginning of the semester: "Design an encapsulated drug (ofyour group's choice) that effectively delivers an appropriate dose for an appropriate number of hours. Select the most .., i f.. in.' method." This problem statement is intentionally both inadequately defined and openended. Defining the problem and the over all goals of the group projects is an aspect of the students' learning. Other educational objectives inherent in the problem state ment include: Consider different types/scales of phenomena (micro scopic, multiple gradient, maximum gradient, macro scopic). Understand what physical models and mathematical models are. Consider unsteady phenomena. Chemical Engineering Education Develop models that cover as many types of phenom ena as are appropriate. Develop the maximum gradient modelfor spherical coordinates. Develop experimental protocol around ..... i,,,i drug delivery with Tootsie Pops. Consider numerical techniques to solve the problem (ODE vs. PDE). The class meets twice weekly (Tuesdays and Thursdays) and was structured so that Tuesday classes during the first third of the semester are in computer labs and students are provided instruction in VBA/Excel and MATLAB. Thursday lectures provided instruction on mathematical principles and modeling methodology. Over the remainder of the semester, Tuesday lecture periods served as times for students to be engaged either in group work or with instructors providing support and mentoring in sessions where specific content could be presented (lab safety discussion, review sessions, Figure 1. Drug (lollipop) dissolution. Vol. 42, No. 1, Winter 2008 presentation skills, etc). The students are initially daunted by the possibilities inherent in the project but also are will ingand in some cases, quite eagerto rise to the challenge that the project represents. Two individual case studies, which are introduced at separate times, provide the basic impetus for the students' project development and direction. Discussions of projects during class periods and project status reporting (memos) provide additional mentoring moments for the in structors over the course of the semester. To accommodate the extra time required in class and to minimize lecturing, a variety of reading protests (administered as online quizzes in Blackboard) were given to ensure stu dents comprehended material prior to class. Also, it allowed the instructors to shorten "lecture" time and increase active learning experiences. The overall approach taken is intended to encourage critical thinking and creativity in the process of problem solving by con necting applications of the mathematical and numerical methods taught in lectures more closely to other engineering research methods. The restructured course also mimics experiences that students should expect to encounter while working towards a thesis (M.S.) or dissertation (Ph.D.) in a graduate program. Homework assignments and exams were administered to students to ensure that they were able to individually learn and demonstrate knowledge in the content. The overall course grade was from exams (60%), individual homework (20%) and the group project (20%). This ensured that the students were learning all of the course objectives. PROJECT GROUPS At the start of the semester, the students were put in groups. The questions in Table 1 were used to ensure heterogeneity in student groups based on experience, gender (when determin able from name), and ethnicity. INTRODUCTION TO LABORATORY METHODS Once the students were in groups, the instructors led an ex perimental session where the students began their project work by examining a dissolution case study. A laboratory was set up that was designated for the students use (see Figure 1). TABLE 1 Questions Used to Group Students Encapsulated Drug Project, Aug. 31, 2006 On the card, write: Your name Your home (i.e., China, Michigan, Antarctica) Your degree (i.e., M.S. Chemical Engineering, Ph.D. Chemistry) Your computer experience (High/Medium/Low) Your experimentation experience (High/Medium/Low) Your transport equation experience (High/Medium/Low) Your numerical method experience (High/Medium/Low) The experiments are driven by the goals and basis outlined in an initial case studyproviding a starting point for the student projects. The intent of the initial lab period is also to provide instruction on lab notebooks and experimental con cerns such as safety and experimental measurement, as well as to outline the challenges that experimental work entails. The class would later discuss and determine the safety rules for the lab with all students signing the resulting safety agree ment as shown in Table 2. A discussion of this nature resulted in a more openended perspective on safety along with a dif ferent rational for and understanding of other users' feelings in a laboratory environment and their sense of safety. CASE STUDIES Case studies served as the primary creative impetus for students' projects. While the students may be eager to undertake the challenge posed by the project they also need a sense of the structure that allows such studies to be successful. The initial case study provides their starting point. Case Study #1 Experimental Investigation and Modeling This case study, from Jensen,[81 is intended for freshman students. The goal is for students to investigate the factors affecting the dissolution of a proposed coating for a new pharmaceutical. The story is written from the perspective of two chemical engineering students in "ChemE 101," a fictional course. The learning objectives for the students reading and working on the case study are: 1) describe the factors affecting mass transfer between phases; 2) collect and evaluate data on mass transfer; and 3) evalu ate a model to describe mass transfer. These objectives remain appropriate for the graduate students' projects as well. The intent of the case study is simply to first determine the appropriateness of the linear model for a lollipop (the students used Tootsie Pops) in order to experimentally investigate a system having an encapsulating material over an inner "drug" material. The students were then tasked with developing and outlining their own project: developing a better model of encapsu lated drug delivery, as can be seen in Table 3. As students were completing the initial experimental aspects of the project, they were allowed to ask for materials to be provided for them to help solve various problems. This exposed the students to a justintime teaching[91 or lecturebydemand strategy. Once students' initial experimental and numerical techniques have been applied to Case Study #1 and they have begun further experiments of their own designs, a second case study is introduced. The purpose of the second case study is to introduce the concepts of com putational modeling and refocus students on the overall 26 aims of encapsulated drug delivery. This case study is dis cussed after the necessary computer instruction (VBA/Excel and MATLAB) is complete. Case Study #2Modeling and Numerical Simulation The first step in any modeling process is to simplify the system of interest into essential aspects that can be math ematically described while still retaining essential features of the larger realworld system of interest. This case study provides an example of a model of a human body interacting with drugs delivered orally. The second case study arises from an assignment previ ously developed and used in an undergraduate Computer TABLE 2 Laboratory Safety/Practices for 307 EN CHE 5743 Class Lab 1. Safety glasses (eye protection) are encouraged 2. Move deliberatelyno running 3. No more than 2 groups and 1 individual in 307 at one time 4. No food/drink/gum 5. Closed shoes and full pants (or skirt) will be worn all times in 307 6. Keep lab clean and equipment stored in a neat fashion 7. Unplug all equipment when done 8. Close door and lock when the last person leaves Name (print) Signature TABLE 3 First Project Memo To: Students of Chemical Engineering 5743 From: The Instructor Date: Aug. 31, 2006 Subject: 1st group memo for encapsulated drug class project. A 12 page memo is due Sept. 7 at the beginning of class that answers/consid ers the following. 1) Initial experimental protocol for your physical model a. Equipment/supplies neededyour group will be responsible for procurement b. Data collection procedure c. Other information 2) Initial computer modeling/numerical methods for mathematical model a. Computer software to be used b. Numerical methods to be used c. Other information 3) Simplifications made to start the initial work 4) Future considerations that more appropriately model the real system 5) Questions for the instructor. 6) A reference list to help in the project. Include at least 2 references per group member. Chemical Engineering Education Programming for Engineers course taught at Oklahoma State University by the coauthor. The case study outlines a model of the human body as a set of three interacting compartments where each represents a broader aspect of human physiology deemed important in drug delivery. The three interacting com partments are referred to as the oral, central, and peripheral compartments. The oral compartment represents primarily the mouth and stomach where a drug is gradually dissolving. The central compartment represents both the blood stream (circulatory system) and those organs that are highly perfused (liver, kid ney, lungs, etc.). The peripheral compartment includes muscle and tissues where diffusion of medications is often slower and where drugs may build up in concentration over time. Case Study #2 also defines the specific interactions among the compartments with governing equations for the transfer of a drug to and from compartments, as shown in Figure 2. These equations include rates/pathways for a drug leaving or entering the central compartment, where k is the rate drugs are eliminated from the body, k, is the drug metabolizing, and k is the drug being further absorbed (to the peripheral compart ment). All of the rates are dependent on the concentration of the drug currently in the central compartment, Co. The central compartment balance also includes terms describing the drug entering from the oral compartment, kCo, and returning from the peripheral compartment, kpCp. The case study tasks students with creating a computer program capable of solving the drug delivery model, inves tigating simulated behaviors, and discussing the results of their efforts. The overall perspectives and intentions of Case Study #2 prove valuable for the graduate course as well. In extending the basic princi ples outlined by Oral the second case dC study, graduate d =kAo students will ex I, & plicitly consider multiple drug L doses, statisti cal uncertainties Central inherent in hu cc /man reactions cdk= E +, )C+C to medications, the limitations Sand assumptions necessary for Peripheral c, A4 Figure 2. = kpCp+kcC Case Study #2: d1 Drug delivery model. Vol. 42, No. 1, Winter 2008 complex system modeling, and applications of numerical methods discussed in the course as tools for solving systems of equations. CASE STUDY USE Case studies served as the primary impetus for students' projects in the course. Through use of the first case study stu dents were provided an initial direction from which they could develop their own experimental goals and were presented with the realities inherent in both mathematical modeling and experimental work. The second case study provided a different perspective for the students where the challenge became more open ended and constructing better modelsin tractable mathematical terms able to capture essential behaviors of the actual system of interestwas a goal. These models and related simulations provided additional paths for examination of complex systems where important factors and influences can be estimated and examined computationally when experimental data or meth ods are difficult at best. The modeling and numerical methods in computer simulation complemented the experimental laboratory work initiated with Case Study #1. The second case also served to remind the students of the overall project goals) as outlined in the original problem statement. Translat ing mathematical and conceptual models into computer simulations also tasks the students with understanding and integrating their previous experimental approaches (those completed as well as those under way) into a more com plex system description. This aspect of applied numerical investigations asks students to determine a better overall "picture" and to hopefully find a more effective resolution for the project goalss. The case studies used in the course are complementary tools in engineering research. The implantation of the cases in the course provided introductions for students to the role of laboratory experiments along with computational numeri cal simulations. PROJECT REPORTING Over the course of the semester, project status, learning, and student assessment (grading) are addressed by asking the student groups to prepare memoranda at appropriate points during the project. Table 4 (next page) lists the additional proj ect status memos that student teams were asked to prepare. FINAL REPORTS AND PRESENTATIONS The written projects were due the second to last day of the semester. Presentations were on the last day of the semester. Four days before the final presentations were given, the in structors provided the students with instruction on creating appropriate Microsoft PowerPoint documents and delivering an effective presentation. STUDENT WORK AND COURSE ASSESSMENT Student Project Examples One group undertook an investigation of the design of a cost effective controlledrelease drug delivery system for Digoxin.1101 In their efforts they proposed a design a four layer delayedrelease pillintended to sustain optimal blood plasma concentration over a 24hour period; the proposed TABLE 4 Project Status Memos 2nd memo due September 25th at the beginning of class. 1) Revise your 1st memo considering that your first experiment should take 1 hour and 15 minutes (class time). You should keep this very simple and consider using the physical and mathemati cal model in case study #1. You need a very detailed plan for your experiment. Where will you get the equipment for the experiments? 2) Develop the mathematical model in the case study using the: a. Himmelblau and Bischoff approach b. Smith, Pike, and Murrill approach 3rd memo due October 6th (4:30 p.m. 423 EN) 1) Develop the mathematical model in the case study using the: a. Himmelblau and Bischoff approach b. Smith, Pike, and Murrill approach Follow Example A approach. 2) Discuss your data and numerical model from your experi ments on Sept. 28. Use the inclass discussions from Oct. 3 as your guide. 4th memo due October 20th (4:30 p.m. 423 EN) 1) Find your initial experimental data from Sept 28. Plot the data r vs. t. Does equation 20.1 of the case study "fit" your data? 2) Rerun your experiment with a solid lollipop. Plot the data r vs. t. Does equation 20.1 "fit" your data? 3) Develop a nextlevel model (mathematical and physical) that gets your group toward the ultimate goal of designing an encapsulated drug. Clearly plot your data from your experiment and show how well it "fits" your data from your experiment (physical model). Discuss the appropriateness of the models (physical and mathematical). Don't get too fancy yet! 5th memo due Friday December 1 at noon in 423 EN 1) Tell us how your group is considering determining "cost effectiveness." 2) What are your aims (i.e., which drug are you using, how are you delivering the drug, where are you delivering it to)? 3) How are you focusing the problem statement into specific goals? 4) Progress on physical model and experiments. 5) Progress on mathematical model. 6) Back up your claims with references to literature. What are you using for your main references? Include an accurate and complete bibliography that you have been compiling since the start of the project. fourlayer coating dissolution is shown in Table 5. The students generated a conceptual model, determined simplifications and assumptions, and created numerical simu lations of their system. An example is shown in Figure 3 (the graph is unedited student work). The work shown represents only one example of the students' thinking and submitted work during the course. In their final report, the economics of the proposed drug delivery method were outlined along with their current in vestigations to date. Overall the work submitted during the course, along with the interactions the instructors had with the students, showed students to be very actively engaged with course materials and found to be meeting or in many instances exceeding the instructors' goals for the reconstructed course. Survey Assessment An endofcourse assessment was given to the 13 graduate students in the course. University Institutional Review Board approval was given for the survey. The students gave consent for the authors to use the results of the survey in presentations and publications. The survey was anonymous and examined after the conclusion of the semester. The survey had three sections. Section 1 Q1. I can appropriately use a lab book. Q2. I am confident with my experimental skills. Q3. I am comfortable working in groups/on teams. Q4. I am able to give effective presentations using Power Point. Q5. I feel confident that I can use VBA/Excel to solve problems in this course. Q6. My numerical methods skills are appropriate to solve problems in this course. Q7. I feel confident that I can use MATLAB to solve problems in this course. Q8. I am able to develop mathematical models that ap propriately consider relevant phenomena. Q9. I understand the need to follow appropriate lab safety procedures. TABLE 5 Student Designs: FourLayer Pill Layer Coating (hrs dissolve) Digoxin (pg) 1 .167 2.35 2 5.83 0.9 3 4.67 0.9 4 4.67 0.9 Chemical Engineering Education Q10. I enjoy a problembased learning environment. Q11. I can search the literature for relevant background material for projects. Q12. I understand the components of a research report / proposal. Q13. I know what goes into a developing a good presenta tion. The students were given a 5point Likert scale to evaluate their agreement with the above statements. Table 6 shows the results for Section 1 of the assessment. All scores listed were above neutrality, indicating a positive agreement of all questions. The students felt the most agree ment with the statement "I enjoy a problembased learning environment." In the authors' opinion the students gained a lot from the active learning experience and were engaged throughout the entire project and course. The next highest agreement was for the statement "I know what goes into developing a good presentation." The instructors felt that this is principally due to the class period four days prior to class end, in which presentation skills were discussed. The students had the lowest level of agreement for the statement "I feel confident that I can use MATLAB to solve problems in this course." This is not surprising since the coauthors spent a majority of the class discussing solution strategies in VBA/Excel. The next lowest level 2.5 of agreement was for the statement "I am able to give effective pre sentations using PowerPoint." This 2 "" was interesting given their strong agreement to knowing what goes into a good presentation. One pos 1 5 sibility is that this result may be due E to language challenges that many of foreign students experienced. The i 1 students also had not given their pre sentation yet. The project was turned in on the same day as the assessment o 5 D Figure 3. Digoxin concentra _ tion in blood plasma. o and the presentation was the next day. Foreign students prob ably had anxiety about giving a presentation based on a project they just turned in and delivering a presentation the next day in a second language. Section 2 Q1. How would you rate your knowledge of MATLAB? Q2. Did you use MATLAB to solve problems in the course? Why or why not? Q3. How would you rate your knowledge of VBA/Excel? Q4. Did you use VBA/Excel to solve problems in the course? Why or why not? Q5. How would you rate your math skills, particularly eigenvalues and matrix calculations? Q6. How would you rate your knowledge of the transport equations (the equations of continuity for energy, mass, momentum)? Q7. How would you rate your knowledge of numerical methods (finite difference.)? Q8. How helpful were the MATLAB help sessions that Dr. High provided? Q9. How helpful were the VBA help sessions that Dr. Maase provided? 5 = Strongly agree, 4 = agree, 3 = neutral, 2 = disagree, 1 = strongly disagree. *Indicates where there is a tie in the ranking. Vol. 42, No. 1, Winter 2008 20 25 Time (hrs) Q10. How helpful was Blackboard to you? All of these questions had room for student comments. Table 7 shows assessment results. The highest rating for the five questions where students rat ed themselves was in their knowledge of numerical methods (Q7). They also felt confident with their math skills (Q5). The lowest was for their knowledge of MATLAB (Q1), again this wasn't surprising considering that VBA was emphasized. For the questions where the students responded to how helpful various resources were, they gave highest score for the VBA sessions and for the Blackboard classroom management system and lower for the MATLAB sessions. For Q2 (did you use MATLAB?), eight students said yes, three said no and two didn't answer. Student comments for why they used MATLAB included: "Only when we were assigned to" "Sometime because it is easier to use" "Its easy to use" And for comments why they did not use MATLAB: "Used VBA (more comfortable with)" "I don't have enough experience to work on it com fortably" For Q4 (did you use VBA?), 12 said yes and one didn't an swer. Student comments for why they used VBA included: "I love to use it." "For numerical methods and solve diff. equations" "Because I was trying to learn VBA." "It's very simple." "Easy to use and easily accessible." "(For) all problems because of ease of interface and prior background." Section 3 This section was where students could write freeform comments to the questions. Student comments are added after each statement. See the Appendix for comments. According to the assessment, it appears that the students appropriately developed their expertise in four areas: math ematical modeling; experimental strategies; numerical methods; Q1 and computer strategies on MAT Value 2.46 LAB and VBA/Excel. It is also Rank 5 clear that they enjoyed the prob lembased environment in which the course was administered. For Q1, Q3, Q5, Q6, Q7: 4=1 For Q8, Q9, Q10: 4=Extrem *Indicates where there is a ti CONCLUDING COMMENTS This project proved invaluable for beginning graduate students to identify the connection between modeling, math ematical descriptions, and experimental reality. It was also intriguing for the students to not only consider mathematical but physical models in which experimentation with the real system is, at best, impracticalas is the case in biological systems. The students gained valuable teamwork skills not common for many graduate students as well as improvement in communication skills. The assessments show that the stu dents were particularly engaged in the course. The approach taken encourages critical thinking and creativ ity in the process of problem solving, while connecting appli cations of mathematical and numerical methods more closely with other engineering research methods. The restructured course also mimics experiences that students can expect to encounter while working toward a thesis (M.S.) or dissertation (Ph.D.) in a chemical engineering graduate program. REFERENCES 1. Smith, K., S. Sheppard, D. Johnson, and R. Johnson, "Pedagogies of Engagement: ClassroomBased Practices," J. Eng. Ed., 94(1), 87 (2005) 2. Woods, D., A. Hrymak, R. Marshall, P Wood, C. Crowe, T. Hoffman, J. Wright, P Taylor, K. Woodhouse, and C. Bouchard, "Developing Problem Solving Skills: The McMaster Problem Solving Program," J. Eng. Ed., 86(2), 75 (1997) 3. Prince, M., "Does Active Learning Work?A Review of the Research," J. Eng. Ed., 93(3), 223 (2004) 4. Smith, C., R. Pike, and P Murrill, Formulation and Optimization of Mathematical Models, International Textbook Company, Scranton, PA (1970) 5. Himmelblau, D., and K. Bischoff, Process Analysis and Simulation, Wiley, NY (1968) 6. Hornbeck, R., Numerical Methods, Prentice Hall, Upper Saddle River, NJ (1982) 7. Riggs, J, An Introduction to Numerical Methods for Chemical Engi neers, Texas Tech University Press, Lubbock, TX (1994) 8. Jensen, J., "Chapter 20 Dissolution Case Study," in A Users Guide to Engineering, Prentice Hall Source, Upper Saddle River, NJ (2004) 9. Novak, G., A. Gavrin, W. Christian, and E. Patteson, JustInTime Teaching: Blending Active Learning with Web Technology, Benjamin Cummings, San Francisco (1999) 10. Weiss, M., and W Kang, "Inotropic Effect of Digoxin in Humans: Mechanistic Pharmacokinetics Pharmacodynamic Model Based on Slow Receptor Binding," Pharmaceutical Research, 21, 235 (February 2004) TABLE 7 Assessment Results for Section Two Q3 Q5 Q6 Q7 Q8 Q9 Q10 2.85 3.00 2.77 3.08 2.77 3.08 3.08 3 2 4 1* 3 1* 1* Extensive, 3=Pretty good, 2=Limited, 1= None ely, 3=Very, 2=Somewhat, 1= Not helpful e in the ranking. Chemical Engineering Education APPENDIX The following are freeform responses to the survey ques tions (comments are verbatim). Q1. Describe how well you thought the project allowed you to understand the use of physical models and experiments to simulate phenomena (particularly biological). "It made me understand physical and experimental drug dissolution." "I never knew how to model physical phenomenon before I enrolled in this course. It helped me improve my computer skills." "Oh great!!! Actually being new to grad studies, it was a welcome breeze of air." "The project was useful in .... ,,,, the concepts of physical models and experiments." "The project gave me some insight how to simulate a physi cal model and to develop mathematical model." "I think this project is very helpful to understand how to construct the physical models, carry out experiments, con struct and solve mathematical models." "It helped me a lot in developing a good understanding about the phenomena and in visualizing the problem." "I believe now I understand the concept of having a physi cal model and fiii ,, mathematical model." "Got a lot of knowledge about the mathematical model." "Very well. P 1, .;,,, pure experimental work to comprehend modeling was good." "The project idea was good. Saw relationship." Q2. Describe how well you thought the project allowed you to understand the use of mathematical models to simulate phenomena (particularly biological). "It was important as we can now understand how important it is to make simple assumptions. They make almostreal models with less effort." "The late equations could be easily coded in VBA, and the experimental results could be studied easily." "Yeah, it helped a lot, but biological systems are complex, so we had to make several assumptions to mathematically represent it." "I think this project helped me to understand how to convert the physical phenomena into mathematical models and know the mechanism of the phenomena to make suitable assumptions and negligibility." "It assisted me tremendously. Now I relate the methods learned to reallife situations." "This seemed a bit vague .1i ....,i ..t. we had some dif ficulty j i.ii;,, physical to mathematical." Q3. Discuss how the project helped you to improve your numerical method skills. "I am not sure it helps me much on my numerical method skills." "Not much, I was good at them." "Gave me a better understanding of Runge Kutta and other methods." "The project gave opportunity to solve numerically the ODEs." "By solving project, I used a lot of numerical methods, like how to solve algebraic equations, differential equations, and regress the parameters." "The project helped me in improving my numerical skills." "I can now understand numerical analysis and relate to reallife situations." "Really helped us to model realtime problems. "I felt like the project was very numericalmethod light." "I don't know how. But!!! It made my numerical method skill to improve!" Q4. Discuss how the project helped you to improve your computational skills. "It helps me use Excel and VBA." "Had homework, projects, and a good discussion in the lab." "May have helped more had I been serious i, J ii,, day 1." "Yes it helped me to improve as we faced the real life prob lems. " "This program needs to use Excel and VBA to solve the program. It's helpful." "Yes, it did improve my computational skills." "I felt like the project required very little computational skills." Q5. Describe your ability to develop a costeffective en capsulated drug. "Cost effectiveness is dependent on how many people need this and how I can optimize." "At this point, on the basis of literature review, I can de termine the factors which can/might affect the cost of drug, and they can be varied according to the requirement." "I can develop a costeffective encapsulated drug consider ing mathematical models, economic evaluation models." "I couldn't actually develop an exact costeffective project." Vol. 42, No. 1, Winter 2008 "No knowledge whatsoever. I wish you had given back ground info about drug analysis or economics." "I search the related information and get a rough knowl edge about the costeffective encapsulated drug." "The costeffective part was difficult to evaluate." Q6. Discuss the usefulness of the initial case study and Dr. Maase's case study to the final project. "The initial case study helped in understanding the aim of the project." "Dr. Maase's case study was very helpful in ,111, the code plus understanding the model." "Very helpful. Great instructors and great interactive course." "Very useful! Especially Dr. Maase's case study which gave me a better perspective of the model. "It was helpful as it gave insight how to simplify the real system to develop mathematical models." "They are all very useful to understand this project." "Initial case study and Dr. Maase's case study both give me an introduction, details, and hints to the final project." "Helpful guide to know how to start." Q7. Other comments. "Give us all a good grade finally, please." 7 "I felt like the project was at times taken to be more dif ficult than it really was. Many of the memo requests seemed identical." a Chemical Engineering Education Random Thoughts... STUDENT RATINGS OF TEACHING: MYTHS, FACTS, AND GOOD PRACTICES RICHARD M. FIELDER North Carolina State University REBECCA BRENT Education Designs, Inc. Myths about student ratings of teaching abound on every campus, usually accompanied by confident assurances that ratings are just popularity contests that reward entertainers and penalize the best teachers. (Re markably, the second group always seems to include the ones doing the assuring.) Some years ago we surveyed the myths and summarized the extensive research that showed most of them to be wrong."1 Now it's 15 years later and a lot more research has been done, with similar outcomes. Unfortunately the myths are still alive and well, so here is the 2008 version of what "everyone knows" about student ratings and how much of that wisdom is supported by research. Myth. Student ratings are not valid assessments of teaching quality. Fact. False. Thousands of research studies have shown that student ratings correlate positively with every other measure of teaching effectiveness, including alumni ratings, peer rat ings, administrator ratings, measures of learning (e.g., stan dardized tests, common exams in multisection courses, and ratings of student portfolios), and student motivation. 21 The magnitude of the observed correlations varies considerably across individual studies and a few studies report contradic tory results, but the weight of the evidence is clear. If students consistently say someone's teaching is good or bad, they're almost certainly right. Myth. The highest i. i, ii % go to the easiest courses. Fact. False. Up to a point, courses rated as more difficult on average get higher ratings than easier courses, with ratings only beginning to fall when courses reach levels of difficulty beyond the backgrounds of most enrolled students. In a recent study of 1,045 engineering, science, and humanities courses at two universities, Dee 31 found that student perceptions of Copyright ChE Division of ASEE 2008 Vol. 42, No. 1, Winter 2008 course workload were not significantly different for courses in the top and bottom quartiles of student ratings, with the marginally higher workload rating going to the courses in the top quartile. Myth. Bad teachers who are easy graders get higher evalu ations than good teachers who are strict graders. Fact. False. Individual instructors who give high grades rela tive to local averages may get higher ratings than they would if their grades were lh i, .,'"1 but no studies have turned up ineffective teachers who got high ratings just by giving high grades.E5] However, the possibility that it could happen sup ports the common recommendation to use multiple sources of assessment data. Myth. They may not like me now because I'm 1 Ii ... t n and maintain high standards, but in afew years they'll appreciate how good a teacher I was. Fact. Generally falseit happens sometimes, but not often.1 2] Alumni ratings correlate significantly with student ratings given previously to the same instructor. If your students think you're a great teacher now, most will still remember you fondly in the future, and if they think you're lousy, don't Richard M. Felder is Hoechst Celanese Professor Emeritus of Chemical Engineering at North Carolina State University. He is co author of Elementary Principles of Chemical Processes (Wiley, 2005) and numerous articles on chemical process engineering r and engineering and science education, and regularly presents workshops on ef fective college teaching at campuses and conferences around the world. Many of his publications can be seen at Rebecca Brent is an education consultant specializing in faculty development for ef f fective university teaching, classroom and computerbased simulations in teacher education, and K12 staff development in language arts and classroom management. She codirects the ASEE National Effective Teaching Institute and has published articles on a variety of topics including writing in un Sdergraduate courses, cooperative learning, Public school reform, and effective university teaching. expect to start getting holiday greetings from them in five or 10 years. Not all common beliefs about evaluations are wrong, of course. It's true that teachers who are enthusiastic and car ing tend to get better ratings than those who are reserved and distant, but so hal Enthusiasm and caring of instructors also correlate with motivation and learning of their students, 21 suggesting that the higher ratings are probably deserved. It's also true that other things being equal, elective courses tend to get higher ratings than required courses, upperlevel courses tend to get higher ratings than lowerlevel courses, small and moderately sized classes get higher ratings than very large classes, student ratings in engineering and the sciences are lower than ratings in other fields, and female instructors in engineering and the physical sciences get lower ratings than male instructors.[2] On average these effects are small, but they exist and should be taken into account when ratings are used to make decisions about such things as reappointment, tenure, promotion, and merit raises. In short, student evaluations have high levels of reliability and validity and should always be part of the process used to evaluate teaching. There are some aspects of a course that students are in no position to evaluate, however, including whether the course learning objectives are appropriate, the content is current with the state of the field, and the course adequately prepares the students for subsequent courses in the curriculum. Those things can only be evaluated by knowledgeable peers. Student ratings should therefore not be the sole source of teaching assessment data but should be supplemented with peer ratings and other measures of teach ing effectiveness.61] If different sources agree, as they usually will, it's a good indication that the overall assessment is a fair one; if they disagree, it's a red flag, and an effort should be made to find out what's going on. Since student ratings will undoubtedly remain central to teaching evaluation (as they should), everything possible should be done to make them as effective as possible. The following recommendations most of which are drawn from the papers cited in Reference 2address that goal. Use a rating form that has been developed with the as sistance of someone knowledgeable about educational measurements. There is a science to survey construc tion in general and educational rating instrument con struction in particular. Either use a form that has been developed and validated elsewhere, such as the IDEA Student Ratings of Instruction system ( or work with an education specialist on your campus or an external consultant. Don't trust i.,1,, collected from fewer than 10 students or less than 2/3 of a class, and don't make personnel decisions based on 1.,11, from a single semester. Use a few global or summary items with Likertscale (15) i ,r1,,, for summative evaluation (evaluation used to help inform personnel decisions), and a longer list of more specific items for formative evaluation (diagnostic evaluation used to help instructors improve their teach ing). Global items correlate more strongly with student learning than more specific items do, and you'll get a higher rate of return if there are fewer questions. When i. i, i1,, .i;,u, i, remember that they may be i, 7/ti affected byfactors other than the quality of instruction, such as the nature and level of the course, the class size, and the gender of the instructor. The IDEA system and TCETools include provisions for taking these factors into account. Try to persuade students that their j.,1,,, will be considered carefully and may have an impact on fac ulty personnel decisions and decisions about teaching assignments. If you can make this case convincingly, most students will take the ratings seriously and you should get a good rate of return. If you can't make the case, there is no reason the students should take the ratings seriously and you should not be surprised if they don't. REFERENCES 1. Felder, R.M., "WhatDo They Know, Anyway?"( ,.... 3 /., 2( ,), 134135 (1992), 2. (a) Cashin, WE., "Student Ratings of Teaching: Recommendations for Use," IDEA Paper No. 22, Center for Faculty Evaluation and De velopment, Kansas State University, January 1990, and Concerns About Student Ratings," New Directionsfor Teaching and Learning, 87, 315 (Fall 2001); (c) McKeachie, W.J., "Student Ratings: Their Validity of Use," American Psychologist, 52(11), 1218 1225 (1997); (d) McKeachie, WJ., Teaching Tips, llth Ed., Boston, Houghton Mifflin Co., 326328 (2002) 3. Dee, K.C., "Student Perceptions of High Course Workloads Are Not Associated With Poor Student Evaluations of Instructor Performance," J. Eng. Ed., 96(1), 6978 (2007) 4. Greenwald, A.G., and G.M. Gillmore, "Grading Leniency is a Remov able Contaminant of Student Ratings," American Psychologist, 52(11), 12091217 (1997) 5. Abrami, PC., WJ. Dickens, R.P Perry, and L. Leventhal, "Do Teacher Standards forAssigning Grades Affect Student Evaluations of Instruc tion?" J. Educational Psychology, 72, 107118 (1980) 6. Felder, R.M., and R. Brent, "How to Evaluate Teaching," Chem. Eng. Ed., 38(3), 200202 (2004), Teaching," ( Teaching Portfolios" ( All of the Random Thoughts columns are now available on the World Wide Web at http://www.ncsu.edu/effective_teaching and at http://che.ufl.edu/~cee/ Chemical Engineering Education curriculum 0 Introducing Stochastic Simulation of Chemical Reactions Using the Gillespie Algorithm and MATLAB: REVISITED AND AUGMENTED A. ARGOTI, L.T. FAN, J. CRUZ, AND S.T. CHOU* Kansas State University Manhattan, KS 66506 I n their contribution, MartinezUrreaga and his collabora Andres Argoti is a research associate in the Department of Chemical tors'11 have presented the stochastic simulation of a chemi Engineering at Kansas State University. He received his B.S. in Chemical cal reactionspecifically, a simple reversible chemical Engineering from Universidad Nacional de Colombia, Bogota, and his M.S. and Ph.D. from Kansas State University, both in Chemical Engineer reaction whose rate law is of the firstorder, or linear. Their ing. His major research interest is in the stochastic analysis, modeling, work is indeed useful and informative: Randomly or stochas and simulation of chemical and biochemical processes. tically behaving processes or systems are ubiquitous in the L.T. Fan, a University Distinguished Professor, holds the Mark H. and chemical or allied industries.[29] Margaret H. Hulings Chair in Engineering and is director of the Institute of Systems Design and Optimization at Kansas State University. He The current contribution involves two aspects and is served as department head of the Department of Chemical Engineering intended to complement and augment the work of Martinez between 1968 and 1998. Currently, his major research interests are in stochastic analysis and modeling and in processnetwork synthesis Urreaga and his collaborators[11 to enhance its usefulness, based on process graphs (Pgraphs). Fan received his B.S. in chemical First, the reacting system (i.e., the simple reversible chemi engineering from National Taiwan University, M.S. in chemical engineer ing from Kansas State University, and M.S. in Mathematics and Ph.D. in cal reaction) illustrated by them is modeled as a stochastic chemical engineering from West Virginia University. process, specifically a birthdeath process the most general prbcss, ifical a ir process he e stnt del ien Juan Cruz is a graduate research assistant in the Department of Chemical subclass of Markovian processes.10 "] The resultant model, in Engineering at Kansas State University where he is currently carrying out turn, gives rise to the master, or governing, equation11, 12] of his doctoral studies. Cruz received his B.S. in chemical engineering from the birthdeath process whose solution renders it possible to Universidad Nacionalde Colombia, Bogota; subsequently, he had a two year stint performing research at University of Puerto Rico, Mayagaez. His obtain the analytical expressions for the process' means and major research interests include nonaqueous enzymology, biomolecular higher moments about the means, e.g., the variances, which engineering, and molecular modeling and simulation. are collectively a manifestation of the process' inherent fluc Songtien Chou obtained his B.S. in chemical engineering from National tuations. Second, the master equation of the birthdeath process Taiwan University, and earned M.S. degrees in both chemical engineering is stochastically simulated via the Monte Carlo method. and statistics, and a Ph.D. in statistics, all from Kansas State University. is stochastically simulated ia the Monte Carlo method1He is teaching in the Department of Finance and Banking at Kun Shan The method is implemented by the timedriven approach[14 University in Taiwan; he was the department chair between 2000 and 5] in addition to the eventdriven approach[10, 1523] adopted by 2002. His research interests include the application of stochastic pro cesses, risk analysis, and environmental engineering. Department of Finance and Banking, Kun Shan University, YungKang City, Tainan Hsien, 71003 Taiwan Copyright ChE Division of ASEE 2008 Vol. 42, No. 1, Winter 2008 3. MartinezUrreaga and his collaborators11l on the basis of the Gillespie algorithm.l0, 17 18] The eventdriven approach entails the determination of the probability distribution of a random waiting time or period of quiescence19, 20]; the simulation clock is advanced according to this random waiting time. Unlike the eventdriven approach, the timedriven approach advances the simulation clock by a fixed time increment,114 15] which can be estimated as a function of the intensities of transition embedded in the master equation of the process. The time driven approach is especially useful when the probability distribution for the random waiting time in the eventdriven approach is exceedingly complex to determine. The means and variances of the random variables characterizing the birthdeath process have been computed in implementing the eventdriven and timedriven approaches. For validation, the analytical solutions from the stochastic model as well as the numerical results from the Monte Carlo simulation are com pared with each other. In addition, these results are validated by comparing the values of the means obtained from the solution of the deterministic model for the simple reversible chemical reaction as presented by MartinezUrreaga and his collaborators.[1] The two approaches for stochastic simulation are further illustrated with the photoelectrochemical disinfection of bacteria whose rate of decaying is linear. The results from simulation are validated by comparing them with the available experimental data[241 as well as with the analytical solutions derived from the corresponding stochastic model. MODEL FORMULATION The system under consideration comprises the populations of molecules of species A and B per unit volume that are be ing chemically transformed into one another according to the simple reversible reaction, A B (1) k2 Clearly, the population of A decreases and that of B increases when one molecule of A transforms into one molecule of B with forward reactionrate constant k1. Similarly, the population of B decreases and that of A increases when one molecule of B transforms into one molecule of A with reverse reactionrate constant k2. The system constitutes, therefore, a special class of Markov processes, the birthdeath process.11" 12] Thus, the numbers of molecules of A and B at time t, NA(t) and NB(t), respectively, can be taken as the random variables of the birthdeath process of interest. A realization of NA(t) is denoted by nA, and that of NB(t) by ng. At any time t, the total number of molecules in the system remains constant; hence, the following relationship holds; NA (0)+N(0)= NA (t)+N,(t) nAo +nBo = NA (t)+ N (t) (2) where NA(O) or nA0 is the number of molecules of A at the outset of the reaction, i.e., at t = 0. Similarly, NB(0) or ni0 is the number of molecules of B at t = 0. These two quantities are constant, thereby indicating that only one of the random variables in Eq. (2) is independent. By denoting the total number of molecules at t = 0, i.e., (nA0+nBO), as nT0, Eq. (2) can be rewritten as nT = NA t)+ NBt) (3) If NA(t) is selected as the independent random variable, NB(t) can be expressed in terms of NA(t), from the above equation, as NB (t) = nTO NA (t) (4) whose realization is naturally nB nTO nA Consequently, the state space for the current system is all the possible numbers of molecules of A, i.e., {nTO, (nT 1), ..., 2, 1,0}. MASTER EQUATION The derivation of the master equation of a stochastic process has been extensively elaborated in our previous contribution to this journal.12s5 For the birthdeath process of interest, the probability balance around any arbitrary state nA of the inde pendent random variable, NA(t), gives rise to1" 12] d P(nA;t)= W+ (nA 1;t)p(n 1;t) dt +W (n +l;t)p(nA +l;t) [W (nA;t)+W (nA;t) p(nA;t) (6) This derivation is also detailed inAppendix A. The term, p(nA; t), in the above equation denotes the probability of nA mol ecules being present at time t. On the basis of the firstorder rate law, the intensity of birth, W+(nA; t), and the intensity of death, W (nA; t), have been defined as"' dn W+ (nA;t) = = k2nB = k2 (nTO nA) (7) dt dn, W (nA;t) dnA kn, ,A dt respectively. It is worth noting that the intensities of birth and death can be defined on the basis of not only linear but also nonlinear rate laws, thereby giving rise to a variety of linear and nonlinear stochastic models. In fact, the formulation of stochastic models for processes obeying nonlinear rate laws has been presented in some of our earlier contributions.J26 28] MEANS AND VARIANCES The solution of the master equation, Eq. (6), renders it pos sible to obtain the mean and higher moments about the mean of the random variables, N (t) and NB(t); see Appendix B. Chemical Engineering Education Among these higher moments, the second moment about the mean, i.e., the variance, is of utmost importance: It signifies the fluctuations, or scatterings, of the random variable about its mean.[29] Eq. (6) in conjunction with Eqs. (7) and (8) yields the expression for the mean of N,(t), E[NA(t)] or (NA(t)), as (NA (t))= n AO (k +k2)k2 exp[(k +k2)t +k2 (9) (k +k2 nTO Note that in the particular case where nA0 = nT0, i.e., where nBO = 0, the above expression reduces to30, 31] (NA (t))= TO {k exp[(k, +k2)t +k2, (10) (kl +k2) In addition, the mean of NB(t), E[NB(t)] or (NB(t)), is, from Eqs. (4) and (9), (NB (t))=nTO 1 I AO (k +k2)k2 exp[(k +k2)t +k2 (11) (k, +k2) nTO J Similarly, the expression for the variance of NA(t), Var[NA(t)] or Ao2(t), is evaluated as ( (t)=k nTO 1exp[(k +k2)t]} 2 (k +k2) [nA k + k2) + k2 exp[(k+ k2) t + klk (12) "TO For the case, nA0 = n, this equation becomes[301 S(kt)= k 1 {1exp(k1 +k2)t]}NA(t)) (13) (7, (t) *kj +k2 where (NA(t)) is given by Eq. (10). The standard deviation, GA(t), is the square root of the variance, A2(t); thus, from Eq. (12), 1/2 A jt nw = TO (1 exp[(k,1 +k2) t]1/2 t (k + k2) nAO k1 k )+k2 exp[(k1+k2)t +klk2 (14) From Eqs. (4) and (12), the variance of NB(t), Var[NB(t)] or (B2(t), is St) nTO exp[(k +k2)t} S(k +k2) SnAO j(kik) + k exp[(k +k2)t +kk2 = (t) (15) From this expression, oB(t) is given by 1/2 TB (t)= nTO {1exp[(k k2)t}/2 (k, + k2) 2 22 [T ( k2) +k2 exp[(k +k2)t+ klk2 Vol. 42, No. 1, Winter 2008 TA(t) Randomly or stochastically behaving processes or systems are ubiquitous in the chemical or allied industries. In general, the stochastic simulation of chemical kinetics can focus on the temporal evolution of the number concentrations of entities comprising a reacting system or the spatial interactions among these entities. The coefficient of variation, CV(t), signifies the relative fluctuations of a random variable about its mean[32]; it is computed as the ratio between the standard deviation, o(t), and the mean, (N(t)). From Eqs. (9) and (14), the coefficient of variation of NA(t), i.e., CVA(t), is CVA (t) A (t) (NA (t)) 1/2 nTk0 1 exp (k +k2) (k + k2)t 1/2 nA k k2 k +k SnT 1 2 *+k TOk, + k) n k k2) k2 exp (k, (k l k 2) n TO 2 CVA (t)= 1 exp k nTO exp (k, + k2)t] kk 1/2 kL k2 k2)t]+k2 k2')t /2 1/2 no (nAk k k) +k exp[ (k1 +k2)t +kk2 TO17) nAO (k k2)k2 exp[(k +k2)t +k2 By substituting the expression for (NA(t)) as given by Eq. (9), the above equation can be rewritten as CVA (t)= A ) [ k2 jexp (NA (t)) k, +k,2 (k,+ k2)t] [k 1 (NA t)) k, +k2 Sk2 J nHAoexp[(k +k2)t] Sk nA exp2(k +k2)t}1/2 k, +_k2 j When n0 = nA0, this expression becomes CVA( t)= 1exp k k, +k2) 2) (NA (t))1/2 1 (NA (t)) where (NA(t)) is given by Eq. (10). In their contribution, MartinezUrreaga and his col laborators'11 indicate that the relative fluctuations of NA(t) about its mean (NA(t)), i.e., the coefficient of variation, CVA(t), are approximately of the order of ((NA(t)))1/2N (t), or( NA(t) 1/2 as revealed through Eqs. (18) and (19). STOCHASTIC SIMULATION The master equation of the birthdeath process, Eq. (3), is stochastically simulated by means of the Monte Carlo method. The stochastic simulation of chemical kinetics by this method has been known for many years.17, 18 21 3336] A number of the works devoted to it, however, fail to deal with the crux of any stochastic simulation, which is its capability to estimate not only the mean but also higher moments about the mean, especially the vari Chemical Engineering Education ance.[25] According to Haseltine and Rawlings,[37 "... stochastic simulation of chemical kinetics has received an increased amount of attention from the modeling community. .. ." In general, the stochastic simulation of chemical kinetics can focus on the temporal evolution of the number concentrations of entities comprising a reacting system or the spatial interactions among these entities. The former can be accomplished by resorting to the classical formulation of the Monte Carlo method,131 and the latter can be carried out by means of Kinetic Monte Carlo (KMC) methods, which take into account the spatial distribution of the entities at the atomic or molecular scale.[38 39] In this work, the Monte Carlo method is deployed for stochastically simulating the temporal evolution of the numbers of molecules of A and B for the simple reversible reaction; thus, the simulation of spatial interactions among these molecules is not considered. The two basic procedures to implement the method, one resorting to the eventdriven approach10 1523] and the other resorting to the timedriven approach,[14 15] are described in detail; these two ap proaches differ in the manner of updating the simulation clock of the process's temporal evolution. EventDriven Approach The eventdriven approach advances the simulation clock by a random waiting time, T, which has an exponential distribution.10, 19] No event takes place during the time interval, (t, t + T), and only one event occurs at the end of this time interval at which the state of the system is specified by the probability of transition corresponding to each event. The Gillispie algorithm[10 17 18] com prises a series of steps to perform the Monte Carlo simulation of Markov processes by the eventdriven approach. This algorithm has been recently extended[351 to accelerate the computing speed of the original formulation.17,18] Herein, Gillespie's original algo rithm, as presented by MartinezUrreaga and his collaborators, '1 has been revised and augmented to include the evaluation of the means, variances, and standard deviations of the birthdeath process of interest as described below. Step 1. Define the total number of molecules per unit volume, nTO, the total number of simulations, or trajectories, Zf, and the length of each simulation, tf. Initialize the simulation counter as Z < 1. Step 2. Initialize clock time t, datarecording time 6,[31] the realizations of NA(t) and NB(t) at time t and simulation Z, i.e., nAz(t) and nBz(t), respectively, and the realizations of NA(6) and NB(6) at time 6 and simulation Z, i.e., nA z(6) and nB z(6), respec tively, as follows: 00 to nAZ (to) nA0 nB.Z (to) TO nAZ (t) nA.Z (0) nAZ(t0) nBZ ) nBZ to) Step 3. Sample a realization u from the uniform random variable, U, on interval (0, 1). Estimate T according to the following expressioni10; (20) where W [nAZ(t);t] = k2[n TnA(t)] and W [nA,(t);t] = knAz(t) are the intensities of birth and death, Eqs. (7) and (8), respectively; see Ap pendices C and D. Step 4. Advance clock time as t < (t + T). Step 5. If (6 < t), then continue to the next step; otherwise, continue to Step 8. Step 6. Compute the sample means, variances, and standard deviations at time 6 as follows: a. Record the value of realizations at 6: nAZ () nA,Z t T) nB.z () nB.Z (t T) b. Store the sum of realizations at 6: z A.z A nAZ (0) Z=1 Z B,z () nBZ () z=1 c. Store the sum of squares of realizations at 6: z 4A,Z Zn) n2 z=1 z Zn,z 0) nB ,z (0) Z=1 d. Store the square of sum of realizations at 6: Z 4'BZK0) ZnB AZ(BI,Z z=l [ A,Z (0)2 B,Z(0) 2 e. Compute the sample means at 6[13 19 20]: 1 z 1 mAZ AZ AAZZ6A Z z1 Z 1 z 1 mBZ )z ()  MBZ()ZEB,Z() =BZ() Z Zl Z (21) (22) f. If 1 < Z < Zf, then compute the sample variances and standard deviations at 0[13 19, 20]: Vol. 42, No. 1, Winter 2008 W+ n (t);t +W n t);t n(lu)  W [nA, (t); t+W_[n,(t);t] sz (e)  nAZ AZZ) Z AZe)1 ) z 1) z znz O (Z 1) Z (23) sB,z (0) Z1 Z (Z n ))  nz(O) z Z1 (Z 1) z z ( (24) SAz(0) s zO (0)1/2 SBZ 0) 5s2z( ) ( 2 Step 7. Advance 6 by a conveniently small AO as 6 < (6+A6). If (6 < t), then return to Step 5; otherwise, continue to Step 10. Step 8. Determine the state of the system at the end of time interval (t, t +T). To accomplish this, sample a realization u' from the uniform random variable, U, on interval (0, 1) and compute the probability of transition corresponding to the birth event, w [nA,(t);t], as follows: W+ [nA, (t);t (2 w [nAZ (t); tl= 27) W+ nA, (t);t+ W [nA,Z (t); tK If {0 population of molecules of A increases by one; thus, nA, (t) [nA, (t T) nBZ (t) [nT nAZ (t) nAZ() nAz (t) nB () nZ (t) Otherwise, a death event occurs, i.e., the population of mol ecules of A decreases by one; thus, nA, (t) nA,Z (t ) nB.Z (t) [nT nA, (t) nAZ() nA,Z (t) nBZ (0) nB (t) Step 9. Repeat Steps 3 through 8 until tf is reached. Step 10. Update simulation counter as Z (Z + 1). Step 11. Repeat Steps 2 through 10 until Zf is reached. Given in Appendix E is the computer code in MATLAB for performing Monte Carlo simulation on the basis of the Gillespie algorithm via the eventdriven approach as presented above. TimeDriven Approach As briefly indicated at the outset of this contribution, the timedriven 40 approach[14 15] differs from the eventdriven ap proach: It advances the simulation clock by a fixed time increment of At, which is sufficiently small so that at most one or no event occurs during time interval (t,t + At). At the end of this interval, the state of the process is determined by the probability of transition corresponding to each event. The series of steps for implementing Monte Carlo simulation of Markov processes in general and the birthdeath process of inter est in particular by the timedriven approach is given below. Step 1. Define the total number of molecules per unit volume, nTO, the total num ber of simulations, or trajectories, Z,, and the length of each simulation, t,. Initialize the simulation counter as Z < 1. Compute time increment At as follows: 1 At=1 (28) WMt nA (hA;t) + wM (nA; t) I where WM(nA;t) and WM(nA;t) are the maximum possible values of the inten sities of birth and death, respectively. Step 2. Initialize time t and the realizations of NA(t) and NB(t) at time t and simulation Z, i.e., nA,(t) and nz(t), respectively, as follows: t to nAZ (t) nAO nB.Z (to) nTO A.Z (t) Step 3. Compute the sample means, vari ances, and standard deviations at time t as follows: a. Record the values of realizations at t: nAZ (t) nA.Z nB. (t) nB.Z b. Store the sum of realizations at t: z Z A.Z (t) nAZ (t) Z=1 Z c. Store the sum of squares of realiza tions at t: z QA.Z (t) Z T Z (t) z=1 Z ^B.Z (t) n n.z (t) z=1 Chemical Engineering Education d. Store the square of sum of realizations at t: z Z Bz (t) n BZ(t) [ Kz(t)2 . Z(t)12 e. Compute the sample means at t: mA.z (t) nA.z () = 7.z (t 7=1 Z z 1 mBz(t) ZnBz(t)= 7, (t) 71 Z Step 7. Repeat Steps 3 through 6 until tf is reached. Step 8. Update the simulation counter as Z < (Z + 1). Step 9. Repeat Steps 2 through 8 until Zf is reached. Given in Appendix F is the computer code in MATLAB for performing Monte Carlo (29) simulation via the timedriven approach as presented above. (30) f. If 1 < Z < Z,, then compute sample the variances and standard deviations at t: z (tt)  Z1) nZzt(t) sB.z (t) n zt) S(zi { B (Z 1) z t1 ) 1 z 2 I Z 1aTz1 = (Z1)L  i nB (t) Z z1 1 ) _ 1) {^B.z(t) (zi~l  Z Az(t) (31) Z1 I Z (t) (32) Z SAz (t) sz ( t)l2 (33) SBz(t) S2z(t)2 (34) Step 4. Advance time as t < (t + At). Step 5. Estimate the probabilities of transition corresponding to the birth and death events as {W+[na.z(t);t]}At and {W [naz(t);t]+W [naz(t);t]}At, respectively. Step 6. Determine the state of the system at the end of time interval (t,t + At). To accomplish this, sample a realization u from the uniform random variable U on interval (0, 1) and compare it with the probabilities of transition for the birth and death events. If u < [W (nA;t)]At, then a birth event occurs; thus, nA, (t) nAz (t At) + 11 nB (t) [nTO nAZ (t) If u < [W (nA;t) + W (nA;t)]At, then a death event occurs; thus, nAz (t) [nAz (t At) 1 nB (t) [nT nA. (t) Otherwise, no event occurs; thus, nAz (t) nz (t At) nB.Z (t) nT nAz (t) Vol. 42, No. 1, Winter 2008 RESULTS AND DISCUSSION The means, variances, and standard de viations of the random variables, NA(t) and NB(t), characterizing the simple reversible reaction have been computed with the values for k1 and k2, and the two sets of values for nA0 and nB0, given by MartinezUrreaga and his collaborators."1 Thus, k1 is 4 103 s 1 and k2 is 1 103 s1; the first set of values for nA0 and n0O comprises 175 molecules of A and 25 molecules of B, and the second set comprises 3,500 molecules of A and 500 molecules of B. The means, (NA(t)) and (NB(t)), have been computed from the stochastic model according to their corresponding analytical expressions, Eqs. (9) and (11); the results are illustrated for the first set of nA0 and nB0 in Figure 1 (next page) and for the second set of nA0 and nB0 in Figure G1 in Appendix G. In these figures, the numbers of molecules of A and B obtained from the solution of the deterministic model as presented by Mar tinezUrreaga and his collaborators1lI are superimposed for comparison. Clearly, the values of (NA(t)) and (NB(t) )obtained from the stochastic model are in excellent accord with the corresponding deterministic values, thereby verifying that the mean component of the stochastic model for the birthdeath process is equivalent to the deterministic model. The standard deviations, oa(t) and o,(t), signify the variability attributable to the in ternal or characteristic noises of the process as predicted by the stochastic model. The values of oa(t) and o (t) have been computed from Eqs. (14) and (16), respectively; they are plotted in Figures 1 and G1 as the stan dard deviation envelopes, [(N (t)) o(t)] for N (t) and [(NB(t)) o,(t)] for NB(t), 41 0 200 400 600 800 1000 t,s Figure 1. Temporal evolution of the numbers of molecules of A and B per unit volume with noA = 175 and nBo = 25: The solutions from the stochastic model are compared with those from the deterministic model. respectively. Note that the variability or dispersion of the process is more pronounced in Figure 1 than in Figure G1: As indicated by MartinezUrreaga and his collaborators"11 and earlier in the current contribution, the relative extent of fluctuations of a random variable N(t) about its mean (N(t)) is approximately of the order ((N(t))' (Nio), or (N(t) "12, i.e., it is inversely proportional to the square root of the popula tion size. In other words, the smaller the population size, the greater the extent of variability or dispersion of the process about its mean and vice versa. The sample means, mA z(6) and m. z(6), have been comput ed through the Monte Carlo method on the basis of the event driven approach by averaging the realizations of NA(t) and NB(t) from Z successive simulations. The results are illustrated for Z = 2 in Figure 2 and Z = 50 in Figure 3 for the first set of nAo and n B; the values of (NA(t)) and (NB(t)) computed from the stochastic model as well as the numbers of molecules of A and B obtained from the solution of the deterministic model are superimposed in both figures for comparison. Clearly, mA z(6) and m z(6) approach (NA(t)) and (NB(t)), respectively, as the number of simulations, Z, varies from 2 to 50. The results are in line with the Weak Law of Large Numbers (WLLN), stating that the sample mean approaches the population mean as the size of the sample becomes large.[29] In stochastically simulating the birthdeath process, the size of the samples, i.e., the numbers of realizations of NA(t) and NB(t), increases as the number of simulations, Z, increases. Similarly, m,(6)) and mn z(6) are essentially identical to (NA(t)) and (NB(t)), respectively, when they are computed by averaging 50 simula tions, i.e., Z = 50, for the second set of nA0 and n0O; see Figure G2 in Appendix G. The sample standard deviation envelopes, [mA, (6) sAz(6)] for NA(t) and [mBz(6) SBz(6)] for NB(t), are also plotted in Figures 2 and 3 as well as in Figure G2 in Appendix G; they are essentially identical to [(NA(t)) o,(t)] 42 0 200 400 600 800 1000 torO,s Figure 2. Temporal evolution of the means and standard deviation envelopes of the numbers of molecules of A and B per unit volume with n o = 175 and nBo = 25: The average of two Monte Carlo simulations via the event driven approach is compared with the analytical solu tions resulting from the stochastic model; the solution of the deterministic model is represented by the solid lines. 0 200 400 600 800 1001 _t or e, s Figure 3. Temporal evolution of the means and standard deviation envelopes of the numbers of molecules of A and B per unit volume with n o = 175 and nBo = 25: The average of 50 Monte Carlo simulations via the eventdriv en approach is compared with the analytical solutions resulting from the stochastic model; the solution of the deterministic model is represented by the solid lines. and [(NB(t)) OB(t)], respectively, for Z = 50. Figures G3 and G4 in Appendix G exhibit mAz(t) and mn.z(t) as well as [mA.z(t) s .z(t)] and [mn.z(t) _ S.z(t)], which have been computed through the Monte Carlo method by resorting to the timedriven approach. For the first set of nAO and n B, the results with Z = 2 are presented in Figure G3, and those with Z = 50 are presented in Figure G4; they are compared with the solutions of the deterministic and stochastic models. Note that mA.z(t) and mBz (t) in Figure G4 Chemical Engineering Education are equivalent to (NA(t)) and (NB(t)), respectively, for Z = 50; this is also the case for the second set of nA0 and nB0 as illustrated in Figure G5 in Ap pendix G. Table 1 lists the computational times, required by the Monte Carlo method via the eventdriven and timedriven approaches, for estimating the sample means, variances, and standard deviations of the random variables, NA(t) and NB(t), characterizing the simple reversible reaction. To perform the Monte Carlo simulations by resorting to these approaches, the MATLAB codes given in Appendices E and F were implemented on a PC (Pentium IV 3.0 GHz with 512 MB RAM;Windows XP). Clearly, the time required by the eventdriven approach was shorter than that required by the timedriven approach. Figures 2 and 3 in conjunction with Figures G2 through G5 in Appendix G reveal a fundamental distinction between the variations attributable to the inherent or internal noises of a random process and those attributable to the numerical method to stochastically simulate it. The former can only be quantified in terms of higher moments about the means of the random variables characterizing the process, especially, the variances or standard devia tions. The latter arise from the generation of samples on the basis of random numbers; these variations tend to vanish as the number of random samples increases, i.e., as the number of simulation runs becomes large. The efficacy of the two approaches for stochas tic simulation via the Monte Carlo method, i.e., the eventdriven and timedriven approaches, are further illustrated with an example of photoelec trochemical disinfection of bacteria whose rate of decaying is assumed to be of the firstorder, or linear. In contrast to a chemical reaction involving molecules that are discrete and microscopic in size; the disinfection of bacteria deals with microorgan isms which are also discrete but mesoscopic in size. In general, the inherent fluctuations of a system comprising mesoscopic entities are appreciably more pronounced than those of a system comprising microscopic entities, especially, when the number concentrations of such entities are minute. Naturally, the system of interest is the population of bacteria per unit volume. Catalyzed photoelec trochemically, these bacteria die one at a time; moreover, the bacteria have ceased to grow and do not reproduce throughout the deactivation. This sys tem constitutes a special instance of the birthdeath processes in which the intensity of birth is absent; hence, it is termed the puredeath process.[10 11 25] Vol. 42, No. 1, Winter 2008 In modeling the photoelectrochemical disinfection of bacteria as a puredeath process, the number of live bacteria at time t, Nc(t), is taken as the random variable of the process; a realization of Nc(t) is denoted by nc. All the possible numbers of bacteria are the states of the process, and the collection of these numbers, {nco, (n0 1), ..., 2, 1, 0}, constitutes the state space, where nco is the initial number of bacteria, i.e., N,(0) or nc at t = 0. For the puredeath process, the probability balance around any arbitrary state nc of the independent random variable, Nc(t), leads to"11 dp(nc;t)=W (nc +1;t)p(nc +1;t)W (nc;t)p(nc;t) dt (35) which is the master equation of the process11, 25]; see Appendix A. On the basis of the firstorder rate law, the intensity of death, W (nc;t) , in the above expression is dnc W (nc;t) dnc k3nc dt where k3 is the firstorder rate constant. For the puredeath process, the expressions of the mean, E[Nc(t)] or (Nc(t)), and the variance, Var[Nc(t)] or oc2(t), are obtained from Eqs. (35) and (36) as[25 40] (Nc (t))= nco exp(k3t) (37) and 7c (t)= nco exp(k3t)[1 exp(k3t)], (38) respectively; see Appendix B. Obviously, oc2(t) can be related to (Nc(t)) as C( (t) = [ exp(k3t)l(Nc (t)) The standard deviation, oc(t), is, from Eq. (38), (Tc (t)= n0 {exp(k3t)1 exp(k3t) }12 (39) (40) TABLE 1 Computational Times for Estimating the Sample Means, Variances, and Standard Deviations of the Random Variables, N (t) and NB(t), Characterizing the Simple Reversible Reaction by the Monte Carlo Method via the EventDriven and TimeDriven Approaches Initial values Eventdriven approach* Timedriven approachW Z=2 Z=50 Z=2 Z=50 n = 175 < s < s Is Is nBO= 25 nA = 3,500 1 s 9s 5s 17s nB0= 500 Notes: * In the eventdriven approach, the sample means, variances, and standard deviations have been recorded every two seconds, i.e., AO = 2 s. In the timedriven approach, the fixed time increment, At, has been com puted according to Eq. 28 in the text with WM(nA;t)=k2nT = k2(nAO+nBO) and W M(nA;t)=k2nAO. Thus, At = 1.111 s with nA = 175 and nB = 25, and At = 0.056 s with n = 3,500 and nB = 500. (36) or, from Eq. (39), (c (t)= lexp (k 3t)12 (Nc (t))1/2 (41) From Eqs. (37) and (40), the coefficient of variation, CVc(t), is computed as 1 1exp(k3t) 242) C exp(k3t) which can be rewritten as CVc (t)= [1 exp(k3t) 2 (Nc (t))2 (43) (Nc (t)) Note that for the puredeath process the relative fluctuations of Nc(t) about its mean (Nc(t)), as given by CVc(t), are ap proximately of the order of ((N(t) ))1 2/ (Nc(t)), or (Nc(t)) 12, as indicated by MartinezUrreaga and his collaborators. 1] The temporal mean of the model, Eq. (37), has been re gressed on the experimental data for the photoelectrochemical disinfection of E.coli[24] near the termination period of the process. By identifying 140 cells per milliliter as the initial population of bacteria, nco, the nonlinear regression by means of the adaptive random search procedure41 42] has resulted in 0.091 min1 for the value of k3. With these values of no and k3, (Nc(t) ) and oc(t) have been computed from Eqs. (37) and (40), respec tively. Figure 4 illustrates the resultant values of (Nc(t)) and [(Nc(t)) oc(t)] in conjunction with the experimental data. 24] In addition, the sample means, mcz(6) and mc.z(t), and standard deviation envelopes, [mcz(0) sc.z(6)] and [mc.z(t) scz(t)], have been computed via the Monte Carlo method by resorting to both the eventdriven and timedriven approaches, respectively. The results from averaging 50 simulations, i.e., Z = 50, are presented in Figure 4 for comparison. Note that the results from Monte Carlo simulation closely approximate the analytical results from the stochastic model. As expected, the deviations or fluctuations of the experimental data[24] are substantially more pronounced than those predicted by the stochastic model: The overall deviations of the experimental data include those attributable not only to the internal noises of the process but also to the external noises arising from inevitable manipulation errors and instrumental limitations that can never be totally suppressed. CONCLUDING REMARKS The contribution of MartinezUrreaga and his collabora torsM11 on the stochastic simulation of chemical reactions, specifically, a simple reversible chemical reaction, has been complemented and augmented in two aspects. First, a sto chastic model for a simple reversible chemical reaction has been derived based on the firstorder, i.e., linear, rate law as a birthdeath process. The master equation arising from the model has rendered it possible to analytically obtain the ex 44 pectedmeans and variances of the process. Second, the master equation of the process has been stochastically simulated through the Monte Carlo method via the timedriven approach in addition to the eventdriven approach on the basis of the Gillespie algorithm. The process's means and variances from numerical simulation are in accord with the solutions of the deterministic and stochastic models. To further reveal their efficacy, the two approaches for stochastic simulation by the Monte Carlo method are illustrated with the photoelectro chemical disinfection of bacteria obeying the firstorder rate law. The mean computed from stochastic simulation are in line with the available experimental data as well as with the analytical results derived from the corresponding stochastic model. As expected, the fluctuations of the data about their mean are large when compared with those computed from the stochastic simulation. ACKNOWLEDGMENT This is contribution No. 071235, Department of Chemical Engineering, Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506. NOMENCLATURE E[NA(t)] mean, expected value, or the first moment of the random variable, Na(t) E[NB(t)] mean of the random variable, NB(t) E[Nc(t)] mean of the random variable, Nc(t) 10 t or 0, min Figure 4. Temporal evolution of the mean and standard deviation envelope near the tailend of photoelectrochemical disinfection of a population ofE. coli per unit volume: The average of 50 Monte Carlo simulations by resorting to the eventdriven and timedriven approaches is compared with the analytical solutions resulting from the stochastic model; the initial population of bacteria, nco, is 140 cells per milliliter. Chemical Engineering Education k, k, k, NA(t) NB(t) Nc(t) nA nB nc nAO nBO nTO nco (NA(t)) (NB(t)) (Nc(t)) p(nA; t) forward reactionrate constant, s 1 reverse reactionrate constant, s 1 firstorder rate constant, min random variable representing the number of mol ecules of A at time t random variable representing the number of mol ecules of B at time t random variable representing the number of live bacteria at time t realization of the random variable, NA(t) realization of NB(t) realization of Nc(t) number of molecules of A at t = 0, NA(0) number of molecules of B at t = 0, NB(0) total number of molecules at any arbitrary t, (n A + nBO) number of live cells at t = 0, Nc(0) E[NA(t)] E[NB(t)] E[Nc(t)] probability of nA molecules being present at time t p(nc; t) probability of nc live bacteria being present at time t t U u Var[NA(t)] Var[NB(t)] Var[Nc(t)] W (nA; t) time uniform random variable on interval (0,1) realization of U variance of the random variable, NA(t) variance of NB(t) variance of Nc(t) intensity of birth for the birthdeath process in state nA at time t W (nA; t) intensity of death for the birthdeath process in state n at time t W (nc; t) intensity of death for the puredeath process in state nc at time t Greek Letters 0 datarecording time in the eventdriven approach oA2(t) Var[NA(t)] oB2(t) Var[NB(t)] oc2(t) Var[Nc(t)] GA(t) standard deviation of the random variable, NA(t) OB(t) standard deviation of NB(t) oc(t) standard deviation of Nc(t) T random waiting time in the eventdriven approach REFERENCES 1. 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Misek, "Stochastic Modeling of Dispersion Formation in Agitated LiquidLiquid Systems," Transactions of the Institution of Chemical Engineers, 60, 48 (1982) 15. Rajamani, K., WT. Pate, and D.J. Kinneberg, "TimeDriven and EventDriven Monte Carlo Simulations of LiquidLiquid Dispersions: A Comparison," Ind. Eng. Chem. Fund., 25, 746 (1986). 16. Kendall, D.G., "AnArtificial Realization of a Simple 'BirthandDeath' Process," J. Roy. Stat. Soc. B, 12, 116 (1950) 17. Gillespie, D.T., "A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions." J. Comput. Phys., 22, 403 (1976) 18. Gillespie, D.T., "Exact Stochastic Simulation of Coupled Chemical Reactions." J. Phys. Chem., 81, 2340 (1977) 19. Shah, B.H., J.D. Borwanker, and D. Ramkrishna, "Monte Carlo Simula tion of Microbial Population Growth," Math. Biosci., 31, 1 (1976) 20. Shah, B.H., D. Ramkrishna, and J.D. Borwanker, "Simulation of Particulate Systems Using the Concept of the Interval of Quiescence," AIChE J., 23, 897 (1977) 21. Rao, C.V., andA.P Arkin, "Stochastic Chemical Kinetics and the Quasi SteadyState Assumption: Application to the Gillespie Algorithm," J. Chem. Phys., 118, 4999 (2003) 22. Cao, Y., D.T. Gillespie, and L.R. Petzold, "Accelerated Stochastic Simulation of the Stiff EnzymeSubstrate Reaction," J. Chem. Phys., 123, 144917(2005) 23. Ullah, M., H. Schmidt, K.H. Cho, and 0. Wolkenhauer, "Deterministic Modelling and Stochastic Simulation of Biochemical Pathways using MATLAB," IEE Proc. Syst. Biol., 153, 53 (2006) 24. Harper, J.C., PA. Christensen, T.A. Egerton, T.P Curtis, and J. Gun lazuardi, "Effect of Catalyst Type on the Kinetics of the Photoelec trochemical Disinfection of Water Inoculated with E. Coli," J. Appl. Electrochem., 31, 623 (2001) 25. Fan, L.T., A. Argoti Caicedo, S.T. Chou, and WY. Chen, "Stochastic Modeling of Thermal Death Kinetics of a Cell Population: Revisited," Chem. Eng. Educ., 37, 228 (2003) 26. Duggirala, S.K., and L.T. Fan, "Stochastic Modeling of NonLinear Sieving Kinetics," Chem. Eng. Comm., 61, 59 (1987) 27. Fox, R.O., and L.T. Fan, "Application of the Master Equation to 45 Coalescence and Dispersion Phenomena," Chem Eng Sci., 43, 655 (1988) 28. Chou, S., L.T. Fan, A. Argoti, R. VidalMichel, andA. More, "Stochastic Modeling of Thermal Disinfection of Bacteria According to the Logistic Law,"AIChE J., 51, 2615 (2005) 29. Casella, G., and R.L. Berger, Statistical Inference, Duxbury, Pacific Grove, CA, pp. 54, 55, 59, 8991, 233 (2002) 30. McQuarrie, D.A., "Kinetics of Small Systems. I," J. Chem. Phys., 38, 433 (1963) 31. Steinfeld, J.I., J.S. Francisco, and W.L. Hase, Chemical Kinetics and Dynamics, Prentice Hall, Englewood Cliffs, NJ, pp. 97102 (1989) 32. Pang, WK., PK. Leung, W.K. Huang, and W Liu, "On Interval Estimation of the Coefficient of Variation for the ThreeParameter Weibull, Lognormal, and Gamma Distribution: A SimulationBased Approach," Eur. J. Oper. Res., 164, 367 (2005) 33. Dixon, D.A., and R.H. Shafer, "Computer Simulation of Kinetics by the Monte Carlo Technique," J. Chem. Educ., 50, 648 (1973) 34. Moebs, W.D.C., "A Monte Carlo Simulation of Chemical Reactions," Math. Biosci., 22, 113 (1974) 35. Gillespie, D.T., "Approximate Accelerated Stochastic Simulation of Chemically Reacting Systems," J. Chem. Phys., 115, 1716 (2001) 36. Rawlings, J.B., and J.G. Ekerdt, Chemical ReactorAnalysis andDesign Fundamentals, Nob Hill Publishing, Madison, WI (2002) 37. Haseltine, E.L., and J.B. Rawlings, "Approximate Simulation of Coupled Fast and Slow Reactions for Stochastic Chemical Kinetics," J. Chem. Phys., 117, 6959 (2002) 38. Drews, T.O., A. Radisic, J. Erlebacher, R.D. Braatz, PC. Searson, and R.C. Alkire, "Stochastic Simulation of the Early Stages of Kinetically Limited Electrodeposition," J. Electrochem. Soc., 153, 434 (2006) 39. Mastny, E.A., E.L. Haseltine, and J.B. Rawlings, "Stochastic Simulation of Catalytic Surface Reactions in the Fast Diffusion Limit," J. Chem. Phys., 125, 194715 (2006) 40. Spilimbergo, S., and D. Mantoan, "Stochastic Modeling of S. Cerevi siae Inactivation by Supercritical CO2," Biotechnol. Prog., 21, 1461 (2005) 41. Fan, L.T., H.T. Chen, and D. Aldis, "An Adaptive Random Search Procedure for LargeScale Industrial and Process Systems Synthesis," Proceedings of the Symposium on Computers in Design and Erection of Chemical Plants, Aug. 31Sep. 4, Karlovy Vary, Czechoslovakia, pp. 279291 (1975) 42. Chen, H.T., and L.T. Fan, "Multiple Minima in a Fluidized Reactor Heater System," AIChE J., 22, 680 (1976). 43. Taylor, H.M., and S. Karlin, Introduction to Stochastic Modeling, Academic Press, San Diego, pp. 25, 357 (1998) 44. Jeffreys, H., and B. Jeffreys, Methods of Mathematical Physics, 3rd Ed., Cambridge University Press, Cambridge, pp. 2223 (1999) 45. McQuarrie, D.A., Mathematical Methods for Scientists and Engineers, University Science Books, Sausalito, CA, pp. 4, 1053 (2003) APPENDICES Seven appendices (Appendices A through G) can be viewed or downloaded from the corresponding author's Web site at be obtained as hard copies by writing to f.Lai kiu \.diu Chemical Engineering Education IM!1 class and home problems Applications of THE PENGROBINSON EQUATION OF STATE Using Mathematica HOUSAM BINOUS National Institute of Applied Sciences and Technology Consider a single component characterized by its critical pressure, temperature, acentric factor, Po T and o. The PengRobinson equation of state[2 4] is given by: RTa P = (1) (Vb) [V(V+b)+b(Vb)] where R T b = 0.07780 R Pc (RTo )2 a= 0.45724 L 1+m ) Pc T Tc m= 0.37464 + 1.54226 0.26992 2. (5) The PengRobinson EOS* is part of a family of equations called cubic because the compressibility factor, Z = PV/RT, is a solution of the following cubic equation derived from Eq. (1) and written for a multicomponent mixture: 3 + ( B)Z2 +(A 3B2 2B)Z + (AB +B2 * Tunis, Tunisia where c c c c A=EEy.yA, or EExxA, I1 ji1 I1 j1i A = AAA 0 (1k ) C C B= yB, or x,B, A 1 = 1 P Pr A =0.45724 a and B =0.07780 ST2 T S For each component, we define the reduced pressure and temperature using P,, =P PP, and T = T/ and a is (4) identified by an equation similar to Eq. (3) given for the pure component case. The binary interaction parameter, k , Housam Binous is a fulltime faculty member at the National Institute of Applied Sciences and Technology in Tunis. He earned a Dipldme d'ingenieur in biotech nology from the Ecole des Mines de Paris and a Ph.D. in chemical engineering from the University of California at Davis. His research interests include the applications Sof computers in chemical engineering. Copyright ChE Division of ASEE 2008 B3= 0 (6) *All these computations are available in the form of notebooks upon requestfrom the author or the Wolfram Research Information Center.11 Vol. 42, No. 1, Winter 2008 The object of this column is to enhance our readers' collections of interesting and novel prob lems in chemical engineering. Problems of the type that can be used to motivate the student by presenting a particular principle in class, or in a new light, or that can be assigned as a novel home problem, are requested, as well as those that are more traditional in nature and that elucidate dif ficult concepts. Manuscripts should not exceed 14 doublespaced pages and should be accompanied by the originals of any figures or photographs. Please submit them to Professor James O. Wilkes (email: wilkes@umich.edu), Chemical Engineering Department, University of Michigan, Ann Arbor, MI 481092136. is obtained from HYSYS 3.2, a major process simulator by Aspen Technology, Inc.,[1] or assumed to be equal to zero if the simulator is not available. The liquid and vapor mole fractions are related by: y, =K,x, with i= 1,2,.., C (11) where K is the equilibrium constant. This equilibrium constant is obtained using the t method as follows: K,= fori= ,2,...,C (12) v where the fugacity coefficients are: (Zv 1) In(ZvB) A B 241B v, = exp 2 yA B 1 B1 (13) 1   In +(+52) A B Z B The vaporphase and liquidphase fugacity coefficients, which measure deviation from ideality, are defined from the liquidphase and gasphase fugacities, fv andfl as follows: v f f l S and (14) y,P x,P The method is a direct result of expressing vapor liquid equilibrium conditions (i.e., temperatures, pressures, and fugacities are equal in both phases) and the definition of fugacity coefficients. A similar expression to Eq. (13) is obtained for the liquid phase fugacity coefficient, i,, by replacing the gasphase compressibility factor, Zv, with its liquidphase counterpart, ZI. These two compressibility factors are the largest and smallest roots of Eq. (6), respectively. For specified values of P and T, the smallest root, Zl allows the determination of the liquid molar volume while the highest root, Z permits the computation of the vapor molar volume. The intermediate root has no physical significance. Finally, one needs the following expression for the en thalpy departure function from ideality in order to compute enthalpy: HD = RT(Z 1) 1+ 24B RT P L +(I + 2)B T d A(RT)2 (RT)2' ( LoZ+ 1 )B dT P PA (15) ADIABATIC FLASH CALCULATIONS FOR MULTICOMPONENT MIXTURES Problem statement A quaternary mixture, at 485 psia and 100 F, is composed of 0.41% hydrogen, 5.71% methane, 70.97% benzene, and 48 22.91% toluene. This mixture is sent to a stabilizer to remove hydrogen and methane. The feed pressure is decreased adia batically from 485 psia to 165 psia by valve and pipe line pressure drop. Find the vaporphase fraction, the temperature and other relevant variables. Solution This problem has been solved using a tedious iterative tech nique by Henley and Seader.[6] The unknowns in this problem are the mole fractions in the two phases, the temperature, the vapor phase fraction, and the compressibility factors. We have 12 nonlinear algebraic equations to solve simultane ously. These equations are three equilibrium relations, three component mass balances, two summation rules, two cubic equations of the compressibility factors, the enthalpy balance, and the famous Rachford and Rice equation given by: zK, ) = 0 (16) Ii+i vap(K 1) where xvap is the fraction vaporized, V/F The Mathematica commands, which allow the determination of the 12 un knowns, are based on the builtin function FindRoot and are given by: Eq[i_]:= y[i] EQ[i_]:= x[i] == z[i] 4 == K[i] x[i] /(1+ vap (K[i] 1)); 4 sEq[1],Eq[2],Eq[3], y[i] sol = FindRootEQI,EQ[2,EQ[3 EQ[I1],EQ[2],EQ[3], Sz[i](K[i] + (K[i]1))== 0, 1,> z[i] (K[i] 1) /(1 (a+ vp (K[i] 1)) == 0, 1I1 Z ^3 + (1+ B,)Z A2 Z (A 3B A2 2B,) Al B+B A 2 + B A3 == 0, Zv 3 +(1+ B) Zv 2 + Zv (A 3Bv 2 2Bv) Av Bv + B 2 + B A3 == 0, HF == vap HV + (1 vap) HL},{x[1],0.0004}, {x[2],0.03},{x[3],0.7},{x[4],0.25}, ,vap0.01o {y[1],0.1}, {y[2],0.85},{y[3],0.02}, {y[4], 0.001}, {Z,,0.05}, {Z,,0.95}, {T,540}, Maxlterations > 1000] / /Chop We find the following output, x[l] 0.00020583, x[2] 0.0259441, x[3] 0.736087, x[4] 0.237763 y[l] 0.106105, y[2] 0.873206, y[3] 0.0185161,y[4] 0.00217289, tvap 0.0367724,Z, 0.0405552,1 Z, 0.981151, T 560.297 Chemical Engineering Education The vaporphase fraction vap and temperature are 0.0367 and 560.297 R, respectively. SOLUBILITY OF METHANOL IN NATURAL GAS Problem statement Methanol is used as a solvent in the sweetening of natural gas (i.e., removal of carbon dioxide and hydrogen sulfide). Loss of methanol by transfer from liquid phase to gas phase can become an important economic consideration. Compute the gas phase mole fraction of methanolP31 in treated natural gas leaving the sour gas absorption unit vs. temperature at a pressure of 145.038 psia. Treated natural gas is composed of 70% methane, 10% nitrogen, 10% carbon monoxide and 10% hydrogen. Solution We perform several isothermal flash computations at vari ous temperatures. The problem's unknowns are the ten mole fractions in the gas and liquid phases, the vapor phase fraction, and the two compressibility factors. The available equations are the five equilibrium relations, the two summation rules, the five mass balances, the two cubic equation of state of the compressibility factors, and the Rachford and Rice equation. The calculation uses the builtin function FindRoot to solve 13 nonlinear algebraic equations simultaneously. The Math ematica commands, which allow the determination of the 13 unknowns, are the following: o E 0 D 0.001 0.00001 350 400 450 500 550 Temperature in R Figure 1. Solubility of methanol in treated natural gas vs. temperature at 145.038 psia. Eq[i_]:= y[i] == K[i] x[i] EQ[i_]:= x[i] == z[i] /(1+ vap (K[i] 1)); For[i = ,i < 15, T= 1329.67,347.67,365.67,383.67,401.67, T= 419.67,437.67,455.67,473.67, 491.67,509.67,527.67,545.67,563.67} [[i]l, sol[i]= FindRoot[ Eq[l],Eq[2],Eq[3],Eq[4], y[i]== 1, EQ[1],EQ[2],EQ[3],EQ[4], x[i]== 1, I=1 Lz[i] (K[i] 1)/(1 + vap (K[i] 1))== 0, I1 Z1A 3 + (I+ 1B) Zl A2 + Z, (A, 3B132 2B1) A, B, +B,^2 +B 3== 0, ZA3+(1+ B)ZA2+ Z (A 3B, A2 2B,) Av B +BA 2 +B A3== 0 {x[1],x[1] /.sol[i 1]},{x[2],x[2] .sol[i1]}, {x[3],x[3] .sol[i 1]}, {x[4],x[4] .sol[i 1]},{x[5],x[5] /.sol[i 1]}, {y[1],y[1] /.sol[i 1]}, {y[2],y[2] .sol[i 1]}, {y[3],y[3]/ .sol[i 1]}, {y[4],y[4] .sol[i 1]}, {y[5],y[5] /.sol[i 1]}, { vap vap/ .sol[i 1]}, {Z1, Z / .sol[i 1]}, {Zv,,v /.sol[i 1]}, Maxlterations > 1000] / Chop,Print[i],i + +}] Figure 1 depicts the computed solubility of methanol in treated natural gas vs. temperature at 145.038 psia. We observe an increase in solubility at higher temperatures. We see that these solubility values can become large enough to present serious economic drawback due to excessive loss of methanol. HIGHPRESSURE CHEMICAL EQUILIBRIUM COMPUTATION Problem Statement Nitrogen and hydrogen react to form ammonia. This reac tion is favored by low temperatures and high pressure. Kinetic considerations, however, oblige us to use high temperature. Thus, reactors are operated at very high pressures to get a reasonably high conversion of reactants. High gasphase pressures imply significant deviation from ideality and the need to take into account the fugacity coefficients.[4] In fact, the equilibrium constant depends on K as follows: a a YNH, 1 X(2X) 1 Ka= 05 15 K = K (17) =1 YN2 YH2 p 1 X 3(1 X) P 2 2 Vol. 42, No. 1, Winter 2008 where Kv is given by: K=v (18) 1.05 15 a, = 4,y, (19) and X is the extent of reaction. Plot K and X vs. pressure at temperatures ranging from 900 OR to 1440 OR. Solution The unknowns in this type of problems are five: the mole fraction in the gas phase, the extent of reaction, and the gas phase compressibility factor. Once again, the calculation uses the builtin function FindRoot to solve five nonlinear algebraic equations simultaneously. The Mathematica com mands, which allow the determination of the five unknowns, are the following: P = 0.001; For[i = 1,i < 17,{so11260= Findo Ka[T] (0.5(1 X)) 0.5(1.5(1 X)) A1F.5 4,[1] 1.5 4,[2] A 0.5== X(2 X)/P 4,[3],y[3](2 X) == X,y[2] (2 X) == 0.5 (1 X), y[l](2 X) == 1.5(1 X), Z, 3 + (1+ B,) Z A 2+ Z, (A 3BA 2 2B,) A, Bv+ B, 2+BA 3== 0} {y[2],0.2},{y[1],0.1},{y[3],0.4},{X,0.6}, {Zv, 0.95},MaxIterations 10000], Pp[i] = P,Kv[i] = v,[3] / ([1] A 1.5 v,[2] A 0.5)/ .so11260,X[i]= X /.so11260, Print[i," ",P],P = P + 1000,i= i +1}] In Figure 2, we plot Kv vs. pressure at various temperatures (900, 1080, 1260 and 1440 R). Values ofKK are significantly different from unity, indicating that this factor must be taken into account at high pressures. The extent of reaction at equi librium vs. pressure, for the same temperatures, is represented in Figure 3. The extent of reaction approaches unity at high pressures. CONCLUSION We have shown through simple examples how one can take advantage of the numerical and graphical capabilities of Mathematica to perform flash calculations for multicom ponent mixtures, solubility determination, and highpressure chemical equilibrium computations. Similar computations can be performed very easily using Matlab. 7 NOMENCLATURE x: liquid mole fraction y: vapor mole fraction Po,: critical pressure (psia) To.: critical temperature (R) Pr,: reduced pressure Tr,: reduced temperature m: acentric factor Z: compressibility factor 4, 4,: fugacity coefficients K: equilibrium constant Pressure in psia Figure 2. K vs. pressure at various temperatures. 1.0 0 S0.8 0.4 900 R 0.2 \ 1080 R 1260 R 1440 R 0.0 0 2000 4000 6000 8000 10000 12000 14000 Pressure in psia Figure 3. Extent of reaction vs. pressure at various temperatures. Chemical Engineering Education 0 2000 4000 6000 8000 10000 12000 14000 HD: k: C: z: R: a: v: vap 0vap : departure from ideality enthalpy (cal/mol) binary interaction parameter number of components feed mole fraction universal gas constant (cal/mol.R) activity of species i (psia) stoichiometric coefficient liquid phase fugacity coefficient vapor phase fugacity coefficient fraction vaporized = V/F REFERENCES 1. 2. Tester, J.W., and M. Modell, Thermodynamics and Its Applications, 3rd Ed., Prentice Hall, Upper Saddle River, NJ (1996) 3. Prausnitz, J.M., R.N. Lichtenthaler, and E.G. deAzevedo, Molecular Thermodynamics of FluidPhase Equilibria, 3rd Ed., PrenticeHall, Englewood Cliffs, NJ (1998) 4. Sandler, S.I., ( /.... .i....t i ........ ring Thermodynamics, Wiley, NY, 3rd Ed. (1999) 6. Henley, E.L., and J.D. Seader, EquilibriumStage Separation Opera tions in Chemical Engineering, Wiley, NY (1981) 7. Vol. 42, No. 1, Winter 2008 This onepage column will present practical teaching tips in sufficient detail that ChE educators can adopt the tip. The focus should be on the teaching method, not content. With no tables or figures the column should be approximately 450 words. If graphics are included, the length needs to be reduced. Tips that are too long will be edited to fit on one page. Please submit a Word file to Phil Wankat CELEBRATING ChE IN SONG ALAN M. LANE The University ofAlabama Every semester I survey my students to learn why they chose to attend UA, why they chose to major in chemi cal engineering, and what kind of career they aspire to. I also ask about their major interests and hobbies. I found that chemical engineering students organized COE Does ART, an official campus organization in which engineering students put on a musical each semester. Several chemical engineering students competed in a recent university "battle of the bands" as members of the rock group Liquid Metal. Others are members of the university's marching band, sing in a musical, write short stories, create art, or make crafts. Engineering is an incredibly creative profession so it should not surprise anyone that their activities outside the classroom reflect very creative personalities. Can we harness this creativ ity for the benefit of chemical engineering education? Many years ago, CEE published the poem "The Reynolds' Number Song"'11 that humorously celebrates the Reynolds Number. Since most current students enjoy rap music, I found it useful to modify the poem slightly and perform it in class as the "Reynolds Rap" (pun intended). Imagine the class pounding out a rhythm on their desks and joining me in the chorus, "Take a 'd' times a 'v and a rho by a mu, put them all together with a little bit of glue, now you got a number gonna see you through, and tell you what the [insert colorful language of your choice] fluid's gonna do." It is certainly hard to forget the equation after that exercise! Copyright ChE Division ofASEE 2008 In the fall of 2006, I recorded the "Reynolds Rap" in my home studio and played it for my 90 freshmen in the Intro duction to Chemical Engineering seminar. During the 2007 summer lab several students asked me to write a song about distillation columns, suggesting the title "Steppin' Off the Stages." Since our endofsummerlab picnic was themed The Fiesta before the Siesta, I set it to a Latin rhythm and played it at the picnic. The students very much enjoyed both songs, and many downloaded them. These songs are available free as MP3 downloads at ing pleasure andif you dareto use in your classroom. I believe there are many wouldbe rock stars in the profession that have also written or will write a chemical engineering themed song. Make a good recording and send me the MP3 file and lyrics by email (alane@eng.ua.edu), and you just might find it posted on this Web site and played on chemical engineers' iPods around the globe! REFERENCES 1. Harriott, P., "The Reynolds' Num berSong,"( i ..... in 8(1),12 (1979) 1 Figure 1. Doobie 'Doghouse' Wilson (AKA Alan Lane) with his dobro. Chemical Engineering Education Alan M. Lane is a professor of chemical and biological engineer ing at The University of Alabama. He conducts research in hetero geneous catalysis and teaches chemical reaction engineering. At night he performs as Doobie "Doghouse" Wilson in local bars and coffee houses Me,= Rteaching tips ) M]n1 class and home problems The object of this column is to enhance our readers' collections of interesting and novel prob lems in chemical engineering. Problems of the type that can be used to motivate the student by presenting a particular principle in class, or in a new light, or that can be assigned as a novel home problem, are requested, as well as those that are more traditional in nature and that elucidate dif ficult concepts. Manuscripts should not exceed 14 doublespaced pages and should be accompanied by the originals of any figures or photographs. Please submit them to Professor James O. Wilkes (email: wilkes@umich.edu), Chemical Engineering Department, University of Michigan, Ann Arbor, MI 481092136. CAN I TRUST THIS SOFTWARE PACKAGE? An Exercise in Validation of Computational Results MORDECHAI SHACHAM BenGurion University of the Negev Beer I,. .., 84105, Israel NEIMA BRAUNER TelAviv University TelAviv 69978, Israel W. ROBERT ASHURST Auburn University Auburn, Alabama 36849 MICHAEL B. CUTLIP University of Connecticut Storrs, CT 06269 M mathematical software packages such as Polymath, Mordechai Shacham is the Benjamin H. Swig professor of chemical MATLAB and Mathcad are widely used for engi processes and former chair at BenGurion University of the Negev in neering problem solving. The application of several Israel. He received his B.Sc. and D.Sc. degrees from the Technion, Israel Institute of Technology. His research interests include analysis, modeling packages to typical chemical engineering problems has been and regression of data, applied numerical methods and prediction, and demonstrated by Cutlip, et al.[' The main characteristic of consistency analysis of physical properties. He is past president and an honorary fellow of the Israel Institute of Chemical Engineers, and the recipi these packages is that they provide a "problem solving envi ent of the 2000 CACHE Award for excellence in computing in chemical ronment (PSE)"[2] rather than just scientific subroutine librar engineering education. ies callable from programming languages that were popular Neima Brauner is a professor at TelAviv University in TelAviv, Israel. She a few years ago. For routine use of the software packages in received her B.Sc. and M.Sc. in chemical engineering from the Technion problem solving, there is no need to be an expert in either Institute of Technology in Haifa, Israel, and her Ph.D. in mechanical problem solving, there no need to be an expert in either engineering from the TelAviv University. Her research interests include numerical methods or programming. In most cases, it is suf hydrodynamics and transport phenomena in twophase flow systems ficient to specify the mathematical model of the problem using and development of interactive statistical and numericalmethods for data analysis in process analysis and design. She is an editor for Reviews in the syntax required by the package. The technical details of Chemical Engineering, and associate editor of Heat Transfer Engineering. the solution are carried out by the package. There are cases, SW. Robert Ashurst is currently an assistant professor of chemical however, where the solution is obviously incorrect and/or the engineering at Auburn University. His research focuses on design of program stops with an error message. molecular precursors for advanced monolayer films, tribology at the micro and nanoscale, and the influence of surface chemical treatments The failure of a software package to reach a correct solu on micro and nanoscale devices. He completed his Ph.D. in chemical tion is commonly due to the presence of errors in the math engineering at the University of California at Berkeley in 2003 under the support of a National Science Foundation Graduate Fellowship. He is an ematical model or the selection of an inappropriate solution active member of AIChE. algorithm and/or inappropriate algorithm parameters (such Michael B. Cutlip has spent his entire academic career at the University of as initial estimates, error tolerances, etc). Validation of the Connecticut, where he served as department head of chemical engineer mathematical models has been discussed in detail by Brauner, ing and director of the university's honors program. He is a former chair and national program chair for the Chemical Engineering Division, and he et al.[3] In this paper, we present an exercise that can be used cochaired the ASEE Summer School for chemical engineering faculty in Copyright ChEDivi2002. He is also coauthor of the POLYMATH software package. SCopyright ChE Diision of ASEE 2008 Vol. 42, No. 1, Winter 2008 5j in educating students to identify difficulties associated with an algorithm for the solution of a problem and help them to select appropriate parameters for the algorithm. This exercise involves numerical solution for a system of ordinary differential equations (ODE's). It can be used at two levels. 1. The first is in courses where students routinely use numerical software for solving ODE's (i.e., Reaction Engineering, Process Control and Process Simula tion). In such courses, this part of the exercise can be used to demonstrate that computer solutions can be inaccurate or even completely incorrect, and therefore it is always necessary to validate a numerical solu tion. 2. At the more advanced level, the second part of the exercise can be used in courses related to mathemati cal modeling and numerical analysis. In such courses, this part of the exercise provides training and demon stration of advanced topics such as error estimation and step size control, error ;1,. *. ,, i.. and stiffness of ODE's. In this work, the mathematical software packages Poly math 6.11 and MATLAB2 are used for the demonstration, but other packages may be used in a similar manner. ANALYZING THE BEHAVIOR OF THE RUNGEKUTTA ALGORITHM FOR AN "INTERACTING TANKS" SIMULATION PROBLEM The problem considered in this exercise involves the dynamic simulation of a system of three interacting tanks, as depicted in Figure 1. The model equations (as were provided Ashurst and Edwards'41) that can be derived from mass balances on each tank are  h D D,2 dhl q, q, = 2 +h, (1) q q 4 H2 dt (1) dh2 q2 + q4 q5 = Te2h2 (2) dt dh3 q5 + q3 q6 = h 3 (3) dt where q represents the in/out volumetric flow rate (m3/s), H the tank height (m), h the liquid height (m), D1,2 the upper/lower diameter ratio of tank 1 (m), and t is the time (s). The volumetric flow rates out of the tanks, through a drainage area, Ao (m2), are given by q4 = Ao 2g(h h2) ifh, > h2 q4 = Ao2g(h2 hi) otherwise (4) q5 = Ao 2g(h h3) ifh2 > h3 q, = Ao2g(h3 h) otherwise (5) q6 = Ao2gh (6) The response of the system to a step change in the inlet flow rates is to be investigated. The model equations and the nu merical data pertinent for a particular case, as they have been entered into the Polymath 6.1 software package, are shown in Table 1. Note that the equations and comments in the Polymath input provide a complete documentation of the problem (in Polymath the comments are indicated by the sign: #). TABLE 1 Polymath Model for the "Train of Interactive Tanks" Problem No. Equations and # Comments 1 # Mass balances on the tanks 2 d(hl)/d(t) = If hl >= H1 Then (0) Else (ql q4) / ((pi / 4) * ((D2 + ((Dl D2) / HI) hi) A 2)) # Liquid level in Tank 1 (m) 3 d(h2)/d(t) = If h2 >= H2 Then (0) Else (q2 + q4 q5) (pi * exp(2 h2) ) # Liquid level in Tank 2 (m) 4 d(h3)/d(t) = If h3 >= H3 Then (0) Else (q5 + q3 q6) (h3 A 2) # Liquid level in Tank 3 5 # Outlet flow rates are functions of potential flow theory. 6 q4 = If hl < h2 Then Ao sqrt(2 g (h2 hl)) Else Ao * sqrt(2* g (hl h2))# Volumetric flow rate out of Tank 1 (m^3/s) 7 q5 = If h2 < h3 Then Ao sqrt(2 g (h3 h2)) Else Ao * sqrt(2 g (h2 h3)) # Volumetric flow rate out of Tank 2 (m^3/s) 8 q6 =Ao sqrt((2 g h3))# Volumetric flow rate out of Tank 3 (m^3/s) 9 # Input variables 10 ql = .05 # Volumetric flow rate into Tank 1(m^3/s) 11 q2 = .05 # Volumetric flow rate into Tank 2(mA3/s) 12 q3 = If t <= 50 Then .05 Else.05 +.05* .1 # Volumetric flow rate into Tank 3(mA3/s) 13 # Constants 14 H1 = 4 # Height of Tank l(m) 15 H2 = 2.5 # Height of Tank 2 (m) 16 H3 = 2.5 # Height of Tank 3 (m) 17 D1 = 4 # Upper diameter of Tank l(m) 18 D2 = 1 # Lower diameter of Tank 2 (m) 19 Ao = pi / 100 # Area of drainage orifice at bottom of tanks (m^2) 20 pi =3.141592654 21 g=9.81 22 # Initial Conditions 23 hl(0) = 1.809 # Liquid level in Tank 1 at steady state(m) 24 h2(0) = 1.68 # Liquid level in Tank 2 at steady state(m) 25 h3(0) = 1.163 # Liquid level in Tank 3 at steady state(m) 26 t(0) = 0 27 t(f)= 2400. 'POLYMATH is a product of Polymath Software (< .. ........ t, ..... corn>) 2MATLAB is product of The Math Works, Inc. ( Chemical Engineering Education The system is started at steadystate values of h, h2 and h3 (see lines 23 through 25), where the inlet flow rates to the different tanks are ql = q2 = q3 = 0.05 m3/s. At t = 50 s, a step change of +10% is introduced into q3. The system is simulated up to the final time of tf = 3600 s. The integration of the model equations using the RKF45 algorithm of Polymath with the default parameters yields the Figure 1. A train of three interactive tanks.!4' TABLE 2 Change of the Variable Values after Introduction of a 10% Step Change in Inlet Flow Rate to Tank 3 Variable Initial value Minimal Maximal Final value value value t 0 0 2400 2400 h, 1.807 1.807 1.854328 1.854328 h, 1.68 1.679873 1.725494 1.725494 1.162 1.162 1.221394 1.221394 q 0.0495908 0.0495908 0.0499622 0.0499475 q, 0.1001531 0.0971087 0.1001531 0.0988002 q 0.1500039 0.1500039 0.1537897 0.1537897 dhl/dt 9.39E05 8.66E06 9.39E05 1.17E05 dlh/dt 6.22E06 6.22E06 3.06E05 1.16E05 dh/dt 1.11E04 5.39E07 2.71E04 7.08E06 0 500 1000 1500 time (s) 2000 2500 3000 error message L i i. sqrt ofn ,... ,a,.' number is not allowed, ,.'p q 2, / 3)," which indicates that the height of the liquid level in the 3rd tank (h3) has somehow reached a negative value dur ing the numerical integration. A quick check can be made on the dynamics by reducing the integration final time to 2400 s. As there is no abnormal behavior or unexpected trends during this numerical integration, as shown in Table 2 and Figure 2, the assignment is to check and verify the error in the q3 integration up to the time H H3 of 3600s. The options that need to be investigated for This problem are: 1. There are possible er H3 rors in the implementa \ h3 q tion of the RungeKutta q5 q6 1 ,, ... 2. The default truncation error tolerance of the program is too large for this problem, and the negative value of h3 is a result of large truncation error and propagated error in some of the variables. 3. The system of equations is stiff. The explicit RKF45 may exhibit oscillatory be havior if the ;i,. inr. t, step size is ... ,..a than that dictated by the stability limit. RECOMMENDED STEPS AND ADDITIONAL INFORMATION FOR SOLUTION 1. Most software packages include some variation of the RungeKutta algorithm  with truncation error estimation and step size control as one of the ODE solver tools. Using the same or a similar algorithm in a different software package can provide ad ditional information regarding the source of the error. 2. The appropriateness of the truncation error tolerance can be tested by rerunning the problem using various error tolerances and comparing the variable values obtained at a specified time (say tf). It is preferable to check the values at the final time, as at this point the error propagated throughout the integration interval is considered. 3. The stiffness of the problem can be determined by defin ing the f vector as dh, dt f= dh (7) dt Figure 2. Change of the liquid level in the third tank after introduction of a 10% step change in its inlet flow rate (Polymath Solution). Vol. 42, No. 1, Winter 2008 where dhl/dt, dh2/dt, and dh3/dt are given by Eqs. (13), respectively. The stiffness of the system can be assessed by calculating the stiffness ratio of the df/ah (Jacobian) matrix. The stiffness ratio (SR) is defined by (see, for example, p. 245 in Rice and Do[5]) max IX, SR  (8) min X, where max X, is a negative eigenvalue of the 8f/ah matrix with maximal absolute value, and min JI, is a negative ei genvalue of the 8f/ah matrix with minimal absolute value. If SR>1000, the system of ODE's is considered a stiff system. If the system is stiff, using the appropriate algorithms (i.e., stiff or stiffbs in Polymath) should prevent obtaining the negative h, value during integration. SOLVING THE PROBLEM WITH A DIFFERENT SOFTWARE PACKAGE Polymath 6.1 can be used to convert the model of the problem into a MATLAB function. The Polymath generated function is shown in lines 21 through 42 of Table 3. Note that Polymath reorders the equations and changes the syntax of the model according to the requirements of MATLAB. Some manual editing of the Polymath generated function was done in order to enable more concise presentation (the comments were put in the same line with the commands and the ifelse statements were put in a single line). The "main program" that runs the Polymath generated function (shown in lines 1 through 19 of Table 3) is avail able as a template in the "Help" section of Polymath. Only 1.21 1.2  S1.19 E Cl, 1.18 J _1 2 1.17 1.16 0 500 1000 1500 2000 2500 3000 3500 Time (s) Figure 3. Change of the liquid level in the third tank after introducing a 10% step change in its inlet flow rate (ode45 Algorithm, RelTol = 103). the function name has to be added (see line 1) and the initial values of the variables had to be copied from the Polymath generated model (see lines 3 and 4). The ode45 library func tion of MATLAB can be used to solve the problem. This function is based on essentially the same algorithm as the RKF45 routine of Polymath. MATLAB solution of the program in Table 3 yields a simi lar solution to that obtained with Polymath, except that the liquid level in the 3rd tank (h,) exhibits oscillatory behavior (see Figure 3). These oscillations indicate that there may be difficulties with the MATLAB solution as well, and suggest that the error tolerances should be examined. CHECKING THE ROLE OF THE TRUNCATION ERROR TOLERANCE In the Polymath RKF45 algorithm, the integration step size is controlled by two parameters: the relative error tolerance (RelTol) and the minimum number of reporting points (RP). The default values of these parameters can be found under the Polymath icon of "Setup preferences and numerical param eters," which is under Ordinary Differential Equations and RKF45: RelTol = 106 and RP = 100. The RKF45 program includes an adaptive step size control algorithm that monitors the estimate of the integration error and adjusts the step size of the integration in order to keep the error below the specified threshold value. The step size is increased only as long as at least RP full steps are carried out inside the integration inter val. Therefore, a solution of a lower accuracy can be obtained if the error tolerance is increased, and, at the same time, the number of reporting points is reduced. For instance, if the error tolerance is increased to RelErr = 104 and the number of reporting points is reduced to RP = 30, difficulties are encountered earlier. At some time t < 2400, the error message ("negative square root") is issued by the program. With adjustment of RelErr = 1010 while holding RP = 30, however, no error message is issued when the integration is carried out up to tf = 3600. This implies that a correct solution has been obtained. In the MATLAB ode45 function, the integration step size is adjusted so as to keep the estimated error below the value of e which is a function of the specified relative error tolerance, RelTol, and the specified absolute error tolerance, AbsTol. = max(h, RelTol, AbsTol) (9) The default values of the ode45 function for the absolute error tolerance and for the relative error toler ance are AbsTol = 106 and RelTol = 10, respectively. Since the values of h1, h2, and h3 are of the order of magnitude of 1 (one), the AbsTol is inactive and the e values are of the order of 103. With the default er ror tolerances in MATLAB, the oscillatory solution (shown in Figure 3) is obtained. Reducing the rela Chemical Engineering Education 4000 TABLE 3 MATLAB Program for the "Train of Interactive Tanks" Exercise No. Commands and % Comments 1 function TankTrain 2 clear, clc, format short g, format compact 3 tspan = [0 3600]; % Range for the independent variable 4 yO = [1.807; 1.68; 1.162]; % Initial values for the dependent variables 5 disp(' Variable values at the initial point ); 6 disp([' t = num2str(tspan(l))]); 7 disp(' y dy/dt ') 8 disp([y0 ODEfun(tspan(1),y0)]); 9 [t,y]=ode45(@ODEfun,tspan,y0); 10 for i= :size(y,2) 11 disp([' Solution for dependent variable y' int2str(i)]); 12 disp([' t y' int2str(i)]); 13 14 plot(t,y(:,i)); 15 title([' Plot of dependent variable y' int2str(i)]); 16 xlabel(' Independent variable (t)'); 17 ylabel([' Dependent variable y' int2str(i)]); 18 pause 19 end 20 %   21 function dYfuncvecdt = ODEfun(t,Yfuncvec); 22 hl =Yfuncvec(l); 23 h2 =Yfuncvec(2); 24 h3 =Yfuncvec(3); 25 Ao = pi / 100; %Area of drainage orifice at bottom of tanks (m^2) 26 g = 9.81; 27 ql = .05; %Volumetric flow rate into Tank i(m^3/s) 28 q2 = .05; %Volumetric flow rate into Tank 2(m^3/s) 29 if (t <= 50), q3 = .05; else q3 = .05 + .05 .1; end %Volumetric flow rate into Tank 3(m^3/s) 30 H1 = 4; %Height of Tank l(m) 31 H2 = 2.5; %Height of Tank 2 (m) 32 H3 = 2.5; %Height of Tank 3 (m) 33 D1 = 4; %Upper diameter of Tank l(m) 34 D2 = 1; %Lower diameter of Tank 2 (m) 35 if (hl < h2), q4 = 0 (Ao sqrt(2 g (h2 hl))); else q4 =Ao sqrt(2 g (h h2));end %Volumetric flow rate out of Tank 1 (m^3/s) 36 if (h2 < h3), q5 = 0 (Ao sqrt(2 g (h3 h2))); else q5 = Ao sqrt(2 g (h2 h3)); end %Volumetric flow rate out of Tank 2 (m^3/s) 37 q6 = Ao sqrt(2 g h3); %Volumetric flow rate out of Tank 3 (m^3/s) 38 if (h >= H1), dhldt = 0; else dhldt = (ql q4) / (pi 4 (D2 + (D1 D2) / H1 hi) A 2); end %Liquid level in Tank 1 (m) 39 if (h2 >= H2), dh2dt = 0; else dh2dt =(q2 + q4 q5) (pi exp(2 h2) ); end %Liquid level in Tank 2 (m) 40 %Liquid level in Tank 3 41 if (h3 >= H3), dh3dt = 0; else dh3dt =(q5 + q3 q6) (h3 A 2); end 42 dYfuncvecdt = [dhldt; dh2dt; dh3dt]; Vol. 42, No. 1, Winter 2008 tive error tolerance to RelTol = 106 yields a more accurate, nonoscillatory solution. Since the latter solution is more ac curate by three orders of magnitude, the difference between the solutions can serve as a measure of the "error" in the less accurate solution. This "error" in h3 is shown in Figure 4 (note that a few points of larger errors, in the region where the liquid level changes more rapidly, are not shown). Note that the maximal errors are of the order of 103 (while h311.2, see Table 2). Thus, the integration algorithm indeed keeps the error below the desired relative error tolerance. The error in the slope dh3/dt is shown in Figure 5 (a few points of larger errors are not shown in this plot). The maximal absolute errors are of the order 104, but since the value of dh3/dt ~ 4x 106, the relative error is of the order of 25. Thus, a reasonably low relative error of 103 in the variable (h) value yields absurd slope values, with up to 2500% error. It is interesting to investigate the reasons for these huge dif ferences in the relative error values. In Table 4, the values of some of the variables are presented at t = 3600 as calculated by the ode45 function with the two error tolerances. Observe that there are four correct decimal digits in the values of h1, q +q3, and q6 when the higher error tolerance of 103 is used. Introducing these numbers into Eq. (3) to calculate dh3/dt I TABLE requires subtraction of two numbers that are very close. This operation causes the relative error to increase by seven orders of magnitude! We conclude that the erroneous oscillations in the ode45 solution and the "negative squareroot" error message of the RKF45 algorithms are caused by the lack of consideration of the slope value errors by the stepsize control logic of the two algorithms. CHECKING THE ROLE OF THE PROBLEM STIFFNESS The cause of the erroneous oscillations and error mes sages was identified in the previous section. To complete the exercise, however, the stiffness of the problem must also be checked. There are several possible ways to find the elements of the af/ah matrix and to calculate the eigenvalues of this matrix at various times. One option is to use the symbolic manipu lation options of MATLAB. The program for derivation of the elements of the Jacobian matrix and calculation of the eigenvalues at t = 3600 s is shown in Table 5. First, the vec tor of functions (Eq. 7) is defined using symbolic variables (lines 2 through 4). Note that Comparison of Some Variables Values as calculated By the ode45 Function With Two Error Tolerances Relative t(s) h3 (m) q5+q3 (m3/s) q6 (m3/s) dh3/i Tolerance 106 3600 1.2003444895 0.1524588034 0.15245874 4.41426 103 3600 1.2000989262 0.1524832835 0.15244314 2.78699 % Error 0.020 0.016 0.010 63036 0.0015 0.001  0.0005 0 0.0005 0.001  0.0015  0 500 1000 1500 Time (s) the expressions for q4, q , and q6 (as functions of hl, h2, and h3) were introduced dt into the functions assuming that hl
e08 time period. The symbolic 
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PAGE 1 Vol. 42, No. 1, Winter 2008 1 Chemical Engineering Education Volume 42 Number 1 Winter 2008 CHEMICAL ENGINEERING EDUCATION (ISSN 00092479) is published quarterly by the Chemical Engineering Division, American Society for Engineering Education, and is edited at the University of Florida. Co r respondence regarding editorial matter, circulation, and changes of address should be sent to CEE, Chemical Engineering Department, University of Florida, Gainesville, FL 326116005. Copyright 2005 by the Chemical Engineering Division, American Society for Engineering Education. The statements and opinions expressed in this periodical are those of the writers and not necessarily 120 days of pu b lication. Write for information on subscription costs and for back copy costs and availability. POSTMA S TER: Send address changes to Chemical Engineering Education, Chemical Engineering Department., University of Florida, PUBLICATIONS BOARD EDITORIAL AND BUSINESS ADDRESS: Chemical Engineering Education Department of Chemical Engineering University of Florida Gainesville, FL 32611 PHONE and FAX : 3523920861 EDITOR Tim Anderson ASSOCIATE EDITOR Phillip C. Wankat MANAGING EDITOR Lynn Heasley PROBLEM EDITOR James O. Wilkes, U. Michigan LEARNING IN INDUSTRY EDITOR William J. Koros, Georgia Institute of Technology CHAIRMAN John P. OConnell University of Virginia VICE CHAIRMAN C. Stewart Slater Rowan University MEMBERS Kristi Anseth University of Colorado Jennifer Curtis University of Florida Rob Davis University of Colorado Pablo Debenedetti Princeton University Diane Dorland Rowan Thomas F. Edgar University of Texas at Austin Richard M. Felder North Carolina State University H. Scott Fogler University of Michigan Carol K. Hall North Carolina State University Jim Henry University of Tennessee, Chattanooga Steve LeBlanc University of Toledo Ron Miller Colorado School of Mines Susan Montgomery University of Michigan Ronald W. Rousseau Georgia Institute of Technology Stan Sandler University of Delaware Donald R. Woods McMaster University DEPARTMENT 10 Chemical Engineering at Tufts University Nakho Sung, and Daniel Ryder EDUCATOR 2 Jennifer Sinclair Curtis of the University of Florida Nicholas Peppas, Gintaras Reklaitis, and Erik Ydstie RANDOM THOUGHTS 33 Student Ratings of Teaching: Myths, Facts, and Good Practices Richard M. Felder and Rebecca Brent CLASSROOM 23 Active Problem Solving and Applied Research Methods in a Graduate Course on Numerical Methods Eric L. Maase and Karen A. High CURRICULUM 35 Introducing Stochastic Simulation of Chemical Reactions Using the Gillespie Algorithm and MATLAB: Revisited and Augmented A. Argoti, L.T. Fan, J. Cruz, and S.T. Chou LABORATORY 17 Chemical Kinetics, Heat Transfer, and Sensor Dynamics Revisited in a Simple Experiment Mara. E. Sad, Mario R. Sad, Alberto A. Castro, and Teresita F. Garetto TEACHING TIPS 52 Celebrating ChE In Song Alan M. Lane CLASS AND HOME PROBLEMS 47 Applications of the PengRobinson Equation of State Using Mathematica Housam Binous 53 Can I Trust This Software Package? An Exercise in Validation Of Com putational Results Mordechai Shacham, Neima Brauner, W. Robert Ashurst, and Michael B. Cutlip inside front cover Editorial: Engineers Deserve A Liberal Education C. Judson King PAGE 2 Chemical Engineering Education 2 NICHOLAS PEPPAS The University of Texas GINTARAS REKLAITIS Purdue University ERIK YDSTIE Carnegie Mellon University Jennifer Sinclair Curtis of the University of Florida ChE Copyright ChE Division of ASEE 2008 P rofessor Jennifer Sinclair Curtis of the University of Florida cel ebrates 20 years of an academic career that has made her not only an internationally known researcher and educator, but also a great role model for younger generations of chemical engineers. During her life, she has faced family adversities and almost insurmountable crises. But through her perseverance and convictions, as well as a deep Christian faith, she has triumphed. She has touched many others with her concern, enthusiasm, the chemical engineering profession. Jennifer is a wonderful educator recognized with numerous teaching awards, a student counselor par excellence and a leading administrator (with two chairmanshipsone in Fresh man Engineering at Purdue University and another now in Chemical Engineering at the University of Floridaas well as service as an associate dean of Engineering at Purdue). She is a national leader in numerous societies, including AIChE where she has been a major motivating force of the Particle Technology Forum, and as a recently elected AIChE director. But Jennifer Curtis is also the internationally known leader in particle technology, an innovative engineer, and a scholar. Along with all these achievements she enjoys a wonderful fam ily life with her husband, Barry Curtis, and her children, Jennette and Derek.THE EARLY YEARS Jennifer is a product of the rich Mid western tradition of hard work and ap preciation for education. She was born Jennifer Lynn Doloresco, the daughter of Jerry Doloresco and Carolyn Stortz Doloresco, on Christmas Day 1960 in Cincinnati, OH. Her father enlisted in the Navy immediately after high school to acquire enough money to go to Miami University of Ohio. He received a B.S. in accounting in three yearsa necessity because he didnt have enough money to go for a lon ger period of time. Nicholas Peppas remembers, Jerry Doloresco was especially happy when Jennifer went on to graduate school and received her Ph.D. from Princeton University. He was sad that he could not afford more education himself but he was so happy his children (Jennifer and her brother, Bryan Doloresco) could. I guess this is a pretty common story from the generation before ours that grew up in the postDepression period. Jerry was a loving father, always proud of Jennifers achievements. We kept frequent correspondence, and he was particularly proud when Jennifer started her academic career at Carnegie Mellon University. Jerry passed away from cancer in 1995, when Jennifer was 35. Jennifers mother, Carolyn, is of German descent and still lives in Cincinnati. This marvelous mixture of Italian and German background has given Jennifer her wonderful enthusiastic spirit and optimism that everything will go well, but also her dedicationbordering on stubborn nessto achieve her goals PAGE 3 Vol. 42, No. 1, Winter 2008 3 Jennifer graduated from Northwest High School in Cincin nati in May 1979 as valedictorian of her class. Having worked in a summer job at Procter & Gamble, she was sure she wanted to become a chemical engineer and enrolled in the School of Chemical Engineering at Purdue in the fall of 1979. While taking Honors Chemistry during her freshman year she met fellow student Gavin Sinclair. They started dating in their sophomore year and were married in 1981. At that time, the School of Chemical Engineering at Pur due had instituted a new advising program for all students. Each professor was in charge of 30 undergraduates with full responsibility for advising and registration. Nicholas Peppas remembers, I was a young associate professor, just in my third year at Purdue, and I had been as at CMET 210A came this shy 19yearold with golden locks, form. Without even stepping into the room she extended her right hand and announced Here is my form, please sign it. I asked her to come in with the explanation that Id like to learn more about her family and educational background, as well as her future goals. We ended up talking for an hour. I was truly impressed with her intellect and enthusiasm. My intrigue was There and then we decided that Jennifer would start doing research with me during her sophomore year. Jennifers performance at Purdue was stellar. She was se lected a top sophomore woman in engineering in 1981, a top junior woman in engineering in 1982, and a top senior woman in engineering in 1983. She received an Olin Fellowship for of Akron Polymer Engineering competition, and a best paper award at the MidWestern regional AIChE competition at Dayton, OH, in 1983. In addition, she received the Best ChE Senior award in 1983. She graduated in May 1983 with high est distinction, the Purdue equivalent of a summa cum laude Next to her was another graduateher husband of two years Gavin Sinclair, who had achieved the impossible, i.e. to be awarded two analysis of drug release mechanisms from swellable polymers. The paper was published in 1983 in the Journal of Membrane Science and has been cited more than 200 times. Not unexpectedly, the question of graduate education came up very early in Jennifers undergraduate studies, and the pos sibility of an academic career had appeal as early as 1980. So, that fall Jennifer started working with Nicholas Peppas on a advanced models of nonFickian and anomalous transport in glassy polymers with consideration of the relaxational effects of the macromolecular chains and the associated anomalous transport. The problem had been addressed by others with the addition of a pseudoconvective term (resulting from the relaxational effects) until Jennifers model set the physics correctly, recognizing the importance of the moving bound ary value problem (glassy/rubbery transition and erosion fronts). The paper, Anomalous Transport of Penetrants in Glassy Polymers, published in Colloid and Polymer Science almost 25 years ago [ 261 404409 (1983)] has received 180 citations and is one of the most cited articles in the history of this journal. Another paper was published in the Proceedings of the International Congress of Rheology of 1984 where the work had been presented. Planning for graduate school included some long evening discussions between Jennifer and Gavin. Jennifer had received an NSF Fellowship and was clearly interested in a university with a scholarly reputation where she could continue working on transport phenomena. The choice came down to Princeton University or the University of Pennsylvaniawhere Doug Lauffenburger had put together a great academic and re search program and was particularly successful in attracting a wonderful group of exceptional women chemical engineers. Finally, Jennifer picked Princeton, but Doug has remained a great supporter and mentor throughout her life. Jennifer in third grade, left, and with her dad, Jerry Dolo resco, when she was nishing high school in 1979, below. Opposite page: Jen nifer with fellow researchers in her lab at the University of Florida. PAGE 4 Chemical Engineering Education 4 GRADUA TE SCHOOL AND FAMILY Jennifer started at Princeton University in 1983, while Gavin was employed a state awayat Air Products in Al lentown, PA! They decided to live in Flemington, NJ, which meant that every day Jennifer had to drive 50 minutes each way to school at Princeton. In the demanding classes there, she met a number of other successful chemical engineers who have remained good friends, including Paul Johnson (now dean at Arizona State University), Harry Ploehn (now at the University of South Carolina), and Spyros Anastasiadis (now at the University of Thessaloniki, Greece). At Princeton, she had many inspiring teachers such as Carol Hall, Bill Russell, and Roy Jackson. Her rapport with the latter was exceptional; he became the main advisor of her Ph.D. thesis, entitled Ver tical Transport of Gas and Solids with Radial Solid Density Variations. It was publishedamong othersin a seminal issue of the AIChE Journal in 1989 [GasParticle Flow in a Vertical Pipe with ParticleParticle Interactions, AIChE J. (1989) 35 14731486]. Its one of the 50 most cited papers in the history of the journal. While she was in graduate school, Jennifer and Gavin decided to start a family. Nicholas Peppas remembers, In October 1985 Jennifer called me with the good news she later that she called me back to share the terrible news that Gavin had been diagnosed with cancer, probably in the lungs, although that was not clear yet. We spent several days and many hours talking about doctors and possibilities and we ended up with the solution of SloanKettering, where Gavin was indeed saved after more than six months of treatment. I will never forget that in all these months, with her graduate research, her commuting almost two hours a day, and her being pregnant, Jennifer did not sound weak even for one minute. I never heard or saw her cry, complain, or give up. This was an incredible lesson for me and for all of us around her. Indeed, Jennifer was able to face these family crises because of her deep faith and Christian beliefs that have given her strength all her life. Fortunately, 1986 was a much better year! A daughter, Jen nette, was born in April and the couple was reunited after the completion of Gavins surgery and many treatments. Gavin went on to write a book, All Things Work for Good (based on Romans 8:28), about his cancer experiences, focusing on the many lessons he learned and how his Christian faith was strengthened. As for Jennifer, these were the days when she began to master the art of juggling family, career, and life. Her major professor in those days, Roy Jackson, a member of the Royal Society, is now retired and lives in North Caro lina. He remembers, As you may know Jennifer was working for her Ph.D. in my group at Princeton at the time her husband, Gavin, was diagnosed with cancer and a very grim prognosis. At the same time she was pregnant with their first child, so the pressures on her were enormous. In addition she was working on the problem of predicting, in detail, the behavior of gasparticle flows in vertical tubes. The behavior of these flows turned out to have an unexpectedly complex (and interesting) structure, whose details were revealed only by meticulously careful computation and interpretation. In this she succeeded brilliantly, despite everything else weighing on her mind at the time. Her resilience and cheerfulness were simply amazing. He continued, After graduation the demands of her career Gavins health. But, in spite of this, her reputation grew steadi ly and I was proud to follow her progress. She is an amazing woman who combines academic ability and administrative competence with a warm and friendly personality that has made her so many friends. Of my many academic children she is, perhaps, the one in whom I take the greatest pride.AN ACADEMIC CAREER IN THE SERVICE OF STUDENTS In August 1987, while still a graduate student at Princeton, Jennifer answered the calling to teach. As the family could not leave the area because of Gavins medical insurance (the dreaded preexisting condition), she accepted an offer from Lafayette College and began teaching as an assistant profes sor in that schools Department of Chemical Engineering. She stayed there until December 1989, one semester after her formal graduation from Princeton. During her years at Lafayette she became the master teacher and educator that she is, winning numerous teaching awards. As an educator, she taught a wide range of courses such as Separations, Transport, Transport Lab, Fluid Mechanics, Heat and Mass Transfer, Control, Fortran, and Differential Equa tions. To these courses she later added Kinetics, Advanced Transport (grad level), Advanced Math (grad level), Fluid Mechanics, and Heat and Mass Transfer at Carnegie Mellon; Winning the 1989 Lafayette College Teaching Award (pictured with students Paul Verderber and Dave McVeigh). PAGE 5 Vol. 42, No. 1, Winter 2008 5 Kinetics, Fluid Mechanics, and Control at the University of Arizona; and Materials and Energy Balances, Particulate Systems, and Transport Phenomena at Purdue University. By 1989, Gavins medical condition had improved and he was able to move to a new locationPittsburghwhere he was employed by PPG Industries. Jennifer interviewed and was offered a position as an assistant professor at Carnegie Mellon University. Nicholas Peppas remembers again: She was truly excited with the opportunities at CMU. In December days in our homeJennifer, Lisa, and I, planning the future, tions on biomedical applications of microand nanoparticles. I could see the future leader of particle technology in all her intellectual glory.AT CARNEGIE MELLON At CMU, Jennifer found a truly exceptional faculty with colleagues who provided the scholarly environment for her professional development. During the interview process she was particularly affected by then Head of the Department John Anderson (now president of IIT in Chicago) as well as by colleague Ignacio Grossmann. Anderson remembers, When I knew she was not only an excellent scholar and teacher but also a future academic leader. She has proven me correct. She leads by example, and her enthusiasm is infectious. The way Jennifer has handled her personal and professional lives provides a role model for all of us. In a short amount of time Jennifer developed new courses Bolio, who is currently working for McKinsey in Siberia, and Christine Hrenya, who is an associate professor at the University of Colorado in Boulder. The students helped her develop a strong research program supported by industry and government. Ignacio Grossman remembers that Jennifer was the strongest and most popular teacher in the department, and received the highest scores on student evaluations. This is especially noteworthy because she was teaching core courses like Mathematics for Chemical Engineers and Heat and Mass Transfer. She also started to give invited seminars and re search presentations outside the department while engaging herself in university and professional service. While chair of the undergraduate committee, she conceived a plan and raised funds for a new computer cluster for undergraduates. The computer cluster is still a very popular and well used place for students to work and meet. While the computers have been upgraded many times since, credit for the initial concept goes back to Jennifers ingenuity. Jennifer managed to do all of this, not only extremely well, but always with a big smile, and while placing the ma jority of her attention on her husband and children, writes Grossmann. Erik and Genevieve Ydstie moved to Pittsburgh in 1992 when he joined the CMU faculty, and they made a point of moving into the same neighborhood as Jennifer and Gavin. We have a daughter in the same age group as Jennette and we wanted to spend time together, Erik Ydstie remembers. We knew we could not go wrong as far as location was concernedGavin and Jennifer had done extensive research and had even written a computer program to take as many considerations into account as possible to help in their hous ingdecision process! We have many fond memories of good family times spent together after church and late evenings discussing politics and all manner of other things. Yet in the mid1990s, after Jennifer received promotion to associate professor at CMU, happy times gave way to worries once again. There was some deterioration in Gavins health, especially due to the cold winters in Pittsburgh. Grossmann closed the door, and let me know she was leaving for Arizona because Gavins health required that he live in an environ ment with dry air. Despite the fact that her departure was a big setback to our department, I admired how she was able to put her familys priorities above her professional ambitions. This to me speaks volumes of Jennifers integrity, and her great dedication to her family and her profession. In 1995, the family moved to Tucson, where Gavin con tinued to work for PPG remotely and Jennifer became an associate professor at the University of Arizonawhose ChE department was under the expert leadership of Tom Peterson, now dean of Engineering. The relocation rounded out happily with the birth of Jennifers and Gavins son Derek in early 1996. Jennette reads Daddys book, All Things Work for Good to Derek (early 1996). PAGE 6 Chemical Engineering Education 6 But while Tucson was a great place for Gavins health, and PPG could not continue and medical insurance problems refrom a new company due to his preexisting condition. And thats when Gavin and Jennifer remembered their roots. THE PURDUE YEARS In summer 1997, Jennifer and Gavin returned to Purdue to launch the next phase of their careersshe as a tenured associate professor and he as an assistant professor in organizational leadership and supervision as well as in agricultural economics. Although he was physically limited due to the many health problems he faced, Gavin had completed a masters degree in economics at Lehigh University while working full time at Air Products, and then completed a Ph.D. in social and decision sciences at CarnegieMellon while working full time at PPG. This dualacademiccareer couple immediately set about having an impact on their respective departments and students. In spring 1998, Jennifer launched a survey course in Particle Technology (described in a CEE article in fall 1999). Beginning in 2000, this course was video broadcast to include industrial audiences, and it was cotaught with Carl Wassgren (M.E.) from 2002 on. The Harris taking up the course when Jennifer no longer could. In 1999, Jennifer was elected trustee of CACHE (Computer Aids for Chemical Engineering), where she championed the introduction of computational her CEE article in spring 1998). It was through her initiative that Rodney Fox and Dick LaRoche were elected to the CACHE board to further promote this domain in chemical engineering research and education. In fall 2000, while Gavin launched his book, Life, Love and Economics based on his very successful introductory economics course, Jennifer launched the administrative component of her rapidly accelerating career: She accepted the position of head of Freshman Engineering, leading a department that was home to some 2,200 beginning engineering students. She took on this major challenge with her usual optimism and cando style, reenergizing the staff, recruiting new factransition to college. She was remarkably adept at soothing the concerns of parents while encouraging their sons and daughters to time for campus activities. She was also very effective at both attracting new resources and capitalizing on those the department had. She worked closely with an Industrial Advisory Council for Freshman Engineering that helped to refocus that departments strategic directions to its central role of preparing students for success in the professional schools. The chair of that council, Bob Davis (Air Products & Chemicals) comments on her effective leadership style: Jennifer added energy to everything she touched. Her real gift was empowering all the members of her department so that the resources of the department were seemingly multiplied. The impact on results and morale was remarkable! Sadly, into this whirlwind of positive pursuits came a devastating downturn: Gavins health deteriorated again, and he was moved back to Tucson because of the excellent medical facilities there. He passed away during Christmas 2000. FORGING ON Jennifers optimistic outlook prevailed. In spite of her many administrative duties and the neverending meetings, she continued growing her research group to 10 solely or jointly advised graduate students. In Fall 2001 she was promoted to full professor and shortly thereafter (March 2002) she was named the associate dean of the College of Engineering with Above, the Sinclair family (Jennette, Jennifer, Gavin, and Derek) on a 1998 Caribbean cruise. An annual cruise has remained a tradition for Jennifer and her family. Initially they were all Caribbean cruises, but then they became Alaskan and more recently Mediterranean journeys as well. Below, Gavin Sinclairs textbook, Life, Love and Economics, which was featured in Newsweek magazine in May 2000 and on the front page of The Wall Street Journal in October 2000. ately set about having an impact on their respective departments and students. In spring 1998, Jennifer launched a survey course in Particle Technology article in fall 1999). Beginning in 2000, this course was video broadcast to include industrial audiences, and it was cotaught with Carl Wassgren (M.E.) from 2002 on. The CACHE (Computer Aids for Chemical Engineering), where she championed the introduction of computational Jennifer added energy to everything she the resources of the department were seemingly multiplied. The impact on results and morale was remarkable! Sadly, into this whirlwind of positive pursuits came a devastating downturn: Gavins health deteriorated again, and he was moved back to Tucson because of the excellent medical facilities there. He passed away during Christmas 2000. FORGING ON PAGE 7 Vol. 42, No. 1, Winter 2008 7 responsibility for Undergraduate Education, while continuing her duties as head of Freshman Engineering. And, of course, the particle technology course had to go on as well! In her associate dean role, Jennifer was instrumental in setting the stage for the reorganization of the Freshman Engineering Department and the subsequent launch of a new Department Engineering Program, the Women in Engineering Program, and the Cooperative Education Program. Her highly effective leadership in engineering education was recognized through the 2003 Sharon Keillor ASEE Award for Women in Engi neering Education. By fall 2003, it was clear that Jennifer was ready to move on to further challengesto the good fortune of the University of Florida. Without question, her Purdue years spanned a very productive phase of her profes sional life as teacher, researcher, and academic leader, but also involved very low as well as high points in her personal life. She remains a valued colleague and has certainly left her impact on Purdue Engineering.A N IN TE RNA T IONALLY KNOWN R E S E AR C H E R Throughout her academic career, the impetus for Jennifers energy and creativity has been her attraction to, and thus seminal contributions toward, key research issues. In this area, she has contributed on manifold fronts both on a national and international scale. Jennifer has an internationally recognized research program. Her work in the development and validation of numerical an interdisciplinary research area of both great technologi encountered in the pharmaceutical, chemical, mining, agri cultural, food processing, and petroleum industries. Jennifer has received various awards for her pioneering work in this area. One of her most notable successes is the adoption of softwarepackage vendors. Both companiesFluent and CFXhave adapted her models for dilute and densephase academic and industrial experts. Jennifer received the NSF Presidential Young Investigator Award and an engineering college research award when she was a professor at Carnegie Mellon University. Also early in her career, she was awarded NSF equipment grants that funded the purchase of important equipment such as a laser Doppler velocimetera purchase that helped her propel her early research program further toward the leading edge of science. 1. John Doney, ChE, M.S., Carnegie Mellon Uni versity, Heat Transfer in Dilute ParticleLaden Turbulent Flows, 1993 2. Eduardo Bolio, ChE, Ph.D., Carnegie Mel lon University, Dilute Turbulent GasSolid Flows with ParticleParticle Interactions, 1994 3. Christopher Alexander, ChE, M.S., Carnegie Mellon University, Pneumatic Transport with a Bimodal Particle Size Distribution, 1995 4. John Doney, ChE, Ph.D., Carnegie Mel lon University, Electrostatic Behavior of Charged Particle Mixtures, 1996 5. Christine Hrenya, ChE, Ph.D., Carnegie Mel lon University, Dense GasSolid Suspension Flow,1996; 1997 recipient of Best Ph.D. in Particle Technology by the International Particle Technology Forum (Dr. Hrenya is now associate professor of chemical and biomolecular engineering at the University of Colorado) 6. Pawan Agarwal, ChE, M.S., The University of Arizona, Modeling GasSolid Flow with a Bimodal PSD, 1997 7. Michael Schabel, Materials Science, M.S., The University of Arizona, Characterization of Particle Traps in Plasmas, 1997 8. Tim Mallo, ChE, Ph.D., Carnegie Mellon University, Modeling Heat Transfer in Dilute and Dense Phase GasSolid Flow, 1997 9. Berend van Wachem, ChE, Ph.D., Delft Univer sity, Flow in a Fluidized Bed with a Bimodal PSD, 2000 (Dr. van Wachem is now associate GRADUATE STUDENTS WHO DID THEIR THESES WITH JENNIFER CURTIS professor at Chalmers University, Gothenburg, Sweden, Department of Applied Mechanics) 10. Cynthia BlakePowell, ChE, M.S., Purdue University, Comparison of AdHoc Theory Predictions and CFD Predictions for Con 11. Agus Sumantri, ChE, M.S., Purdue Universi ty, Comparative Analysis of Computational Fluid Dynamic Models for GasSolid Flow in Risers, 2000 12. Edward (Nick) Jones, ChE, Ph.D., Purdue University, The Effect of PSD on GasSolid Flows, 2001 13. Chie Min Chung, ChE, M.S., Purdue Univer sity, CFD Modeling of a Fluidized Bed with a Wide Particle Size Distribution, 2002 14. Stephanus Budilarto, ChE, Ph.D., Purdue University, Experimental Study of Velocity Ratio, Particle Size and Size Distribution Effects in ParticleLaden Jet Flow, 2003 15. Kim Hayden Henthorn, ChE, Ph.D., Purdue University, The Effect of Particle Rough ness on the Flow Behavior of Fine Powders, NSF Graduate Fellow, 2004 (Dr. Henthorn is now assistant professor in chemical and biological engineering at University of Mis souriRolla) 16. Kunn Hadinoto, ChE, Ph.D., Purdue Uni versity, DensePhase Flow Measurements using LDV, 2004 (Dr. Hadinoto is now assistant professor in chemical and biomo lecular engineering at Nanyang Technologi cal University) 17. Michael Lasinski, ChE, Ph.D., Purdue University, Particle Clustering in Granular Flows, 2004 (coadvised student) 18. Bill Ketterhagen, ChE, Ph.D., Purdue University, DEM Simulations of Granular Discharge from Hoppers, 2006 (coadvised student) 19. Robert Hamilton, ChE, Ph.D., Purdue Uni versity, Evolving Particle Size Distribution in the Flow of GasSolid Mixtures, (1999 ) (coadvised student) 20. Benjamin James, ChE, Ph.D., University of Florida, Effect of Particle Shape on ParticlePhase Stress, (2004 ) 21. Mark Pepple, ChE, Ph.D., University of Florida, LDV Measurements of LiquidSolid Flow, (2004 ) 22. Anshu Anand, ChE, Ph.D., University of Florida, Effects of Particle Cohesion and Shape on Particle Segregation in Hopper Flows, (2005 ) 23. Julio Cesar Castro Vazquez, ChE, Ph.D., Uni versity of Florida, Effect of Wall Roughness on Wall Friction Angle, (2006 ) 24. Juan Pedro Marval, Departamento de Ener gtica, Universidad Nacional Experimental Francisco de Miranda, Punto Fijo, Edo. Falcn, Venezuela, Numerical Simulation of Saltating Particles using Granular Kinetic Theory, (2006 ) (coadvised student) 25. Byung Chu, ChE, M.S., University of Florida, Experimental Measurements of Sifting Segregation in Hoppers, (2005 ) PAGE 8 Chemical Engineering Education 8 The importance of her research has been validated through multiyear funding from major corporations such as Chevron, Industrial Technology Consortium on Multiphase Phase Flow. This consortium involved collaboration between industry, academia, and national labs. It received national recognition, and as part of it Jennifer gave a panel presentation before Congress in 1999. inspirational talks. In her academic career, she has given more than 180 invited lectures. in various positions. For example, she served on the Board of Directors for the American Chemical SocietyPetroleum Research Fund (ACSPRF), as a member of an NSForga nized delegation to South Africa promoting collaboration in particle technology, as a member of an external review panel to review the entire Chemical and Transport Division of the NSF, as a member of the editorial advisory board of the Journal of Powder Technology and the Journal of Phar maceutical Development and Technology, and since 2007 as associate editor of the AIChE Journal Within AIChE, she served as the national meeting cochair, along with Gintaras Reklaitis, for the 2002 Annual AIChE Meeting that was held in Indianapolis. She was elected to the executive committee of the Particle Technology Forum for eight years and to the AIChE Fluid Mechanics Programming Committee for four years, and was chair of the Transport and Energy Processes Division in 2006. A unique contribution to the annual chemi cal engineering meetings has been her continuous leadership and support (with Rosanne Ford of the University of Virginia) of the Christian ChE Fellowship meetings that are held on Tuesday or Wednesday mornings of the AIChE meetings. These meetings were initiated in 1989.AN EDUCA T OR, ADVISOR, AND MENT OR Jennifer is a highly effective teacher and mentor. She re ceived departmental and university teaching awards while at Lafayette College and the University of Arizona. At Purdue, she received the departmental Kimberly Clark Mentoring Award and was named to the universitywide Teaching for Tomorrow Program. In 2001, Jennifer served, along with then Purdue Engineering Dean Linda Katehi, on a 10mem ber invited NSF panel on the Future of Engineering Educa tion. From 2003, she also served on the NAEs Engineering Education Committee. She is very active in training new chemical engineering professors. She has participated as an instructor for new chemical engineering educators at the Ameri can Society for Engi neering Educations Chemical Engineer ing Summer School, In 1997, she served as a lecturer at this work shop. In 2002 and 2007, she organized and conducted a very well attended workshop on CFD. In August 2000, when she was named head of Purdues Freshman Engineering Department, she embarked upon attending a series of workshops that better prepared her for administration. For example, in 2001 she attended the twoweek Management Development Program at Harvard University and in 2002 the American Council on Educations Chairs and Deans Workshop. As a Ph.D.student advisor, Jennifer has excelled by trans ferring to her students the same style and the same warm concern that Roy Jackson, her Ph.D. advisor at Princeton, had transferred to her. Her former Ph.D. student Kimberly Hayden Henthornnow an assistant professor in chemical engineering at the University of MissouriRollaremarks, I think the thing that impressed me most about Jennifer was her leadership styleshe led by example. She kept her graduate students motivated and working hard by showing us that she was working just as hard as (and sometimes harder than) we were. Right: At Carnegie Mellon University with John Anderson and Ignacio Grossman in 1995. Below: In Australia with a friend in 2005. Facing page: Jennifer, Derek, and Barry celebrating Jennettes graduation from her parents alma mater, Purdue University, in May 2007. PAGE 9 Vol. 42, No. 1, Winter 2008 9 Naturally, Jennifer has not only been an exceptional engineer, researcher, and educator, but also a wonder ful role model for the young generation of women chemical engineers. More than 20 years ago, she had ChE and NAE member Carol Hall, then at Princeton. And many years later, Jennifer is now recognized as an example for others. Again, Kim Hayden Henthorn notes: Having a strong, female role model like Jen nifer has made a tremendous impact on me. She has been through so much, but has kept an upbeat, positive attitude throughout. When I feel overwhelmed trying to balance everything on my plate, I always remind myself that Jennifer has been through all of this before, and I give her a call. She is still willing to listen and give good, thoughtful advice about everything from family life to being a woman in academia. Christine Hrenya, another former Ph.D. student of Jennifers and now an associate professor at the Uni versity of Colorado, admires Jennifers ability to focus and be successful. Jennifer is incredibly productive, but she has an unfair advantage over the rest of usshe this is her longtime average! Along the same lines, she is an expert at multitasking. Once when I was visiting her house for dinner, she promptly pulled out the iron I was overstaying my welcome!). This is the love and admiration shown by her students. And Hrenya concludes: On a more serious note, Jennifer was an outstanding advisor, and continues to be a role model, friend, and sounding board for all topics under the sun. I am extremely fortunate to have spent so many of my formative years under her guidance.A N E W LIF E A T T H E U NIV E RSI T Y OF F LORIDA In 2001 Jennifer met Barry Curtis, another Purdue chemical engineer, who had graduated two years before her (in 1981), hav Advisory Board dinner in Houston. Jeff Hemmer then hooked us up on a strategic planning project for Freshman Engineering, when I was head of that department. The rest is history. They were married in 2002 and moved to Gainesville in 2005. Barry has been a wonderful father to Jennette and Derek and a most supportive and admiring husband and partner to Jennifer. Just before the move to Florida though, the family took yet another tripto Australia! Jennifer was a visiting professor at the Chemi cal Engineering Department of Monash University in Melbourne in 2005, received the Eminent Overseas Lectureship Award from the Institution of Engineers in Australia, and visited a number of Australian universities for seminars. In 2005, Jennifer Curtis assumed a new responsibility as the chair of the Department of Chemical Engineering at the University of Florida. In a very short time, she has taken an already nationally recognized program (achieved under the continuous and imaginative leadership of the editor of this journal, Tim Anderson) and further improved its national ranking. Here is how Anuj Chauhan, associate professor in this depart ment, sums up the thoughts of the younger generation of professors there: Jennifer runs the department in a very fair, transparent, and juniors, and she deserves a lot of credit for the rapid growth of the department and the improvement in rankings. At the University of Florida Jennifer keeps an active research program and collaborates with the nationally known ERC program on particle technology. Meanwhile, the Sinclair Curtis family is thriving. Derek is now 11 years old and in sixth grade. Jennette attended Purdue in the School of Managementthe same school from which her father graduated 24 years earlierand received her B.S. in management in May 2007 with highest distinction (a perfect 4.00). She is now a graduate student at Michigan State University, specializing in organizational behavior. Like her parents, she wants to become a professor. Looking back at Jennifer Curtiss life and accomplishments, one cannot help but admire her concern for others, her high standards, and her dedication to education and advising. But one cannot forget the adversities she has faced in her life. As Nicholas Peppas likes to point out: At some time or another all of us have had crises or adversities in our lives. How Jennifer was able to cope with them has been an inspiration for all of us. I think Jennifer, Barry, and the children have found the golden rule of a good, inspiring, and productive life. They are a wonderful example that all things work for good . PAGE 10 Chemical Engineering Education 10 BY NAKHO SUNG AND DANIEL RYDER T ufts College was chartered in 1852 as the 163rd institution of higher learning in United States. Two University is a diverse ensemble of undergraduate, graduate, and professional schools that includes the Schools of Arts and Sciences, Engineering, the Cummings School of Veterinary Medicine, the Schools of Dental Science and Medicine, the Sackler School of Graduate Biomedical Sciences, the Friedman School of Nutrition Science and Policy, the Tissue Engineering Resource Center, and the Fletcher School of International Law and Diplomacy. Fulltime students total 8,830, of which 4,950 are undergraduates. ChE at Tufts University ChE Copyright ChE Division of ASEE 2008 The atrium of the Tufts Science and Technology Center, home to Chemical and Biological Engineering. PAGE 11 Vol. 42, No. 1, Winter 2008 11 The School of Engineering includes the Department of Chemical and Biological Engineering as well as the Depart ments of Biomedical Engineering, Civil and Environmental Engineering, Electrical Engineering and Computer Science, and Mechanical Engineering. Chemical Engineering debuted at Tufts in 1898 to become Chemistry Department until February 1949; its undergraduate program was initially accredited three years later. An M.S. program was added in 1962 and a Ph.D. program in 1965. (now a professor in the department of Chemical Engineer ing and Materials Science at the University of California, Davis) and Wayne Sanborn (who worked for Matheson Gas Donald Merchant, Joseph DelPico (who worked for Polaroid Corporation), and Edward Denk, Jr. (now a consultant at Evergreen Solar). This year the department will award some 15 graduate 20 B.S.ChE degrees, enrollment in each of the continuing undergraduate classes exceeds 30 students. A perennial hightech incubation zone, the greater Boston area has long been one of the nations hubs for biotechnology R&D. Within this community, numerous professionals are Tufts Chemical Engineering alumni. Many remain actively involved with the department, which offers teaching and research programs in biological engineering and has contrib uted nationally to development of the relevant undergraduate curricula. For the past 15 years, the department has been assembling a core group of biological engineering faculty. Graduate research is divided within three main areasbiological en gineering, energy and environment, and advanced materials. The current plan is to focus on expanding clean energy and environmental sustainability along with systems engineering research and teaching.TUF T S SC I E N CE AND T EC HNOLO G Y C E N TE R The department is currently housed in the Tufts Science and Technology (SciTech) Center, which was constructed ment of Energy. At that time, the Department of Chemical Engineering was the only academic department housed at the facility. The Tufts High Energy Physics Laboratory, Biotech nology Engineering Center, ElectroOptic Research Center, and Laboratory for Materials and Interfaces (LMI) were also housed at the site. Both the Biotechnology Center and LMI were associated research facilities of the ChBE Department. Later, through the generous support of Raytheon Corporation, the Pollution Prevention Projects Laboratory was established within the departments facilities. In addition to the Depart ment of Chemical and Biological Engineering, the complex currently houses the recently established Department of Biomedical Engineering and the Gordon Institute for Engi neering Management.CHEMICAL AND BIOLOGICAL ENGINEERING The department has a longstanding history relating to the development of educational programs in bioengineering. In 1986, the then Chemical Engineering Department initiated a Massachusettssponsored continuing education program to train technical staff for biotechnology business. The Bio as a series of handson, introductory, technical courses and remains a vital link to local industry, has led to the develop ment of a graduatedegree program. Today, the department offers graduate degrees in both chemical engineering and biotechnology engineering, including courseonly M. Eng. degrees attractive to working professionals, as well as tradi tional thesisbased M.S. and Ph.D. programs. In 2004 the department hosted a series of NSFsponsored workshops focused on the development of chemical and bio logical engineering curricula. Participants from academia and local bioindustries helped to shape our current undergraduate program (see Table 1), which has evolved to include intro ductory courses on biology, statistical thermodynamics and molecular structure (Physical Chemistry II), and biochemistry. Simultaneously, the chemical engineering science core has incorporated biological systems and processes, while the upperlevel course offerings have been supplemented with applied biologybased electives (see Table 2). We are currently redesigning the senior laboratory courses to comprise indus trysponsored research projects in biotechnology, metabolic engineering, energy sustainability, systems engineering, and advanced materials.F ACUL TY The current department faculty includes nine tenuretracked members and two jointly appointed members: Professor Linda M. Abriola currently dean of Engineering, does research in the mathematical modeling of the transport and fate of organic chemical contaminants in porous media. scribe the interphase mass partitioning and nonaqueous phase migration of organic liquid contaminants in the subsurface. Her recent research involves the use of models and laboratory ing the persistence of organics and controlling the effective ness of aquifer remediation technologies. She is also a member of the Department of Civil and Environmental Engineering, and a member of the National Academy of Engineering. PAGE 12 Chemical Engineering Education 12 Professor Maria FlytzaniStepha nopoulos spear heads clean energy technology and catalysis research in the School of Engineering. Her work focuses on catalytic fuel pro cessing, including hydrogen generation, for fuel cells; and new materials for air pollution abatement and fuel gas desulfuriza tion. Her groups fundamental research on catalyst structure and function has very active oxidation catalysts based on Au, Pt, and Cu on cerium oxide. An exciting new development in the via reversible adsorption. FlytzaniStephanopouloss research group presently comprises one postdoctoral fellow, seven Ph.D. and four M.S. research assistants, and several undergraduates. Her research is supported by the NSF, the DOE, the Army Research Laboratory, and industry. Since 2002, she has served as the editor of Applied Catalysis B: Environmental Professor Christos Georgakis established the Systems Research Institute for Chemical and Biological Processes in 2005. Modeled on the Chemical Process Modeling and Con trol Research Center at Lehigh University, the Institute fosters interdisciplinary research in modeling, optimization, and operation of chemical and biological systems and processes in collaboration with leading pharmaceutical and chemical Consortium on Batch Manufacturing of Pharmaceuticals and Chemicals (BMPC)began operation in 2006. The consor tium is a cooperative research program that will focus on improving the development, design, and operation of batch pharmaceutical and chemical processes, with the ultimate Initial research areas encompass designing batch reactors for multistep synthesis; enantiomeric and polymorphic crys tallizations; plantwide simulations, scaleup, and process sensitivities; and statistical process monitoring and process control that will effectively contribute to the implementa tion of the FDAs Process Analytical Technology Initiative. Oct. 25, 2006. Professor David L. Kaplan s group applies biopolymer engineering to elucidate structurefunction relationships, polysaccharides. They apply physiological and genetic engi neering approaches to control biopolymer chemistry in order to then explore selfassembly and function from a materials science and engineering perspective. The biopolymer sys tems are studied for potential biomaterials applications, such as matrices or scaffolds to support cell and tissue growth, for controlled release of pharmaceuticals, and as biosen sor platforms. Selective processing, surface chemistry, or genetic engineering redesigns are employed to functionalize biopolymer material surfaces or control morphology and structure, in order to enhance biomaterial interactions with inorganic and biologic interfaces. In combination with human stem cells and complex bioreactor systems, the biopolymer biomaterial systems are employed in the generation of func tional human tissues including bone, cartilage, and adipose systems. These systems are studied both in vitro and in vivo for tissue regeneration, as well as to serve as human disease models with which to understand disease progression, and subsequently for pharmaceutical discovery. Transport issues are addressed via vascularization strategies in vitro and in vivo to promote improved functional tissue outcomes. Kaplans lab collaborates with many colleagues within and outside of Tufts. He also serves as the chair of the Department of Biomedical Engineering. Professor Kyongbum Lee s research in systems biology, metabolic engineering research in systems biology, and meta bolic engineering and tissue engineering is aimed at better understanding the chemical design of living systems through the development and application of analytical platforms, mod eling tools, and other enabling technologies. He is especially interested in characterizing and manipulating cellular metabo lism for a variety of basic and applied studies. His research is therapies for obesity and type 2 diabetes. Current work focuses on the metabolism of liver and white fat cells. Some of the ChBE faculty. Top, LR: Christos Georgakis, Blaine Pfeifer, Kyongbum Lee, Jerry Meldon, Daniel Ryder, Hyunmin Yi. Bottom, LR: Ramin Haghgooie, Nakho Sung, Eric Anderson. Faculty member Martin Sussman. PAGE 13 Vol. 42, No. 1, Winter 2008 13 There is increasing evidence that the functions and struc tural features of a biochemical network are intertwined. Thus, a useful approach to studying the design of these complex systems has been to characterize their layout, or wiring. Lees research aims to develop a modeling framework to integrate the analysis of the structural and functional layout of biochemical networks. For example, he has recently de veloped and applied a series of algorithms for the rational decomposition of metabolic networks based on experimen tally derived reaction activity data. In the context of metabolic engineering, Lees group uses mathematical models of metabolic pathways in conjunction with experimental biochemistry techniques to quantitatively study the metabolic basis of adipose tissue development and growth. In addition to fundamental insights into adipocyte metabolic regulation, they seek to obtain: 1) novel drug tar gets, for example enzymes that catalyze the key controlling steps in adipocyte lipid accumulation; and 2) robust, sensitive, and easily measured metabolite biomarkers for diagnosis of peutics. To these ends his group has performed in silico and metabolic intervention, and demonstrated the feasibility of achieving a global shift in lipid accumulation through the forced expression of a single protein. Lee also uses tissue engineering to design, build, and optimize model systems whereby important cellular func tions may be studied under wellcontrolled conditions while maintaining a high degree of physi ological relevance. The primary focus here is on engineering the cellular component and its medium (as opposed to the materials or reactor housing). Current projects leverage developments in incubators that afford spatial control over the chemical en vironment of the cultured cells. This incubator is used for a number of ongoing studies involving heterogeneous cell components ( e.g. adipocytepreadipocyte coculture) and solution gradients of exogenously added chemicals ( e.g. drug transformation by hepatocytes). Professor Jerry H. Meldon s research interests are in mass transfer and separation processes, particularly when revers ible chemical reactions control the rate and selectivity of a diffusional separation. His primary focus is on membrane processes, gas scrubbing, and mathematical modeling of mass transfer with chemical reaction. Recent work includes theoreti cal and experimental investigations of hydrogen transport in palladium membranes and its role in catalytic membrane reac tors for the production of pure hydrogen. One Ph.D. student is building and modeling a novel spinning disk system for reaction and separation; another is modeling NOx scrubbing. Professor Blaine A. Pfeifer s research group seeks to the synthesis of therapeutic products. For example, they ap ply molecular and process engineering in conjunction with molecular biology, microbiology, analytical chemistry, and bacterial genetics to develop microbial bioprocesses and products that can potentially target cancer, bacterial infections, diabetes, and other diseases. ways to generate biological products. One approach is to Left, one of several students in the senior projects lab who got handson experience when the departments distillation column needed repair. Below, Danny Pierre, a graduate student in Pro fessor FlytzaniStephanopouloss catalysis lab. Above, Gregory Botsaris gives a presentation at the 2005 sympo sium in his honor. PAGE 14 Chemical Engineering Education 14 transplant genetic material responsible for an important thera peutic product into a convenient and processfriendly bacterial microorganism (such as Escherichia coli) for eventual product scaleup and development. Current projects include: 1) cellu lar and metabolic optimization for production of the antibiotic erythromycin; 2) production of new and established anticancer agents; 3) development of new biological vaccine systems; and 4) modeling and experiments to identify and elucidate cellular bottlenecks in natural product biosynthesis. Professor Daniel F. Ryder s primary research lies in the processing and characterization of ceramic and organically for application in high frequency devices; chemical process ing of superconducting oxide materials; the development of and chemical vapor deposition methods for IIIV nitride semiconductors. Professor and Chair Nakho Sung s research in the area of structurepropertyprocessing relationships of polymers and composites includes development of in situ monitoring techniques for characterization of reactions in polymers and oped for various matrix polymers for advanced composites to follow cure reactions as well as water uptake using distalend insensitive, such as those based on polyimides, epoxies, and cence monitoring. Another area of research has been to develop a methodol ogy for tailor designing composite interphases. Using a lowtemperature plasma of various monomers and inert gases, as adhesion strength and longterm stability against tempera ture and moistureinduced degradation. Sorption/diffusion behavior of organic molecules in multiphase polymers is investigated using a model composite consisting of glassy and rubbery domains where morphology is systematically varied. These model composites facilitate elucidation of complex diffusion behavior and development of a predictive diffusion model. Recently, Sungs group has been developing lowcost membranes for fuel cell application and hydrogen for lowtemperature direct methanol fuel cells. Professor Hyunmin Yi s research interests lie in biochemi cally driven nanometer scale fabrication (nanobiofabrication) of high throughput biosensors, biophotonic devices, and nanocatalysts for biomedical, environmental, and energy ap plications using smart biopolymers and viral nanotemplates. His primary research areas are nanobiofabrication with geneti fabrication with smart biopolymers. Yis group has developed nucleic acid hybridizationbased mosaic viruses through their own genomic mRNA. These biologically derived nanotubeswith the precise dimensions of 300 nanometer (nm) length,18 nm outer diameter and 4 nm inner channelcan serve as nanotemplates for covalent coupling of functional nanoparticles. Building upon these facile assembly strategies of these potent nanotemplates, his group is working toward the development of highthroughput biosensors and BioMEMS devices for environmental and biological threat detection. In closely related collaboration with FlytzaniStephanopoulos, they are also developing na Left, Zheng Zhou, a graduate student in Professor FlytzaniStephanopouloss catalysis lab. Right, graduate student Lisa Schupmann works with a multireactor system in Professor Georgakiss Systems Research Institute. PAGE 15 Vol. 42, No. 1, Winter 2008 15 in turn arterial blood pressure is the main objective. ematical model of transport in the tubular and microcirculatory systems of the renal medulla is in development in order to ex amine how interactions between oxygen, nitric oxide (a potent vasodilator), and superoxide affect tubular sodium reabsorp blood pressure. Mathematical simulations of ion concentration changes within the bulk cytoplasm and microdomains are under devel opment to elucidate contractil ity of resistance vessels, such as medullary descending vasa recta, regulated by variation of intracellular Ca2+ concentration ([Ca2+]cyt) in both endothelium and smooth muscle. This model accounts for the characteristics of the channels and transporters that exchange ions between the plasma membrane, SR stores, and cytosol. Edwardss group is investigating which factors are most likely to serve as principal determinants of [Ca2+]cyt and delineate the mechanisms through which the bulk cytosol, microdomains, and SR communicate. Their goal is to ultimately relate local medullary NO concentrations to [Ca2+]cyt changes and to blood vessel diameter variations. The early transition years of any department present a series and determination of its faculty. The accomplishments of these early efforts are often taken for granted with the passage of time, but whatever level of success that this department presently enjoys is due in large part to a small group of faculty that has served Tufts throughout their professional career. Martin V. Sussman joined the department in 1961 and served on the faculty for 37 years before retiring due to ill ness in 1998. His popular Technology as Culture course, which offered liberal art students insights into the world of technology, made him a household name across the Tufts community. He earned his doctorate in chemical engineer ing from Columbia University and was a Fellow of both the American Institute of Chemical Engineers and the American Chemical Society. He was the author of two books on ther modynamics as well as numerous technical articles. Martins A closeup of the Tufts Chemical and Biological Engineering Departments Xray diffractometer. noscale gold particlebased catalysts for energy applications using the viruses as nanotemplates with high capacity and precise nanoscale spacing. In his second research area, Yi seeks to develop biocom patible highthroughput biosensing platforms and implant able biophotonic devices for biomedical and environmental applications. They exploit the stimuliresponsive properties of smart biopolymers to fabricate nanoscale patterns and waveguides with high spatial, temporal, and orientational control. Example biopolymers include structural proteins such as gelatin and silk as well as polysaccharides such as agarose and chitosan. Recently his research group has shown that the thermoresponsive morphology transition of common means for consistent manufacturing of nanometerscale sur face diffraction gratings under mild processing conditions. They are currently working on building allbiopolymeric biosensors and nanoimprinted biopolymeric gratings. Research Professor Aurlie Edwards s research falls broadly in the domain of biological transport phenomena, developing theoretical models of transport in the renal medullary micro circulation, both at the macroscale and the cellular scale. The microcirculation in the renal medulla plays an essential role in term maintenance of arterial blood pressure. Understanding PAGE 16 Chemical Engineering Education 16 T ABLE 2 Biological Engineering Electives ChBE 160 Biochemical Engineering ChBE 161 ChBE 162 Introduction to Biotechnology ChBE 163 Recombinant DNA Techniques Laboratory ChBE 164 Biomaterials & Tissue Engineering ChBE 166 Principles of Cell and Microbial Cultures ChBE 167 Metabolic and Cellular Engineering ChBE 168 Biotechnology Processing Projects Laboratory T ABLE 1 Chemical & Biological Engineering Curriculum at Tufts University First Year Fall Term Spring Term EN 1 (Intro. to Computers in Engineering) EN 2 (Intro. to Engineering Communication) Math 11 (Calculus 1) Math 12 (Calculus 2) Chem 1 (Chem. Fundamentals w/ Lab) Chem 2 (Chem. Principles w/ Lab) English 1 (Expository Writing) Humanities/Social Sciences Elective EN Elective1 (Intro. to Engineering) Physics 11 (General Physics w/ Lab) Second Year Fall Term Spring Term ChBE 10 (Thermo & Process Calc I) ChBE 11 (Thermo & Process Calc II) Math 13 (Calculus 3) Math 38 (Differential Equations) ES 11 (Intro. to Biology) ChBE 39 (Appl. Math and Numerical Methods) ES 10 (Intro. to Materials Science) Chem 32 (Physical Chemistry II) Chem 33 (Physical Chemistry Lab I) Humanities/Social Sciences Elective Third Year Fall Term Spring Term ChBE 21 (Fluid Dynamics & Heat Transfer) ChBE 22 (Mass Transfer) Chem 51 (Organic Chem I) ChBE 102 (Reactor Design) Chem 53 (Organic Chem Lab I) Bio 152 (Biochemistry & Cellular Metabolism) ES 3 (Intro to Electrical Engineering) Advanced Chemistry Elective Humanities/Social Sciences Elective ChBE Concentration Elective Free Elective Fourth Year Fall Term Spring Term ChBE 45 (Separation Processes) ChBE 60 (Chemical Process Design) ChBE 51 (ChBE Unit Ops Lab) ChBE 52 (ChBE Projects Lab) ChBE 109 (Process Dynamics & Control) ChBE Concentration Elective ChBE Concentration Elective Advanced Chemistry Elective Humanities/Social Sciences Elective Humanities/Social Sciences Elective of materials science and biotechnol ogy. He was author to some 20 pat ents that included the Incremental Draw Process for the processing of tissue culture matrix. The department was saddened by his passing in 2005, but his legacy and connection with Tufts survive through the generous funding of the Jeanne and Martin Sussman Endowed Fellowship. Gregory D. Botsaris joined the department in 1965 immediately after earning his Ph.D. in chemical engi in 2004 after 39 years of service, but remains on campus to continue his research in crystallization processes. He is the author of numerous technical ration processes. His contributions to crystallization technology were recognized by his peers through the establishment of the Gregory Botsaris Lecture series. Ken Van Wormer is the new est emeritus faculty member. Van Wormer joined the department as an instructor in 1954 while completing his doctoral studies at M.I.T. His continuous service to the department for an impressive 53 yearsincluding 10 years as chairis a testament to his dedication to the profession.ACKNOWLEDGMENTS Sources for this feature article came primarily from the Kenneth Van Wormers Speech on the Occasion of the 85th Anniversary of Chemical Engineering at Tufts, April 1986 was an especially valuable source for the departments early history. His speech contained references to Russell F. Miller s Light on the Hill: A History of Tufts College (Boston: Beacon Press, 1966). Authors acknowledge the help from Joanna Adele Huckins in drafting this article. PAGE 17 Vol. 42, No. 1, Winter 2008 17 S ince its beginnings, chemical engineering education has incorporated practical activities into the curricula (known as laboratory practices) in varying degrees (number of teaching hours, number of credits) according to the type of course or level of advancement in a given program. The historical evolution of this type of activity in chemical engineering programs has been recently discussed in literature. [1,2] It seems quite clear that, with the exception of certain courses and subjects directly involved with standard practical experiments ( e.g. general and instrumental analytic chemistry), there is ample choice in their design and imple mentation. On the other hand, new developments both in the processing of experimental data and in process control and automation techniques make it possible to design and imple ment laboratory practices in practically every area at any important investments in time and money that, unfortunately, many schools are not willing to undertake. A valid alternative, then, is to redesign these practical activi ties to maximize the use of resources. This work describes a simple, inexpensive lab experiment that was originally de signed for qualitative demonstration on the existence of hot from the viewpoint of the processing and interpretation of the data collected through the use of a relatively sophisti cated mathematical model and the independent validation ChE CHEMICAL KINETICS, HEA T TRANSFER, AND SENSOR DYNAMICS REVISITED MARA. E. SAD, MARIO R. SAD, ALBERTO A. CASTRO, TERESITA F. GARETTO Universidad Nacional del Litoral Santa Fe, Argentina knowledge has been broadened (development and validation of mathematical models, heat transfer and sensor dynamics) at practically null costs (considering time and economic resources). In what follows, both the experimental procedure and the development and validation of the model are described. Re sults illustrate this type of activities performed by students. The subject to which this activity belongs is Chemical Reac Copyright ChE Division of ASEE 2008 PAGE 18 Chemical Engineering Education 18 tion Engineering I, which also has other practical activities of different design that range from pilot scale experiments with semiindustrial equipment to computational simulations of real processes. CONTENTS: THEIR RELEVANCE AND DESCRIPTION Several authors have reported studies with adiabatic reac tors using the strongly exothermic reaction between sodium this reaction was used to experimentally assess theoretical tors and steady state multiplicity and hysteresis or oscillations [37] More recently, this reacting system was used to evaluate some new techniques in control and optimization engineering and in risk management. [818] Heat transfer in chemical reactors is an important topic both from theoretical and practical points of view. A common practice in chemical reaction engineering courses (which may be taken in parallel with energy transfer courses by the students) is to assume very simple models of this phenom enon to avoid further complications in the classical chemical reactor models. Another aspect added to thermal kinetic data processing is the incorporation of the time lag concept in the temperature measurement. This concept naturally arises in the course of process control, which students take normally afterward, but lag to help introduce students to the idea of noninstanta neous measurement. In this case, chemical reaction is very rapid, and we have chosen a sensor (digital thermometer) with a deliberately high thermal mass so the lag effect is more noticeable. THEORETICAL FUNDAMENT ALS AND MA THEMA TICAL MODEL The reaction between sodium thiosulphate and hydrogen 147 kcal/mol Na 2 S 2 O 3 ) and practically irreversible. The cur rently accepted stoichiometry for the reaction is: 24 41 24 22 32 22 36 24 2 Na SO HO Na SO Na SO HO AB () C CD E 4 The kinetic behavior of this system has been well studied by several authors [36] of the reagents: rk CC rg mo lN aS Oc ms AB .. [] /. () 22 3 3 2 The dependence of k with T will be taken as the usual Ar rhenius form: kA e ER T ( ) /. 3 even though some slightly different form was reported in earlier studies. For a batch reactor the mathematical model is made up of two differential equations resulting from mass and energy balances. Concentration of reagents sodium thiosulphate (C A ), hydrogen peroxide (C B ), and temperature (T) can be described by using these balances, the stoichiometry ratio between both reagents, and the initial conditions: dC dk CC VC pd Td kC A T AB v T A /. ( ) .. (/ ). () () 4 C CH AU Tt CC CC B R t BB AA .( ). .( )( ) .( ) ( 5 2 6 0 0 ) ) () () () () T C CC C AA BB 0 0 0 0 0 0 7 The thermal mass of the beaker used as a reactor has been neglected in the energy balance (Eq. (5)), due to its low mass (less than 4g), to avoid further complications in the proposed model. This set of differential equations can be solved using an appropriate numerical method (Euler, RungeKutta, etc.) in order to obtain the evolution of products concentration and temperature with time. It must be noted that this system may exhibit a stiff [17,18] behavior, so the numerical integration method should be a robust one. [19] For exothermic reactions such as this one, with heat ex change resulting from an external cooling media, the tem At this point: kC CH AU Tt dT d TA B R t () .. .( ). .( )/ () 08 After this point the temperature will decrease with a di minishing slope, as the reaction rate tends asymptotically to be the same as that of the surrounding media: [. .. () ] ( ) /( () kC CH dT dA U TA B R t 0 9 / /. ). () () VC pT tT t v 10 pendently estimate the value of U (from At.U/V.Cp v ) using values of temperature/time in this time domain. For the vessel used as a reactor, values of V and A t were 100 cm 3 and ca. 120 cm 2 The value of Cp v was 1 cal/g.K. [8] These values were The time lag of the sensor (digital thermometer) was taken [2224] dT dT T R R /( /) .( ) ( ) 1 1 1 point in the sensor probe, which is at the read temperature T R Both the high chemical reaction rate and the deliberate choice of the new dependent variable T R in the model with its ad PAGE 19 Vol. 42, No. 1, Winter 2008 19 EXPERIMENT AL The equipment used is simple, inexpensive and is comprised of the following elements: a 100 mL disposable, lightweight polystyrene beaker, a magnetic stirrer, a digital thermometer of the type used for frozen meat control at coldstorage plants (273 K/423 K range, 0.1K resolution) and a 500 mL crystal lizer to be used as an ice bath. A few millimeters of clearance was left between the bottom of the beaker and the stirrer plate to achieve a more homogeneous external heat transfer envi ronment. The chemical reactants employed are of analytical grade, but commercial hydrogen peroxide (1020 vol, with stabilizer) or commercial thiosulfate have been used with no substantial result alteration. A series of experiments is synthetically described below, for typical work performed by students that can be easily EXPERIMENT #1 This experiment consists of loading the reactor with 50 mL of a Na 2 S 2 O 3 ~ 0.8M solution, placing the thermometer 2 O 2 ~1.2M as rapidly as pos sible. Then, temperature readings should immediately start at appropriate time intervals. Figure 1 shows typical results obtained by students conducting this experiment at 303 K (room temperature). It was observed that the temperature increased rapidly at the beginning, went through a wellmarked maximum for about one minute, and then decreased increasingly at a slower pace. This results were obtained because when the limiting reactant is practically exhausted, the reaction rate becomes negligible and the system behaves as a simple vessel with exchanges heat by convection (internally forced, externally [20,21] EXPERIMENT #2 This experiment consists of following the liquid cooling (preferably using the same solution used in Experiment #1 after its completion) under the same conditions previously liquid from the previous experiment, preheated up to that temperature level, is left to evolve spontaneously under the same stirring conditions. Many time/temperature values are then registered (10 15 values) to estimate a value of use the asymptotic integrated form of the energy balance differential Eq. (5): ln () /( )( ./ .) .. () Tt Tt AU VC p t v 0 12 The slope of a straight line in the graph of ln[(T 0 t)/(T t)] of an experiment of this type together with statistical analysis of the data. EXPERIMENT #3 This experiment should also be performed with to a convenient temperature level (343353 K) and existing under the same conditions (stirring, etc.). In this case, the thermometer (previously in thermal equilibrium at room temperature) is introduced and registered temperature evolution is read as a function of time (given the ordinary response times of tools such as the thermometer employed, the readings should be carried out at a frequency of about 25 s). An essential condition for this experiment is that the temperature of the liquid mass be constant during the experiment. Since the time consumed is short, this condition is practically accomplished. The integrated form of differential Eq. (11) yields the classical step Figure 1 Reaction between Na 2 S 2 O 3 and H 2 O 2 in a batch reactor with heat exchange by natural convection (room temperature: 300 K). PAGE 20 Chemical Engineering Education 20 ln / ( /) .( ) TT TT RF RF 0 1 1 3 by regression of the experimental data. The value of T F is the (T F = T R  ). Figure 3 shows typical The results obtained with a typical kinetic experiment may be correlated using a simple model (Model I) inte grated by the mass balance (Eq. (4)) and energy (Eq. (5)) differential equations with proper boundary conditions and can also be correlated by a more so phisticated model (Model II) including the differential equation accounting for thermometer time response with the ad the number of parameters (and hence the error associated with its determina tion) are reduced to 2 in both models analysis of the results of Experiments # 2 and 3, are taken as known. Figure 1 shows the adjustment of both types parameters), and Table 1 summarizes the results obtained as well as the compari son between the parameters obtained and those from independent experi ments or reported in the literature. No tice the better adjustment of the model in Figure 1, including the time lag of the thermometer readings. Likewise, in Table 1, a good correspondence can be observed between the parameters de termined from the kinetic experiment, those from independent experiments, and those reported in the literature. This correspondence is more marked in the case of the model including the thermometer time lag. The interpreta tion and analysis of the results is usually performed using any available software capable of performing the regression of the temperaturetime experimental values, by way of numerical integra tion of the differential equations (with methods adequate for stiff systems) using any parameter search engine such as simplex, steepest descent, Monte Carlo, etc. There is a lot of general software available for these ends (Matlab, Matematica, MathCAD, to mention just a few) and some directoriented free applications (such as Scientist from Micromath) to solve the problem of statistical determination Figure 2 Determination of parameter dened in Eq. (12). Figure 3 Determination of the thermometer time lag. PAGE 21 Vol. 42, No. 1, Winter 2008 21 of the parameters and/or to simulate the system. It is not the cal and statistical methods employed in the correlation and modeling of experimental data, so only the results are shown as long as alumni are not compelled to use any particular software. The analysis of data presented in this paper was done using software with Episode (stiff) integrator, which includes the analytical calculus of the Jacobian matrix and a least square method (minimizing relative error) for the parameter adjustments. An additional experiment to assess the usefulness of the model to predict the behavior of the system can be done by modifying the heat transfer scenario ( i.e. using an ice bath as external transfer medium). In this case the values of kinetics parameters (A and E) and the sensor time constant are taken as the previously determined ones, and the heat transfer param eter is separately determined in an individual cooling experi is expected to change for the ice bath surrounding media) or estimated from temperaturetime values for higher times. Typical results of such experiences are not shown here, but in Fig. 4 a good concordance can be seen between predicted and the usefulness of the model for predictive purposes. CONCLUSIONS Some very simple laboratory experiments can be inexpen a more open analysis and correlation of obtained data and methods that strictly pertain to another (but closely related) area of knowledge. Modeling and simulation, two fundamen tal tools in chemical engineering, should be used despite the modest quality of equipment and obtained data to illustrate important aspects of the formulation and validation of mod els, design of complementary experiences for estimation of parameters, and the utilization of current numerical analysis and statistical procedures and software. Rather surprisingly in this case, as long as only one variable is measured, the results of parameter estimation and simulation are good and constitute an additional stimulus for students. NOMENCLA TURE A Arrhenius preexponential factor [cm 3 /mol 1 /s 1 ] A t Effective heat transfer area [cm 2 ] C Molar concentration [mol/cm 3 ] Cp v Volumetric heat capacity [cal/cm 3 /K 1 ] E Activation energy [cal/mol 1 ] k Kinetic constant [cm 3 /mol 1 /s 1 ] r Reaction rate [mol/cm 3 s 1 ] R Gas constant [cal/mol 1 /K 1 ] T Temperature of system, real [K] t Temperature of surrounding media [K] T 0 Initial temperature [K] T F Constant temperature of medium in Experiment #3 [K] T R Digital thermometer temperature reading [K] T R0 Initial digital thermometer temperature reading [K] 1 /cm 2 /K 1 ] V Reactor volume [cm 3 ] t .U/V.Cp v [s 1 ] R Enthalpy change of reaction [cal.mol 1 ] REFERENCES 1. Feisel, L.D., and G.D. Peterson, The Challenge of the Laboratory in Engineering Education, J. Eng. Ed. 91 (4), 367 (2002) T ABLE 1 Comparison of Parameter Values PARAMETER VALUE OBTAINED MODEL I VALUE OBTAINED MODEL II VALUE OBTAINED INDEPENDENT EXPERIMENT VALUE REPORTED IN THE LITERATURE A 2.10 10 2.10 10 2.10 10 6.85.10 11 E (Cal/mol) 8120 8238 82389200 U (KCal/s m 2 K) 0.0031 0.0035 0.0036 0.0020.02 1 ) 0.1259 0.1259 Figure 4. Testing the predictive model in ice bath experience. PAGE 22 Chemical Engineering Education 22 2. Feisel, L.D., and A.J. Rosa, The Role of the Laboratory in Undergradu ate Engineering Education, J. Eng. Ed. 94 (1), 121 (2005) 3. Root, R.B., and R.A. Schmitz, An Experimental Study of Steady State Multiplicity in a Loop Reactor, AIChE J. 15 (5), 670 (1969) 4. Vetjasa, S.A., and R.A. Schmitz, An Experimental Study of Steady State Multiplicity and Stability in an Adiabatic Stirred Reactor, AIChE J. 16 (3), 410 (1970) 5. Chang, M., and R.A. Schmitz, An Experimental Study of Oscillatory States in a Stirred Reactor, Chem. Eng. Sci. 30 21 (1975) 6. Guha, B.K., G. Narsimhan, and J.B. Agnew, An Experimental Study of Transient Behaviour of an Adiabatic ContinuousFlow Stirred Tank Reactor, Ind. Eng. Chem. Process Des. Dev. 14 (2), 146 (1975) 7. Schmitz, R.A., R.R. Bautz, W.H. Ray, and A. Uppal, The Dynamic Behavior of a CSTR: Some Comparisons of Theory and Experiment, AIChE Journal 25 (2), 289 (1979) 8. Grau, M.D., J.M. Nougues, and L. Puigjaner, Batch and Semibatch Reactor Performance for an Exothermic Reaction, Chem. Eng. and Proc. 39 (2), 141 (2000) 9. Ferrouillat, S., P. Tochon, D. Della Valle, and H. Peerhossaini, Open Loop Thermal Control of Exothermal Chemical Reactions in Multi functional Heat Exchangers, Int. J. of Heat and Mass Transfer 49 (1516), 2479 (2006) 10. Bhanot, S., S. Kumar, and M.K. Vasantha, Online Instantaneous and Delayed Control of a Fast Tubular Reactor, TENCON 1998 IEEE Region 10 International Conference on Global Connectivity in Energy, Computer, Communication and Control Vol. 2 New Delhi, India, 448 (1998) 11. Palanki, S., and J. Vemuri, EndPoint Optimization of Batch Chemi cal Processes, Proceedings 42nd IEEE Conference on Decision and Control 5 4759 (2003) 12. Kulkarni, A.J., S.V. Patil, and V.K. Jayaraman, Memory Based Local Learning: Application to Process Engineering Problems, Int. Journal of Chem. Reactor Eng. 1 A23 (2003) 13. Prat, L., A. Devatine, P. Cognet, M. Cabassud, C. Gourdon, S. Elgue, and F. Chopard, Performance Evaluation of a Novel Concept Open Plate Reactor Applied to Highly Exothermic Reactions, Chem. Eng. & Tech. 28 (9), 1028 (2005) 14. Ausikaitis, J., and A.J. Engel, The Behavior of the Iron(III)Catalyzed Oxidation of Ethanol by Hydrogen Peroxide in a FedBatch Reactor, AIChE J. 20 256 (1974) 15. Ruiz, D., Fault Diagnosis in Chemical Plants Integrated to the Infor mation System, Thesis, Universitat Politecnica de Catalunya (2001) 16. Shukla, P.K., and S. Pushpavanam, Parametric Sensitivity, Runaway, and Safety in Batch Reactors Experiments and Models, Ind. & Eng. Chemistry Research 33 (12), 3202 (1994) 17. Gao, Q.Y., Y.L. An, and J.C. Wang, A Transition from Propagating Medium, Phys. Chem. 6 (23), 5389 (2004) 18. Karer G., G. Musi, and B. Zupan, Predictive Control of Temperature in a Batch Reactor with Discrete Inputs, Proceedings of the 1 3 th Mediterranean Conference on Control and Automation Limassol Cyprus, 855, June 2729 (2005) 19. Solodov A., and V. Ochkov, Stiff Differential Equations in Differen tial Models. An Introduction with Mathcad Springer Berlin Heidelberg (2005) 21. Perry R.H., and C.H. Chilton (Eds.), Chemical Engineers Handbook, 5th Ed., McGrawHill Book Company, Chapt. 10 (1973) 22. Sutton, C.M., In Situ Measurement of Resistance Thermometer SelfHeating and Response Time, Meas. Sci. Technol. 5 896 (1994) 23. Dally, J.W., W.F. Riley, and K.G. McConnell (Eds.), Instrumentation for Engineering Measurements, 2nd Ed., Iowa State Univ. (1993), 11.6 Dynamic Response of Temperature Sensors. PAGE 23 Vol. 42, No. 1, Winter 2008 23 F have implemented ways of better engaging students, including active and cooperative learning, learning communities, service learning, cooperative education, inquiry and problembased learning (PBL), and team projects. [1] Peda gogies of these types have been shown to clearly be effective in allowing students to improve their learning. It is no longer important to consider simply the content covered in a course, but how the material is going to be taught. Researchers at McMaster University [2] found that to solve problems successfully students need both comprehension of chemical engineering and what are called general prob lemsolving skills. The researchers developed workshops to explicitly develop skills deemed appropriate for improving Prince, [3] in his article on active learning, provides additional motivation for incorporating problembased learning in the part of the students. The professor is no longer solely respon sible for delivering course content. Prince provides extensive and credible evidence suggesting that faculty should consider nontraditional models for promoting academic achievement and positive student attitudes. Based on the above evidence of the value of PBL and active problem solving, the instructors reconstructed their graduate course in Chemical Engineering Modeling. In Fall 2006, the instructors introduced components to the course exposing ChE Copyright ChE Division of ASEE 2008 ACTIVE PROBLEM SOLVING AND APPLIED RESEARCH METHODS ERIC L. MAASE AND KAREN A. HIGHOklahoma State University Stillwater, OK 74078 PAGE 24 Chemical Engineering Education 24 students to case studies, active problem solving, teamwork, and experimentation with an the aim of promoting creative and critical thinking as students identify and practice the art of modeling. MODELING COURSE The Chemical Engineering Modeling course has been structured to give the students experiences to meet the fol lowing objectives. At the end of the semester, the students should be able to: (a) Develop mathematical models describing chemical engineering phenomena. (b) Evaluate the assumptions, limitations, and restrictions necessary to solve practical problems by mathematics. (c) Use classical numerical techniques to solve the equa tions that result from model formulation (ordinary and partial differential equations, and linear and nonlinear simultaneous algebraic equations). (d) Become familiar with available computational tools cally in MATLAB and VBA/Excel). The course topics include: (b) Matrices (d) Nondimensionalization (e) Taylors Series (f) Ordinary Differential Equations (g) Linearization (h) Linear/Nonlinear Equations (i) Finite Differences (j) Initial Value Problems (IVPODEs) (k) Boundary Value Problems (BVPODEs) (l) Partial Differential Equations (PDEs) (m) Applications of Modeling/Numerical Analysis All students in the class are graduate status and, more im The course is a requirement for all chemical engineering graduate students at Oklahoma State University. Students are concurrently enrolled in graduatelevel thermodynamics and diffusional operations courses during this semester as well. The diffusional course covers the equations of continuity and focuses on analytical solutions to equations. Activities and instructional materials teaching program ming and problemsolving principles using Visual Basic for Applications (VBA) linked through Microsoft Excel and MATLAB have developed during previous semesters as course materials. These computer tools are employed by the students in solving algebraic, ordinary differential equations (initial value and boundary value) and partial differential equations (parabolic and elliptic). The instructors predominantly use two references during the semester that help students develop and set up mathematical models of physical phenomena. The Smith, Pike, and Murrill (SPM) [4] methodology employs an IOGA (input, output, generation, and accumulation) approach of building models using a control volume. The Himmelblau and Bischoff (HB) [5] methodology relies on beginning with equations of continuity situation. The course includes homework assignments in mathematical theory (paper solutions), computer exercises in numerical methodsboth programming with VBA and MATLABand exams evaluating students understanding. In past years students have worked on projects that have them evaluate the effectiveness of the ODE boundary value problem (BVP) strategies of matrix method, shooting method, and the successive overrelaxation method (see Hornbeck [6] and Riggs [7] ). The students are given the opportunity to pick chemical phenomena that give rise to ODE BVPs. Prior students have stated that they learned a great deal from such projects but also have wondered if there were anyway to be given a project that allowed them to model actual experimental phenomena. AN ACTIVE LEARNING ENVIRONMENT a semesterlong group project where the students perform experimental work in a lab, report and discuss their efforts via memoranda to the instructors, and apply mathematical theory, numerical methods, and developing modeling skills The students were presented with the following problem statement at the beginning of the semester: Design an encapsulated drug (of your groups choice) that effectively delivers an appropriate dose for an appropriate number of hours. Select the most costeffective method. This problem statement is intentionally both inadequately all goals of the group projects is an aspect of the students learning. Other educational objectives inherent in the problem state ment include: Consider different types/scales of phenomena (micro scopic, multiple gradient, maximum gradient, macro scopic). Understand what physical models and mathematical models are. Consider unsteady phenomena. PAGE 25 Vol. 42, No. 1, Winter 2008 25 Develop models that cover as many types of phenom ena as are appropriate. Develop the maximum gradient model for spherical coordinates. Develop experimental protocol around simulating drug delivery with Tootsie Pops. Consider numerical techniques to solve the problem (ODE vs. PDE). The class meets twice weekly (Tuesdays and Thursdays) third of the semester are in computer labs and students are provided instruction in VBA/Excel and MATLAB. Thursday lectures provided instruction on mathematical principles and modeling methodology. Over the remainder of the semester, Tuesday lecture periods served as times for students to be engaged either in group work or with instructors providing could be presented (lab safety discussion, review sessions, presentation skills, etc). The students are initially daunted by the possibilities inherent in the project but also are will ingand in some cases, quite eagerto rise to the challenge that the project represents. Two individual case studies, which are introduced at separate times, provide the basic impetus for the students project development and direction. Discussions of projects during class periods and project status reporting (memos) provide additional mentoring moments for the in structors over the course of the semester. To accommodate the extra time required in class and to minimize lecturing, a variety of reading pretests (administered as online quizzes in Blackboard) were given to ensure stu dents comprehended material prior to class. Also, it allowed the instructors to shorten lecture time and increase active learning experiences. The overall approach taken is intended to encourage critical thinking and creativity in the process of problem solving by con necting applications of the mathematical and numerical methods taught in lectures more closely to other engineering research methods. The restructured course also mimics experiences that students should expect to encounter while working towards a thesis (M.S.) or dissertation (Ph.D.) in a graduate program. Homework assignments and exams were administered to students to ensure that they were able to individually learn and demonstrate knowledge in the content. The overall course grade was from exams (60%), individual homework (20%) and the group project (20%). This ensured that the students were learning all of the course objectives. PROJECT GROUPS At the start of the semester, the students were put in groups. The questions in Table 1 were used to ensure heterogeneity in student groups based on experience, gender (when determin able from name), and ethnicity. INTRODUCTION T O LABORA T ORY METHODS Once the students were in groups, the instructors led an ex perimental session where the students began their project work by examining a dissolution case study. A laboratory was set up that was designated for the students use (see Figure 1). T ABLE 1 Questions Used to Group Students Encapsulated Drug Project, Aug. 31, 2006 On the card, write: Your name Your home ( i.e., China, Michigan, Antarctica) Your degree ( i.e., M.S. Chemical Engineering, Ph.D. Chemistry) Your computer experience (High/Medium/Low) Your experimentation experience (High/Medium/Low) Your transport equation experience (High/Medium/Low) Your numerical method experience (High/Medium/Low) Figure 1. Drug (lollipop) dissolution. PAGE 26 Chemical Engineering Education 26 The experiments are driven by the goals and basis outlined in an initial case studyproviding a starting point for the student projects. The intent of the initial lab period is also to provide instruction on lab notebooks and experimental con cerns such as safety and experimental measurement, as well as to outline the challenges that experimental work entails. The class would later discuss and determine the safety rules for the lab with all students signing the resulting safety agree ment as shown in Table 2. A discussion of this nature resulted in a more openended perspective on safety along with a dif ferent rational for and understanding of other users feelings in a laboratory environment and their sense of safety. CASE STUDIES Case studies served as the primary creative impetus for students projects. While the students may be eager to undertake the challenge posed by the project they also need a sense of the structure that allows such studies to be successful. The initial case study provides their starting point. This case study, from Jensen, [8] is intended for freshman students. The goal is for students to investigate the factors affecting the dissolution of a proposed coating for a new pharmaceutical. The story is written from the perspective of two chemical engineering students in ChemE 101, a reading and working on the case study are: 1) describe the factors affecting mass transfer between phases; 2) collect and evaluate data on mass transfer; and 3) evalu ate a model to describe mass transfer. These objectives remain appropriate for the graduate students projects as well. The intent of the case study is model for a lollipop (the students used Tootsie Pops) in order to experimentally investigate a system having an encapsulating material over an inner drug material. The students were then tasked with developing and outlining their own project: developing a better model of encapsu lated drug delivery, as can be seen in Table 3. As students were completing the initial experimental aspects of the project, they were allowed to ask for materials to be provided for them to help solve various problems. This exposed the students to a justintime teaching [9] or lecturebydemand strategy. Once students initial experimental and numerical techniques have been applied to Case Study #1 and they have begun further experiments of their own designs, a second case study is introduced. The purpose of the second case study is to introduce the concepts of com putational modeling and refocus students on the overall aims of encapsulated drug delivery. This case study is dis cussed after the necessary computer instruction (VBA/Excel and MATLAB) is complete. system of interest into essential aspects that can be math ematically described while still retaining essential features of the larger realworld system of interest. This case study provides an example of a model of a human body interacting with drugs delivered orally. The second case study arises from an assignment previ ously developed and used in an undergraduate Computer T ABLE 3 First Project Memo To: Students of Chemical Engineering 5743 From: The Instructor Date: Aug. 31, 2006 Subject: 1st group memo for encapsulated drug class project. A 12 page memo is due Sept. 7 at the beginning of class that answers/consid ers the following. 1) Initial experimental protocol for your physical model a. Equipment/supplies neededyour group will be responsible for procurement b. Data collection procedure c. Other information 2) Initial computer modeling/numerical methods for mathematical model a. Computer software to be used b. Numerical methods to be used c. Other information 4) Future considerations that more appropriately model the real system 5) Questions for the instructor. 6) A reference list to help in the project. Include at least 2 references per group member. T ABLE 2 Laboratory Safety/Practices for 307 EN CHE 5743 Class Lab 1. Safety glasses (eye protection) are encouraged 2. Move deliberatelyno running 3. No more than 2 groups and 1 individual in 307 at one time 4. No food/drink/gum 5. Closed shoes and full pants (or skirt) will be worn all times in 307 6. Keep lab clean and equipment stored in a neat fashion 7. Unplug all equipment when done 8. Close door and lock when the last person leaves Name ________________________________________ (print) Signature______________________________________ PAGE 27 Vol. 42, No. 1, Winter 2008 27 Programming for Engineers course taught at Oklahoma State University by the coauthor. The case study outlines a model of the human body as a set of three interacting compartments where each represents a broader aspect of human physiology deemed important in drug delivery. The three interacting com partments are referred to as the oral, central, and peripheral compartments. The oral compartment represents primarily the mouth and stomach where a drug is gradually dissolving. The central compartment represents both the blood stream (circulatory system) and those organs that are highly perfused (liver, kid ney, lungs, etc.). The peripheral compartment includes muscle and tissues where diffusion of medications is often slower and where drugs may build up in concentration over time. the compartments with governing equations for the transfer of a drug to and from compartments, as shown in Figure 2 These equations include rates/pathways for a drug leaving or entering the central compartment, where k e is the rate drugs are eliminated from the body, k r is the drug metabolizing, and k c is the drug being further absorbed (to the peripheral compart ment). All of the rates are dependent on the concentration of the drug currently in the central compartment, C c The central compartment balance also includes terms describing the drug entering from the oral compartment, k a C o and returning from the peripheral compartment, k p C p The case study tasks students with creating a computer program capable of solving the drug delivery model, inves tigating simulated behaviors, and discussing the results of their efforts. The overall perspectives and intentions of Case Study #2 prove valuable for the graduate course as well. In extending the basic princi ples outlined by the second case study, graduate students will ex plicitly consider multiple drug doses, statisti cal uncertainties inherent in hu man reactions to medications, the limitations and assumptions necessary for complex system modeling, and applications of numerical methods discussed in the course as tools for solving systems of equations. CASE STUDY USE Case studies served as the primary impetus for students dents were provided an initial direction from which they could develop their own experimental goals and were presented with the realities inherent in both mathematical modeling and experimental work. The second case study provided a different perspective for the students where the challenge became more open ended and constructing better modelsin tractable mathematical terms able to capture essential behaviors of the actual system of interestwas a goal. These models and related simulations provided additional paths for examination of complex systems examined computationally when experimental data or meth The modeling and numerical methods in computer simulation complemented the experimental laboratory work initiated with Case Study #1. The second case also served to remind the students of the overall project goal(s) as outlined in the original problem statement. Translat ing mathematical and conceptual models into computer simulations also tasks the students with understanding and integrating their previous experimental approaches (those completed as well as those under way) into a more com plex system description. This aspect of applied numerical investigations asks students to determine a better overall for the project goal(s). The case studies used in the course are complementary tools in engineering research. The implantation of the cases in the course provided introductions for students to the role of laboratory experiments along with computational numeri cal simulations. PROJECT REPORTING Over the course of the semester, project status, learning, and student assessment (grading) are addressed by asking the student groups to prepare memoranda at appropriate points during the project. Table 4 (next page) lists the additional proj ect status memos that student teams were asked to prepare. FINAL REPORTS AND PRESENT A TIONS The written projects were due the second to last day of the semester. Presentations were on the last day of the semester. structors provided the students with instruction on creating appropriate Microsoft PowerPoint documents and delivering an effective presentation. C E RCCPPAo dC kkkCkCkC dt P PPCC dC kCkC dt o Ao dC kC dt Peripheral Oral Central Figure 2. Case Study #2: Drug delivery model. PAGE 28 Chemical Engineering Education 28 STUDENT WORK AND COURSE ASSESSMENT One group undertook an investigation of the design of a cost effective controlledrelease drug delivery system for Digoxin. [10] In their efforts they proposed a design a fourlayer delayedrelease pillintended to sustain optimal bloodplasma concentration over a 24hour period; the proposed fourlayer coating dissolution is shown in Table 5. The students generated a conceptual model, determined lations of their system. An example is shown in Figure 3 (the graph is unedited student work). The work shown represents only one example of the students thinking and submitted work during the course. delivery method were outlined along with their current in vestigations to date. Overall the work submitted during the course, along with the interactions the instructors had with the students, showed students to be very actively engaged with course materials and found to be meeting or in many instances exceeding the instructors goals for the reconstructed course. An endofcourse assessment was given to the 13 graduate students in the course. University Institutional Review Board approval was given for the survey. The students gave consent for the authors to use the results of the survey in presentations and publications. The survey was anonymous and examined after the conclusion of the semester. The survey had three sections. Section 1 Q1. I can appropriately use a lab book. Q3. I am comfortable working in groups/on teams. Q4. I am able to give effective presentations using Power Point. problems in this course. Q6. My numerical methods skills are appropriate to solve problems in this course. problems in this course. Q8. I am able to develop mathematical models that ap propriately consider relevant phenomena. Q9. I understand the need to follow appropriate lab safety procedures. T ABLE 5 Student Designs: FourLayer Pill Layer Coating (hrs dissolve) 1 .167 2.35 2 5.83 0.9 3 4.67 0.9 4 4.67 0.9 T ABLE 4 Project Status Memos 2nd memo due September 25th at the beginning of class. should take 1 hour and 15 minutes (class time). You should keep this very simple and consider using the physical and mathemati cal model in case study #1. You need a very detailed plan for your experiment. Where will you get the equipment for the experiments? 2) Develop the mathematical model in the case study using the: a. Himmelblau and Bischoff approach b. Smith, Pike, and Murrill approach 3rd memo due October 6th (4:30 p.m. 423 EN) 1) Develop the mathematical model in the case study using the: a. Himmelblau and Bischoff approach b. Smith, Pike, and Murrill approach Follow Example A approach. 2) Discuss your data and numerical model from your experi ments on Sept. 28. Use the inclass discussions from Oct. 3 as your guide. 4th memo due October 20th (4:30 p.m. 423 EN) 1) Find your initial experimental data from Sept 28. Plot the 2) Rerun your experiment with a solid lollipop. Plot the data r 3) Develop a nextlevel model (mathematical and physical) that gets your group toward the ultimate goal of designing an encapsulated drug. Clearly plot your data from your experiment (physical model). Discuss the appropriateness of the models (physical and mathematical). Dont get too fancy yet! 5th memo due Friday December 1 at noon in 423 EN 1) Tell us how your group is considering determining cost effectiveness. 2) What are your aims ( i.e., which drug are you using, how are you delivering the drug, where are you delivering it to)? goals? 4) Progress on physical model and experiments. 5) Progress on mathematical model. 6) Back up your claims with references to literature. What are you using for your main references? Include an accurate and complete bibliography that you have been compiling since the start of the project. PAGE 29 Vol. 42, No. 1, Winter 2008 29 Q10. I enjoy a problembased learning environment. Q11. I can search the literature for relevant background material for projects. Q12. I understand the components of a research report / proposal. Q13. I know what goes into a developing a good presenta tion. The students were given a 5point Likert scale to evaluate their agreement with the above statements. Table 6 shows the results for Section 1 of the assessment. All scores listed were above neutrality, indicating a positive agreement of all questions. The students felt the most agree ment with the statement I enjoy a problembased learning environment. In the authors opinion the students gained a lot from the active learning experience and were engaged throughout the entire project and course. The next highest agreement was for the statement I know what goes into developing a good presentation. The instructors felt that this is principally due to the class period four days prior to class end, in which presentation skills were discussed. The students had the lowest level of agreement for the problems in this course. This is not surprising since the coauthors spent a majority of the class discussing solution strategies in VBA/Excel. The next lowest level of agreement was for the statement I am able to give effective pre sentations using PowerPoint. This was interesting given their strong agreement to knowing what goes into a good presentation. One pos sibility is that this result may be due to language challenges that many of foreign students experienced. The students also had not given their pre sentation yet. The project was turned in on the same day as the assessment and the presentation was the next day. Foreign students prob ably had anxiety about giving a presentation based on a project they just turned in and delivering a presentation the next day in a second language. Section 2 Q1. How would you rate your knowledge of MATLAB? Q2. Did you use MATLAB to solve problems in the course? Why or why not? Q3. How would you rate your knowledge of VBA/Excel? Q4. Did you use VBA/Excel to solve problems in the course? Why or why not? Q5. How would you rate your math skills, particularly eigenvalues and matrix calculations? Q6. How would you rate your knowledge of the transport equations (the equations of continuity for energy, mass, momentum)? Q7. How would you rate your knowledge of numerical Q8. How helpful were the MATLAB help sessions that Dr. High provided? Q9. How helpful were the VBA help sessions that Dr. Maase provided? T ABLE 6 Assessment Results for Section One Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Value 4.08 4.23 4.31 3.69 4.23 4.31 3.38 4.00 4.31 4.46 4.23 4.31 4.38 Rank 10 7* 3* 12 7* 3* 13 11 3* 1 7* 3* 2 5 = Strongly agree, 4 = agree, 3 = neutral, 2 = disagree, 1 = strongly disagree. *Indicates where there is a tie in the ranking. Figure 3. Digoxin concentra tion in blood plasma. PAGE 30 Chemical Engineering Education 30 Q10. How helpful was Blackboard to you? All of these questions had room for student comments. Table 7 shows assessment results. ed themselves was in their knowledge of numerical methods lowest was for their knowledge of MATLAB (Q1), again this wasnt surprising considering that VBA was emphasized. For the questions where the students responded to how helpful various resources were, they gave highest score for the VBA sessions and for the Blackboard classroom management system and lower for the MATLAB sessions. For Q2 (did you use MATLAB?), eight students said yes, three said no and two didnt answer. Student comments for why they used MATLAB included: Only when we were assigned to Sometime because it is easier to use Its easy to use And for comments why they did not use MATLAB: Used VBA (more comfortable with) I dont have enough experience to work on it com fortably For Q4 (did you use VBA?), 12 said yes and one didnt an swer. Student comments for why they used VBA included: I love to use it. For numerical methods and solve diff. equations Because I was trying to learn VBA. Its very simple. Easy to use and easily accessible. (For) all problems because of ease of interface and prior background. Section 3 This section was where students could write freeform comments to the questions. Student comments are added after each statement. See the Appendix for comments. According to the assessment, it appears that the students appropriately developed their expertise in four areas: math ematical modeling; experimental strategies; numerical methods; and computer strategies on MAT LAB and VBA/Excel. It is also clear that they enjoyed the prob lembased environment in which the course was administered.CONCLUDING COMMENTS This project proved invaluable for beginning graduate students to identify the connection between modeling, math ematical descriptions, and experimental reality. It was also intriguing for the students to not only consider mathematical but physical models in which experimentation with the real system is, at best, impracticalas is the case in biological systems. The students gained valuable teamwork skills not common for many graduate students as well as improvement in communication skills. The assessments show that the stu dents were particularly engaged in the course. The approach taken encourages critical thinking and creativ ity in the process of problem solving, while connecting appli cations of mathematical and numerical methods more closely with other engineering research methods. The restructured course also mimics experiences that students can expect to encounter while working toward a thesis (M.S.) or dissertation (Ph.D.) in a chemical engineering graduate program. REFERENCES 1. Smith, K., S. Sheppard, D. Johnson, and R. Johnson, Pedagogies of Engagement: ClassroomBased Practices, J. Eng. Ed. 94 (1), 87 (2005) 2. Woods, D., A. Hrymak, R. Marshall, P. Wood, C. Crowe, T. Hoffman, J. Wright, P. Taylor, K. Woodhouse, and C. Bouchard, Developing Problem Solving Skills: The McMaster Problem Solving Program, J. Eng. Ed. 86 (2), 75 (1997) 3. Prince, M., Does Active Learning Work? A Review of the Research, J. Eng. Ed. 93 (3), 223 (2004) 4. Smith, C., R. Pike, and P. Murrill, Formulation and Optimization of Mathematical Models International Textbook Company, Scranton, PA (1970) 5. Himmelblau, D., and K. Bischoff, Process Analysis and Simulation Wiley, NY (1968) 6. Hornbeck, R., Numerical Methods Prentice Hall, Upper Saddle River, NJ (1982) 7. Riggs, J, An Introduction to Numerical Methods for Chemical Engi neers Texas Tech University Press, Lubbock, TX (1994) 8. Jensen, J., Chapter 20Dissolution Case Study, in A Users Guide to Engineering Prentice Hall Source, Upper Saddle River, NJ (2004) 9. Novak, G., A. Gavrin, W. Christian, and E. Patteson, JustInTime Teaching: Blending Active Learning with Web Technology Benjamin Cummings, San Francisco (1999) 10. Weiss, M., and W. Kang, Inotropic Effect of Digoxin in Humans: Mechanistic Pharmacokinetics/ Pharmacodynamic Model Based on Slow Receptor Binding, Pharmaceutical Research 21 235 (February 2004) T ABLE 7 Assessment Results for Section Two Q1 Q3 Q5 Q6 Q7 Q8 Q9 Q10 Value 2.46 2.85 3.00 2.77 3.08 2.77 3.08 3.08 Rank 5 3 2 4 1* 3 1* 1* For Q1, Q3, Q5, Q6, Q7: 4=Extensive, 3=Pretty good, 2=Limited, 1= None For Q8, Q9, Q10: 4=Extremely, 3=Very, 2=Somewhat, 1= Not helpful *Indicates where there is a tie in the ranking. PAGE 31 Vol. 42, No. 1, Winter 2008 31 APPENDIX The following are freeform responses to the survey ques tions (comments are verbatim). Q1. Describe how well you thought the project allowed you to understand the use of physical models and experiments to simulate phenomena (particularly biological). It made me understand physical and experimental drug dissolution. I never knew how to model physical phenomenon before I enrolled in this course. It helped me improve my computer skills. Oh great!!! Actually being new to grad studies, it was a welcome breeze of air. The project was useful in correlating the concepts of physical models and experiments. The project gave me some insight how to simulate a physi cal model and to develop mathematical model. I think this project is very helpful to understand how to construct the physical models, carry out experiments, con struct and solve mathematical models. It helped me a lot in developing a good understanding about the phenomena and in visualizing the problem. I believe now I understand the concept of having a physi Got a lot of knowledge about the mathematical model. Very well. Relating pure experimental work to comprehend modeling was good. The project idea was good. Saw relationship. Q2. Describe how well you thought the project allowed you to understand the use of mathematical models to simulate phenomena (particularly biological). It was important as we can now understand how important it is to make simple assumptions. They make almostreal models with less effort. The late equations could be easily coded in VBA, and the experimental results could be studied easily. Yeah, it helped a lot, but biological systems are complex, so we had to make several assumptions to mathematically represent it. I think this project helped me to understand how to convert the physical phenomena into mathematical models and know the mechanism of the phenomena to make suitable assumptions and negligibility. It assisted me tremendously. Now I relate the methods learned to reallife situations. This seemed a bit vague throughout . we had some dif Q3. Discuss how the project helped you to improve your numerical method skills. I am not sure it helps me much on my numerical method skills. Not much, I was good at them. Gave me a better understanding of Runge Kutta and other methods. The project gave opportunity to solve numerically the ODEs. By solving project, I used a lot of numerical methods, like how to solve algebraic equations, differential equations, and regress the parameters. The project helped me in improving my numerical skills. I can now understand numerical analysis and relate to reallife situations. Really helped us to model realtime problems. I felt like the project was very numericalmethod light. I dont know how. But!!! It made my numerical method skill to improve! Q4. Discuss how the project helped you to improve your computational skills. It helps me use Excel and VBA. Had homeworks, projects, and a good discussion in the lab. May have helped more had I been serious starting day 1. Yes it helped me to improve as we faced the real life prob lems. This program needs to use Excel and VBA to solve the program. Its helpful. Yes, it did improve my computational skills. I felt like the project required very little computational skills. Q5. Describe your ability to develop a costeffective en capsulated drug. Cost effectiveness is dependent on how many people need this and how I can optimize. At this point, on the basis of literature review, I can de termine the factors which can/might affect the cost of drug, and they can be varied according to the requirement. I can develop a costeffective encapsulated drug consider ing mathematical models, economic evaluation models. I couldnt actually develop an exact costeffective project. PAGE 32 Chemical Engineering Education 32 No knowledge whatsoever. I wish you had given back ground info about drug analysis or economics. I search the related information and get a rough knowl edge about the costeffective encapsulated drug. Q6. Discuss the usefulness of the initial case study and Dr. The initial case study helped in understanding the aim of the project. Dr. Maases case study was very helpful in writing the code plus understanding the model. Very helpful. Great instructors and great interactive course. Very useful! Especially Dr. Maases case study which gave me a better perspective of the model. It was helpful as it gave insight how to simplify the real system to develop mathematical models. They are all very useful to understand this project. Initial case study and Dr. Maases case study both give me Helpful guide to know how to start. Q7. Other comments. I felt like the project was at times taken to be more dif identical. PAGE 33 Vol. 42, No. 1, Winter 2008 33 Random Thoughts . M yths about student ratings of teaching abound on assurances that ratings are just popularity contests that reward entertainers and penalize the best teachers. (Re markably, the second group always seems to include the ones doing the assuring.) Some years ago we surveyed the myths and summarized the extensive research that showed most of them to be wrong. [1] Now its 15 years later and a lot more research has been done, with similar outcomes. Unfortunately the myths are still alive and well, so here is the 2008 version of what everyone knows about student ratings and how much of that wisdom is supported by research. * Myth. Student ratings are not valid assessments of teaching quality. Fact. False. Thousands of research studies have shown that student ratings correlate positively with every other measure of teaching effectiveness, including alumni ratings, peer rat ings, administrator ratings, measures of learning ( e.g. stan dardized tests, common exams in multisection courses, and ratings of student portfolios), and student motivation. [2] The magnitude of the observed correlations varies considerably across individual studies and a few studies report contradic tory results, but the weight of the evidence is clear. If students consistently say someones teaching is good or bad, theyre almost certainly right. Myth. The highest ratings go to the easiest courses. Fact. average get higher ratings than easier courses, with ratings beyond the backgrounds of most enrolled students. In a recent study of 1,045 engineering, science, and humanities courses at two universities, Dee [3] found that student perceptions of STUDENT RA TINGS OF TEACHING: MYTHS, FACTS, AND GOOD PRACTICESRICHARD M. FELDER North Carolina State UniversityREBECCA BRENT Education Designs, Inc. Copyright ChE Division of ASEE 2008 in the top and bottom quartiles of student ratings, with the marginally higher workload rating going to the courses in the top quartile. Myth. Bad teachers who are easy graders get higher evalu ations than good teachers who are strict graders. Fact. False. Individual instructors who give high grades rela tive to local averages may get higher ratings than they would if their grades were lower, [4] but no studies have turned up ineffective teachers who got high ratings just by giving high grades. [5] However, the possibility that it could happen sup ports the common recommendation to use multiple sources of assessment data. Myth. They may not like me now because Im rigorous and maintain high standards, but in a few years theyll appreciate how good a teacher I was. Fact. Generally falseit happens sometimes, but not often. [1, 2] given previously to the same instructor. If your students think youre a great teacher now, most will still remember you fondly in the future, and if they think youre lousy, dont PAGE 34 Chemical Engineering Education 34 All of the Random Thoughts columns are now available on the World Wide Web at 10 years. * Not all common beliefs about evaluations are wrong, of course. Its true that teachers who are enthusiastic and car ing tend to get better ratings than those who are reserved and distant, but so what? Enthusiasm and caring of instructors also correlate with motivation and learning of their students, [2] suggesting that the higher ratings are probably deserved. Its also true that other things being equal, elective courses tend to get higher ratings than required courses, upperlevel courses tend to get higher ratings than lowerlevel courses, small and moderately sized classes get higher ratings than very large classes, student ratings in engineering and the sciences are engineering and the physical sciences get lower ratings than male instructors. [2] On average these effects are small, but they exist and should be taken into account when ratings are used to make decisions about such things as reappointment, tenure, promotion, and merit raises. In short, student evaluations have high levels of reliability and validity and should always be part of the process used to evaluate teaching. There are some aspects of a course that students are in no position to evaluate, however, including whether the course learning objectives are appropriate, the adequately prepares the students for subsequent courses in the curriculum. Those things can only be evaluated by knowledgeable peers. Student ratings should therefore not be the sole source of teaching assessment data but should be supplemented with peer ratings and other measures of teach ing effectiveness. [6] If different sources agree, as they usually will, its a good indication that the overall assessment is a fair Since student ratings will undoubtedly remain central to teaching evaluation (as they should), everything possible should be done to make them as effective as possible. The following recommendationsmost of which are drawn from the papers cited in Reference 2address that goal. Use a rating form that has been developed with the as sistance of someone knowledgeable about educational measurements. There is a science to survey construc tion in general and educational rating instrument con struction in particular. Either use a form that has been developed and validated elsewhere, such as the IDEA Student Ratings of Instruction system ( PAGE 35 Vol. 42, No. 1, Winter 2008 35 I n their contribution, MartnezUrreaga and his collabora tors [1] have presented the stochastic simulation of a chemi work is indeed useful and informative: Randomly or stochas tically behaving processes or systems are ubiquitous in the chemical or allied industries. [29] The current contribution involves two aspects and is intended to complement and augment the work of MartnezUrreaga and his collaborators [1] to enhance its usefulness. First, the reacting system ( i.e. the simple reversible chemi cal reaction) illustrated by them is modeled as a stochastic subclass of Markovian processes. [10, 11] The resultant model, in turn, gives rise to the master, or governing, equation [11, 12] of the birthdeath process whose solution renders it possible to obtain the analytical expressions for the process means and higher moments about the means, e.g. the variances, which tuations. Second, the master equation of the birthdeath process is stochastically simulated via the Monte Carlo method. [13] The method is implemented by the timedriven approach [14, 15] in addition to the eventdriven approach [10, 1523] adopted by Department of Finance and Banking, Kun Shan University, YungKang City, Tainan Hsien, 71003 Taiwan REVISITED AND AUGMENTED ChE A. ARGOTI, L.T. FAN, J. CRUZ, AND S.T. CHOU Kansas State University Manhattan, KS 66506 Copyright ChE Division of ASEE 2008 PAGE 36 Chemical Engineering Education 36 MartnezUrreaga and his collaborators [1] on the basis of the Gillespie algorithm. [10, 17, 18] The eventdriven approach entails the determination of the probability distribution of a random waiting time or period of quiescence [19, 20] ; the simulation clock is advanced according to this random waiting time. Unlike the eventdriven approach, the timedriven approach advances [14, 15] which can be estimated as a function of the intensities of transition embedded in the master equation of the process. The timedriven approach is especially useful when the probability distribution for the random waiting time in the eventdriven approach is exceedingly complex to determine. The means and variances of the random variables characterizing the birthdeath process have been computed in implementing the eventdriven and timedriven approaches. For validation, the analytical solutions from the stochastic model as well as the numerical results from the Monte Carlo simulation are com pared with each other. In addition, these results are validated by comparing the values of the means obtained from the solution of the deterministic model for the simple reversible chemical reaction as presented by MartnezUrreaga and his collaborators. [1] The two approaches for stochastic simulation are further illustrated with the photoelectrochemical disinfection of bacteria whose rate of decaying is linear. The results from simulation are validated by comparing them with the available experimental data [24] as well as with the analytical solutions derived from the corresponding stochastic model. MODEL FORMULA TION The system under consideration comprises the populations of molecules of species A and B per unit volume that are be ing chemically transformed into one another according to the simple reversible reaction, AB k k 1 2 1 Clearly, the population of A decreases and that of B increases when one molecule of A transforms into one molecule of B with forward reactionrate constant k 1 Similarly, the population of B decreases and that of A increases when one molecule of B transforms into one molecule of A with reverse reactionrate constant k 2 The system constitutes, therefore, a special class of Markov processes, the birthdeath process. [11, 12] Thus, the numbers of molecules of A and B at time t, N A (t) and N B (t), respectively, can be taken as the random variables of the birthdeath process of interest. A realization of N A (t) is denoted by n A and that of N B (t) by n B At any time t, the total number of molecules in the system remains constant; hence, the following relationship holds; NN Nt Nt or nn Nt Nt A B A B AB A B 00 00 2 where N A (0) or n A0 is the number of molecules of A at the outset of the reaction, i.e. at t = 0. Similarly, N B (0) or n B0 is the number of molecules of B at t = 0. These two quantities are constant, thereby indicating that only one of the random variables in Eq. (2) is independent. By denoting the total number of molecules at t = 0, i.e. (n A0 +n B0 ), as n T0 Eq. (2) can be rewritten as nN tN t TA B 0 3 If N A (t) is selected as the independent random variable, N B (t) can be expressed in terms of N A (t), from the above equation, as Nt nN t B T A 0 4 whose realization is naturally nn n BT A 0 5 Consequently, the state space for the current system is all the possible numbers of molecules of A, i.e. {n T0 (n T0 1), ..., 2, 1, 0}. MASTER EQUA TION The derivation of the master equation of a stochastic process has been extensively elaborated in our previous contribution to this journal. [25] For the birthdeath process of interest, the probability balance around any arbitrary state n A of the inde pendent random variable, N A (t), gives rise to [11, 12] d dt pn tW nt pn t A A A ; ; ; 11 ;; Wn tp nt A A 11 ;; ; Wn tW nt pn A A A t t 6 This derivation is also detailed in Appendix A. The term, p(n A ; t), in the above equation denotes the probability of n A mol rate law, the intensity of birth, W + (n A ; t), and the intensity of death, W (n A [1] Wn t dn dt kn kn n A A B T A ; 22 0 7 and Wn t dn dt kn A A A 1 8 respectively. It is worth noting that the intensities of birth and nonlinear rate laws, thereby giving rise to a variety of linear and nonlinear stochastic models. In fact, the formulation of stochastic models for processes obeying nonlinear rate laws has been presented in some of our earlier contributions. [2628] MEANS AND V ARIANCES The solution of the master equation, Eq. (6), renders it pos sible to obtain the mean and higher moments about the mean of the random variables, N A (t) and N B (t); see Appendix B. PAGE 37 Vol. 42, No. 1, Winter 2008 37 Among these higher moments, the second moment about the mean, i.e. the variance, is about its mean. [29] Eq. (6) in conjunction with Eqs. (7) and (8) yields the expression for the mean of N A (t), E[N A (t)] or N A (t) as Nt n kk n n kk k A T A T 0 12 0 0 12 2 2 1 22 ex pk kt k 9 Note that in the particular case where n A0 = n T0 i.e. where n B0 = 0, the above expression reduces to [30, 31] Nt n kk kk kt k A T 0 12 1 1 22 10 ex p In addition, the mean of N B (t), E[N B (t)] or N B (t) is, from Eqs. (4) and (9), Nt n kk n n kk B T A T 0 12 0 0 12 1 1 kk kt k 2 1 22 ex p 11 Similarly, the expression for the variance of N A (t), Var[N A 2 (t), is evaluated as A T A t n kk kk t n n 2 0 12 2 12 0 1 ex p T T kk kk 0 1 2 2 2 2 2 ex p 1 12 12 12 kt kk For the case, n A0 = n T0 this equation becomes [30] A t k kk kk t 2 1 12 12 1 ex p Nt A 13 where NA(t) (t), is the square root of 2 (t); thus, from Eq. (12), A T t n kk kk t 0 12 12 12 12 1 / / ex p n n n kk k A T 0 0 1 2 2 2 2 2 ex xp / kk tk k 12 12 12 14 From Eqs. (4) and (12), the variance of N B (t), Var[N B 2 (t), is B T A t n kk kk t n n 2 0 12 2 12 0 1 ex p T T kk kk 0 1 2 2 2 2 2 ex p 1 12 12 2 15 kt kk t A (t) is given by B T t n kk kk t 0 12 12 12 12 1 / / ex p n n n kk k A T 0 0 1 2 2 2 2 2 ex xp / kk tk kt A 12 12 12 16 Randomly or stochastically behaving processes or systems are ubiquitous in the chemical or allied industries. PAGE 38 Chemical Engineering Education 38 about its mean [32] mean, N(t) A (t), i.e. CV A (t), is CV t t Nt n kk kk A A A T 0 12 12 12 1 / ex p t n n kk k A T 12 0 0 1 2 2 2 2 2 / ex pk kt kk 12 12 12 0 12 0 0 12 2 / n kk n n kk k T A T ex pk kt k o 12 2 r r CV t n kk t n A T A 1 1 0 12 12 12 0 / / ex p n n kk kk T0 1 2 2 2 2 2 1 ex p kt kk n n A T 21 2 12 0 0 / kk kk kt 12 2 1 2 ex p k 2 17 By substituting the expression for N A (t) as given by Eq. (9), the above equation can be rewritten as CV t Nt k kk kk A A 1 2 12 12 ex p t t k kk 1 12 N Nt k kk nk kt A A 2 12 0 1 2 ex p k kk nk kt A 1 12 0 1 2 2 ex p 12 18 / When n A0 = n T0 this expression becomes CV t k kk kk t A 1 12 12 1e xp 12 12 19 / / Nt Nt A A where N A (t) is given by Eq. (10). In their contribution, MartnezUrreaga and his col laborators [1] A (t) about its mean N A (t) i.e. the A (t), are approximately of the order of ( N A (t) ) 1/2 /N A (t), or N A (t) 1/2 as revealed through Eqs. (18) and (19). ST OCHASTIC SIMULA TION The master equation of the birthdeath process, Eq. (3), is stochastically simulated by means of the Monte Carlo method. The stochastic simulation of chemical kinetics by this method has been known for many years. [17, 18, 21, 3336] A number of the works devoted to it, however, fail to deal with the crux of any stochastic simulation, which is its capability to estimate not only the mean but also higher moments about the mean, especially the vari In general, the stochastic simulation of chemical kinetics can focus on the temporal evolution of the number concentrations of entities comprising a reacting system or the spatial interactions among these entities. PAGE 39 Vol. 42, No. 1, Winter 2008 39 ance. [25] According to Haseltine and Rawlings, [37] . . stochastic simulation of chemical kinetics has received an increased amount of attention from the modeling community. . . In general, the stochastic simulation of chemical kinetics can focus on the temporal evolution of the number concentrations of entities comprising a reacting system or the spatial interactions among these entities. The former can be accomplished by resorting to the classical formulation of the Monte Carlo method, [13] and the latter can be carried out by means of Kinetic Monte Carlo (KMC) methods, which take into account the spatial distribution of the entities at the atomic or molecular scale. [38, 39] In this work, the Monte Carlo method is deployed for stochastically simulating the temporal evolution of the numbers of molecules of A and B for the simple reversible reaction; thus, the simulation of spatial interactions among these molecules is not considered. The two basic procedures to implement the method, one resorting to the eventdriven approach [10, 1523] and the other resorting to the timedriven approach, [14, 15] are described in detail; these two ap proaches differ in the manner of updating the simulation clock of the processs temporal evolution. The eventdriven approach advances the simulation clock by a [10, 19] one event occurs at the end of this time interval at which the corresponding to each event. The Gillispie algorithm [10, 17, 18] com prises a series of steps to perform the Monte Carlo simulation of Markov processes by the eventdriven approach. This algorithm has been recently extended [35] to accelerate the computing speed of the original formulation. [17, 18] Herein, Gillespies original algo rithm, as presented by MartnezUrreaga and his collaborators, [1] has been revised and augmented to include the evaluation of the means, variances, and standard deviations of the birthdeath process of interest as described below. Step 1. volume, n T0 the total number of simulations, or trajectories, Z f and the length of each simulation, t f Initialize the simulation counter as Z 1. Step 2 [31] the realizations of N A (t) and N B (t) at time t and simulation Z, i.e. n A,Z (t) and n B,Z (t), respectively, and the realizations of N A B and simulation Z, i.e. n A,Z B,Z tively, as follows: 00 00 0 0 0 t nt n nt nn t n AZ A BZ TA Z , A AZ AZ BZ BZ nt nn t , , 0 0 0 0 Step 3. Sample a realization u from the uniform random to the following expression [10] ; 1 1 Wn tt Wn tt nu AZ AZ , ; ; 20 where W + [n A,Z (t);t] = k 2 [n T0 n A,Z (t)] and W [n A,Z (t);t] = k 1 n A,Z (t) are the intensities of birth and death, Eqs. (7) and (8), respectively; see Ap pendices C and D. Step 4. Advance clock time as t Step 5. continue to Step 8. Step 6. Compute the sample means, variances, and standard nn t nn t AZ AZ BZ BZ , , AZ AZ Z Z BZ BZ Z Z n n , , 1 1 AZ AZ Z Z BZ BZ Z Z n n , , 2 1 2 1 AZ AZ Z Z AZ n , 1 2 2 B BZ BZ Z Z BZ n , 1 2 2 [13, 19, 20] : m Z n Z AZ AZ Z Z AZ , 1 1 21 1 m Z n Z BZ BZ Z Z BZ , 1 1 22 1 f then compute the sample variances [13, 19, 20] : PAGE 40 Chemical Engineering Education 40 s Z n Z n AZ AZ AZ Z Z , 2 2 1 1 1 1 2 1 1 1 Z Z AZ Z 1 1 23 Z AZ s Z n Z n BZ BZ BZ Z Z , 2 2 1 1 1 1 2 1 1 1 Z Z BZ Z 1 1 24 Z BZ ss AZ AZ , / 2 12 25 ss BZ BZ , / 2 12 26 Step 7. f ), then return to Step 5; otherwise, continue to Step 10. Step 8. Determine the state of the system at the end of time interval u from the uniform random variable, U, on interval (0, 1) and compute the probability of transition corresponding to the birth event, w + [n A,Z (t);t], as follows: wn tt Wn tt Wn tt AZ AZ AZ , ; ; ; Wn tt AZ ; 27 If {0< u PAGE 41 Vol. 42, No. 1, Winter 2008 41 d. Store the square of sum of realizations at t: AZ AZ Z Z AZ tn tt , 1 2 2 B BZ BZ Z Z BZ tn tt , 1 2 2 e. Compute the sample means at t: mt nt Z t AZ AZ Z Z AZ , 1 1 29 mt nt Z t BZ BZ Z Z BZ , 1 1 30 f then compute sample the variances and standard deviations at t: st Z nt Z nt AZ AZ Z Z AZ Z Z , 2 2 1 1 1 1 1 2 1 1 Z t AZ 1 1 31 Z t AZ st Z nt Z nt BZ BZ Z Z BZ Z Z , 2 2 1 1 1 1 1 2 1 1 Z t BZ 1 1 32 Z t BZ st st AZ AZ , / 2 2 12 33 st st BZ BZ , / 2 12 34 Step 4. Advance time as t Step 5. Estimate the probabilities of transition corresponding to the birth and death events as {W + [n A,Z + [n A,Z (t);t]+W [n A,Z Step 6. To accomplish this, sample a realization u from the uniform random variable U on interval (0, 1) and compare it with the probabilities of transition for the birth and death events. If u < [W + (n A birth event occurs; thus, nt nt t nt nn t AZ AZ BZ TA Z , , 1 0 If u < [W + (n A ;t) + W (n A nt nt t nt nn t AZ AZ BZ TA Z , , 1 0 Otherwise, no event occurs; thus, nt nt t nt nn t AZ AZ BZ TA Z , , 0 Step 7. Repeat Steps 3 through 6 until t f is reached. Step 8. Update the simulation counter as Z (Z + 1). Step 9. Repeat Steps 2 through 8 until Z f is reached. Given in Appendix F is the computer code in MATLAB for performing Monte Carlo simulation via the timedriven approach as presented above. RESUL TS AND DISCUSSION The means, variances, and standard de viations of the random variables, N A (t) and N B (t), characterizing the simple reversible reaction have been computed with the values for k 1 and k 2 and the two sets of values for n A0 and n B0 given by MartnezUrreaga and his collaborators. [1] Thus, k 1 is 4 10 s and k 2 is 1 10 s for n A0 and n B0 comprises 175 molecules of A and 25 molecules of B, and the second set comprises 3,500 molecules of A and 500 molecules of B. The means, N A (t) and N B (t) have been computed from the stochastic model according to their corresponding analytical expressions, Eqs. (9) and (11); the results A0 and n B0 in Figure 1 (next page) and for the second set of n A0 and n B0 in Figure G1 in Appendix of A and B obtained from the solution of the deterministic model as presented by Mar tnezUrreaga and his collaborators [1] are superimposed for comparison. Clearly, the values of N A (t) and N B (t) obtained from the stochastic model are in excellent accord with the corresponding deterministic values, thereby verifying that the mean component of the stochastic model for the birthdeath process is equivalent to the deterministic model. A B (t), signify the variability attributable to the in ternal or characteristic noises of the process as predicted by the stochastic model. The A B (t) have been computed from Eqs. (14) and (16), respectively; they are plotted in Figures 1 and G1 as the stan dard deviation envelopes, [ N A (t) A (t)] for N A (t) and [ N B (t) B (t)] for N B (t), PAGE 42 Chemical Engineering Education 42 respectively. Note that the variability or dispersion of the process is more pronounced in Figure 1 than in Figure G1: As indicated by MartnezUrreaga and his collaborators [1] and earlier in the current contribution, the relative extent of N(t) is approximately of the order ( N(t) ) 1/2 / N(t) or N(t) 1/2 i.e. it is inversely proportional to the square root of the popula tion size. In other words, the smaller the population size, the greater the extent of variability or dispersion of the process about its mean and vice versa. The sample means, m A,Z B,Z ed through the Monte Carlo method on the basis of the eventdriven approach by averaging the realizations of N A (t) and N B (t) from Z successive simulations. The results are illustrated n A0 and n B0 ; the values of N A (t) and N B (t) computed from the stochastic model as well as the numbers of molecules of A and B obtained from the solution of the deterministic model are A,Z and m B,Z N A (t) and N B (t) respectively, as the number of simulations, Z, varies from 2 to 50. The results are in line with the Weak Law of Large Numbers (WLLN), stating that the sample mean approaches the population mean as the size of the sample becomes large. [29] In stochastically simulating the birthdeath process, the size of the samples, i.e. the numbers of realizations of N A (t) and N B (t), increases as the number of simulations, Z, increases. Similarly, m A,Z and m B,Z N A (t) and N B (t) respectively, when they are computed by averaging 50 simula tions, i.e. Z = 50, for the second set of n A0 and n B0 ; see Figure G2 in Appendix G. The sample standard deviation envelopes, [m A,Z A,Z A (t) and [m B,Z B,Z B (t), are also plotted in Figures 2 and 3 as well as in Figure G2 in Appendix G; they are essentially identical to [ N A (t) A (t)] t, s 0 200 400 600 800 1000 Number of molecules 0 25 50 75 100 125 150 175 200 Deterministic Model Stochastic Model Molecules of A Molecules of B Mean of N A (t), N A (t) Mean of N B (t), N B (t) N B (t) B (t) N A (t) A (t) Figure 1. Temporal evolution of the numbers of molecules of A and B per unit volume with n A0 = 175 and n B0 = 25: The solutions from the stochastic model are compared with those from the deterministic model. t or s 0 200 400 600 800 1000 Number of molecules 0 25 50 75 100 125 150 175 200 Stochastic model Mean of N A (t), N A (t) Mean of N B (t), N B (t) Monte Carlo simulation, Eventdriven Sample means, m A,2 ( ) or m B,2 ( ) [m A,2 ( ) s A,2 ( )] or [m B,2 ( ) s B,2 ( )] N B (t) B (t) N A (t) A (t) Figure 2. Temporal evolution of the means and standard deviation envelopes of the numbers of molecules of A and B per unit volume with n A0 = 175 and n B0 = 25: The average of two Monte Carlo simulations via the eventdriven approach is compared with the analytical solu tions resulting from the stochastic model; the solution of the deterministic model is represented by the solid lines. t or s 0 200 400 600 800 1000 Number of molecules 0 25 50 75 100 125 150 175 200 Mean of N A (t), N A (t) Mean of N B (t), N B (t) N B (t) B (t) Monte Carlo simulation, Eventdriven Sample means, m A,50 ( ) or m B,50 ( ) [m A,50 ( ) s A,50 ( )] or [m B,50 ( ) s B,50 ( )] Stochastic model N A (t) A (t) Figure 3. Temporal evolution of the means and standard deviation envelopes of the numbers of molecules of A and B per unit volume with n A0 = 175 and n B0 = 25: The average of 50 Monte Carlo simulations via the eventdriv en approach is compared with the analytical solutions resulting from the stochastic model; the solution of the deterministic model is represented by the solid lines. and [ N B (t) B (t)], respectively, for Z = 50. Figures G3 and G4 in Appendix G exhibit m A,Z (t) and m B,Z (t) as well as [m A,Z (t) s A,Z (t)] and [m B,Z (t) s B,Z (t)], which have been computed through the Monte Carlo method n A0 and n B0 the results with Z = 2 are presented in Figure G3, and those with Z = 50 are presented in Figure G4; they are compared with the solutions of the deterministic and stochastic models. Note that m A,Z (t) and m B,Z (t) in Figure G4 PAGE 43 Vol. 42, No. 1, Winter 2008 43 are equivalent to N A (t) and N B (t) respectively, for Z = 50; this is also the case for the second set of n A0 and n B0 as illustrated in Figure G5 in Ap pendix G. Table 1 lists the computational times, required by the Monte Carlo method via the eventdriven and timedriven approaches, for estimating the sample means, variances, and standard deviations of the random variables, N A (t) and N B (t), characterizing the simple reversible reaction. To perform the Monte Carlo simulations by resorting to these approaches, the MATLAB codes given in Appendices E and F were implemented on a PC (Pentium IV 3.0 GHz with 512 MB RAM; Windows XP). Clearly, the time required by the eventdriven approach was shorter than that required by the timedriven approach. Figures 2 and 3 in conjunction with Figures G2 through G5 in Appendix G reveal a fundamental distinction between the variations attributable to the inherent or internal noises of a random process and those attributable to the numerical method to stochastically simulate it. The former can only be means of the random variables characterizing the process, especially, the variances or standard devia tions. The latter arise from the generation of samples on the basis of random numbers; these variations tend to vanish as the number of random samples increases, i.e., as the number of simulation runs becomes large. tic simulation via the Monte Carlo method, i.e. the eventdriven and timedriven approaches, are further illustrated with an example of photoelec trochemical disinfection of bacteria whose rate linear. In contrast to a chemical reaction involving molecules that are discrete and microscopic in size; the disinfection of bacteria deals with microorgan isms which are also discrete but mesoscopic in size. comprising mesoscopic entities are appreciably more pronounced than those of a system comprising microscopic entities, especially, when the number concentrations of such entities are minute. Naturally, the system of interest is the population of bacteria per unit volume. Catalyzed photoelec trochemically, these bacteria die one at a time; moreover, the bacteria have ceased to grow and do not reproduce throughout the deactivation. This sys tem constitutes a special instance of the birthdeath processes in which the intensity of birth is absent; hence, it is termed the puredeath process. [10, 11, 25] In modeling the photoelectrochemical disinfection of bacteria as a puredeath process, the number of live bacteria at time t, N C (t), is taken as the random variable of the process; a realization of N C (t) is denoted by n C All the possible numbers of bacteria are the states of the process, and the collection of these numbers, {n C0 (n C0 1), ..., 2, 1, 0}, constitutes the state space, where n C0 is the initial number of bacteria, i.e. N C (0) or n C at t = 0. For the puredeath process, the probability balance around any arbitrary state n C of the independent random variable, N C (t), leads to [11] d dt pn tW nt pn tW nt pn C C C C C ; ; ;; ; 11 t t 35 which is the master equation of the process [11, 25] ; see Appendix A. On (n C ;t) in the above expression is Wn t dn dt kn C C C ; 3 36 where k 3 the expressions of the mean, E[N C (t)] or N C (t) and the variance, Var[N C C 2 (t), are obtained from Eqs. (35) and (36) as [25, 40] Nt nk t C C 0 3 37 ex p and C C tn kt kt 2 0 3 3 1 3 8 ex pe xp C 2 (t) can be related to N C (t) as C C t k tN t 2 3 1 3 9 ex p C (t), is, from Eq. (38), C C tn kt kt 0 12 3 3 12 1 4 0 / / ex pe xp TABLE 1 Computational Times for Estimating the Sample Means, Variances, and Standard Deviations of the Random Variables, N A (t) and N B (t), Characterizing the Simple Reversible Reaction by the Monte Carlo Method via the EventDriven and TimeDriven Approaches Initial values Eventdriven approach Timedriven approach Z = 2 Z = 50 Z = 2 Z = 50 n A0 = 175 n B0 = 25 < 1 s < 1 s 1 s 1 s n A0 = 3,500 n B0 = 500 1 s 9 s 5 s 17 s Notes: In the eventdriven approach, the sample means, variances, and standard deviations have been recorded every two seconds, i.e. puted according to Eq. 28 in the text with W + M (n A ;t)=k 2 n T0 = k 2 (n A0 +n B0 ) and W M (n A ;t)=k 2 n A0 A0 = 175 and n B0 = 25, A0 = 3,500 and n B0 = 500. PAGE 44 Chemical Engineering Education 44 or, from Eq. (39), C C t k tN t 1 4 1 3 12 12 ex p / / C (t), is computed as CV t n kt kt C CO 1 1 12 3 3 1 / ex p ex p / / 2 42 which can be rewritten as CV t k t Nt Nt C C C 1 4 3 3 12 12 ex p / / of N C (t) about its mean N C (t) as given by CV C (t), are ap proximately of the order of ( N C (t) ) 1/2 / N C (t) or N C (t) 1/2 as indicated by MartnezUrreaga and his collaborators. [1] The temporal mean of the model, Eq. (37), has been re gressed on the experimental data for the photoelectrochemical disinfection of E.coli [24] near the termination period of the process. By identifying 140 cells per milliliter as the initial population of bacteria, n C0 the nonlinear regression by means of the adaptive random search procedure [41, 42] has resulted in 0.091 min 1 for the value of k 3 With these values of n C0 and k 3 N C (t) C (t) have been computed from Eqs. (37) and (40), respec tively. Figure 4 illustrates the resultant values of N C (t) and [ N C (t) C (t)] in conjunction with the experimental data. [24] In addition, the sample means, m C,Z C,Z (t), and standard deviation envelopes, [m C,Z C,Z C,Z (t) s C,Z (t)], have been computed via the Monte Carlo method by resorting to both the eventdriven and timedriven approaches, respectively. The results from averaging 50 simulations, i.e. Z = 50, are presented in Figure 4 for comparison. Note that the results from Monte Carlo simulation closely approximate the analytical results from the stochastic model. As expected, [24] are substantially more pronounced than those predicted by the stochastic model: The overall deviations of the experimental data include those attributable not only to the internal noises of the process but also to the external noises arising from inevitable manipulation errors and instrumental limitations that can never be totally suppressed. CONCLUDING REMARKS The contribution of MartnezUrreaga and his collabora tors [1] on the stochastic simulation of chemical reactions, complemented and augmented in two aspects. First, a sto chastic model for a simple reversible chemical reaction has i.e. linear, rate law as a birthdeath process. The master equation arising from the model has rendered it possible to analytically obtain the ex pected means and variances of the process. Second, the master equation of the process has been stochastically simulated through the Monte Carlo method via the timedriven approach in addition to the eventdriven approach on the basis of the Gillespie algorithm. The processs means and variances from numerical simulation are in accord with the solutions of the deterministic and stochastic models. To further reveal their Monte Carlo method are illustrated with the photoelectro law. The mean computed from stochastic simulation are in line with the available experimental data as well as with the analytical results derived from the corresponding stochastic mean are large when compared with those computed from the stochastic simulation. ACKNOWLEDGMENT This is contribution No. 071235, Department of Chemical Engineering, Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506.NOMENCLA TURE E[N A random variable, N A (t) E[N B(t)] mean of the random variable, N B (t) E[N C(t)] mean of the random variable, N C (t) Figure 4. Temporal evolution of the mean and standard deviation envelope near the tailend of photoelectrochemical disinfection of a population of E. coli per unit volume: The average of 50 Monte Carlo simulations by resorting to the eventdriven and timedriven approaches is compared with the analytical solutions resulting from the stochastic model; the initial population of bacteria, n C0 is 140 cells per milliliter. PAGE 45 Vol. 42, No. 1, Winter 2008 45 k 1 forward reactionrate constant, s k 2 reverse reactionrate constant, s k 3 N A(t) random variable representing the number of mol ecules of A at time t N B(t) random variable representing the number of mol ecules of B at time t N C(t) random variable representing the number of live bacteria at time t n A realization of the random variable, N A (t) n B realization of N B (t) n C realization of N C (t) n A0 number of molecules of A at t = 0, N A (0) n B0 number of molecules of B at t = 0, N B (0) n T0 total number of molecules at any arbitrary t, (n A0 + n B0 ) n C0 number of live cells at t = 0, N C (0) N A (t) E[N A (t)] N B (t) E[N B (t)] N C (t) E[N C (t)] p(n A; t) probability of n A molecules being present at time t p(n C; t) probability of n C live bacteria being present at time t t time U uniform random variable on interval (0,1) u realization of U Var[N A(t)] variance of the random variable, N A (t) Var[N B(t)] variance of N B (t) Var[N C(t)] variance of N C (t) W + (n A; t) intensity of birth for the birthdeath process in state n A at time t W (n A; t) intensity of death for the birthdeath process in state n A at time t W (n C; t) intensity of death for the puredeath process in state n C at time t A 2(t) Var[N A (t)] B 2(t) Var[N B (t)] C 2(t) Var[N C (t)] A(t) standard deviation of the random variable, N A (t) B(t) standard deviation of N B (t) C(t) standard deviation of N C (t) REFERENCES 1. 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Misek, Stochastic Modeling of Dispersion Formation in Agitated LiquidLiquid Systems, Transactions of the Institution of Chemical Engineers 60 48 (1982) 15. Rajamani, K., W.T. Pate, and D.J. Kinneberg, TimeDriven and EventDriven Monte Carlo Simulations of LiquidLiquid Dispersions: A Comparison, Ind. Eng. Chem. Fund. 25 746 (1986). Process, J. Roy. Stat. Soc. B 12 116 (1950) 17. Gillespie, D.T., A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. J. Comput. Phys. 22 403 (1976) 18. Gillespie, D.T., Exact Stochastic Simulation of Coupled Chemical Reactions. J. Phys. Chem. 81 2340 (1977) 19. Shah, B.H., J.D. Borwanker, and D. Ramkrishna, Monte Carlo Simula tion of Microbial Population Growth, Math. Biosci. 31 1 (1976) 20. Shah, B.H., D. Ramkrishna, and J.D. Borwanker, Simulation of Particulate Systems Using the Concept of the Interval of Quiescence, AIChE J. 23 897 (1977) 21. Rao, C.V., and A.P. 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McQuarrie, D.A., Mathematical Methods for Scientists and Engineers University Science Books, Sausalito, CA, pp. 4, 1053 (2003) APPENDICES Seven appendices (Appendices A through G) can be viewed or downloaded from the corresponding authors Web site at be obtained as hard copies by writing to fan@ksu.edu. PAGE 47 Vol. 42, No. 1, Winter 2008 47 C onsider a single component characterized by its critical pressure, temperature, acentric factor, P c T c and The PengRobinson equation of state [24] is given by: P RT Vb a VV bb Vb () () 1 where b RT P a RT P mT c c c c r 0 07780 2 0 45724 11 2 ( ) 2 3 4 () () T T T r c and m 0 37464 1 54226 0 26992 5 2 .. .. () The PengRobinson EOS is part of a family of equations called cubic because the compressibility factor, Z = PV/RT, is a solution of the following cubic equation derived from Eq. (1) and written for a multicomponent mixture: ZB ZA BB ZA BB B 3 2 2 2 3 1 3 2 0 6 () where Ay yA xx A A ij ij j C i C ij ij j C i C i 1 1 1 1 7 or () j ji j i j ii i C ii i C AA k By Bx B 05 1 1 1 8 () () or () 9 2 Aa P T B P T i i r r i r r i i i i 0.45724 and 0.07780 ( () 10 temperature using P/ / r i i i i PP an dT TT cr c and a i is pure component case. The binary interaction parameter, k ij The object of this column is to enhance our readers collections of interesting and novel prob lems in chemical engineering. Problems of the type that can be used to motivate the student by presenting a particular principle in class, or in a new light, or that can be assigned as a novel home problem, are requested, as well as those that are more traditional in nature and that elucidate dif (email: wilkes@umich.edu), Chemical Engineering Department, University of Michigan, Ann Arbor, MI 481092136. ChE THE PENGROBINSON EQUA TION OF ST A TE HOUSAM BINOUS National Institute of Applied Sciences and Technology Tunis, Tunisia *All these computations are available in the form of notebooks upon request from the author or the Wolfram Research Information Center. [1] Copyright ChE Division of ASEE 2008 PAGE 48 Chemical Engineering Education 48 is obtained from HYSYS 3.2, a major process simulator by Aspen Technology, Inc., [5] or assumed to be equal to zero if the simulator is not available. The liquid and vapor mole fractions are related by: yK x ii i with i1 2,..., C ( 11) where K i is the equilibrium constant. This equilibrium constant is obtained using the method as follows: K i l v i i fo ri 1, 2,..., C ( 12) v v i v ji j j i i Z B B ZB A B yA A B B ex p ln 1 22 2 ln ZB ZB v v 12 12 () 13 liquidphase and gasphase fugacities, fa nd f i v i l as follows: v i v i l i l i i i f yP f xP an d ( ) 14 The method is a direct result of expressing vaporliquid equilibrium conditions ( i.e. temperatures, pressures, A similar expression to Eq. (13) is obtained for the liquidl i by replacing the gasphase compressibility factor, Z v with its liquidphase counterpart, Z l These two compressibility factors are the largest and of P and T, the smallest root, Z l allows the determination of the liquid molar volume while the highest root, Z v permits the computation of the vapor molar volume. The intermediate Finally, one needs the following expression for the en thalpy departure function from ideality in order to compute enthalpy: HR TZ B RT P Lo g ZB ZB D () 1 1 22 12 12 T d dT A RT P A RT 2 2 P P () 15 ADIABA TIC FLASH CALCULA TIONS FOR MUL TICOMPONENT MIXTURES of 0.41% hydrogen, 5.71% methane, 70.97% benzene, and 22.91% toluene. This mixture is sent to a stabilizer to remove hydrogen and methane. The feed pressure is decreased adia batically from 485 psia to 165 psia by valve and pipe line pressure drop. Find the vaporphase fraction, the temperature and other relevant variables. This problem has been solved using a tedious iterative tech nique by Henley and Seader. [6] The unknowns in this problem are the mole fractions in the two phases, the temperature, the vapor phase fraction, and the compressibility factors. We have 12 nonlinear algebraic equations to solve simultane ously. These equations are three equilibrium relations, three component mass balances, two summation rules, two cubic equations of the compressibility factors, the enthalpy balance, and the famous Rachford and Rice equation given by: zK K ii va pi i c 1 11 0 1 6 1 () where va p is the fraction vaporized, V/F. The Mathematica commands, which allow the determination of the 12 un knowns, are based on the builtin function FindRoot and are given by: Eq iy iK ix i EQ ix iz i [_ ]: [] [] [] [_ ]: [] [] /( 1 va p Ki so lF indRoot Eq Eq Eq ([ ] )); [ [] ,[ ], 1 12 [ [] ,[ ], [] ,[ ], [] 3 1 12 3 1 4 yi EQ EQ EQ i xi zi Ki Ki i va p [] ,[ ]( [] )/ (( [] 1 1 1 1 4 10 31 23 2 1 4 )) ^( )^ (^ i l l l l ll Z B ZZ AB 2 2 23 0 31 2 B AB BB Z B ZZ l ll l l v v v v ) ^^ ^( )^ ( (^ ) ^^ AB B AB BB HF H vv v vv v v va p 32 2 23 0 V VH Lx xx va p () ,[ ], ., [] ,. ,[ 1 1 0 0004 20 03 3 30 74 02 5 00 11 01 ], ., [] ,. ,. ,[ ], x y va p ,[ ], ., [] ,. [] ,. ,, yy y Z l 20 85 30 02 40 001 0 00 50 95 54 0 10 ., ,. ,, ZT Ma xI te ra tions v 0 00 ]/ /C hop x x x [] ., [] ., [] 10 00020583 20 0259441 30 73608 7 74 0 237763 10 106105 20 ,[ ]. [] ., [] x y y .. [] ., [] ., 873206 30 0185161 40 00217289 y y va p p l v Z Z T 0 0367724 0 0405552 0 981151 560 ., ., ., .297 PAGE 49 Vol. 42, No. 1, Winter 2008 49 The vaporphase fraction va p and temperature are 0.0367 S OLU B ILI T Y OF M ET HANOL IN NA T URAL G AS Methanol is used as a solvent in the sweetening of natural gas ( i.e. Loss of methanol by transfer from liquid phase to gas phase can become an important economic consideration. Compute the gas phase mole fraction of methanol [3] in treated natural gas leaving the sour gas absorption unit vs. temperature at a pressure of 145.038 psia. Treated natural gas is composed of 70% methane, 10% nitrogen, 10% carbon monoxide and 10% hydrogen. ous temperatures. The problems unknowns are the ten mole fractions in the gas and liquid phases, the vapor phase fraction, and the two compressibility factors. The available equations compressibility factors, and the Rachford and Rice equation. The calculation uses the builtin function FindRoot to solve 13 nonlinear algebraic equations simultaneously. The Math ematica commands, which allow the determination of the 13 unknowns, are the following: Eq iy iK ix i EQ ix iz i [_ ]: [] [] [] [_ ]: [] [] /( 1 va p Ki Fo ri i T ([ ] )); [, ., 1 11 5 329 67 347 67 7 365 67 383 67 401 67 419 67 437 67 455 6 ,. ,. ,. ., ., .7 7 47 36 7 491 67 509 67 527 6 ,. ., ., 7 7 545 67 563 67 1 ,. ,. }[ [ ]], [] [[ i so li Fi ndRoot Eq ] ], [] ,[ ], [] ,[ ], Eq Eq Eq yi EQ i 23 4 1 1 5 [ [] ,[ ], [] ,[ ], [] [] ([ 12 34 1 1 5 EQ EQ EQ xi zi K i i i K i Z B va p i l l ]) /( ([ ]) ), ^( 11 10 31 1 5 ) )^ (^ ) ^^ ZZ AB B AB BB Z l l ll l ll l l 23 22 23 0 v v v v v vv v vv v BZ ZA BB AB B ^( )^ (^ ) ^ 31 23 22 2 B xx so li x v ^ [] ,[ ]/ .[ ], [] 30 11 12 ,, [] /. [] [] ,[ ]/ .[ ], [ xs ol i xx so li x 21 33 1 4 44 15 51 ], [] /. [] ,[ ], [] /. [] xs ol ix xs ol i y yy so li yy so li [] ,[ ]/ .[ ], [] ,[ ]/ .[ ] 11 12 21 [] ,[ ]/ .[ ], [] ,[ ]/ .[ yy so li yy so li 33 1 44 1] ], [] ,[ ]/ .[ ], ,/ .[ yy so li so li va pv ap 55 1 1 1 1 ], ,/ .[ ], ,/ .[ ], ZZ so li ZZ so li ll vv M Ma xI te ra tions Chop ii 1000 ]/ /, Pr int[ ], ] Figure 1 depicts the computed solubility of methanol in treated natural gas vs. temperature at 145.038 psia. We observe an increase in solubility at higher temperatures. We see that these solubility values can become large enough to present serious economic drawback due to excessive loss of methanol.HIGHPRESSURE CHEMICAL EQUILIBRIUM COMPUT A TION Nitrogen and hydrogen react to form ammonia. This reac tion is favored by low temperatures and high pressure. Kinetic considerations, however, oblige us to use high temperature. Thus, reactors are operated at very high pressures to get a reasonably high conversion of reactants. High gasphase [4] In fact, the equilibrium constant depends on K v as follows: Ka y yy P K XX X a i i C NH NH v i 1 05 15 3 22 1 2 1 .. 2 2 31 2 1 17 05 15 . () X P K v 350 400 450 500 550 0.00001 0.001 0.1 Temperaturein R Mole fraction ofmethanol Figure 1. Solubility of methanol in treated natural gas vs. temperature at 145.038 psia. PAGE 50 Chemical Engineering Education 50 where K v is given by: K ay v NH NH ii i 3 22 05 15 18 19 .. () () and X is the extent of reaction. Plot K v and X vs. pressure at temperatures ranging from 900 R to 1440 R. fraction in the gas phase, the extent of reaction, and the gas phase compressibility factor. Once again, the calculation uses the builtin function FindRoot algebraic equations simultaneously. The Mathematica com are the following: P Fo ri is ol Fi ndRoot Ka T 0 001 11 7 1260 .; [, [ [] ]( .( )) ^. (. () ) ^. [] ^. [ 05 10 51 51 15 11 52 X X v v ] ]^ () /[ ], [] () 05 23 3 2 XX Py X Xy v [ [] () .( ), [] () .( ), ^ 22 05 1 12 15 1 X X yX X Z v 3 3 12 3 22 2 () ^( ^) ^ BZ ZA BB AB BB vv vv v v vv v v v yy yX ^, [] ,. ,[ ], ., [] ,. 3 0 20 21 01 3 04 ,, ., ,. }, ], [ 06 09 5 10000 ZM ax It er ations Pp i v ] ], [] [] /( [] ^. [] ^. )/ PK vi so l vv v 3 11 52 05 12 260 1260 1000 ,[ ]/ ., Pr int[ ," ", ], Xi Xs ol iP PP ,, }] ii 1 In Figure 2, we plot K v vs. pressure at various temperatures v different from unity, indicating that this factor must be taken into account at high pressures. The extent of reaction at equi librium vs. pressure, for the same temperatures, is represented in Figure 3. The extent of reaction approaches unity at high pressures.CONCLUSION We have shown through simple examples how one can take advantage of the numerical and graphical capabilities ponent mixtures, solubility determination, and highpressure chemical equilibrium computations. Similar computations can be performed very easily using Matlab. [7]NOMENCLA TUREx: liquid mole fraction y: vapor mole fraction P c,i: critical pressure (psia) T c,i: critical temperature (R) P r,i: reduced pressure T r,i: reduced temperature : acentric factor Z: compressibility factor l v K i: equilibrium constant Figure 2. K v vs. pressure at various temperatures. Figure 3. Extent of reaction vs. pressure at various temperatures. 0 2000 4000 6000 8000 10000 12000 14000 0.0 0.2 0.4 0.6 0.8 1.0 Pressure in psia Figure 2 K v versus pressure at various temperature. 900 R 1080 R 1260 R 1440 R v K 0 2000 4000 6000 8000 10000 12000 14000 0.0 0.2 0.4 0.6 0.8 1.0 Pressure in psia Extentofreaction Figure 3 Extent of r eaction versus pressure at various temperature. 900 R 1080 R 1260 R 1440 R PAGE 51 Vol. 42, No. 1, Winter 2008 51 H D: departure from ideality enthalpy (cal/mol) k ij: binary interaction parameter C: number of components z: feed mole fraction R: universal gas constant (cal/mol.R) a i: activity of species i (psia) v i li vi va p : fraction vaporized = V/FREFERENCES 1. PAGE 52 Chemical Engineering Education 52 ChE the column should be approximately 450 words. If graphics are included, the length needs to be Copyright ChE Division of ASEE 2008 CELEBRA TING C h E IN SONGALAN M. LANE The University of Alabama Figure 1. Doobie Doghouse Wilson (AKA Alan Lane) with his dobro. E very semester I survey my students to learn why they chose to attend UA, why they chose to major in chemi cal engineering, and what kind of career they aspire to. I also ask about their major interests and hobbies. I found that chemical engineering students organized COE Does students put on a musical each semester. Several chemical engineering students competed in a recent university battle of the bands as members of the rock group Liquid Metal. Others are members of the universitys marching band, sing in a musical, write short stories, create art, or make crafts. Engineering is an incredibly creative profession so it should not surprise anyone that their activities outside the classroom Many years ago, CEE published the poem The Reynolds Number Song [1] that humorously celebrates the Reynolds Number. Since most current students enjoy rap music, I found it useful to modify the poem slightly and perform it in class as the Reynolds Rap (pun intended). Imagine the class pounding out a rhythm on their desks and joining me in the chorus, Take a d times a v and a rho by a mu, put them all together with a little bit of glue, now you got a number gonna see you through, and tell you what the [insert colorful language of It is certainly hard to forget the equation after that exercise! In the fall of 2006, I recorded the Reynolds Rap in my home studio and played it for my 90 freshmen in the Intro duction to Chemical Engineering seminar. During the 2007 summer lab several students asked me to write a song about distillation columns, suggesting the title Steppin Off the Stages. Since our endofsummerlab picnic was themed The Fiesta before the Siesta, I set it to a Latin rhythm and played it at the picnic. The students very much enjoyed both songs, and many downloaded them. These songs are available free as MP3 downloads at ing pleasure andif you dareto use in your classroom. I believe there are many wouldbe rock stars in the profession that have also written or will write a chemical engineeringthemed song. Make a good recording and send me the MP3 engineers iPods around the globe!REFERENCES 1. Harriott, P., The Reynolds Num ber Song, Chem. Eng. Ed. 8 (1), 12 (1979) PAGE 53 Vol. 42, No. 1, Winter 2008 53 M athematical software packages such as Polymath, MATLAB and Mathcad are widely used for engi neering problem solving. The application of several packages to typical chemical engineering problems has been demonstrated by Cutlip, et al. [1] The main characteristic of these packages is that they provide a problem solving envi ronment (PSE) [2] ies callable from programming languages that were popular a few years ago. For routine use of the software packages in problem solving, there is no need to be an expert in either numerical methods or programming. In most cases, it is suf the syntax required by the package. The technical details of the solution are carried out by the package. There are cases, however, where the solution is obviously incorrect and/or the program stops with an error message. The failure of a software package to reach a correct solu tion is commonly due to the presence of errors in the math ematical model or the selection of an inappropriate solution algorithm and/or inappropriate algorithm parameters (such as initial estimates, error tolerances, etc). Validation of the mathematical models has been discussed in detail by Brauner, et al. [3] In this paper, we present an exercise that can be used The object of this column is to enhance our readers collections of interesting and novel prob lems in chemical engineering. Problems of the type that can be used to motivate the student by presenting a particular principle in class, or in a new light, or that can be assigned as a novel home problem, are requested, as well as those that are more traditional in nature and that elucidate dif (email: wilkes@umich.edu), Chemical Engineering Department, University of Michigan, Ann Arbor, MI 481092136. ChE CAN I TRUST THIS SOFTWARE P ACKAGE? MORDECHAI SHACHAMBenGurion University of the Negev BeerSheva 84105, Israel NEIMA BRAUNER TelAviv University TelAviv 69978, Israel W. ROBERT ASHURST Auburn University Auburn, Alabama 36849MICHAEL B. CUTLIP University of Connecticut Storrs, CT 06269 Copyright ChE Division of ASEE 2008 PAGE 54 Chemical Engineering Education 54 an algorithm for the solution of a problem and help them to select appropriate parameters for the algorithm. This exercise involves numerical solution for a system of ordinary differential equations (ODEs). It can be used at two levels. 1. numerical software for solving ODEs ( i.e., Reaction Engineering, Process Control and Process Simula tion). In such courses, this part of the exercise can be used to demonstrate that computer solutions can be inaccurate or even completely incorrect, and therefore it is always necessary to validate a numerical solu tion. 2. At the more advanced level, the second part of the exercise can be used in courses related to mathemati cal modeling and numerical analysis. In such courses, this part of the exercise provides training and demon stration of advanced topics such as error estimation and step size control, error propagation, and stiffness of ODEs. In this work, the mathematical software packages Poly math 6.1 1 and MATLAB 2 are used for the demonstration, but other packages may be used in a similar manner. ANALYZING THE BEHAVIOR OF THE RUNGEKUTT A ALGORITHM FOR AN INTERACTING TANKS SIMULA TION PROBLEM The problem considered in this exercise involves the dynamic simulation of a system of three interacting tanks, as depicted in Figure 1. The model equations (as were provided Ashurst and Edwards [4] ) that can be derived from mass balances on each tank are qq Dh DD H 14 21 12 1 2 4 d dh dt qq qe dh dt qq qh h 1 24 5 2 2 53 63 2 1 2 2 () () d dh dt 3 3 () where q j 3 /s), H i the tank height (m), h i the liquid height (m), D 1,2 the upper/lower diameter ratio of tank 1 (m), and t is the time drainage area, Ao (m 2 ), are given by qA og hh if hh qA og hh othe rw i 4 1 21 2 4 2 1 2 2 () s se qA og hh if hh qA og hh o () () () 4 2 2 5 2 32 3 5 3 2 t ther wise qA og h () () 5 2 6 6 3 rates is to be investigated. The model equations and the nu merical data pertinent for a particular case, as they have been entered into the Polymath 6.1 software package, are shown in Table 1. Note that the equations and comments in the Polymath input provide a complete documentation of the problem (in Polymath the comments are indicated by the sign: #). 1 POLYMATH is a product of Polymath Software ( PAGE 55 Vol. 42, No. 1, Winter 2008 55 The system is started at steadystate values of h 1 h 2 and h 3 different tanks are q 1 = q 2 = q 3 = 0.05 m 3 /s. At t = 50 s, a step change of +10% is introduced into q 3 The system is simulated f = 3600 s. The integration of the model equations using the RKF45 algorithm of Polymath with the default parameters yields the error message Error: sqrt of negative number is not allowed, sqrt(2*g*h 3 ), which indicates that the height of the liquid level in the 3rd tank (h 3 ) has somehow reached a negative value dur ing the numerical integration. A quick check can be made on As there is no abnormal behavior or unexpected trends during this numerical integration, as shown in Table 2 and Figure 2, the assignment is to check and verify the error in the integration up to the time of 3600s. The options that need to be investigated for this problem are: 1. There are possible er rors in the implementa tion of the RungeKutta algorithm. 2. The default truncation error tolerance of the program is too large for this problem, and the negative value of h 3 is a result of large truncation error and propagated error in some of the variables. 3. The system of equations is stiff. The explicit RKF45 may exhibit oscillatory be havior if the integration step size is greater than that dictated by the stability limit. RECOMMENDED STEPS AND ADDITIONAL INFORMA TION FOR SOLUTION 1. Most software packages include some variation of the RungeKutta algorithm with truncation error estimation and step size control as one of the ODE solver tools. Using the same or a similar algorithm in a different software package can provide ad ditional information regarding the source of the error. 2. The appropriateness of the truncation error tolerance can be tested by rerunning the problem using various error tolerances and comparing the variable values obtained at a f ). It is preferable to check the values at the integration interval is considered. ing the f vector as f dh dt dh dt dh dt 1 2 3 7 () Figure 1. A train of three interactive tanks. [4] T ABLE 2 Change of the Variable Values after Introduction of a 10% Step Change in Inlet Flow Rate to Tank 3 Variable Initial value Minimal value Maximal value Final value t 0 0 2400 2400 h 1 1.807 1.807 1.854328 1.854328 h 2 1.68 1.679873 1.725494 1.725494 h 3 1.162 1.162 1.221394 1.221394 q 4 0.0495908 0.0495908 0.0499622 0.0499475 q 5 0.1001531 0.0971087 0.1001531 0.0988002 q 6 0.1500039 0.1500039 0.1537897 0.1537897 dh 1 /dt 9.39E05 8.66E06 9.39E05 1.17E05 dh 2 /dt 6.22E06 6.22E06 3.06E05 1.16E05 dh 3 /dt 1.11E04 5.39E07 2.71E04 7.08E06 Figure 2. Change of the liquid level in the third tank after introduction of a 10% step change in its inlet ow rate (Polymath Solution). PAGE 56 Chemical Engineering Education 56 where dh 1 /dt, dh 2 /dt, and dh 3 /dt are given by Eqs. (13), respectively. The stiffness of the system can be assessed by 245 in Rice and Do [5] ) SR i i ma x min () 8 where max i with maximal absolute value, and min i is a negative ei the system is stiff, using the appropriate algorithms ( i.e., stiff or stiffbs in Polymath) should prevent obtaining the negative h 3 value during integration. SOLVING THE PROBLEM WITH A DIFFERENT SOFTWARE P ACKAGE Polymath 6.1 can be used to convert the model of the problem into a MATLAB function. The Polymath generated function is shown in lines 21 through 42 of Table 3. Note that Polymath reorders the equations and changes the syntax of the model according to the requirements of MATLAB. Some manual editing of the Polymath generated function was done in order to enable more concise presentation (the comments were put in the same line with the commands and the ifelse statements were put in a single line). The main program that runs the Polymath generated function (shown in lines 1 through 19 of Table 3) is avail able as a template in the Help section of Polymath. Only the function name has to be added (see line 1) and the initial values of the variables had to be copied from the Polymath generated model (see lines 3 and 4). The ode45 library func tion of MATLAB can be used to solve the problem. This function is based on essentially the same algorithm as the RKF45 routine of Polymath. MATLAB solution of the program in Table 3 yields a simi lar solution to that obtained with Polymath, except that the liquid level in the 3 rd tank (h 3 ) exhibits oscillatory behavior (see Figure 3). These oscillations indicate that there may be that the error tolerances should be examined. CHECKING THE ROLE OF THE TRUNCA TION ERROR TOLERANCE In the Polymath RKF45 algorithm, the integration step size is controlled by two parameters: the relative error tolerance ( RelTol ) and the minimum number of reporting points (RP). The default values of these parameters can be found under the Polymath icon of Setup preferences and numerical param eters, which is under Ordinary Differential Equations and RKF45: RelTol = 10 6 and RP = 100. The RKF45 program includes an adaptive step size control algorithm that monitors the estimate of the integration error and adjusts the step size of threshold value. The step size is increased only as long as at least RP full steps are carried out inside the integration inter val. Therefore, a solution of a lower accuracy can be obtained if the error tolerance is increased, and, at the same time, the number of reporting points is reduced. For instance, if the error tolerance is increased to RelErr = 10 4 and the number of reporting points is reduced to RP are encountered earlier. At some time t < 2400, the error message (negative square root) is issued by the program. With adjustment of RelErr = 10 10 while holding RP = 30, however, no error message is issued when the integration is carried out up to t f = 3600. This implies that a correct solution has been obtained. In the MATLAB ode4 5 function, the integration step size is adjusted so as to keep the estimated error below the i error tolerance, RelTol tolerance, AbsTol i i hT ol Ab sT ol max( ,) () Re l 9 The default values of the ode45 function for the absolute error tolerance and for the relative error toler ance are AbsTol = 10 6 and RelTol = 10 3 respectively. Since the values of h 1 h 2 and h 3 are of the order of magnitude of 1 (one), the AbsTol is inactive and the i values are of the order of 10 3 With the default er ror tolerances in MATLAB, the oscillatory solution (shown in Figure 3) is obtained. Reducing the rela Figure 3. Change of the liquid level in the third tank after introducing a 10% step change in its inlet ow rate ( ode45 Algorithm, RelTol = 10 3 ). PAGE 57 Vol. 42, No. 1, Winter 2008 57 T ABLE 3 MATLAB Program for the Train of Interactive Tanks Exercise No. Commands and % Comments 1 function TankTrain 2 clear, clc, format short g, format compact 3 tspan = [0 3600]; % Range for the independent variable 4 y0 = [1.807; 1.68; 1.162]; % Initial values for the dependent variables 5 disp( Variable values at the initial point ); 6 disp([ t = num2str(tspan(1))]); 7 disp( y dy/dt ); 8 disp([y0 ODEfun(tspan(1),y0)]); 9 [t,y]=ode45(@ODEfun,tspan,y0); 10 for i=1:size(y,2) 11 disp([ Solution for dependent variable y int2str(i)]); 12 disp([ t y int2str(i)]); 13 disp([t y(:,i)]); 14 plot(t,y(:,i)); 15 title([ Plot of dependent variable y int2str(i)]); 16 xlabel( Independent variable (t)); 17 ylabel([ Dependent variable y int2str(i)]); 18 pause 19 end 20 %21 function dYfuncvecdt = ODEfun(t,Yfuncvec); 22 h1 = Yfuncvec(1); 23 h2 = Yfuncvec(2); 24 h3 = Yfuncvec(3); 25 26 g = 9.81; 27 28 29 30 H1 = 4; %Height of Tank 1(m) 31 H2 = 2.5; %Height of Tank 2 (m) 32 H3 = 2.5; %Height of Tank 3 (m) 33 D1 = 4; %Upper diameter of Tank 1(m) 34 D2 = 1; %Lower diameter of Tank 2 (m) 35 1 (m^3/s) 36 2 (m^3/s) 37 38 39 40 %Liquid level in Tank 3 41 42 dYfuncvecdt = [dh1dt; dh2dt; dh3dt]; PAGE 58 Chemical Engineering Education 58 tive error tolerance to RelTol = 10 6 yields a more accurate, nonoscillatory solution. Since the latter solution is more ac curate by three orders of magnitude, the difference between the solutions can serve as a measure of the error in the less accurate solution. This error in h 3 is shown in Figure 4 (note that a few points of larger errors, in the region where the liquid level changes more rapidly, are not shown). Note that the maximal errors are of the order of 10 3 (while h 3 ~1.2, see Table 2). Thus, the integration algorithm indeed keeps the error below the desired relative error tolerance. The error in the slope dh 3 /dt is shown in Figure 5 (a few points of larger errors are not shown in this plot). The maximal absolute errors are of the order 10 4 but since the value of dh 3 /dt ~ 4 6 the relative error is of the order of 25. Thus, a reasonably low relative error of 10 3 in the variable (h 3 ) value yields absurd slope values, with up to 2500% error. It is interesting to investigate the reasons for these huge dif ferences in the relative error values. In Table 4, the values of some of the variables are presented at t = 3600 as calculated by the ode45 function with the two error tolerances. Observe that there are four correct decimal digits in the values of h 3 q 5 +q 3 and q 6 when the higher error tolerance of 10 3 is used. Introducing these numbers into Eq. (3) to calculate dh 3 /dt requires subtraction of two numbers that are very close. This operation causes the relative error to increase by seven orders of magnitude! We conclude that the erroneous oscillations in the ode45 solution and the negative squareroot error message of the RKF45 algorithms are caused by the lack of consideration of the slope value errors by the stepsize control logic of the two algorithms. CHECKING THE ROLE OF THE PROBLEM STIFFNESS The cause of the erroneous oscillations and error mes exercise, however, the stiffness of the problem must also be checked. at various times. One option is to use the symbolic manipu lation options of MATLAB. The program for derivation of the elements of the Jacobian matrix and calculation of the eigenvalues at t = 3600 s is shown in Table 5. First, the vec (lines 2 through 4). Note that the expressions for q 4 q 5 and q 6 (as functions of h 1 h 2 and h 3 ) were introduced into the functions assuming that h 1 PAGE 59 Vol. 42, No. 1, Winter 2008 59 CONCLUSIONS The exercise presented here demonstrates a systematic integration algorithm used to reach a correct solution of a particular problem. The procedure starts with checking whether a similar algorithm in a different package yields the same solution, and whether the default algorithm parameters are appropriate. Then, the stiffness of the problem is assessed, and a comparison of solutions obtained by stiff and nonstiff algorithms is carried out. In courses where students routinely use numerical software for solving ODEs, carrying out the two parts of the exercise can strate that results of ODE solvers of error tolerances and problem solution by different packages accurate solutions. The Polymath package, in particular, is wellsuited for such tests as it enables exporting the model equation to other packages. For students with appropriate numerical analysis background, this twopart exercise provides a good demonstration of the concepts of error estimation, step size control, error propagation and stiffness in systems of ODEs using a reallife example. This exercise demonstrates that a complete analysis and validation of the results are important in problem solving and lar software package. While students may use software packages for routine problem solving with a limited knowl edge of the numerical methods utilized by the package, there is also a need to include numerical methods within the curriculum. This knowledge is needed numerical problem solving. ACKNOWLEDGMENTS The authors wish to thank Auburn University chemical engineering stu this problem to our attention. REFERENCES 1. Cutlip, M.B., J.J. Hwalek, H.E. Nuttall, M. Shacham, J. Brule, J. Wid man, T. Han, B. Finlayson, E.M. Rosen, and R. Taylor, A Collection of Ten Numerical Problems in Chemical Engineering Solved by Various Mathematical Software Packages, Comput. Appl. Eng. Educ. 6 (3), 169 (1998) 2. Enright, W.H., The Design and Implementation of Usable ODE Software, Numerical Algorithms 31 125 (2002) 3. Brauner, N., M. Shacham, and M.B. Cutlip, Computational Results: How Reliable Are They? A Systematic Approach to Modal Validation, Chem. Eng. Ed. 30 (1), 20 (1996) 4. Ashurst, W.R., and R.J. Edwards, personal communications (2006) 5. Rice, R.G., and D.D. Do, Applied Mathematics and Modeling for Chemical Engineers John Wiley (1995) T ABLE 5 No. Commands 1 syms q1 q2 q3 g h1 h2 h3 H1 Ao D1 D2 g 2 f(1)= (q1 (Ao sqrt(2 g (h1 h2)))) / ((pi / 4) ((D2 + ((D1 D2) / H1) h1) ^ 2)); 3 f(2)= (q2 + (Ao sqrt(2 g (h1 h2)))(Ao sqrt(2 g (h2 h3)))) / (pi exp(2 h2) ); 4 f(3)= ((Ao sqrt(2 g (h2 h3))) + q3 ((Ao sqrt((2 g h3))))) / (h3 ^ 2); 5 for i=1:3 6 df(i,1)=diff(f(i),h1); 7 df(i,2)=diff(f(i),h2); 8 df(i,3)=diff(f(i),h3); 9 end 10 Ao = pi / 100; g = 9.81; q1 = 0.05; q2 = 0.05; q3=0.05 + 0.05 0.1; 11 H1 = 4; H2 = 2.5; H3 = 2.5; D1 = 4; D2 = 1; 12 h1=1.8656; h2= 1.7367; h3=1.2282; 13 df_numeric=subs(df); 14 format long g 15 Lambda=eig(df_numeric) Figure 5. Error in the liquid level slope: dh 3 /dt ( ode45 Algorithm, RelTol = 10 3 ). PAGE 60 Chemical Engineering Education 60INSERT SCANNED POST OFFICE FORM HERE 