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Mathl. Cornput. Modellin9 Vol. 17, No. 3, pp. 3-11, 1993Printed in Great Britain. AII rights reserved

0895-7177193 6.00 + 0.00Copyright@ 1993 Pergamon Press Ltd

THE VITAL ROLE OF MATHEMATICAL MODELLING INCHEMICAL ENGINEERING EDUCATIONS. S. E. H. ELNASHAIE~, F. M. ALHABDAN

Modehing, Sinudetion, and Optimization Group (MSOG)Chemicai Engineering Department, College of EngineeringKing Saud University, Biyadh, P.O. Box 800, Saudi Arabia

S. S. ELSHISHINIChemistry Department, King Saud University, Fbyadh, andChemical Engineering Department, Cairo University, Egypt

(Received Augurt 1992)Abstract-It is argued, in this paper, that the fast expansion of the domain of chemical engineeringnecessitates a new approach to chemicai engineering education. It is suggested, and discussed, thatthis new approach should be based upon system theory and mathematical modehing in order toachieve a higher level of generaiization and economy of information, which will ahow the training ofthe chemical engineers for the diverse chemical and related industries without violating their naturalincompressibility. The impact of the suggested approach on the training of chemical engineers forresearch work is also briefly discussed.

INTRODUCTIONChemical engineering has evolved into a very demanding discipline extending to a wide spectrumof inducstries and requiring a wide spectrum of knowledge in many diversified fields. The easydays of chemical engineers (or chemical technologists), being simply mechanical engineers withsome background in industrial chemistry have long gone. Norton, at MIT, started in 1888 whatmay be called the first chemical engineering curriculum. However, it was more of a chemicaltechnology than a chemical engineering curriculum as we have known it for the last three tofour decades. It consisted mostly of descriptive courses in industrial chemistry and chemicaltechnology. In about 1923 came the preliminary steps in the direction of classifying chemicalengineering equipment and processes, on a higher level of generality. This was represented by theconcept of unit operations, where, for example, distillation or extraction are taught as unifiedcourses not necessarily related to a specific industry.

This conceptual approach spread all over the world and started to give birth to more ofthese generalized subjects. Systematic, though simple, methods of design were developed (e.g.,McCabe-Thiele diagrams) and the concept of equilibrium stages became well established. Theemphasis at this stage, which extended to the 1960s, was on the overall behavior of the chemicalequipment without real involvement into the details of the micro-scale processes.

The second milestone in chemical engineering came in 1960 with the publication of lhnsportPhenomena by Bird, Stewart and Lightfoot [l]. Their new approach emphasized the micro-scaleprocesses and the analogy between mass, heat, and momentum transfer in different processes.

This stage witnessed an explosion of changes in chemical engineering both in research and edu-cation, giving birth to new well defined fields. A prime example is chemical engineering kinetics,that gradually evolved into the rich discipline under the title of Chemical Reaction Engineering,which emphasizes the design, analysis, and optimization of different types of chemical reactors.*Invited paper for the Department Modeking-A Personal Viewpoint.Author to whom 811correspondence should be addressed.

Typeset by A,+@-lj3

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4 S.S.E.H. ELNASHAIE ef al.The chemical engineering discipline started to bifurcate faster and faster year after year. Pro-cess Dynamics and Control became an important branch of chemical engineering. Plant design,economics and other specialized disciplines within chemical engineering started to grow at a rapidrate.

These developments were accompanied with a steady growth in productivity, sophistication,and the level of competition in the chemical industry. In order to meet these demands, a typicalchemical engineering curriculum had to be crowded with many subjects. In addition to basic sci-ence courses (mathematics, physics, chemistry, biology, thermodynamics, etc.) the student hadto be taught the new chemical engineering disciplines: mass and heat balance, mass and heattransfer, chemical reaction engineering, multistage operations, process dynamics and control,plant design and economics, etc., in addition to supplementary subjects from other engineeringdisciplines such as mechanical, civil, and electronic engineering. The training of chemical engi-neers left little room for what James Wei calls the Third Paradigm which should emphasize aGlobal outlook to the relation between Engineering and Society [2].

Despite this crowding-up of the chemical engineering curriculum, a practicing engineer or anexperienced professor will look at any curriculum and will find a lot to be desired. Students arelearning about chemical reaction engineering principles, which is a must, but they dont learnmuch about actual industrial reactors; they learn classical control theory, which is a must, butthey know very little about digital control, in a time where most plants are discarding analogcontrol and installing digital control systems, etc. The list is endless and gives a strong temptationto add more and more courses, but since the students are incompressible (as Levenspiel once putit [3]), it is very difficult to add more courses in response to the legitimate desires of experiencedindustrialists and professors.

The situation becomes even more acute with the expansion of chemical engineering into newfields, especially biotechnology and the electronic industries. In additon to that, chemical engi-neers are qualified and obliged to play a leading role in the environmental challenge that is facingthe human society in a dangerous and complicated manner.

In a nutshell, we do have a problem that cannot be solved by quantitative measures; it is soacute that it needs, actually, a change of concept.THE PRESENT STATUS OF CHEMICAL INDUSTRY

AND UNDERGRADUATE CHEMICAL ENGINEERING EDUCATIONThe chemical industry has revolutionized human life to such an extent that it invaded every do-main of modern life in our society. The industrial development and the innovative work of many

pioneering chemical engineering researchers, coupled with the advancement in computer technol-ogy, makes the training of chemical engineers for the future quite a challenging task. A chemicalengineer graduating today is expected in his career to deal with a wide range of problems thatneed a sound fundamental basis as well as an arsenal of practical knowledge. Of course, much ofthe practical knowledge is acquired in the industry after graduation; however, the undergraduatetraining is the critical factor that determines the degree of success of the trajectory of the chemicalengineers after graduation. The socio-economic, safety, and environmental challenges, togetherwith the fast expansion in the use of digital computers, make the task even more difficult. Is itplausible today to produce a chemical engineer who is not fluent in the application of computerpower to chemical engineering problems? Is it possible to be satisfied with a graduate who knowsenough about chemical engineering, but who is illiterate with regard to some of the basic com-puter software and hardware necessary for computer controlled experimentation or operation ofequipment? Considering the socio-economic implications of chemical industry, is it wise to pro-duce chemical engineers who are not sufficiently socially aware of the problems of their societyand the connections between these problems and their professeion? Of course, we should be awarethat the four or five years of chemical engineering education is not and should not be a substitutefor the whole life experience that he or she will acquire before the university education or aftergraduation, However, these 4-5 years are crucial for the production of our hoped-for ChemicalEngineer Citizen, a chemical engineer who can grasp the growing fascinating opportunities ofour modern society as well as coping with its difficulties and problems.

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5

A PRELUDE TO OUR MATHEMATICAL MODELLING ARGUMENTThe challenges facing the chemical engineering profession in its process of healthy and fruitful

intercourse with society, as well as with first and second natures, cannot be advanced to higherlevels with one single idea. Certainly, man has learned that life is much more complicated thanthe deterministic views of the 18th and lgth centuries. A more complex view of nature, society,and man-made processes is emerging, from simple monotony, to complex bifurcations which werethought to represent the highest degree of complexity before the revolutionary discovery of chaos,strange attractors, and fractals structures. Scientific development in the last few years shouldteach us that, most probably, nobody will say the last word in anything and that the most thatanybody can hope for, which is very honorable, is to achieve his part in a successful iteration of thecontinuous human rise to higher levels of civilization and intellect with all its dynamical beauty,and its enjoyable new challenges and difficulties. Our argument regarding the role of mathematicalmodelling in undergraduate chemical engineering education will, hopefully, represent one smallpart of a complex set of changes that are needed to take the chemical engineering science forward,through a tortuous route of developments to match the revolutionary changes the human societyis witnessing scientifically, technologically, socially, and politically. We argue, in this paper, thatthe extraordinary expansion in the domain of chemical engineering requires a new step in thedirection of generalized classification that will open new and faster expanding horizons for thisimportant discipline.

This can be achieved through a radical change of the undergraduate chemical engineeringsyllabus, in order to make it based on system theory and the mathematical modelling approach,which is a very effective step forward in organizing knowledge and achieving a much higherlevel of economy of information. Before getting deeper into the subject, it may be worthwhileto present a simple and concise description of system theory and mathematical modelling inchemical engineering.

SYSTEMS AND MATHEMATICAL MODELSIn this part, we discuss very briefly the basic principles of systems and mathematical modelling

theories with special emphasis on chemical engineering problems. System theory is the moregeneral, more abstract part, whereas mathematical modelling is more applied and less abstract.

A system is a whole consisting of elements or subsystems. The concept of systems-subsystemsand elements is relative and depends upon the level of analysis. The system has a boundarythat distinguishes it from the environment. The system may exchange matter and/or energywith the environment depending upon the type of system from a thermodynamical point ofview. A system (or subsystem) is described by its elements (or subsystems), the interactionbetween the elements, and its relation with the environment. The elements of the system canbe material elements distributed topologically within the boundaries of the system and givingthe configuration of the system, or they can be processes taking place within the boundariesof the system and defining its function. They can also be both, together with their complexinteraction. An important property of the system wholeness is related to the principle of theirreducibility of the complex to the simple, or of the whole to its elements; the whole system willpossess properties and qualities not found in its constituent elements. This does not mean thatcertain information about the behavior of the system cannot be deduced from the properties ofits elements, but it rather adds something to them.

Systems can be classified on different bases. The most fundamental classification is that basedon thermodynamic principles and, on this basis, systems can be classified into the followingclasses.

- ISOLATED SYSTEMS. These are systems that exchange neither energy nor matter with theenvironment. The simplest chemical engineering example is the adiabatic batch reactor.These systems tend toward their thermodynamic equilibrium which is characterized bymaximum entropy (highest degree of disorder).

- CLOSED SYSTEMS. They are systems that exchange energy with the environment throughtheir boundaries, but do not exchange matter. The simplest example is a non-adiabatic

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6 S.S.E.H. ELNA~HAIEt d.batch reactor. These systems also tend toward a thermodynamic equilibrium characterizedby maximum entropy (highest degree of disorder).

- OPEN SYSTEMS. They are systems that exchange both energy and matter with the envi-ronment through their boundaries. The most common chemical engineering example is thecontinuous stirred tank reactor. These systems do not tend toward their thermodynamicequilibrium, but rather toward stationary non-equilibrium state and are characterizedby miniumum entropy production.

It is clear from the above classification that batch processes are usually of the isolated or closedtype while the continuous processes are usually of the open type.

AS can be seen from the above definitions, the system theory is very abstract, and in general,it treats any system regardless of whether a mathematical model for this system can be builtor not; mathematical modelling, on the other hand, is less abstract and more applied than thesystem concept.

For continuous processes, a classification of systems from a mathematical point of view is veryuseful for both model formulation, and algorithms for model solution. According to this basis,systems can be classified as follows.

- LUMPED SYSTEMS. These are systems where the state variables describing the systemare lumped in space (invariant in all space dimensions). The simplest example is theperfectly mixed continuous stirred tank reactor. These systems are described at steadystate by algebraic equations, while the unsteady state is described by initial value ordinarydifferential equations where time is the independent variable.

- DISTRIBUTED SYSTEMS. These are systems where the state variables are varying in onedirection or more of the space coordinates. The simplest example is the plug flow reac-tor. These systems are described at steady state, either by ordinary differential equations(where the variation of the state variables is only in one direction of the space coordi-nates, i.e., one-dimensional systems, and the independent variable is this space direction),or partial differential equations independent (where the variation of the state variables isin more that one direction of space coordinates, i.e., two or three-dimensional systems,and the independent variables are these space directions). The ordinary differential equa-tions describing the steady state of the one-dimensional distributed systems can be eitherinitial value differential equations (e.g., plug flow systems) or two-point boundary valuedifferential equations (e.g., systems with superimposed axial dispersion). The equationsdescribing the dynamic behavior of distributed systems are invariably partial differentialequations.

Another classification of systems, which is important for deciding the algorithm for model solution, is that of linear and non-linear systems. The equations of linear systems can usually besolved analytically, while the equations of non-linear systems are almost always solved numeri-cally. In this respect, it is important to recognize the significant fact that physical systems arealmost always non-linear, and that linear systems are either an approximation that should bejustified, or are intentionally linearized in the neighborhood of a certain state of the system andare strictly valid only in this neighborhood.

A third classification, which is relevant and important in chemical engineering, is the classifcation based on the number of phases involved within the boundaries of the system. Accordingto this classification, systems are divided as follows.

- HOMOGENEOUS SYSTEMS. These are systems where only one phase is involved in theprocesses taking place within the boundaries of the system. In reaction systems, thebehavior of these systems is basically governed by the kinetics of the reactions takingplace, without the interference of any diffusion processes.

- HETEROGENEOUS SYSTEMS. These are systems where more than one phase is involvedin the processes taking place. In reaction systems, the behavior of these systems is notgoverned only by the kinetics of the reactions taking place, but also by the complexinteraction between the kinetics and diffusion processes. When the system does not involve

chemical reaction, then the system behavior is not governed by processes taking ein one phase, but rather by the totality of the processes taking place in the different

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phases and the interaction between them. The modelling and analysis of these systems isobviously much more complicated than for homogeneous systems.

MATHEMATICAL MODEL BUILDING-GENERAL CONCEPTSBuilding a mathematical model for a chemical engineering system depends to a large extent onthe knowledge of the physical and chemical laws governing the processes taking place within the

boundaries of the system. This includes the different rates of mass, heat, and momentum transfer,rates of reactions and rates of adsorption-desorption, etc. It also includes the thermodynamiclimitations that decide the feasibility of the process to start with, as well as heat production andabsorption. Mass and heat transfer rates are both dependent on the proper description of thefluid flow phenomena in the system. The ideal case is when all these processes are determinedseparately and then combined into the systems model in a rigorous manner. However, very oftenthis is quite difficult in experimental measurement; therefore, special experiments need to bedevised, couples with the necessary mathematical modelling, in order to decouple the differentprocesses implicit in the measurements.

In the last two decades, there has been a considerable advancement in the development ofmathematical models of different degrees of sophistication for chemical engineering processes.These models are taking their part in directing design procedures as well as in directing scientificresearch. It is important in this respect to recognize the fact that most mathematical modelsare not completely based on rigorous mathematical formulation of the physical and chemicalprocesses taking place in the system. Every mathematical model contains a certain degree ofempiricism. The degree of empiricism limits the generality of the model and, as our knowledge ofthe fundamentals of the processes taking place increases, the degree of empiricism decreases andthe generality of the model increases. The existence of models at any stage, with their appropriatelevel of empiricism, help greatly in the advancement of the knowledge of the fundamentals,and, therefore, helps to decrease the degree of empiricism and increase the level of rigor in themathematical models. Models will always contain certain simplifying assumptions which arebelieved by the model-builder not to affect the predictive nature of the model in any manner thatsabotages the purpose of it.

Different models with different degrees of sophistication can be built. Models which are toosimplified will not be reliable and will not serve the purpose, while models which are too so-phisticated will present an unnecessary and sometimes expensive overburden. In undergraduatechemical engineering education, the concept of model sophistication and its relation to the pur-pose of the model building should be emphasized.

AN OUTLINE OF THE PROCEDURE FOR MODEL BUILDINGThe procedure can be summarized in the following steps.

1. The identification of the system configuration, its environment, and the modes of interac-tion between the system and its environment.

2. The introduciton of the necessary justifiable simplifying assumptions.3. The identification of the relevant state variables that describe the system.4. The identification of the processes taking place within the boundaries of the system.5. The determination of the quantitative laws governing the rates of the processes in terms

of the state variables. These quantitative laws can be obtained from information given inthe literature and/or through an experimental research program coupled with the mathe-matical modelling program.

6. The identification of the input variables acting on the system.7. The formulation of the model equations based on the principles of mass, energy, and

momentum balances appropriate to the type of system.8. The development of the necessary algorithms for the solution of the model equations.9. The checking of the model against experimental results (laboratory, pilot plant, or com-

mercial units) to ensure its reliability, and carrying out a reevaluation of the simplifyingassumptions. This re-evaluation process may result in imposing new simplifying assump-tions or relaxing some of them.

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8 S.S.E.H. ELNASHAIE et alI t is clear that these steps are interactive in nature and the rsults of each step should lead

to a reconsideration of the results of all previous ones. In many instances, Steps 2 and 3 areinterchanged in the sequence, depending on the nature of the system and the degree of knowledgeregarding the processes taking place.

MODELLI NG AND SIMUL ATIONAris [4], in his 1990 Dankwarts Memorial Lecture entitled Manners Makyth Modellers, dis-

tinguishes between modelling and simulation in a special manner which follows the reasoning ofSmith [5] in ecological models:

I t is an essential quality in a model that it should be capable of having a life of itsown. I t may not, in practice, need to be sundered from its physical matrix. I t maybe a poor thing, an il l-favoured thing when it is by itself. But it must be capable ofhaving this independence. Thus Lil jenroth (1918) in his seminal paper on multiplicity ofthe steady states, can hardly be said to have a mathematical model, unless a graphicalrepresentation of the case is a model. He works out the slope of the heat removal linefrom the ratio of numerical values of a heat of reaction and a heat capacity. Certainlyhe is dealing with a typical case, and his conclusions are meant to have applicationbeyond this particularity, but the mechanism for doing this is not there. To say this isnot to detract from Lil jenroths paper, which is a landmark of the chemical engineeringliterature, it is just to notice a matter of style and the point at which a mathematicalmodel is born. For in the next papers on the question of multiple steady states, those ofWagner (1945), Denbigh (1944,1947), Denbigh et al (1948) and Van Heerden (1953),we do not find more general structures. How powerful the life that is instinct in a truemathematical model can be seen from the Fouriers theory of heat conduction where themathematical equations are fecund of all manner of purely mathematical developments.At t he other end of t he scal e a model can cease t o be a model by becoming t oo lar geand too detai led a simulat ion of a situat ion w hose natural li ne of development is tothe part icular rat her than t he general N ceases t o have a li fe of i ts own by becomingdependent for it s vi tal it y on it s physical reali zat ion (The emphasis is ours.) MaynardSmith (1974) was, I believe, the first to draw the distinction in ecological models betweenthose that aimed at predicting the population level with greater and greater accuracy(simulation) and those that seek to disentangle the factors that affect population growthin a more general way (model). The distinction is not a hard and fast one, but it isuseful to discern these alternatives [4].

The basis of the classification given by Aris is very interesting, true and useful. I t is typicalof the Minnesota group founded by Amundson some fifty years ago. This important researchgroup in the history of chemical engineering has almost never verified the models (or the vari-ety of interesting new phenomena resulting from them) against experiments or industrial units.Experimental veri fications of the new and interesting steady state and dynamic phenomena dis-covered by the Minnesota group were carried out at other universities, mostly by graduates fromMinnesota. The most interesting outcome is the fact that not a single phenomenon, which wasdiscovered theoretically using mathematical models by the Minnesota group, was not experi-mentally confirmed later on. This demonstrates the great power of the mathematical modellingdiscipline as expressed by Aris [4], where the model is stripped of many of its details in orderto investigate the most fundamental characteristics of the system. I t also demonstrates the deepinsight into physical systems that can be achieved using mathematics, as Ian Stewart puts it:Perhaps mathematics is effective in organizing physical existence because it is inspired by phys-ical existence . . . . The pragmatic reality is that mathematics is the most effective method thatwe know for understanding what we see around us [6].In this paper, a different, more pragmatic definition for mathematical modelling and simulationthan that of A [4] wil l be adopted. The definiton we adopt is that mathematical modellingwill involve the process of building up the model itself, while simulation involves simulating theexperimental unit using the developed model. Thus, simulation in this sense is closely linked

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to the verification of the model against experimental and industrial units. However, it will alsoinclude the use of the verified models to simulate a certain practical situation specifc to a unit,a part of a production line or an entire production line.

However, since universities are responsible also for the preparation of the students for researchcareers, the syllabus should also include this element. In this case, the definition of Aris [4]must come strongly into play. Since much of the chemical engineering research work today isinterdisciplinary in nature, especially in the relatively new fields such as biotechnology and mi-corelectronics, it should be clear that mathematical modelling is the most suitable and efficientlanguage of communication between the different disciplines involved. Therefore, chemical en-gineering education based on system theory and mathematical modelling seems to be the bestapproach to prepare the chemical engineers for this interdisciplinary research. The statement ofEl Naschie: Today it must be difficult to find a scientist of stature who denies the influence of thebroad sweep of development in science, philosophy or even art on his specialized research [14],emphasizes our point in an elegant fashion. His observation that H. Poincare, the undisputedfirst discoverer of Chaos was trained first in engineering, following the Napoleonic traditions . . . helps to emphasize the need for a broad education of the engineering researchers of tomorrow,which is hard to achieve through the present structure of engineering syllabi.A very general definiton of models is given by Stephen Hawking [S], who relates models totheories of the universe:

A theory is just a model of the universe, or a restricted part of it, and a set of rulesthat relate quantities in the model to observations that we make. It exists only in ourminds and does not have any other reality (whatever that might mean). A theory is agood theory if it satisfies two requirements: it must accurately describe a large class ofobservations on the basis of the model that contains only a few arbitrary elements, andit must make definite predictions about the results of future observations.

One of the major findings of the Minnesota group, using mathematical modelling, is the diicov-ery of a wide variety of static and dynamic bifurcation phenomena in chemical reactors. Althoughtheoretical studies on the bifurcation behavior have advanced considerably during the last threedecades, the industrial appreciation of these phenomena remains very limited. It is of great im-portance that practically oriented chemical engineers dealing with the mathematical modellingof industrial units, become aware of them. These phenomena are not only of theoretical andacademic importance, but they have very important practical implications (e.g., for industrialFluid Catalytic Cracking Units [g-12]).

THEAMUNDSONREPORTANDTHENEEDFOR MODERN CHEMICAL ENGINEERING EDUCATION

The report of the committee on Chemical Engineering Frontiers: Research Needs and Op-portunities [13], generally known as the Amundson report, outlines beautifully the picture ofchemical engineering in the next decades. The report is quite optimistic about the future of theprofession, and we strongly share this optimism. Chemical engineering played an important partin human development in the last few decades, and it is expected to play an even larger role inthe future.

Although the Amundson report emphasizes the challenges facing the American chemical andrelated industries, the report should not be looked upon from this narrow point of view. In fact,the report is far more reaching than that and is to a great extent relevant to the worldwidechemical engineering profession. The report, without ignoring classical chemical engineeringproblems, stresses a number of relatively new chemical engineering fields, namely:

1. Biotechnology and Biomedicine2. Electronic, Photonic, and Recording materials and devices3. Polymers, Ceramics, and Composites4. Processing of energy and natural resources5. Environmental protection, Process safety, and Hazardous waste management

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10 S.S.E.H. ELNASHAIE et al.6. Computer-assisted process and Control Engineering7. Surfaces, Interfaces, and Microstructures.

Emphasizing these fields without being able to ignore classical chemical engineering problemshelps only to emphasize the view that chemical engineering is entering a new era. In addition, thechemical engineering community cannot ignore its fundamental scientific responsibilities towardsthe revolution in scientific knowledge created by the discovery of chaos, strange attractors, andfractal structures, especially that many manifestations of these phenomena are evident in typicalchemical engineering systems.

These considerations lead to the inevitable conclusion that a higher level of organization ofthinking and economy of knowledge is needed in undergraduate chemical engineering education.

SYSTEM THEORY AND MATHEMATICAL MODELLI NGAS TOOLS FOR MORE EFF ICIENT

UNDERGRADUATE CHEMI CAL ENGINEERING EDUCATIONIn order to achieve this higher level of organization and economy of knowledge, it will be

extremely useful that the students are introduced early in their undergraduate education tothe basic principles of system theory. The best and most general classification of systems isbased upon thermodynamic principles. Therefore, it is possible that the students, after passingtheir basic science courses, be exposed to a course in thermodynamics that emphasizes the basicconcepts of thermodynamics. The basic principles of system theory can either be integrated intothis course or may be taught in a separate course. This course should emphasize not only thebasic principles of system theory, but also its relevance to chemical engineering systems. In fact,such a course can be used as an elegant and efficient tool to introduce the student to chemicalengineering systems. The student should learn how to classify chemical engineering systems intotheir main categories, isolated, closed, and open systems. S/he should learn how to divide thesystem into its subsystems or elements depending on the level of analysis. S/he should learn notonly that distillation is a unit operation regardless of the specific kind of distillation, as the unitoperation paradigm teaches us, but to extend her/his mind further and learn that continuous,heterogeneous, multistage processes are systems that are open, formed of more than one phaseand more than one stage, that there are certain processes taking place within their boundariesthat need to be expressed in terms of state variables, and that input variables should be specifiedand parameters should be identified (regardless of whether the process is disti llation, extraction,drying, or multistage catalytic reactors).

Of course, at this early stage, the student wil l not be able yet to develop specific models forspecific processes because s/he would not have studied yet the laws governing the rates of theseprocesses and their form of dependence upon the state variables. S/he even may not be able, atthis stage, to identify completely the state variables of the system. However, this early trainingin system theory wil l orient the student mind and her/his further education in the framework ofthe system approach.

The student can then be ready to take her/his other chemical engineering courses in a newlight; those courses should also be changed to emphasize the system approach, where all therate processes are treated in a unified fashion, emphasizing the laws governing the rates of theseprocesses and its dependence upon state variables, rather than dividing it into mass transfer,heat transfer, momentum transfer, and rates of reactions. The transport phenomena paradigmcan be easily extrended in this direction. This wil l also allow higher emphasis on the interactionbetween these processes and the effects resulting from these interactions, e.g., the interactionbetween mass transfer and chemical reaction and its implication for the behavior of the system.

It is natural that, at this stage, the student will be introduced to the formulation of somesimple mathematical models (design equations) for certain systems.

When the student is familiar with the laws governing the rates of different processes in termsof the state variables, then s/he wil l be ready for an intensive applied course on mathematicalmodelling of chemical engineering systems. This course should cover the basic principles of math-ematical modelling theory, the main classi6cations of mathematical models, the procedures forbuilding mathematical models, application to the development of mathematical models for a large

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number of chemical engineering processes in the petrochemical, petroleum refining, biochemical,electronic industries, as well as some mathematical models for biological systems.

So far, we have emphasized the practical usefulness of the approach. In this last sentence, weshould also emphasize that the approach is beautiful and elegant: He gets full marks who mixesthe useful with the beautiful [14].

CONCLUSIONIn this preliminary article, we have argued that chemical engineering is expanding very quickly

and that a new approach based on system theory and mathematical modelling is needed for ahigher level of organization of thinking and economy of information. The concept has been intro-duced and discussed and the influence of this approach on the structure of chemical engineeringteaching is discussed in a limited fashion, only to show how it can be integrated into the syllabusand how it does affect the approach to teaching rate processes. No attempt is presented here togive a full description of an alternative syllabus; this is deferred to a later stage, after the mainideas are matured through wide discussion among the chemical engineering community. Theauthors will be extremely interested on comments, criticisms, and opposing views on the ideaspresented and discussed in this article.

REFERENCES1. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, (1960).2. 3. Wei, Educating chemical engineers for the future, In Chemical Engineeri ng in a Changing Environment,

Engineering Foundation Conference, J anuary 17-22, Santa Barbara, California, (Edited by S.L. Sandlerand B.A. Finlayson), pp. l-12, (1988).

3. 0. Levenspiel, Private discussion during AIChE annual meeting, San Francisco, (November 1989).4. R. A&, Manners makyth modellers, Chem. Engng. Sci. 46, 1535-1455 (1991).5. J .M. Smith, Models in Ecology, Cambridge University Press, Cambridge, (1974).6. I. Stewart, Does God Play Dice: The New Mathematics of Chaos, Penguin Books, London, (1989).7. MS. El Naachie, Editorial, Chaos, Solit ona and Fr actala 1, l-2 (1991).8. S.W. Hawking, A B f Hi story of Time, From the Big Bang to Black Hol ea, Bantam Press, London, (1989).9. L. Iscol, The dynamics and stability of a fluid catalytic cracker, Joint Automatic Control Conference,

Georgia, pp. 602-807, (1970).10. S S E H Elnashaie and I.M. El-Hennawi, Multiplicity of the steady states in fluidized bed reactors, IV. Fluid

catalytic cracking, Chem. Engng. Sci. 34, 1113-1121 (1979).11. W.M. Edwards and H.N. Kim, Multiple steady states in FCC unit operations, Chem. Engng. Sci. 43,

1825-1830 (1988).12. S.S. E lshishini and S.S.E.H. Elnashaie, Digital simulation of industrial fluid catalytic cracking units, Bifur-

cation and its implications, Chem. Engng. Sci. 45, 553-559 (1990).13. Amundson, Frontiers in Chemical Engineering, Research needs and opportunities, National Academy Press,

Washington, D.C., (1988).14. R. Aris, A quotation from Horace, Comments made during a discussion on Chemical Engineering Education,

1 Ok International Symposium on Chemical Reaction Engineering (ISCREIO), Baael, Switzerland, (1988).