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OnStream Computers, An Example and Some Generalities, by C.G. Laspe ( PDF )
Optimization Theory in the Chemical Engineering Curriculum, by Douglass J. Wilde ( PDF ) Fundmaental Problems in Heterogeneous Catalysis, by Max Peters ( PDF ) Planning Experiments for Engineering Kinetics Data, by H.M. Hulburt ( PDF ) 
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CHEMICAL ENGINEERING EDUCATION THE AMEPICAN SOCIETYY nnr I~ t 1 IIJ September 1963 CHEMICAL FNSINEERING EDUCATTION Septem!,r 1963 Chemical Engineering Division ,aerican Society for Engineering Education CONTENTS jnStream Computers, An Example and Some Generalities, by C. G. Laspe            1 Optimization Theory in the Chemical Engineering Curriculum, by Douglass J. Wilde           9 Fundamental Problems in Heterogeneous Catalysis, by Max Peters            14 Planning Experiments for Engineering Kinetics Data, by H. M. Hulburt            23 Joseph J. MN George Burne J. B. West Chemical Engineering Division American Society for Engineering Education Officers 196364 martin (Michigan) Chairm it (Iowa State) Vice C (Oklahoma State) Secret Ian chairman aryTreasurer CHEMICAL ENGINEERING EDUCATION R Journal of the Chemical Engineering Division, American Society for Engineering Education. Published Quarterly, in March, June, September and December, by Albert H. Cooper, Editor. Publication Office; University of Connecticut P.O. Box 445, Storrs, Connecticut Subscription Price, $2.00 per year. ONSTREAM COMPUTERS  AN EXAMPLE A IND SOME EIERLITIES C. G. Laspe ThompsonRamoWooldridge Products Co. Beverly Hills, Calif. There are many excellent articles (1,2,3,4,5,6,8) giving the general reasons for and the philosophy behind the use of digital computers in the controlling of production processes. Some of the pioneering efforts by Stout and Laspe (1,2,3), as early as 1957, formed the basis for many of the digital control systems in use today. Subsequent papers by Roberts (6), Stout (6), Brandon (5), Madigan (12,13,14), Freilich (19), and Leape (15) presenting the results of actual case studies have bolstered the technical literature on computer control systems. It is not the intent of the present paper to dwell heavily upon the objectives of computer systems, nor upon their design, for these aspects of the problem have received adequate atten tion in the aforementioned papers. It is the purpose of this paper to present the broad aspects of on stream computer control and especially as these aspects affect the curricu la of chemical engineering. In attempting to accomplish this objective, I would like to present a few generalizations, followed by a survey of the ex tent to which computers are used in online control. Next, as an example, the computer control of an ammonia plant will be discussed to illustrate the various branches of engineering and science required in its design. Finally, the most important point, as touching upon this present session, will be a discussion of those fundamental things which a student in an engineering school should learn to understand the use of online control computers. In order to orient our thinking along the lines of onstream control computers, a few generalities should be considered. What are the ingredients that go into making up a computer controlled process? Once these individual ingredients are recognized, then we are in a better position to determine the engineering talents required for the design, construction and operation of such a system. Let us briefly review a few fundamentals. All manufacturing processes are designed and intended to be operated to produce a profit. This profit results from the creation, by a combination of physical and chemical trans formations, of a product or products whose value exceeds the cost of the raw materials and their processing. A major goal of management in plant oper ation is the maximization of this profit. This statement may be considered as the process objective. Therefore, the purpose of computer control is to provide the latest and the most efficient means of reaching this process ob jective. It is realized that, since physical and chemical transformation are involved, many variables influence the realization of the process objective. When one or more of these variables are beyond the control of the operator, such variables are classed as disturbance variables. Compensation for these disturbances is the prime justification for any control scheme. For without disturbances, control would not be required. When there is but a single disturbance variable involved in the opera tion of a plant, then it is possible to derive a unique solution to the con trol problem. In other words, for any given value of the disturbance varia ble, there is only one set of control variables which will meet the desired objective. In this instance, simple relationships may be found which will relate the manipulated variables to the disturbance variables then in effect. On the other hand, when two or more variables are beyond the control of the operator an interesting situation arises. In this case there are generally two or more feasible solutions. Of these feasible solutions, one will prob ably yield the greatest economic return and, therefore, is the desired opti mum solution. It is in these areas where computer control may be justified. Recently, an excellent article by Elliott and Longmire (21) gives the dollar incentives for computer control. The results of their studies on six differ ent production processes is presented. The March issue of Control Engineering presented a survey of onstream control computers. At that time the score card showed 35 closed loop com puting control installations, either online or scheduled to be online by early 1962. Of these 35 installations, nearly half were to be used in either chemical or petroleum processes. In September of 1961 Freilich (19) presented another survey of process control computers in use. Freilich shows a total of 63 process control computers, of which 20 are used in the chemical and petro leum industries. The latest figures available from the May 1962 issue of Control Engineering (20) show a total of 159 control computer sales, of whichh 43 are ins allied In the chemical and petroleum fields. 1 2 CHEMICAL ENGINEERING EDUCATION September 1963 Although the preceding statistics are both encouraging and interesting, they do not tell the complete story. Table I summarizes the known instal lations of digital computers in process control or those units known to be on order. In the petroleum field, several installations have been reported on catalytic cracking plants and on crude distillation units. Single installa tions have been reported for catalytic polymerization, alkylation, and ther mal cracking. In the chemical field ethylene and ammonia appear to be good candidates for computer control by reporting several installations on each. In addition other computer controlled chemical processes include vinyl chlor ide, styrene, acrylonitrile, acetaldehyde, ethylene oxide, and the exotic "alfol" plant of the Continental Oil Company. As can be seen from a study of this list, the gamut of applicable processes is limited only by economic necessity and the imagination of the system designer. From the above list we have chosen as the working example to be dis cussed here today, the computer controlled ammonia plant of Monsanto Chemical Company at Luling, Louisiana. A fairly complete description of this particu lar installation has already been given by Eisenhardt and Williams (17) in the November 1960 issue of Control Engineering. For purposes of description, the ammonia process can be conveniently divided into three separate operations. The first of these is concerned chiefly with the preparation of raw synthesis gas. The second section is concerned with purification and compression, and the third and final section is the synthesis unit itself. In the gas preparation area three chemical re action stages are involved. The primary reformer, the secondary reformer, and the CO converter. The feed to the primary reformer consists of natural gas and steam which in the presence of a catalyst reacts to produce hydrogen, carbon monoxide, and some carbon dioxide. External heat is applied to this unit from a reformer furnace burning natural gas. Essentially 90% of the in coming gas is converted. Steam reforming is the principal reaction involved, although the water gas or CO conversion reaction accounts for some of the hy drogen produced. The variables over which control can be exercised are the flow rates of the natural gas, the fuel gas and the process steam. The secondary reformer serves two specific functions. Firstly, to pro vide additional reaction volume for continuation of the reforming and CO con version reactions initiated in the primary, secondarily, to serve as the in jection point at which nitrogen is introduced into the system. Atmospheric air is used as the source of nitrogen. In the secondary reformer the oxygen from the air which has been introduced reacts with some of the methane and hydrogen in the feed to form water plus CO and C02. The only independent variable over which control can be exercised is the flow of process air. Note that at this particular point in the process, essentially all of the natural gas has been converted into product gases. The residual methane con tent is in the order of 0.3 of one percent. There is also an appreciable a mount of carbon monoxide. The effluent from the secondary reformer flows directly to the CO converter. The sole purpose of the CO converter is to produce additional hydrogen from the incoming carbon monoxide by means of the water gas reaction. rd ditional water is injected at this point in the form of low pressure steam. Because of fundamental thermodynamic and kinetic considerations, the carbon monoxide is not completely consumed. The exit concentration is in the order of three percent. At this point in the process the hydrogen to nitrogen ra tio is fixed and remains constant throughout the remainder of the operations. At this particular point in the process, carbon dioxide represents approxi mately 1i% of the entire raw synthesis gas. This along with the carbon mon oxide must be removed before the synthesis gas can be charged to the final ammonia synthesis reaction stage. Carbon dioxide is removed from this raw gas stream by passing it through a standard Girbotol unit. Monoethanol amine is used as the absor bent. A compression plant consists of several parallel reciprocating com pressors. Each compressor is equipped with five stages of compression. The gas pressure is boosted from approximately 20 pounds per square inch at the inlet to 5,500 pounds per square inch gage at the outlet. The entire gas stream, however, does not pass through all five stages. At the outlet of the fourth stage, the process gas is diverted to the high pressure purifica tion unit. Themain function of this unit is to remove carbon monoxide, plus any residual carbon dioxide. Copper format is used as the absorbent for CO. The residual C02 is removed by a final caustic wash. After removal of impurities, the purified synthesis gas is then directed to the last stage of compression from whence the gas flows to the synthesis unit. n analysis n4 the exit gas shows an essentially pure hydrogen, nitrogen mixture in the io of approximately 3 to I. CHEMICAL ENGINEERING EDUCATION Table 1 SURVEY OF PROCESS COMPUTER APPLICATIONS CompanyLocation Computer 1. Allied Chemicnl Corp. RW300 South Point, Ohio 2. American Oil Company IBM1710 Whiting, Indiana 3. B.A.S.F. RW300 Ludwigshafen, Germany 4. B.F. Goodrich Chemical RW300 Calvert City, Kentucky 5. Celanese Corporation Bay City, Texas H290 Bishop, Texas RW300 6. Continental Oil Company RW300 Lake Charles, La. 7. Dow Chemical Company GE312 Midland, Michigan 8. Dupont Beaumont,Texas (2) ISI609 Florence, S.C. ISI609 Circleville, Ohio ISI609 Gibbstown, N.J. IBM1710 9. DXSunray RW300 Tulsa, Oklahoma 10. Gulf Oil Company RW300 Philadelphia, Pa. 11. Imperial Chemical Ind. Ferranti England 12. Monsanto Chemical Co. Luling, Louisiana RW300 Chocolate Bayou, Texas(4) H290 13. OwensCorning Fiberglass ISI609 Aiken, S. Carolina 14. Petroleum Chemicals,Inc. RW330 Lake Charles, La. 15. Phillips Chemical Co. Borger, Texas Recomp II Bartlesville, Oklahoma TRW330 16. Shell Development Company PB250 Emeryville, Calif. 17. Sun Oil Company IBM1710 Marcus Hook, Pa. 18. Standard Oil Co.(N.J.) Linden, N.J. LGP30 Baton Rouge, La. LGP30 19. Standard Oil Co.(Calif.) El Segundo, Calif. IBM1710 Richmond, Calif. Recomp 20. Tennessee Eastman GE312 Kingsport, Tennessee 21. Texaco RW300 Port Arthur, Texas 22. Tidewater Oil Company ISI609 Delaware City, Delaware 23. Union Carbide Corp. Charleston, W. Va. RW300 Seadrift, Texas RW300 Seadrift, Texas Daystrom 24. Universal Oil Products Daystrom Des Plaines, Illinois Delivery 1961 1961 1961 1959 1962 1962 1961 1960 1960 1961 1961 1961 1961 1959 1962 1960 1963 1959 1962 1961 1959 1961 1959 1960 1961 1958 Process Ammonia Crude distillation Chemical process Vinyl chloride and acrylonitrile 2Ethly hexanol Vapor phase oxidation Alfol process" Styrene Chemical process Chemical process Chemical Process Acrylonitrile pilot plant Crude distillation Catalytic cracking Soda Ash Ammonia Chemical process Logger Ethylene Portable Logger Portable logger Logger Catalytic cracking logger Pilot plant logger Catalytic cracking logger Catalytic cracking Product run calculations Chemical process Catalytic polymerization Naphthalene Pilot plant logger Chemical process Ethylene Pilot plant logger September 1963 CHEMICAL ENGINEERING EDUCATION Table II CRITERIA FOR JUSTIFYING MONSANTO'S COMPUTER CONTROL SYSTEM 1. Maintain maximum gas flow in spite of changing weather and process conditions. 2. Maintain an optimum hydrogentoniyrogen ratio. 3. Maintain an optimum methane concentration at the shift converter exit unless in conflict with 1 or 2. 4. Maintain maximum shift efficiency if not in conflict with objectives 1, 2, or 3. 5. Maintain objectives 2, 3, and 4 under reduced flow conditions. 6. Reduce raw materials flow immediately and safely to compensate for any loss of compression. 7. Log out all important process variables. 8. Provide the plant operator with messages in case of abnormal process or instrument conditions. 9. Provide failsafe features such that instrument or computer malfunctions are detected, alarmed, and prevented from affecting the process. .0. Control the fuel and air to the reformer furnace. .1. Maintain a specified steamtodry gas ratio at the exits of the secondary reformer and the CO converter. The synthesis plant feed is combined with a recycle stream to form the feed to the synthesis reactors. Because of the low conversion per pass, (sp proximately 12%) a high recycle ratio is required. Ammonia is recovered in the reactor effluent gases by condensation. In order to prevent excessive buildup of inerts in the system, purging or venting is required. Now let us turn our attention to some of the factors involved in making this particular plant a good candidate for computer control. As has been pointed out by Eisenhardt and Williams (17). "While there may be a tendency to overdesign some plant equipment an enginecompressor system is usually conservatively sized be cause it represents a major fraction of the capital cost of an ammonia plant. The engine compressor system is thus likely to be one ofthe first units to bottleneck the plant as production increases. At Luling the highest possible production rate and therefore the maximum economic return is obtained by operating the compressor system at maximum possible capacity." Due to the very definite influence of ambient conditions, particularly temperature, upon internal combustion engine compressor efficiencies, the allowable horsepower to be expended by the compressor is not a fixed or arbitary constant. In fact the compressor capacity varies not only as the ambient temperature changes, but also as the mechanical condition of the compressors themselves are changed. In the case of a production limited plant, such as the one we are now discussing, it can readily be seen that the maximum plant throughput is never fixed, but varies in accordance with compression capacities. The purpose of the computer control system now be comes apparent. Its main function is to keep the plant running at maximum capacity by determining the maximum as limited by the capacity in the com pression section. Once knowing the maximum allowable flow of purified syn thesis gas, the computer can then go about its business of setting the many flow controllers in the reforming section. It is not enough that the com pressors be fully loaded at all times. The synthesis gas must also have the proper quality as measured by the hydrogen and nitrogen ratio. Since ech of the processes in the reforming area, namely, the primary reformer, the secondary reformer, and the shift converter involve chemical conversions, complicated kinetic and thermodynamic equations must be solved in arriving at the proper flow settings. To keep the plant properly balanced as well ,s keeping the compression section fully loaded then becomes the major job of the control computer. Table II lists the criteria for justifying Monsanto's computer control system. This table was taken directly from Eisenhardtts and Williams' article. A question which immediately comes to mind is whether or not the pre dicted economic gains have been fully realized. Of course, basic economic figures are considered as proprietary information. However, in a qualitative way we can answer in the affirmative as Mr. Eisenhardt and Illiams have said September 1963 September 1963 CHEMICAL ENGINEERING EDUCATION 5 "Immediately after placing the computer on control, the gains in controlability became evident. When the process is not on computer control, the operator makes minor changes in control ler setpoints trying to hold process temperatures within lim its and maintain gas composition as required by the synthesis loop. Superimposed on top of these minor changes are larger step changes in throughput which are required to compensate for those uncontrolled variables affecting the plant capacity. At best these larger changes are made only .several times a shift. Under computer control, however, the plant throughput is adjusted every eight minutes to obtain maximum possible pro duction as uncontrolled variables changed. Qualitatively,, one can observe from the computer log sheet the steadying of gas compositions and temperatures under computer control as com pared with the irregular control obtained by even the best op erator." This now brings us to a consideration of the topic of engineering fund amentals as related to an understanding of computer control processes. The design of a computer controlled process requires the systems engineering ap proach. This means that the person in charge of the overall project must view the project in its entirety. He should not be burdened with the many small details that go into the system design. But on the other hand, he should be fully aware of the many fundamental engineering sciences which are involved in such a project. In this sense the system engineering approach may be synonymous with the common core approach in chemical engineering ed ucation. In Table III are listed some of the fundamentals required of chem ical engineering students for understanding of digital computer control pro cesses. Topping the list of required fundamentals is an understanding of eco nomics. Since computer control is a tool to assist management in meeting the process objectives, which is normally the maximization of operating profit, the importance of a thorough understanding of economics can hardly be overemphasized. Methods for pricing intermediate products, an analysis of profit from incremental production, and a working knowledge of payout criteria for capital investment are especially important. The basic curricula of chemical engineering will permit a reasonable understanding of the workings of most of chemical processes. However, in the computer control design area perhaps a little more emphasis should be placed upon chemical thermodynamics, chemical kinetics, and stoichiometry. For these are the basic sciences involved in deriving the predictive mathe matical models used in the control computer. Mechanical engineering enters into the design of computer control sys tems firstly, in the determination of the operating characteristics of the mechanical equipment involved. For example, in the ammonia plant system pre viously discussed, compressor capacity calculations were made. In addition to these, certain mechanical equipment limitations had to be evaluated. These limitations, which in computer parlance are called constraints, oftentimes determine or limit the area in which the process variables may be operated. In the field of electrical engineering, as applied to the design of com puter control systems, a main consideration is the communication between the computer proper and the process itself. A working knowledge of the basic AC/DC theory, transmission lines, impedance matching, and the filtering of electrical noise will go a long way in the understanding of the electrical requirements of computer control systems. It is not necessary to become too deeply involved in the computer circuitry itself. However, to deepen the ap preciation for the entire control system, some instruction in this area would be profitable. The ultimate success of any computer control system depends very heav ily upon the ingenuity and mathematical ability of the process analysts. Most computer control systems operate on the basis of mathematical models which simulate or represent the operation of the existing plant. These models may be derived from fundamental theoretical considerations or possi bly from regression analyses of plant data. In either event, considerable mathematical skill must be exercised in obtaining an accurate Pnd represen tative set of equations. Once the mathematical model has been developed, the job is only half done. From here various optimizing techniques must be explored in order that the model may be used most efficiently in reaching the process objective. CHEMICAL ENGINEERING EDUCATION September 1963 Table III FUNDAMENTALS REQUIRED OF CHEMICAL ENGINEERS FOR UNDERSTANDING OF DIGITAL COMPUTER CONTROLLED PROCESSES I. Economics A. Process objectives B. Investment and payout criteria C. General knowledge of market prices D. Methods for pricing intermediate products E. Analysis of incremental profit F. General knowledge of utility costs, etc. II. Chemical Engineering A. Unit operations B. Chemical thermodynamics C. Chemical Kinetics D. Stoichiometry III. Mechanical Engineering A. Operating characteristics of mechanical equipment B. Constraints imposed thereupon IV. Electrical Engineering A. Basic 4C & DC theory B. Transaission lines C. Im dance matching D. Filtering of noise V. Mathematics A. Methods of correlation analysis 1. Regression techniques 2. Curve fitting B. Optimising techniques 1. Maximization by calculus 2. Gradient methods 3. Linear programming 4. Nonlinear programming 5. Dynamic programming 6. Calculus of variations VI. Control System Theory A. Linear feedback systems B. Nonlinear feedback systems C. Sampled aata systems D. Laplace transforms E. Z transforms VII. Instrumentation A. Hardware 1. Types a. Pneumatic b. Electric c. H draulic d. Other 2. Measurement equipment a. Temperature b. pressure c. Flow d. etc. 3. Analytical instruments a. Chromatographs b. Infrared c. Physical properties d. etc. 4. Controllers a. Proportional b. Derivative c. Integral d. Other B. Methods of interconnection C. Reliability and accuracy VIII Computer Fundamentals A. Types and characteristics 1. Digital 2. Analog 3. DDA B. Applicability of computers C. Basic understanding theory of operation D. Programming a. Flow charting b. Coding c. Machine language d. Instructions e. Routines and subroutines September 1963 CHEMICAL ENGINEERING EDUCATION 7 As a part of the applied mathematics curricula, considerable attention should be paid to control system theory. Here such subjects as linear feed back systems, nonlinear feedback systems, sampled data systems, Laplace transforms, and Z transforms should be studied. Since the onstream compu ter is connected, as it were, to a live process, attention must be paid to the process dynamics. All computer systems must recognize these dynamics. Obviously, control actions must not be taken too frequently that the plant is always in a state of jitters, nor must they be taken too infrequently or else the full benefit of computer control will not be realized. The study of instrumentation is essential for a complete understand ing of the computer control process. By instrumentation we think of the hardware involved the sensing elements, the transducers, and the control equipment. It is these items that allows the computer to recognize or sense the state of the process. It is also these items that allows the com puter to take corrective action upon the process. In a sense the primary measuring elements represent the sensors of a living organism. The trans mission lines are the nerves. The control valves and controllers are the muscles, while the computer controlled systems possess the same attribute via the local feedback control loops. Finally, some instruction should be given in computer fundamentals. The differences between the characteristics of digital, analog and the DDA computers should be carefully noted. A basic understanding of the theory of operation of these computers is profitable, though not essential. Some programming instruction should also be given with exercises in flow charting, coding, and in the use of routines and subroutines. In closing I might say that it is impossible for any single man to understand completely all the workings of a digital computer control system. The design of such a system is the work of a team of experts. And yet, this complexity does not preclude its use as an effective and efficient produc tion tool. Let me give you an example. There are not many people that com pletely understand the entire working of a television set or even, for that matter, of the automobile in which we drive to and from work. And yet, there are millions of these machines in everyday use. By the same token the digital computer controlled process, though its design is complex, its oper ation can be made simple enough for a single operator to comprehend. The efficiency of this production tool, this manmachineprocess combination will be increased in the future through continued research efforts in all the areas that have gone into its design. BIBLIOGRAPHY General 1. Stout,T.M., and Laspe, C.G. I.E.C. Vol. 49, July 1957. 2. Laspe, C.G.,"Digital Computers in Refinery Process Control' Ref.Eng. Sept. 1957. 3. Stout, T.M., "System Considerations in Computer Control of Semicontinuous Processes", Proceedings AIEE Computer Control Systems Conference,Oct. 1957. F. Manildi, J.F., "Modern Developments in Control",Automation, July 1959. 5. Brandon, D.B., "Let A Digital Computer Run Your Alkylation Plant",I.E.C. 52, No. 10. 6. Roberts, S.M. and Stout, T.M., "Some Applications of Computer Control in the Iron and Steel Industry", Iron and Steel, March 1960. 7. Mears, F.C., "Organisation of the Computer Control Project", ISA Preprint 139LA61. 8. Buster, A.A., "Application of Controlling Computers to Fractionating Units", C.E.P. Symposium Series, Vol. 57, 1961. 9. Stout, T.M., "Evaluating Control System Poyout from Process Data", Control Engineering, Feb. 1960. 10. Grabbe, E., "Digital Computer Control Systems an Annotated Bibliography", lst Congress International Federation of Automatic Control, Moscow,June 1960. 11. Roberts, S.M.,"Computer Control of Chemical Processes with Stochastic Properties", Proceedings 5th International Instrument and Measurement Conference, Stockholm, Sweden, September 1960. 12. Madigan, J.M., "Online Computer Control of A Chemical Process", 6th Annual Data Processing Conference, Amer. Management Assoc., New York, March 1960. 13. Madigan, J.M., "How B.F. Goodrich Applies Computer Control", Chicago A.I.Ch.E. Feb. 23, 1960. 14. Madigan, J.M., "Computer Controlled Processing", C.E.P., May 1960. 15. Laspe, C.G. & Roberts, S.M., "Online Computer Control of Thermal Cracking , I.E.C.,53, May 1961. 16. Adams, '.F., "Closed Loop Control of an 85,000 Bbl Crude Oil Unit", Oil & Gas Journal, May 21, 1962 17, Eisenhardt, R.D. & Williams, T.J., "Closed Loop Computer Control at Luling", Control Engineering, Nov. 1960. 18. "Computer Control Case History TexacO", Paper submitted to Award Committee for 1961 Kirkpatrick Award Sponsored by Chemical Engineering Magazine. Survey Type References 19. Freilich, A., "Whatts Doing in Computer Control", I.S.A. Jour., Sept. 1961. 20. "Industry Accepts Computing Control", Control Engineering editorial ,May 1962. . Economic 21. Eliot, T.Q., & Longmire, D.R.,"Dollar Incentives for Computer Control", Chem. Eng., Jan. 8, 1962. CHEMICAL ENGINEERING EDUCATION September 1963 OPTIMIZATION THEORY IN THE CHEMICAL ENGINEERING CURRICULUM Douglass J. Wilde Department of Chemical Engineering University of Texas Introduction This article is intended to acquaint teachers of chemical engineering with the theory of optimization, which has developed so rapidly in eleven years that it is now finding its way into the practice, literature, and curricula of our profession. Optimization theory is composed of technical methods for computing the greatest (or least) value of some criterion of value or effectiveness measuring the performance of a system being studied. Since optimization involves, roughly speaking, .finding the best way to do things, it has obvious applications in the chemical industry, where small changes in efficiency can spell the difference between success and failure. Today as always many important decisions can be made simply by choosing some measure of effectiveness and then optimizing it. To justify the inclusion of this new material into the already crowded curriculum we cannot be content with describing the technical content of optimization theory. We must also show why the profession needs it and how it might be fit into existing graduate and undergraduate programs. More over, we must assess the pedagogical value of optimization theory in devel oping engineering judgment, scientific objectivity, and intellectual cre ativity in our students. To begin we shall advance two arguments suggesting the chemical en gineers' need for optimization theory. The first of these will be specu lative, analyzing the role of optimization in engineering decisionmaking. The second will be historical, showing how our very lack of attention to optimization theory has created demands for operations analysts and manage ment scientists to solve problems which could well be handled by engineers. In the heart of the article we shall combine a review of optimiza tion theory with description of a threehour, one semester course already given to chemical engineering seniors having no special preparation. This resume will include references to recent developments of pedagogical in terest. As each phase of the theory is discussed, its most important in dustrial applications will be mentioned so that the "why" may be unfolded at the same time as the "what" and "how". It is quite possible that a pro fessor may not wish to offer an entire course in optimization theory, pre ferring instead to incorporate parts of the theory into various existing courses. Indeed, one would hope that eventually optimization theory would be absorbed into the curriculum in this way. To facilitate such gradual integration we shall indicate where each of the three main parts of op timization theory may be pertinent to such existing courses as plant de sign, kinetics, control, economics, and mathematics. Thus optimization theory may be introduced suddenly or gradually depending on local circum stances. Finally we shall show how study of optimization theory gives a stu dent a set of rules potentially valuable to him in making complex decisions. The rules and procedures are of course worthwhile in themselves, but equally important is the fact that their study reinforces the student's confidence in the rational scientific approach to problems. Optimization theory gives typicod trail engineeg in the analysis of functions of many variables (via multidthit such mensional geometry) and in the use of precise, logical reasoning. Moreover, perimental data to den ke s mdge alone; ultimately the Information the very novelty of the mathematics and the newness of the theory ant, repeat stimulants to the creativity and imagination of the students. The Need or optimization Theory Let us examine two arguments tending to justify the need of the chem ical engineering profession for optimization theory. First consider the typical engineering project. Theoretical principles are combined with ex perimental data to describe the system under study. It is rare thnt such a study is made for the sake of knowledge alone; ultimately the information is to be used for making some sort of decision  build a new plant, replace a heater, or change a catalyst. Without optimization theory, such decIsions must often be made impetuously, or at best, after lbortious case studies, despite the good engineering that went'into the study itself. Such a situ ation is intellectually (and often economically) unsatisfying. 10 CHEMICAL ENGINEERING EDUCATION September 1963 Secondly, consider the rapid growth of the new profession of "opera tions research" or "management science", defined by most of their practi tioners as "the scientific preparation of decisions". This sounds auspio iously like engineering, and on examination of their methods for making de oisions, we find three steps: (1) rational (preferably mathematical) de scription of the system, (2) choice of a measure of effectiveness, and (3) optimization of that measure. Now in most Industrial problems, rational description is precisely the job ofthe engineer, while the choice of a measure of effectiveness is either obvious or impossible. Thus the only difference between industrial operations research and engineering is usual ly that the former profession has better optimization techniques. We sub mit then that the rise of operations research has been due not only to the ability and imagination of its own pioneers, who contributed much to the theory of optimization, but also to the failure of the engineers to study optimization problems. Our loss has been their gain. Optimization Theory Before 1951, optimization had hardly been studied at all since the development of the calculus of variations two centuries earlier and today most engineers know only one method for finding an optimum  the differ ential calculus. By this method one expresses the criterion of effective ness as a function of the independent variables, equates the first deriva tions to zero and then solves the resulting equations. But in industrial problems it is rarely possible to perform all these steps, and even when it is, the "solution" is often unattainable because of physical restric tions on the process. We shall distinguish three branches of optimization theory here, classifying them according to the very obstacles preventing their solution by the differential calculus. The three types of problems are: (1) experimental problems in which the measure of effectiveness is unknown and must be determined by direct experiment, (2) feasibility pro blems in which the apparent optimum lies outside the physical constraints on the system, and (3) interaction problems in which there are so many var iables that the problem must be decomposed and solved in pieces. Each type of problem can be covered in one semesterhour of undergrad uate work, either all at once in a single three hour course or as parts of other existing courses. There are optimization problems which do not fit into these three categories, but we are limiting ourselves here to material that can be taught to a senior engineering student in one semester and be of use to him when he graduates. After each type of problem is described, its historical development will be traced and references of either research or pedagogical interest cited. Then applications will be mentioned and finally, possible locations in the curriculum will be suggested. Experimental problems In an experimental problem one knows almost nothing about the depend ence of the measure of effectiveness on the independent variables, and the only way to obtain information about this dependence is to take measure ments. Kiefer (1.) has described a highly efficient way to carry out the search when there is but one independent variable and no experimental er ror, as for example in the calculation of the optimal number of stages in a distillation column or evaporator. A description of this Fibonacci search procedure in engineering terms is given in (2.) It is interesting perhaps that with this technique one could find the best case out of a possible twenty after only seven case studies. Unfortunately the elegant Fibonacci technique cannot be extended to sit uations with more than one independent variable, and in 1951, the year of revival of interest in optimization, Box and Wilson (3.) suggested their method of steepest ascent for multivariable problems. Recently newer ap proaches to this problem have been advanced  the geometric techniques of Buehler, Shah, and Kempthorne (4.) and the author (5.), as well as the log ical methods of Hooke and Jeeves (6.) Pnd Mugele (7.). The presence of experimental error requires different methods, known in general as stochastic approximation procedures. Dvoretzky (8.) has gen eralized the early methods of Robbins and Munro (9.) and Kiefer and Wolf owitz (10.), an acceleration technique has been proposed by Kasten (11.), and multivariable extensions have been developed by Blum (12.). Some of these procedures have been reviewed from the chemical engineers' point of view by Lapidus et. al. (13.). September 1963 CHEMICAL ENGINEERING EDUCATION 11 These methods are applicable to design and operating problems involv ing either complicated computations or significant measurement error. The Fibonacci technique could conceivably fit into a plant design or economics course, or even into the exposition of staged unit operations. Mltivaria ble procedures are more appropriate in plant design courses, and the In sight they give into multidimensional geometry could well suit them for in clusion in an advanced mathematics course. Stochastic approximation, since it depends on some probability theory, would be appropriate in an engineer ing statistics or probability course. In our experience the theory of ex perimental search for an optimum has been extremely stimulating to stu dents, who seem to be inspired by it to surprisingly original contributions. The author is presently completing a monograph on experimental opti mization, reviewing and explaining all these developments, hopefully in language that an engineering senior can understand. Engineering professors can obtain a free preliminary draft of this material by writing the author, who would be grateful for suggestions and corrections. Feasibility Problems When, as is often the case in the industrial world, the ranges of var iation of the independent variables are limited, it is sometimes physically impossible to attain the conditions where the first derivatives of the ef ficiency criterion all vanish. Such restrictions give rise to feasibility problems because only feasible conditions, those respecting all the con straints, can be considered. The technical term "mathematical programming" (not to be confused with the "programming" of computers) is often applied to such problems. The year 1951 also marks the beginning of the theory of mathematical programming. In that year Dantzig published his "simplex meth od" for solving the linear case. Since that time literally hundreds of articles have appeared on applications of the simplex method, and many petro leum companies have justified the installation of large electronic computers on the improvements in refinery scheduling and product blending made possi ble by mathematical programming. It is traditional in operations research curricula to spend a great deal of time on mathematical programming, especially the linear case, which is the simplest. Much of this time is consumed in introducing the student to matrix algebra. While matrix algebra is interesting in its own right, we have found that one can profitably develop mathematical programming with out it and save considerable time. This is achieved by treating feasibility problems as simple extensions of the classical optimization problem solvable by the differential calculus. Since engineering students are more adept at manipulating derivatives than matrices, this approach has proven quite successful, and it has been possible to take a class through linear and quadratic programming, as well as the decomposition principle to be dis cussed later, in only six weeks. This differential approach, which we think has great pedagogical value, is illustrated in (2.) end justified theoret ically in (14.) Discussion of feasibility problems is appropriate in any economics or design course. The subject may also be used in applied mathematics courses as an application of matrix theory; Lapidus has used this approach in his new book (15.) With the differential approach, mathematical programming can be covered in any engineering calculus course, almost as an exercise in partial differentiation. Interaction Problems Sometimes the criterion of effectiveness depends on so many factors that it is impractical or impossible to find the optimum by classical meth ods. Often such problems are generated by the interaction of smaller sys tems with each other. In such cases it is occasionally possible to decom pose the large problem into smaller ones, solve the subproblems, and re combine these suboptimal systems in such a way that the interactions are properly taken into account. This exploitation of the structure of a sys tem is advantageous because the number of calculations tends to increase as the cube of the number of variables. Thus doubling the number of variable s will ordinarily increase the computation load by a factor of eight. It the problem can be split in two, however, the number of calculations will only double or triple. 12 CHEMICAL ENGINEERING EDUCATION September 1963 Bellman (16.) has shown how to decompose a series of decisions, each depending on the one preceding, by the method he calls "dynamic programming.' This technique, which might also be called "serial optimization", has many applications to such long range planning problems as capital investment, pro duction scheduling, and maintenance planning. Application of dynamic pro gramming to the design of chemical reactors has been described in Aris' re cent monograph (17.) Nemmhauser has given a very clear example of design of a straightline chemical plant by dynamic programming (18.) The conventional exposition of this subject using functional equations is often confusing to students, and we have found the block diagram approach of reference (2.) to be helpful in the classroom. The solar systemsatellite structure of many multiplant scheduling problems lends itself to analysis by Dantzig and Wolfe's decomposition prin ciple when all the equations are linear (19.) A numerical example of the application of this principle to centralized planning is available (20.) This example illustrates the power and clarity of the differential approach mentioned earlier, and senior students have had little difficulty absorbing this material, considered quite abstruse by many operations analysts. References (2), (16), and (17) give many applications of dynamic pro gramming, whose ability to handle timedependent problems makes the process dynamics and control course an attractive place for its introduction. Aris' work suggests that the kinetics and reactor design course would also be suit able. Again, plant design and economics offerings can be used to introduce serial optimization techniques. Related to dynamic programming is Pontry agin's maximum principle (21). The decomposition principle should be dis cussed as an extension of linear programming rather than as a separate topic for the differential formulation makes this extension relatively painless. Pedagogical Values In describing the outline of a course in optimization theory we have indicated how it is a good vehicle for developing mathematical maturity and respect for the scientific method on engineering students. But aside from the technical material, the decision rules themselves can build sound en gineering judgment in the student that will help him make up his mind intel ligently even when there is no time for detailed and rigorous analysis. Study of the onevariable experimental optimization problem gives insight into the important minimaxconcept and the somewhat startling concept of ran domization. Analysis of multivariable problems unearths some rather dis turbing facts about graphical reasoning and the paradoxes that can arise from failing to realize that engineers often work in nonEuclidean space (2.) Linear programming shows that it is sometimes economical to give a customer higher quality than he asks for at no increase in price. The classic "law of diminishing returns" is illustrated quite clearly in the study of quadra tic programming. Anyone's point of view is affected by insight into the farsighted philosophy of dynamic programming, which begins by analyzing the last rather than the first decision in a sequence. Perhpas the most sur prsing decision rule of all comes from study of the decomposition principle, which shows that a central planning board should ask branch managers for non optimal production plans. This is particularly significant because few or gahizations presently operate this way, at least intentionally. Concluding Summary In this review we hope we have given information upon which chemical engineering professors can decide why and how they might introduce optimi zation theory, or parts of it, to their students. The demands of industry have made this necessary; research has made it possible; and pedagogical advances have made it practical. The rest is up to the profession itself. September 1963 CHEMICAL ENGINEERING EDUCATION 13 REFERENCES 1. Kiefer, J., "Sequential Minimax Search for a Maximum", Proo. Amer. Math. Soc, (1953), pp. 502506. 2. Wilde, D.J., "Optimization Methods", a chapter in Advances in Chemical En ineerin g, Vol. III (T.B. Drew, J.W. Hoopers, Jr., and T. Vermeulen, eds.) Academic Press, New York 1962. 3. Box, G.E.P., and Wilson, K.B., "The Experimental Attainment of Op timum Conditions", J. Roy, Stat. Soc., BB (1951) pp. 120 4. Shah, p.V., Buehler, R.J., and Kempthorne, 0., "The Method of Parallel Tangents (PARTAN) for Finding an Optimum", Iowa State Univ. Statistics Lab. Technical Report No. 2, Ames Iowa (April 1961) 5. Wilde, D.J., "The Contour Tangent Optimization Method", to appear in AIChE Journal. 6. Hooke, R., and Jeeves, T.A., "Direct Search Solution of Numerical and Statistical Problems", J. Assoc. Computing Mpch. 8, 2 (April 1961), pp. 212229. 7. Mugele, R.A. "A Monlinear Digital Optimizing Program for Process Con trol Systems", ProcWestern Joint Computer Conference (Spring 1962). 8. Dvoretzky, A., "On Stochastic Approximation", Proc. 3rd Berkeley Symp, Math. Stat. Prob. (J.Neyman, ed.), Berkeley, U. or California Press 119kb), pp. 3955. 9. Robbins, H.,dnd Munro, S., "A Stochastic Approximation Method", Ann. Math. Stat., 23 (1951), . 400407. 10. Kiefer, J., and Wolfowitz, J., "Stochastic Estimation of the Maximum of a Regression Function", Ann. Math. Stat., 23 (1952) pp. 4626 . 11. Kesten, H., "Accelerated Stochastic Approximation", Ann. Mpth. Stat. 29, (1958), pp. 4159. 12. Blum, J.R., "Multidimensional Stochastic Approximation Methods", Ann. Math. Stat., 25 (1954), pp. 73744. 13. Lapidus, L, Stillman, R., Shapiro, S., and Shapiro, E., "Strategy for Optimizing Chemical Engineering Systems", PIChE Journal (Summer 1961). 14. Wilde, D.J., "Differential Calculus in Nonlinear Programming", to ap pear in Operations Research. 15. Lapidus, L. Digital Computation for Chemical Engineers, McGrawHill, New York (1962). 16. Bellman, R., Dynamic Programming, Princeton U. Press, Princeton (1957) 17. Aria, R., Optimal Design of Chemical Reactors  Studies in Dynamic Programming. Academic Press, New York (1961). 18. Nenmhauser, G., Ph.D. dissertation Northwestern University 19. Dantzig, G.B., and Wolfe, P., "A Decomposition Principle for Linear Programs", Operations Research, 8_(Jan. 1960.) 20. Wilde, D.J., "Production Planning of Large Systems by the Decomposition Principle", to be presented at the AIChE annual meeting, Chicago, Dec. 1962. 21. Chang, S.S.L., Synthesis of Optimal Control Systems, McGrawHill, New York (1961). FUNDAMENTAL PROBLEMS IN HETEROGENEOUS CATALYSIS Max. S. Peters University of Colorado Boulder, Colorado At the Chemical Engineering Teachers Summer School in 1939, K.M. Watson presented a paper in which he discussed chemical reaction kinetics for engineers and made an appeal for including more of this type of train ing in the undergraduate chemical engineering program. At the next Chem ical Engineering Teachers School, held in 1948, Watson once again talked on the subject, but this time his emphasis on necessary training was in the area of kinetics of catalytic reactions (25, 26). Since that time, kinet ics courses have been included in most of the undergraduate and graduate chemical engineering curricula in the United States. With the increased emphasis on the chemical aspects of chemical engineering, teaching and aca demic research in the area of chemical engineering kinetics will become even more essential as a component of any adequate undergraduate and grad uate program. The area of heterogeneous catalysis presents one of the most chal lenging opportunities for new approaches from the viewpoints of both re search and teaching. The teaching of heterogeneous catalysis for under graduate or graduate engineers offers an ideal opportunity to impress on the students the importance of recognizing the practical limits of purely theoretical analyses while simultaneously emphasizing the necessity for understanding the fundamental theoretical concepts. The literature is full of examples of controversial claims in catalytic kinetics based on putting too much emphasis on a given theory without adequate evaluation of other possible interpretations. One example is the critical article by Weller on the engineers' use of the LangmuirHinshelwood approach when a power series of the Freundlich type would be equally applicable (29). A rebuttal to the Weller article based on an analysis of real and ideal sur faces is presented by Boudart (3). Another example is the critical analysis of leastsquares determinations of rate constants presented by Chow (10). The controversial nature of many of the currently applied techniques of interpretation in heterogeneous catalysis illustrates vividly the need for more research. The teacher of the subject is dealing with a field which cannot be presented as ancient history with all problems solved. In stead, the teaching can be made extremely interesting by bringing out the fact that many of the past engineering techniques in heterogeneous catalysis are questionable. If the course is taught correctly, perhaps the students can catch some of the excitement of being part of an important engineering area which is ripe for the development of new ideas and new approaches. The recent survey by Hougen on Engineering Aspects of Catalysis emphasizes these points (15). In the past, there has been some tendency to teach chemical engineer ing kinetics strictly from the viewpoint of design techniques, leaving the fundamentals to the area of chemical kinetics. This approach is complete ly inadequate for heterogeneous catalysis because of the many assumptions required for normal design applications. Consequently, to give the engi neer a background for intelligent understanding of the design aspects of heterogeneous catalysis, it is essential to discuss the basic considera tions and, from these, go through the assumptions necessary to give a workable final design equation. In this paper some of the basic consid erations in heterogeneous catalysis normally passed over in undergraduate engineering courses will be discussed. Surface Characteristics An obvious starting point for the teaching of heterogeneous cataly sis is to present an analysis of surface characteristics for solids with, an initial approach through discussion of physical adsorption and chemisorp tion. Because the energy of activation for physical adsorption is small, the activation energies for reactions involving physically adsorbed mole cules cannot be far different than for the homogeneous reaction. Conse quently, physical adsorption does not play an important direct role in catalysis, but it does become important in considering fractions of surface area available on a solid catalyst. Conventionally, the engineering approach to teaching about physical adsorption and chemisorption is to immediately develop the Langmuir adsorp tion isotherm for a unimolecular adsorbed layer in a form such as g= h (1) September 1963 CHEMICAL ENGINEERING EDUCATION 15 where 9 represents the fraction of the surface covered by an adsorbed mono molecular layer at equilibrium, h is the adsorption equilibrium constant, and p represents the partial pressure of the gas adsorbed. From this point, with a few brief comments about multilayer adsorption, the Brunauer, Emmett, and Teller equation for estimation surface area is normally presented in the following form (7): p 1 (cl)p V(POP) V5 OVp( 2) where c is a constant for the system and temperature, po is the saturation vapor pressure for the gas at the temperature used, and vm is the volume of the gas adsorbed to cover the surface completely with a monomolecular layer Here is represented the type of engineering approach many educators find objectionable. If it is worth presenting Fquation 2 at all, then it certainly is necessary to present the limitations on this Equation. This starting point would be a good spot to impress the students with the need for understanding any equations they may use, and, despite the time it might take, a relatively complete derivation of Equation 2 could get an en gineering course in heterogeneous catalysis off to good start. The development of Equation 2 is based on the existence of multimol ecular layers of physically adsorbed molecules with a derivation similar to that of the Langmuir isotherm for unimolecular layers. At equilibrium, the rate of condensation on the bare surface must equal the rate of evap oration of molecules in the first layer. Similarly, for each succeeding layer, the rate of evaporation must equal the rate of condensation on the preceding layer. Therefore, aipso = bisieEi/RT (3) a2psi b2s2eE2/RT (4) ai pii = bisi5eEi/RT where inca b are constants, s represents the surface area covered only by the subscriptindicated layers of molecules, ind 3 represents the heat of adsorption for the indicated layer. Thus, the total surface "rea of the catalyst and the total volume adsorbed are, respectively: n A =f si (6) i m 0 V is. (7) A i: 0 where n represents the number of layers. As indicated in Equations 3, 4, and 3, t:e heat of assorption is sn volved exponentially in each of the equilibrium expressions "or the rite. The assumption is made that, for ll layers except the ;rst. the e.:t c adsorption is constant and equal to the :he:t of evapor ton. It ai e  assumed that the ratio of b/a is constant for all 1d ees xc pt the t. In other words, the Van der 1,aals forces of physical dsc'pton re involve only in the first layer of molecules on the sur" ae. With these assumptions the summation express'ns n 'e 'eve. rectly to give a value of V/Vm as follows: b2 .b bi (.1 a2 a3 ai E2 = E3 Ei = Ev ! From Equations 3, 4, and 5 2 J1 si 11 ( ) s ()) l = c (, C " gale where J= geEv/RT rnd ( ':v)/R Therefore, n i cso i = 1 ( Vm so cso n (E )i J 16 CHEMICAL ENGINEERING EDUCATION September 1963 i= 1 1 Because i (P)1 converges to p/J and ( ) converges j (1op/) 2 verges to ,P/J the summations represented in Equation 11 re such that the equation reduces to n 1 y cp/J 1 (n 1) (p/J) n (p/J) Vm 1p/J 1 (c 1) (p/J) c (p/J)n 1 (12) When p equals the saturation pressure of the gas, po, complete con densation can occur and V must approach infinity. This can only occur if J is equal to po. For the case where p po or for the case where n is a large number, Equation 12 can be rearranged to give Equation 2, and a com plete derivation of Equation 2 showing all assumptions has been presented. The question immediately arises as to the desirability of presenting this much detail in an undergraduate chemical engineering course on heter ogeneous catalysis. Actually, the derivation as presented here is relative ly simple and the insight it gives the students into the limitations of the final equation plus the added understanding of the types of forces involved in condensation and adsorption make the time spent on such a derivation worth while. With this background, students making the standard plots of p/V(pop) versus P/Po to determine catalyst surface area will have a real understanding of what they are doing. The value of the constant c now has physical significance and could be used to give an approximation ofEiEv since gal/bi is normally close to unity. Langmuir Adsorption Isotherm The Langmuir adsorption isotherm, as presented in Equation 1, can be developed directly from Equation 3 as follows to give a physical meaning to the constant h: 5o s oal aEe/RT (13) soo 1 P al eEp where h al e In catalytic studies, use is regularly made of this relationship to develop a model for the reaction. An ideal surface is assumed so that h, and consequently the heat of adsorption Ei, remain constant. For any real surface, there is no valid reason to assume Ei will remain constant inde pendent of the amount adsorbed. This explains one reason why experimental values of h obtained from reaction rate data often do not agree with the oretically equivalent adsorption constants obtained from direct adsorption measurements. Despite these limitations, chemical engineers have been highly suc cessful in using the Langmuir adsorption isotherm as an approach for cor relation of rate data. Because of this success, some novices are convinced that the relationships are completely sound theoretically and attempt to develop detailed mechanisms on this basis. This approach has been attacked by Weller who proposed that real and ideal surfaces were so different that the practical engineer was wasting his time in attempting to use the Lang muirHinshelwood approach (29). He proposed instead the far simpler Freundlich type of expression for expressing reaction rate. For example, for the gas reaction A + B C catalyzed by a solid, the Freundlich powerseries type of rate equation would be rate k(PA) (pB)m (p) ..... () while the LsngmuirHinshelwood result might be in the form of rate k ApB 2 (15) (1 K;A pB Kcpc) September 1963 CHEMICAL ENGINEERING EDUCATION 17 The mathematics of handling Equation 14 is far simpler than for Equation 15; however, a specific reaction mechanism can be applied for Equation 15 while no mechanism can be given for Equation 14. Weller op poses the use of the I engmuirHinshelwood approach on the grounds that real and ideal surfaces are not similar and the results obtained tend to be mis leading. Boudart has responded to this by proposing that discrepancies in the constancy of the adsorption energy can be accounted for by approximating a temperature dependence of this energy as Ei = Eoi zT (16) where Eoi and z are constants (3). The type of approximation represented by Equation (14) would be an excellent method to use for initial analysis of reaction rate data. How ever, for any general application of kinetic results, a mechanism must be satisfied. Accordingly, the information obtained by a preliminary analysis of the Equation 14 type should be used to attempt to predict a mechanism, and this should then be confirmed by a detailed analysis of the type repre sented by Equation 15 plus recognition of the factors introduced by surface nonideality. The overzealous ambition of chemical engineers to prove a mechanism on the basis of an incompletely understood theory has caused many ridicu lous claims to be made in the literature. This illustrates the need for careful development of the theory combined with discussion of its limitation in the teaching of heterogeneous catalysis. The preceding analysis hPs been concerned almost completely with the thermodynamics of adsorption as affected by the heat of adsorption with no attention being given to the concept of entropy relationships. Because gas molecules are free to move in three dimensions and ndsorbed molecules are restricted to no more than two dimensions of motion, a reduction of entropy must occur on adsorption as pointed out by Brunauer (6). Entropy calcula tions can give information on whether the adsorbed molecule is free to move about the surface or is localized with no free rotation. Such information is important in considerations of catalytic kinetics, and variation in en tropies of adsorption can be used, at least partially, to explain changes in heats of adsorption. The concept of entropy has only recently been giv en serious consideration as being important in supplementing the heat of adsorption type of analysis for thermodynamics of surface processes (5, 19) ActiveSite Theory Despite the limitations of the simple adsorption theory with its as sumption of ideal surfaces, this concept has been extended by chemical en gineers to include models whereby catalytic surface reactions take place by means of molecules or atoms adsorbed on socalled activee sites" on the sur face. Thus, the reaction could occur between an adsorbed reactant molecule and a gasphase molecule or between two adjacently adsorbed molecules. In presenting this simple model, authors have recognized the limitations of the theory and have consequently tended to present derivations that in clude incorrect statements which are eventually corrected by the empireel nature of the results. For teaching purposes, errors of this type, even though not important in the final analysis, should be avoided. One example of an error is presented in F classic pioneer public t o by Hougen and Watson (16) which was corrected in their later work. '*, was the statement that the dualsite surface corcentratlon of v cent s'tea adjacent to sites containing adsorbed material A is CV 5 1/2 SC CV (17) where S represents the number of equidistant active sites cent tc an average active site and the subscripts ,. V, 'nd I stand Ic :,'terrls vacant sites, and total sites. The error in Equ tion 17 1ies :a the 1/2 which should be unity. A similar error was aI .. :..v"th (2l) t would be more accurate to use CTCV in p',ce 0 f CT. Any argument tiat these two errors are vaL!' 1 n 'e i:t.ad ty Sn extremely simple mathematical analysis ir. 4,h n a sua.t ca' t r. r ie a tIe total possible dual sites on an ideal surface contain. r, orli vucant ste.' (Cv) and adsorbed sites (Ct). For rdjscent similar sites, tLe duelsite concentration must be onehall of the total number of possible .djaccat. sites times the fraction of total sites occupied by the m teail1. in case, the onehalf is necessary because each pair of' adj cent inoleco, 18 CHEMICAL ENGINEERING EDUCATION September 1963 counted twice. If the adjacent sites are not occupied by similar materials, the onehalf should not be included. For this case, therefore, Total dual sites = SCT = Sites VV Sites pV Sites AA (18) r SCT BCvC SCACV SCA (19) 2 20T CT 20T (19) SC S(C CA) 2 SC2 (20) 2 T 20 T The identity in Equation 20 can only occur if Equation 17 is incor rect and if CT is used in the denominator. Obviously, these two errors as discussed are not important but, for teaching purposes, analysis of this situation and discussion of past errors of this type serve to stimulate class interest and give the students more insight into the type of problems involved in trying to specify surface characteristics. Solid State Analysis and Electrochemical Potential In recent years, hope has been raised that catalytic activity might be explained by a new and quantitative approach involving electronic chem ical potential and based on the principles of solidstate physics. Un fortunately, this approach has not yet produced any significant results, and it is doubtful if it should have more then passing mention in an un dergraduate course on chemical engineering kinetics. In this approach, the assumption is made that chemisorption is ac companied by a transfer of electrons between the adsorbing solid surface and the adsorbent. The adsorbent acts either as a donor or acceptor depend ing upon the direction of electron transfer. Because of the high density of electrons on a metal surface, exact quantitative analysis of the charge transfer process is difficult. However, the surfaces of semiconductors have a far lower concentration of electrons or free carriers than metals and, consequently, are much more satisfactory for theoretical analyses of the chargetransfer process and the resulting energylevel changes (4, 23) . The Fermi level has been proposed as one means for explaining cata lytic activity. At equilibrium conditions, there is a probability (desig nated as the FermiDirac function) that a given quantum state of energy will be occupied by an electron. The energy at which this probability is one half is designated as the Fermi level (20). The Fermi level of a semicon ductor may increase or decrease depending on the concentration of impuri ties and the temperature. Semiconductors are characterized by densities of electrons (n) and of socalled holes (p). Thus an ntype semiconductor has a conductivity due to excess electrons and is a donor while a ptype semi conductor is an acceptor. In applying the electrochemical potential theories to catalytic kinet ics, one can consider that a gas molecule can be adsorbed on a surface in different forms. For example, isopropyl alcohol might be adsorbed on chromium oxide catalyst as a donor through a hydrogen atom, as an acceptor through a hydroxyl group, or with no net charge transfer. If the most re active group were the form adsorbed with no net charge transfer, then a max imum would be expected in the reaction rate if the catalyst were changed gradually from a p to an ntype semiconductor. If the catalyst is initial ly a ptype material it might be changed to an ntype by increasing the pressure of hydrogen in the gas since the hydrogen would release electrons to the solid catalyst. In other words, there would be a shift in the Fermi level and a maximum catalyst activity would be expected at one Fermi level. Figure 1 presents results on the rate of dehydrogenation of isopropyl alcohol catalyzed by chromium oxide from experiments carried out at the Institute of Physical Chemistry of Madrid (13, 14). Under normal conditions a regular decrease in rate with increase in hydrogen partial pressure would be expected as indicated by the dashed line in Figure 1. The experimental results are indicated by the solid line. One possible explanation for this apparent discrepancy could be based on the analysis presented in the pre ceding paragraph wherein there is a Fermi level shift through the level corresponding to maximum activity as the solid catalyst changes from a ptype to ntype semiconductor. Sentnkr AA N*luGINEERING EDUCUATIOJl 19 Rate \ PN1 Figure 1. Rate of Dehydrogenation of Isopropanol Catalyzed by Chromium Oxide at Constant Total Pressure with Varying Partial Pressures of Hydrogen in Isoproppnol (13). Heat and Mass Transfer 0 For the design of catalytic reactors, basically three relationships are needed as follows: (e) The design equation. (b) The rate expression. (c) An energy balance. In the simplest possible case of a plugflow reactor operated adiabatically the design can be accomplished with (a) The Design Equation: Fdx rdW 21) (b) The Rate Expression: r = f (x,T) M22) (c) The Energy Balance: ( H) Fdx = FioCpdT (23) where x represents the fraction of reactant converted, F is the feed rate of reactant, r is the reaction rate based on an amount of catalyst W,AH is the heat of reaction, opis the heat capacity, and subscript i indicates the individual components. Heat and mass transfer considerations must be taken into account when the plugflow assumption is not valid. The same general approach is used for both heat and mass transfer; so this discussion will be limited to mass transfer and development of the appropriate design equations. The conventional approach is to correct for radial and longitudinal diffusivity by using an effective diffusivity for each representing the actual pecked bed as being replaced by a fictitious homogeneous material with the ap propriate diffusivity. Radial bulk flow is neglected. With these assump tions, the following equations can be developed by material balances: (a) For the case of radial diffusion only with constant De/u, J (uC) De 1 J(uC) J (uC) r B= C (24)  u r Jr2 B (b) For the case of longitudinal diffusion only with constant DL/U, DL J2(uC) J(uC) rC = C2) u jZ7 JZ C J (2) (c) For the case of both longitudinal diffusion and radial diffusion J(uC) De 1 J(uC) J2 (uC) DIL 2 (uC) r JZ u r =T 7 r j, B 5 (JZ) JC In vector notation, the preceding equation reduces to div(uG) div D Grad (uC) r B = (27) where u is velocity, C is concentration, r is radius, Z is reactor length eB is bulk catalyst density, 9 is time, and the Dis represent diffuslv' eS ptember 1963 CIM L E 20 CHEMICAL ENGINEERING EDUCATION September 1963 Development of the preceding design equations along with the corres ponding energy balances are straightforward and should be presented in un dergraduate courses on heterogeneous catalysis. However, the important engineering function comes in applying these equation, and the solution of the equations can become quite involved. An example of the problems in volved in selecting boundary conditions along with the need for clear think ing is presented in the following for the case of Equation 25. For a firstorder irreversible reaction, r = kC, and steady state, Equation 25 becomes a secondorder, linear, ordinary differential equation with a solution in the form of C (Const) e mZ where m is a constant. Smith presents limits as follows to give a very simple solution (22): Boundary conditions proposed by Smith: C= Co at Z. 0 C= 0 at Z oa Solution is C = eZ where 1 1 k1 D (26) 71r u= Danckwerts and others (11, 27) have proposed much more reasonable limits based on a concentration gradient existing at the entrance to the reactor. The resulting solution is far more complex and more realistic than that shown by Equation 28. Boundary conditions proposed by Danckwerts and Damkohler: C Co DL dC at Z = 0 dC at Z I L The resulting solution is uZ ua (LZ) usa C e2 fl (ZL) to e 2(1 a')eL Z(la)e (29) ua'L ua L (1 a ') e (1a')2 e where a' = 1 4kDL B A rather detailed discussion of the disadvantages of oversimplifying, as illustrated by Smith's boundary conditions, can stir up much interest among the students and will simultaneously serve to illustrate the need for the students to think on their own rather than merely blindly accept any thing they see in print. The fact that both Equations 28 and 29 reduce to the basic plugflow expression as DL approaches zero can be used to illus trate the fact that one proof of a given theory is not always sufficient to establish its validity. Computer Solutions We have now reached the point where use of computers for problem so lution should be completely acceptable for undergraduates in Chemical En gineering. The kinetics course is an ideal place for including outside problemsafor solution on the computer, and at least one problem of this type should be part of the course. A simple example is given in the fol lowing of a typical problem in kinetics which can readily be solved on a small analog computer of a type similar to the pace TR10. For the case of the following consecutive reactions carried out isothermally A k B k2 C the rate equations for a constantvolume reactor are dCA klCA (30) d k / (31) dCB k1CtA k2CB (31) Hr k2C September 1963 CHEMICAL ENGINEERING EDUCATION 21 Analytical solutions of the first order linear differential equations are ek9 Ca = CAo (32) CB klCAo (ek1 k2) (33) k2kI CC CAo CACB (34) where Cls represent concentration and subscript o represents initial con centration of pure A at zero time. From Equations 30, 31, 34, plots of either CA, Cg. or CC versus 9 could be obtained directly with a small analog computer by use of the un scaled diagram shown in Figure 2. A typical concentrationtime plot tht would result on the xy plotter from the computer is also shown in Figure 2. This simple example can be used effectively with undergraduates to familiarize them with the use of the analog computer. It is particularly appropriate because the students can easily celoulate analytically from Equations 32, 33, 34 the concentrationtime values for direct conporison to the computer results. Volts Figure 2. Analog Computer CA Potentiometer Diagram S For Solving IC CA A" B C YAxis and k, Resultant XT Plot Integrator From Computer Inverter IC C13 k S CC Axis Summer 6 CA de Ic S Volts XAxis Volts R A A B ka.C Con.. C\ 9, Time 03 " CHEMICAL ENGINEERING EDUCATION September 1963 If the students have the background and facilities for solving prob lems on the digital computer, programs are available for solution of Lrng muirHinshelwood types of rate expressions as presented in Equation 15 (12, 38). Other programs are available for the digital computer which could e uaed for problems relating to operational characteristics of iso therm.l tubular flow reactors (2), isothermal batch chemical reactors(17). or solution of the Brunauer, Emmett, and Teller Equation 2 (8). Conclusion The subjects discussed in this paper represent some of the special problems in heterogeneous catalysis that would be appropri te for presenta tion in an undergraduate course. Obviously, there are many standard sub jects, all of which also involve fundamental problems, which should be in cluded. Among these would be analyses of the various resistances involved in the catalytic kinetic processes, experimental techniques, interpreta tion of ex perimental results, types of reactors including special prob lems of construction and operation, optimization techniques, poisoning ef fects, and many others. Intraparticle transport is another fundamental problem which has re ceived inadequate attention in many chemical engineering courses on kinet ics. Fore diffusion and catalytic effectiveness are often completely neglected even though these are important factors in as much as perhaps eighty per cent of all catalytic processes. The work of Thiele (24), Wheeler (30), Aris (1), Weisz (28), and Hougen (15) are significant in show ing the advances being made in this area, and an excellent summary of the current situation on this fundamental problem is presented by Carberry (9). BIBLIOGRAPHY 1. Aris, R., Chem. Eng. Sci., 6, 262 (1957). 2. Bailie, R.r. and. Fpn, "Computer Program Abstract 041Operational Characteristics of Isothermal Tublar Flow reactors," Chem. Eng. Prog., 56, No. 2, 92 (1960). 3. I oudart, M., A.I.Ch.E. Journal, 2, 62 (1959). Boudart, M., .p ., 7,= ,3556 (1952). Boudart, M., "T l face Chemistry of Metals and Semiconductors," 409, Edited by H. C. Getos, John Wiley and Sons, New York, 1960. 6. Brunauer, S., "Physical Adsorption," Princeton University Press, 1943. 7. Brunauer, S., P. H. Emmett, and E. Teller, J. Am. Chem. Soc., 60, 309(1938). 8. Brunauer, Emmett, and Teller Equation CompuTerPFogram Aistracr027, to be published in Chem. Eng. Prog. (1962). 9. Carberry, J. 77"Transpor henomena and Heterogeneous Catalysis," Forth coming publication. 10. Chow, C. H., Ind. Eng. Chem., 50, 799 (1958). 11. Danokwerts, P. V. ,Chem. _ng. !ci., 2, 1 (1953). 12. Dricokmann, W. L., "Computer Program Abstract 033 Polynomial Equation Pitting," Chem. Eng. Prog., 56, No. 8, 86 (1960). 13. Garcia de TlaBania, J F.., and G. K. Orlandini, Technical Note No. 1, AP61 (514)1330, Instituto de Quimica Fisica, C.S.I.C., Madrid, Nov. 1958. 14. GaroiaMoliner, F., "Trapping in Semiconductors," Personal communication, Dept. of Physics, Univ. of Ill., Urbana, Ill., Feb., 1962. 15. Hougen, 0. A., Ind. Ag. Chem., 53, 509 (1961). 16. Hougen, 0. A., 3 KTR. Watson,Tnd. Eng. Chem., 35, 529 (1943). 17. Jeng, B. J., and T.T. Fun, "Computer Program Abstract 056 .Design of Iso thermal Batch Reactors," Chem. Eng. Prog., 56, No. 6, 90 (1960). 18. Kaufman, D. J., and C. VW."Vo, computerr Program Abstract 060 Solution of Simultaneous Linear Equations," Chem. Eng. Progr., $6, No. 9, 78 (1960). 19. Scholten, J. J. F., and P. ZweiteringTTrans. ar. oc., 53, 1363 (1957). 20. Shockley, W., "Electrons and Holes in SemiconducorTT" D.Van Nostrand Co., 1950. 21. Smith, J. M., "Chemical Engineering Kinetics," P. 243, McGrawHill Book Co., Inc., New York, 1956. 22. Ibid., page 365. 23. Stone, p. S., "Chemistry of the solid State," 367, Edited by W. E. Garner, Butterworths Scientific Publications, London (1955). 24. Thiele, E. W., Ind. Eng. Chem., 31, 916 (1939). 25. Watson, K. M., '"RCemical eacltion'Kinetics for Chemical Engineers, Collected Papers on the Teaching of Chemical Engineering," Proc. of Oh. Eng. Div. of A.S.E.E., 2nd Ch. Eng. Summer School, page 175, Publ. by A.I.Ch.E., 1940. 26. Watson, K. M., "Kinetics of Catalytic Reactions," Proc. of Ch. Eng. Div. of A.S.E.E., 3rd Ch. Eng. Summer School, 1948. 27. Wehner, J. F., and R. H. Wilhelm, Chem. Eng. Sci., 6, 89 (1956). 28. Weisz, P. B. and C. D. Prater, "AdVanesTn Cal78yss," vol. VI, 1954. 29. Weller, S., A.I.Ch.E. Journal, 2, 59 (1959). Wheeler, A., Cataysis Vl. YI., P.R. Emmett, Editor, Reinhold Co., :iew York, 1955. PLANNING EXPERIMENTS FOR ENGINEERING KINETIC DATA H. M. Hulburt American Cyanamid Company Central Research Division Stamford, Connecticut Since chemical kinetics is not as yet a predictive science, one of the tasks commonly faced by the engineer in process development is the accumulation of rate data to be used in reactor design. This paper will discuss some of the principles and techniques which can be used to establish useful design data for complex reactions of obscure mechanism which occur under illdefined conditions and yield incompletely characterized products  in other words, the usual case of practical interest. It is a truism so obvious it is usually not taught in physical chemistry courses that one should first establish the stoichiometry of the reaction he is studying. Yet in process development this is often difficult. Analytical methods for the mixtures being produced may be timeconsuming or unavailable and expensive to develop. There is a strong temptation to determine the prin cipal product and most troublesome byproduct and ignore the rest. Gross mater ial balances can often be made on the basis of elemental analyses without de tailed knowledge of molecular composition. Yet these incomplete analyses oftep fail to give adequate kinetic data. When reactions are not well understood, it is not unusual to discover the appearance of a new product when conditions are changed. If the design has been based on data taken in ignorance of this pro duct and operation extrapolated beyond the pilot plant range, serious trouble can ensue. The first step, therefore, is a qualitative survey of the reaction stoichiometry over as wide a range of conditions as possible. The objective of this survey is to establish the main features of the reaction mechanism. For design purposes, the molecular mechanism will never need to be known in de tail, but enough of its properties must be determined to formulate a kinetic model of the reaction for the range of conditions of design interest. The more nearly this model reflects the actual mechanism, the more confidently can it be applied over a wide range of conditions. Nevertheless, at some stage, the en gineer must be satisfied to work with the data he has, recognizing that he has not established a mechanism fully. Is the principal reaction product the ultimate product of reaction; or T6es it disappear in side or subsequent reactions under some conditions? Are the byproducts formed as or from intermediates en route to the main products, or are they formed by independent reaction routes? Do some or all of the pro ducts reach equilibrium or steadystate concentrations which are insensitive to residence time? Is there a phase separation in the course of the reaction? Not every autoclave that is loaded with a homogeneous solution and delivers a homogeneous product solution has had homogeneous contents throughout the course of the run. Are mass or heat transfer rates comparable to or slower than the chemical reaction rates? These are all questions that can receive qualitative answers by comparing the results of a few wellplanned runs. Consider a hypothetical example in which Qacid is made by catalytic con version of electamine and carbon dioxide in a fluid bed. Under reaction con ditions, Qacid is volatile in an atmosphere of carbon dioxide but it is found that organic matter accumulates on the catalyst to a degree depending on the temperature and feed ratio of electamine to CO2. Some undesirable electamnnic acid is found both in the product vapor and in the organic residue on the cpt alyst which, however, is largely unidentified material. The acid could be formed by carboxylation of the amine, but there Is no evidence for the Pctual mechanism. First experiments might be to vary the gas residence time in the con verter, determining the spacetime yield and purity of the elpctmilne product. For experimental convenience in these survey runs, the process Is not run in continuous steadystate. Instead, electamine is added to the catalyst to a predetermined loading at a temperature below that nt which Qacid is formed. The temperature is then raised and Qacid stripped off in a stream of C0, suf ficient to fluidize the bed. The concentration of acid in the product'strer is followed during the stripping process. it was found that the rounds per hour of product recovered is directly proportional to the CCa flow rate, other conditions being fixed, but is nearly indsperdeent of t'e electanine remaining on the catalyst. This was established by a set of runs at two temperatures In which C02 rate and initial electamine loading were varied. 23 A2 CHEMICAL ENGINEERING EDUCATION September 1963 At each temperature, the partial pressure of Qacid in the offgas was nearly constant throughout each run. However, with high initial loading of ele ar n, this partial pressure was less than with low initial loading. With oth,.' conditions comparable, the partial pressure of Qacid was higher at high er temperature. Because of initial transients during the stripping runs as the temperature was being raised, the earliest steady data could be obtained only after 20% to 40% of the Qacid had been stripped off. These results give a strong presumption that Qacid is being formed in vaporsolid equilibrium with the organic matter on the catalyst, since as much conversion was obtained in a runas in runs with twice the residence time. However, runs which differ in initial loading gave somewhat different apparent vapor pressures. Hence, we must conclude that there is additional complexity in the mechanism. An addi tional complication in a fluid bed is the possibility of poor contacting of vapor reactant with catalyst when large "bubbles" can form. Since this by passing increases with gas velocity, a lower yield at high gas velocity would be expected from this cause alone if it were operative, even though the dense phase reaches equilibrium. Pour more runs under strictly continuous steady operation should settle the qualitative nature of the mechanism. In these, temperature and feed com position are fixed but feed rate and bed height are varied as follows: Run V (cu.ft./hr.) H (ft.) *C (hr.) 1 2 Q 2Q L L AL/Q AL/2Q, Conversion of electamine to Qacid product is the measured response. The following logic ensues: Sequence Compare If Then A 1. Runs 1 and 2 Same conversion Equilibrium is att B 1. Runs 1 and 2 Different con Either kinetic con version trol or bypassing B 2. Runs 1 and 3 (Same conversion fNo bypassing Different con tBypassing occurs version From this logic, Run 4 appears superfluous. However, adding it gives a 2 x 2 factorial experiment in bedheight and residence time. gained 1 S L 2 AL/2Q 2 4 AL/Q 1 3 In this balanced design the following effects are measurable. Kinetics: K = YI Y2 +Y3 Y4 Bed Height: H = Y4 Y2 + Y3 Y1 Bypassing: B = YI Y2 Y3 +Y4 The bypassing effect appears as an interaction between the two main effects and the techniques of statistical analysis can be used to get the most out of the data. At the cost of an extra run, considerable additional confidence can be obtained in the conclusions. In this example, we see the diagnostic value of a few wellchosen runs. Even more insight can be gained by abandoning the pilot reactor and studying .he reaction in an altogether different configuration. Electamine and carbon dioxide might be loaded into a pressure cell adapted to an ultraviolet spec trometer. The product Qacid vapor as well as the byproduct, electaminic acid be followed readily by its UV absorption. In this small batch reactor, 3 2Q 2L AL/Q 4 4Q 2L AL/2Q September 1963 CHEMICAL ENGINEERING EDUCATION 25 the approach to steady vapor pressure of Qacid could be followed conveniently at times close to the onset of reaction and at a series of temperatures. By interrupting the run at a given time and analyzing the catalyst and its organ ic contents, it is possible to associate the composition of organic solids with the progress of the main reaction. In this way, the puzzling dependence of rate on feed composition can be resolved. In the case on which this hypothetical example is based, it was found that a nonvolatile dimeric product of electamine forms rapidly on the catalyst with evolution of C02. In a second step, dimer reacts with C02 and forms a solid, "preQ", which rapidly develops a steady vapor pressure of Qacid. At the same time dimer reacts slowly with C02 to form the byproduct electaminic acid. Thus in the initial runs in which the loading of electamine was low, little dimer was formed and the conversion to Qacid proceeded rapid ly, being governed .by the rate of evaporation of preQ. In the runs in which the loading of electamine was high, dimer formed rapidly and the conversion of dimer to Qacid was slower, being governed by the rate of conversion of dimer to preQ. Thus a kinetic model might be written: E 4 Dimer + C02 (1) Dimer + C02 + PreQ (2) Dimer + C02  Electaminic acid (3) PreQ 4 Qacid vapor (4) Reaction 1 is supposed reversible but not instantaneous. Reaction 2 is irreversible and comparable in speed to reaction 1. Reaction 3 is irrever sible and slow. Reaction 4 is reversible and very rapid. Thus far, although rates have been measured, no use has been made of their quantitative magnitudes. The arguments have depended upon relative values, the shapes of timeconcentration curves and similar qualitative data. As a result, however, a reaction model has been reached in terms of which rate data can be quantitatively analyzed. In many cases, such a model is previously known or is sufficiently prob able that one can proceed to the quantitative phase with few preliminaries. In such a case, however, the experimental design should permit testing the fit of the data to the model as well as evaluation of the rate constants and acti vation energies. When a model has been proposed as the basis for further kinetic study, the methods of statistical experimental design can greatly reduce the number of runs required to determine the rate constants and reaction orders. These methods are now quite readily available in the literature 3 and it is not pro posed to discuss them in detail here. They must be used with insight, however, and are no substitute for thought. Some examples will illustrate the power and some of the precautions necessary in this approach. Srini Vasan and the writer4 studied the kinetics of the watergas shift reaction over a commercial iron oxide catalyst. The literature and previous experience suggested a kinetic model, due to Temkin: 7 R  dp = kPoa PH20b (5)  \PPH A Although the reaction is reversible, conditions were chosen to minimize the extent of backreaction. Taking logarithms, log R = log A E + a log pCO + b log PH20 b log p2 (6) RT This is a linear form in I/T and the log p. standard methods for the de swm and analysis of experiments permit the determination of best values for the parameters E/R, log A, a, and b as well as a test of the goodnessoffit of Equation (6) if fairly general conditions on the errors of measurement are satisfied. At no increased labor, the model could be generalized to include all of the possible components with arbitrary exponents: a b c d R = kPcO PH20 PH2 PCO2 26 CHEMICAL ENGINEERING EDUCATION September 1963 The geometrical interpretation of the experimental design is quite help ful. We may think of R as a function of l/T and the log pi which can be plotted in ndimensional Euclidean space with the independent variables as coordinate axes. We seek a mathematical representation of the hypersurface log R(1/T, log Pi). If Equation (6) is such a representation, log R will be a plane parallel to the axis of log PC02. Furthermore, its intercepts on the on the log PH20 and log PH2 axis will be equal but opposite in sign. If Equa tion (7) holds, the log R surface will still be planar, but there will be no a prior constraints on its orientation. The experimental design problem is now to test whether log R is indeed a plane, and, if so, to find its equation. With five factors, a 5plane in sixdimensional space is determined by five points. By determining log R at additional points, the deviation from plan arity can be tested. In the example cited, sixteen points were determined and it was decisively shown that Equation (6) could not represent the data within the reproducibility of experiments. If the log R surface is not planar, what shape is it? Standard methods now exist to determine the best quadratic approximation to log R. It is tempting to use this approximation, since the experiments already done to test the planar hypothesis are the core of the design which determines the quadratic approximation. Before succumbing to temptation, however, one should considerMia objectives carefully. Equation (6) or (7) has a theoretical in terpretation in that a molecular mechanism can be formulated which leads to It. Of course, the possibility of deriving a kinetic equation from a mechanics does not necessarily make it valid. However, the general quadratic form is not derivable from any mechanism and hence cannot possibly be valid except S an empirical interpolation formula. If the design studies which are contem plated will never go outside the range of the data taken, then an interp cs tion formula may be a sufficient representation of the kinetics. However, if extrapolation is necessary, much greater confidence can be had in kinetic  els based on the best mechanism which can be proposed. A second consideration is often important, however. Complex mechanisms involve many parameters. The general Langmuir mechanism 5 for the catalytic irreversible reaction A + B Products has the form kpA P5 (8) C +KA pA+ KB PB7 Each of the parameters k, KA, and KB is exponentially temperature dependent: kj = Aj eEa/RT (9) There are thus nine parameters to determine. Usually they will differ wide ly in magnitude, but theory will not predict in advance which ones may turn out to be negligible. It is a fact of the imperfect world that the more parameters that must be determined, the more difficult it usually is to estimate them. When the models are linear in the parameters, experimental designs may be found which will, in theory, allow good estimates to be obtained. The upper limit on precision is determined largely by the magnitude of the experimental error, the number of runs to be made and the experimental range of the variables. In practice, however, the operable region may be such that balanced designs are made impossible by interdependence of the supposedly independent variables That is, the process may not remain operable unless a change in one variable is compensated for in part by a change in another. In this way correlations tend to creep in to reduce the precision of the estimates as the number of parameters and variables increase. The only partial counter measures are (a) the difficult course of attempting to reduce the magnitude of the experimental errors or (b) an increase in the range of the variables. Chemical processes, however, always have finite restrictions on the operable range of the design variables. The workable temperature range is finite; permissable feed com positions may be limited by phase changes or explosion limits. September 1963 CHEMICAL ENGINEERING EDUCATION 27 When podels are non linear in the parameters, the same difficulties exist except that correlations between estimated parameters are more apt to exist. First the optimum theoretical designs in an unhampered experimental region whi d would maximize the precision of the estimates are usually ex tremely difficult to find mathematically and secondly, the nature of the function itself may make a high dependence of the estimates unavoidable. Hence in any actual case, there will be a maximum number of kinetic parameters which can be determined with precision from experimental data. The more care ful and precise the data, the larger this number becomes, but it rarely exceeds five or six. One must, therefore, scrutinize complex theoretical mechanisms to discover which of the many parameters are likely to be buried in the exper imental error. The quadratic empirical surface can be of great help in this process. 1,6 Thus, if Equation (8) is rewritten and expanded into the form of Equation (7), it will be discovered that KA and Kg occur only in quadratic terms involving I/T and the log p and log pB' respectively. Hence, if the empirical quadratic surface lacks terms in 1/T x log PA and log p, x log p one can be confident that PA can be omitted from the denominator of Equation (8) without worsening the fit. In fact, KA could not be determined with precision from the data that fixed the quadratic surface. In this way, complex mechanisms can be rationally simplified without overstepping the limits of reliability of the data. One may find, of course, that the data can be more simply represented by a new choice of coordinates. Thus Equation (8) is more naturally represented in the form (l/R)l/n 1 KA KB (10) k1 P/n B/n k pa/n1 /n k1 pa/n b/n1 This suggests 1/pA and 1/pB as better variables than log PA and log PB for testing this mechanism. In this space, the original experimental points which are wellspaced in log Pi may be poorly placed to give the best determination of the R1/n surface. New data may be required to determine the parameters of Equation (10) with precision. The availability of highspeed computing capacity modifies some of these possibilities. It may no longer be necessary to linearize the rate expres ,sion for computational reasons. Techniques for nonlinear estimation2 permit working directly with the proposed model. However, the number of parameters which can be determined simultaneously is limited as before and most theoret ical mechanisms must be simplified to make nonlinear estimation feasible. In some cases the ultimate design problem may be able to accommodate an integral kinetic form rather than a differential one. If, for instance, it is clear that a batch reactor will be used with no internal concentration gradients, only the total volume or residence time will be required for the design basis. Rather than design a differential reactor or differentiate in tegral data, one may then propose a kinetic model in which time is an explicit factor. Our previous remarks about the maximum complexity of a useful model are especially pertinent. These points are illustrated in the case of a study undertaken recently in connection with the purification steps of a commercial process. A minor impurity is removed by precipitation with aqueous ammonia. Complication arises out of the base catalyzed hydrolysis of the principal product, which represents a loss. The kinetic study was designed to locate the conditions under which yield at required purity could be maximized. However, since in design it might become necessary to modify some of the conditions, a kinetic model valid over a range of conditions was required. The change in concentra tion of the impurity is small and small hydrolysis of the principal product is expected, even though a large single stage integral converter is ultimately envisioned. As is often the case, some older data were available for which the ex perimental precision was only moderate. New data in both small and large reactors were obtained to test the assumed independence of yields on reactor size. In all, four sets of data were available: 1. New small reactor data 2. Old small reactor data 3. New large reactor data 4. Old large reactor data 28 CHEMICAL ENGINEERING EDUCATION September 1963 Four independent variables had been studied: 1. Temperature (T) 2. Weight per cent product initially in the impure mixture (P) 3. Weight per cent ammonia initially (N) 4. Reaction time (9) The response in each run was the precipitate produced, measured as y, per cent of the initial product p. Since it was desired to make use of the unplanned older data, an ortho gonal factorial design or central composite second order design was not pos sible. As a preliminary survey, a full quadratic model in four variables was fitted by least squares to four groups of data: a. Sets 1 and 2 b. Sets 1, 2 and 3 c. Sets 1, 3 and 4 d. Sets 1, 2, 3, and 4 An additional block variable was added for data in sets 3 and 4 to test the effect of reactor size. This effect proved to be nonsignificant and compar ison of residual mean squares of each group of data. showed no significant difference between groups. Hence all of the data was used in the final anal ysis. A full quadratic model in four variables has fifteen coefficients, four of which only serve to locate the origin with respect to which linear terms vanish. By rotating axes about this origin, the six crossterms can be elim inated, leaving only the four squared terms and the constant. This reduction to canonical form is done by proper choice of four new independent linear combinations of the independent variables. The results of this analysis are: y .65362 + .49654 z2 .09111 z2 .02671 zi .00083 z2 (11) where zI = .95990 TI .27634 P' + .02143 N' + .04214 9' (12) z2 = .08732 T' + .37390 P' + .92333 N' .00672 9' (13) z3 = .26183 T' + .688518 P' .38296 N' + .03520 9' (14) z4 = .04916 T' + .01703 P' .01892 N' .99847 9' (15) Here the primes refer to scaled values of the independent variables. For example, T' = (T(C.) 141.85)/27 (16) From Equation (11) we see that z1 is by far the most important term. By Equation (12), zI is nearly independent of N and 9 and measures T and p most strongly. Reaction time, 9, is almost identical with z4. Hence we feel justified in treating its small effect independently of the other var iables. Since many of the twentyone coefficients are smell and probably nonsignificant, one suspects that there are many alternatives to the quad ratic form which fit the data as well. Therefore, a reasonable mechanism was postulated as a guide to a simpler kinetic expression. Suppose the hydrolysis to be catalyzed by hydroxyl ion. Then one might have p + OH  POH (kl) (17) NH3 + H20 NH4+ OH" (k2) (18) dfPOH_7 = kl J .7j _ff (19) d9 September 1963 CHEMICAL ENGINEERING EDUCATION 29 ZU7 : K2 LH37 = K2 No 517 ffOHJ (20) i _7 ZH7 ofH3j P_7 Po ZPOH (21) Y : JOH:7 / Po (22) whence d dL0H ' = klk2 1 ( HE:7/No) (y Por o) (y) 23) d@ P0 d 12 [fHl7/No + (y Po/lo) since y remains much less than unity throughout all the tests, and 1/2 1/2 SZB o/H o = K2 /No (24) which is also small, Equation (23) can be simplified to SklK2N(1y Po/No) Po 7 (25) : kK2No klK2 (26) 'hen y is very small, the first term dominates and the Initial conversion should be given approximately by y = (k, K2No/Po)1/2 0 (27) This suggests a slightly generalized empirical kinetic expression log (y C) = log A g p log P + n log N5+ ( log 9 (28) T A preliminary estimate of the coefficients with their confidence limits gave the data in Case I of Table I. This confirms our earlier conclusion thet 9 could be treated independently as shown in Equetion (27). The data were the2 refitted fixing o( = p n = 0.5, their theoretical values in Equation (27) This gave Casell in Table I. Table I Case I Case II 5% Cony. Coefficient Limits Coefficient 95% Conf. Limits in A + 16.64 4.75 + 16.16 n + 8257.87 1889.65 +7648.17 192t.25 p 0.562 t 0.344 (0.50) 0 n + 0.261 t 0.567 (+0.50) 0 W + 0.978 t 0.279 (*0.50) 0 There is no significant difference in the fit between Equation 28 with four constants (Case I) and Equation 11 with fifteen. Note that while n and p are not significantly different from their theoretical values, v( is significantly larger than 0.5. Nevertheless, the fit forced with 0.5 (Case II) is not tremendously worse. 30 CHEMICAL ENGINEERING EDUCATION September 1963 Examination of the residuals shows that the fit is poorest at low No. In fact, some hydrolysis occurs even in the absence of added ammonia. The hypothetical mechanism does not allow for this. The empirical models based on it force the fit by averaging up the reaction order with respect to am monia. A more realistic model might arise by adding a term to Equation (26) which is proportional to 1y. This, when integrated, leads' to y = Cel + c2No ln ll No y (29) where Cl, c2 and 03 might each have exponential temperature dependence. The difficulty of fitting Equation (29) is much increased by its nonlinear form. For small y, Equation (29) reduces to Equation (28). Hence it will be essen tial to use the nonlinear form if any improvement is to be expected. In this example, an empirical quadratic form has again given insight in to what mechanistic terms should be retained in formulating a kinetic model with fewer constants to determine. It should be clear that there are a large number of kinetic models that will represent a given set of data. Unless these data are of very high precision, the fact that the engineer has found one such set lends very little support to the corresponding mechanistic in terpretation. However, qualitative features can be discerned and more sensi tive experimers are suggested by the analysis which may test the mechanistic assumptions in a less equivocal way. Nevertheless, the mechanistically in spired empirical kinetic form will usually be simpler and reliable over a wider range of variables than a pure linear or quadratic form in the original experimental variables. Thus the requirements of engineering data for design purposes can be met without sacrificing the best theoretical knowledge avail able. REFERENCES 1. Box, G. E. P. and P. V. Youle, Biometrics, 11, 287323 (1955). 2. Box, G. E. P. and G. A. Coutie, Proc. Inst. Elec. Eng., 103B, Suppl. 1, 100107 (1956). 3. Davies, 0. L., Ed., "Design and Analysis of Industrial Experiments," Oliver and Boyd, London (1954). 4. Hulburt, H. M. and Srini Vasan, C. D., A.I.Ch.E. Journal, 7, 143147 (1961). 5. Laidler, K.J., "Chemical Kinetics," McGrawHill, N.Y. (1950). 6. Pinchbbeck, P. H., Chem. Eng. Soi., 6, 105 (1957). 7. Temkin, M. I. and Kul'kova, N. V., Zhur. Fis. KhIm., 23, 695713 (1949). 
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