THE AMEPICAN SOCIETYY nnr I~ t 1 IIJ
CHEMICAL FNSINEERING EDUCATTION
Chemical Engineering Division
,aerican Society for Engineering Education
jn-Stream 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
J. B. West
Chemical Engineering Division
American Society for Engineering Education
martin (Michigan) Chairm
it (Iowa State) Vice C
(Oklahoma State) Secret
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
Thompson-Ramo-Wooldridge 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 afore-mentioned 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 on-line 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 on-line control computers.
In order to orient our thinking along the lines of on-stream 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 on-stream
control computers. At that time the score card showed 35 closed loop com-
puting control installations, either on-line or scheduled to be on-line 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.
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. Mono-ethanol amine is used as the absor-
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. The-main 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
SURVEY OF PROCESS COMPUTER APPLICATIONS
1. Allied Chemicnl Corp. RW-300
South Point, Ohio
2. American Oil Company IBM-1710
3. B.A.S.F. RW-300
4. B.F. Goodrich Chemical RW-300
Calvert City, Kentucky
5. Celanese Corporation
Bay City, Texas H-290
Bishop, Texas RW-300
6. Continental Oil Company RW-300
Lake Charles, La.
7. Dow Chemical Company GE-312
Beaumont,Texas (2) ISI-609
Florence, S.C. ISI-609
Circleville, Ohio ISI-609
Gibbstown, N.J. IBM-1710
9. DX-Sunray RW-300
10. Gulf Oil Company RW-300
11. Imperial Chemical Ind. Ferranti
12. Monsanto Chemical Co.
Luling, Louisiana RW-300
Chocolate Bayou, Texas(4) H-290
13. Owens-Corning Fiberglass ISI-609
Aiken, S. Carolina
14. Petroleum Chemicals,Inc. RW-330
Lake Charles, La.
15. Phillips Chemical Co.
Borger, Texas Recomp II
Bartlesville, Oklahoma TRW-330
16. Shell Development Company PB-250
17. Sun Oil Company IBM-1710
Marcus Hook, Pa.
18. Standard Oil Co.(N.J.)
Linden, N.J. LGP-30
Baton Rouge, La. LGP-30
19. Standard Oil Co.(Calif.)
El Segundo, Calif. IBM-1710
Richmond, Calif. Recomp
20. Tennessee Eastman GE-312
21. Texaco RW-300
Port Arthur, Texas
22. Tidewater Oil Company ISI-609
Delaware City, Delaware
23. Union Carbide Corp.
Charleston, W. Va. RW-300
Seadrift, Texas RW-300
Seadrift, Texas Daystrom
24. Universal Oil Products Daystrom
Des Plaines, Illinois
Vinyl chloride and
Vapor phase oxidation
Pilot plant logger
Pilot plant logger
Pilot plant logger
CHEMICAL ENGINEERING EDUCATION
CRITERIA FOR JUSTIFYING MONSANTO'S
COMPUTER CONTROL SYSTEM
1. Maintain maximum gas flow in spite of changing weather and process
2. Maintain an optimum hydrogen-to-niyrogen 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
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 steam-to-dry 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
build-up 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 engine-compressor 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 CHEMICAL ENGINEERING EDUCATION 5
"Immediately after placing the computer on control, the gains
in control-ability 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-
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 over-all 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-
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
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
FUNDAMENTALS REQUIRED OF CHEMICAL ENGINEERS FOR
UNDERSTANDING OF DIGITAL COMPUTER CONTROLLED PROCESSES
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
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
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
c. H draulic
2. Measurement equipment
3. Analytical instruments
c. Physical properties
B. Methods of interconnection
C. Reliability and accuracy
VIII Computer Fundamentals
A. Types and characteristics
B. Applicability of computers
C. Basic understanding theory of operation
a. Flow charting
c. Machine language
e. Routines and sub-routines
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 on-stream 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 sub-routines.
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 man-machine-process combination
will be increased in the future through continued research efforts in all the
areas that have gone into its design.
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 Semi-continuous
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
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., "On-line 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., "On-line 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.
21. Eliot, T.Q., & Longmire, D.R.,"Dollar Incentives for Computer Control",
Chem. Eng., Jan. 8, 1962.
CHEMICAL ENGINEERING EDUCATION
OPTIMIZATION THEORY IN THE CHEMICAL ENGINEERING CURRICULUM
Douglass J. Wilde
Department of Chemical Engineering
University of Texas
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 decision-making.
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 three-hour, 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-
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 multidthi-t 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.
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 semester-hour 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.
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.
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
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 sub-problems, and re-
combine these sub-optimal 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 straight-line 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 system-satellite 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 time-dependent 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.
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 one-variable 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 non-Euclidean 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
far-sighted philosophy of dynamic programming, which begins by analyzing the
last rather than the first decision in a sequence. Perhpas the most sur-
pr-sing 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.
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
1. Kiefer, J., "Sequential Minimax Search for a Maximum", Proo. Amer.
Math. Soc, (1953), pp. 502-506.
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. 1-20
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
6. Hooke, R., and Jeeves, T.A., "Direct Search Solution of Numerical and
Statistical Problems", J. Assoc. Computing Mpch. 8, 2 (April 1961),
7. Mugele, R.A. "A Monlinear Digital Optimizing Program for Process Con-
trol Systems", Proc-Western 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. 39-55.
9. Robbins, H.,dnd Munro, S., "A Stochastic Approximation Method", Ann.
Math. Stat., 23 (1951), --. 400-407.
10. Kiefer, J., and Wolfowitz, J., "Stochastic Estimation of the Maximum
of a Regression Function", Ann. Math. Stat., 23 (1952) pp. 462-6 .
11. Kesten, H., "Accelerated Stochastic Approximation", Ann. Mpth. Stat.
29, (1958), pp. 41-59.
12. Blum, J.R., "Multidimensional Stochastic Approximation Methods", Ann.
Math. Stat., 25 (1954), pp. 737-44.
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, McGraw-Hill,
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,
21. Chang, S.S.L., Synthesis of Optimal Control Systems, McGraw-Hill,
New York (1961).-
FUNDAMENTAL PROBLEMS IN HETEROGENEOUS CATALYSIS
Max. S. Peters
University of 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-
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 Langmuir-Hinshelwood 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 least-squares 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.
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 (c-l)p
V(PO-P) 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 = bisie-Ei/RT (3)
a2psi b2s2e-E2/RT (4)
ai pi-i = bisi5e-Ei/RT
where inca b are constants, s represents the surface area covered only
by the subscript-indicated 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:
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
si 1-1 ( ) s ()) l = c (, C "
where J= ge-Ev/RT rnd ( ':v)/R
Therefore, n i
cso i = 1 (
Vm so cso n (E )i
16 CHEMICAL ENGINEERING EDUCATION September 1963
i= 1 1
Because i (P)1 converges to p/J and ( ) converges
j (1o-p/) 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 1-p/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(po-p) 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 ofEi-Ev
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 eE-p
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
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-
muir-Hinshelwood 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
power-series type of rate equation would be
rate k(PA) (pB)m (p) ..... ()
while the Lsngmuir-Hinshelwood 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 engmuir-Hinshelwood 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
The over-zealous 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)
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 so-called activee sites" on the sur-
face. Thus, the reaction could occur between an adsorbed reactant molecule
and a gas-phase 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 dual-site 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 CT-CV 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 duel-site
concentration must be one-hall of the total number of possible .djaccat.
sites times the fraction of total sites occupied by the m teail-1. in
case, the one-half 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 one-half should not be included. For this case, therefore,
Total dual sites = SCT = Sites V-V Sites p-V Sites A-A (18)
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 solid-state 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 charge-transfer process and the resulting energy-level 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 Fermi-Dirac 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 so-called holes (p). Thus an n-type semiconductor has
a conductivity due to excess electrons and is a donor while a p-type 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 n-type semiconductor. If the catalyst is initial-
ly a p-type material it might be changed to an n-type 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
p-type to n-type semiconductor.
Sentn-kr AA N*luGINEERING EDUCUATIOJl 19
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 plug-flow 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 plug-flow 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 J2(uC) J(uC) rC = C2)
u jZ-7 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)
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 first-order irreversible reaction, r = kC, and steady state,
Equation 25 becomes a second-order, 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)
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 (L-Z) usa
C e2 fl (Z-L)
to e 2(1 a')eL -Z(l-a)e (29)
ua'L -ua L
(1 a ') e (1-a')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 plug-flow 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.
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 TR-10.
For the case of the following consecutive reactions carried out
A k B k2 C
the rate equations for a constant-volume reactor are
dCA klCA (30)
d k / (31)
dCB k1CtA -k2CB (31)
September 1963 CHEMICAL ENGINEERING EDUCATION 21
Analytical solutions of the first order linear differential equations
Ca = CAo (32)
CB klCAo (e-k1 -k2) (33)
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 concentration-time plot th-t
would result on the x-y plotter from the computer is also shown in Figure
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 concentration-time values for direct conporison to
the computer results.
Volts Figure 2.
CA Potentiometer Diagram
S For Solving
IC CA A-" B C
k, Resultant X-T Plot
Integrator From Computer
S CC -Axis
6 CA -de Ic
S Volts X-Axis
R A A B k-a.C
" 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-
muir-Hinshelwood 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).
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).
1. Aris, R., Chem. Eng. Sci., 6, 262 (1957).
2. Bailie, R.r. and. Fpn, "Computer Program Abstract 041--Operational
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 CompuTer-PFogram Aistracr-027, to be
published in Chem. Eng. Prog. (1962).
9. Carberry, J. 77"Transpor -henomena and Heterogeneous Catalysis," Forth-
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. Garoia-Moliner, 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- KT-R. 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 Sem-icondu-corTT" D.-Van Nostrand Co.,
21. Smith, J. M., "Chemical Engineering Kinetics," P. 243, McGraw-Hill 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, "AdVa-nes-Tn Ca-l78yss," 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
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 ill-defined 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 time-consuming 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
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 steady-state 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 well-planned runs.
Consider a hypothetical example in which Q-acid is made by catalytic con-
version of electamine and carbon dioxide in a fluid bed. Under reaction con-
ditions, Q-acid 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
First experiments might be to vary the gas residence time in the con-
verter, determining the space-time yield and purity of the elpctmilne product.
For experimental convenience in these survey runs, the process Is not run in
continuous steady-state. Instead, electamine is added to the catalyst to a
predetermined loading at a temperature below that nt which Q-acid is formed.
The temperature is then raised and Q-acid stripped off in a stream of C0, suf-
ficient to fluidize the bed. The concentration of -acid in the product'stre-r
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.
A2 CHEMICAL ENGINEERING EDUCATION September 1963
At each temperature, the partial pressure of Q-acid in the off-gas 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 Q-acid 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 Q-acid had been stripped off. These results give a
strong presumption that Q-acid is being formed in vapor-solid equilibrium with
the organic matter on the catalyst, since as much conversion was obtained in a
run-as 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:
Conversion of electamine to Q-acid 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 by-passing
B 2. Runs 1 and 3 (Same conversion fNo bypassing
Different con- tBypassing occurs
From this logic, Run 4 appears superfluous. However, adding it gives
a 2 x 2 factorial experiment in bed-height and residence time.
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 well-chosen 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 Q-acid vapor as well as the byproduct, electaminic acid
be followed readily by its UV absorption. In this small batch reactor,
September 1963 CHEMICAL ENGINEERING EDUCATION 25
the approach to steady vapor pressure of Q-acid 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 non-volatile 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, "pre-Q", which rapidly develops a steady vapor pressure of
Q-acid. 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 Q-acid proceeded rapid-
ly, being governed .by the rate of evaporation of pre-Q. In the runs in which
the loading of electamine was high, dimer formed rapidly and the conversion
of dimer to Q-acid was slower, being governed by the rate of conversion of
dimer to pre-Q.
Thus a kinetic model might be written:
E 4 Dimer + C02 (1)
Dimer + C02 ----+ Pre-Q (2)
Dimer + C02 ----- Electaminic acid (3)
Pre-Q 4 Q-acid 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 time-concentration 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-
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 water-gas 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 back-reaction. Taking logarithms,
log R = log A E + a log pCO + b log PH20 b log p2 (6)
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 goodness-of-fit
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 n-dimensional 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 5-plane in six-dimensional 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 e-Ea/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-
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
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/n-1 /n k1 pa/n b/n-1
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 well-spaced in log Pi may be poorly placed to give the best determination
of the R-1/n surface. New data may be required to determine the parameters of
Equation (10) with precision.
The availability of high-speed computing capacity modifies some of these
possibilities. It may no longer be necessary to linearize the rate expres-
,sion for computational reasons. Techniques for non-linear 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 non-linear 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
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 non-significant 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-
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 cross-terms 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)
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.
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 twenty-one coefficients are smell and probably
non-significant, 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)
September 1963 CHEMICAL ENGINEERING EDUCATION 29
ZU-7 : K2 LH37 = K2 No 517- ffOH-J (20)
i _7 ZH7 ofH-3j
P_7 Po Z-POH (21)
Y : JOH:7 / Po (22)
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
SZB o/H o = K2 /No (24)
which is also small, Equation (23) can be simplified to
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)
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.
Case I Case II
Coefficient Limits Coefficient 95% Conf.
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 1-y. 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 non-linear form.
For small y, Equation (29) reduces to Equation (28). Hence it will be essen-
tial to use the non-linear 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-
1. Box, G. E. P. and P. V. Youle, Biometrics, 11, 287-323 (1955).
2. Box, G. E. P. and G. A. Coutie, Proc. Inst. Elec. Eng., 103B, Suppl.
1, 100-107 (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, 143-147
5. Laidler, K.J., "Chemical Kinetics," McGraw-Hill, 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, 695-713 (1949).