Analog Computers in Process Design (Application Bulletin No. 3)
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Analogue Computers In Process Design
Analog Computer
APPLICATION BULL
The THRey 10 Progressive Engineering
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electronic Associates, inc.
Manufacturers of ma AICTE] Precision Analog Computing Equipment
THE AUTHOR
CHARLES W. WORLEY
Mr. Worley studied electrical engineering at the Ohio State Uni-
versity where he received his B.S. degree in Automatic Control Theory.
He was employed by the Minneapolis-Honeywell Regulator Company,
Brown Instrument Division where he specialized in the field of process
control. In this field, he pioneered in the use of feedback control theory
as applied to process problems.
Originally employed as an Applications Engineer at Electronic Asso-
ciates’ Princeton Computation Center, Mr. Worley’s work covered the
development of industrial applications for the analog computer. He js
a licensed professional engineer and has authored several technical
papers on the subiect of process control.
In his present capacity as Manager of Market Development, he is
concerned with the use of the Analog Computer in solving engineering
and economic problems associated with the automization of industrial
processes.
Analog Computers in Process Design
ABSTRACT — The analysis of industrial processes requires the numerical
solution of a large number of linear and non-linear different equations.
The analog computer provides the process engineer with a practical
tool with which these solutions can be quickly and easily obtained.
However, use of analog computers for process analysis is presently
limited by lack of familiarity with them by many engineers. This paper
will discuss the use of the general purpose analog computer in the
study of process control systems and design of industrial processes.
the Key to progressive engineering
Introduction
The utility of the analog computer in the aircraft
and missile industries is an accepted fact. Almost all
major aircraft companies have large analog computer
installations which have proven to be extremely valuable
for the solution of problems associated with the design
of modern aircraft and guided missiles.
The use of the general purpose analog computer in
the processing industries has increased tremendously
during the past two years and indications are that they
can become an even more valuable tool for the solution
of problems in the processing industries. This seems
obvious to the thinking engineer since problems associa-
ted with the various branches of engineering are basically
the same. The problems associated with the guidance of a
ballistic missile are essentially the same as that of the
control of an industrial process.
Theory of Operation
It is of interest to note that the basic theory of oper-
ation and application of the modern analog computer is
not new. The similarity of the mathematical laws which
govern mechanical motion, heat transfer, fluid flow,
chemical kinetics, and the flow of electrical current
serves as a basis upon which analogies can be made
between different physical systems. Early papers by
Baker and Pachkis' have done much to develop the
direct analogy method of obtaining analytical and ex-
perimental solutions to enginering problems. Although
the direct analog has been and will continue to be of
use to the system engineer in visualizing complete
physical systems, its use as a general purpose equation
solver suffers due to some rather serious problems. Pas-
sive element analogs of complete physical systems re-
quire rather complex electrical circuits and consequently
it becomes difficult to eliminate purely electrical circuit
problems. There is also the problem of sizing and
selection of circuit parameters to fit the physical system
being investigated. For this reason active circuit ele-
ments, i.e. operational amplifiers, are used in the analog
computer of today.
The modern analog computer depends upon only one
analogous relationship and that is the integrating char-
acteristic of the charge on an electrical capacitor. Thus,
the analog computer performs basically as a mathema-
tical equation solver. Yet it retains the capabilities
of direct simulation which is its biggest attraction to
the engineer.
“This paper presented at ISA 1958 National Symposium for Chemical and
Petroleum Instrumentation, February |
2
)
and 4, 1958, Wilmington, Delaware.”
Electronic
Associates Ine.
The general purpose analog computer has the ability
to handle a wide variety of engineering problems, all
expressible in terms of differential or algebraic equations.
Unlike digital computers, analog computers employ dis-
tinct computing elements for each mathematical opera-
tion required to solve a given problem. This type of
operation, known as parallel operation, is an essential
reason for the very high computing speeds possible with
the d-c analog computer; most problems, regardless of
complication, are solved within seconds by general pur-
pose analog computers and within fractions of a second
by repetitive machines. Parallel operation, however, im-
poses some practical limits on the complexity of prob-
lems which can be solved on analog computers. The
average problem solved today employs from 15 to 40
integrations, 20 to 60 summations, 10 to 25 multiplica-
tions and divisions of two variables, and from 2 to 10
function generations. Problems involving from 40 to
150 integrations, 60 to 400 summations, 25 to 125 mul-
tiplications and divisions of two variables, and from 10
to 50 function generations are not uncommon. Accuracies
obtained vary from 0.05% to 0.50$¢ depending upon
the components used and the characteristics and com-
plexity of the problem. Basic component accuracies of the
modern analog computer are 0.01%.
Accordingly, it is customary to obtain more compli-
cated mathematical operations through combinations of
a limited number of simple operations, performed by
basic computing elements. It has been proved explicitly?
that a wide range of problems can be solved conveniently
by the application of only the following computing
elements:
1. Devices which multiply by positive or negative
constant coefficients.
. Devices which sum two (or more) variables.
Devices which produce the product of two variables.
. Devices to generate functions of variables.
. Devices that generate the time integral of a vari-
able.
WB Le bo
The machine variables which these components oper-
ate on are d-c voltages made proportional to the variables
of given problems. The inputs and outputs of each of
these components are collected at a central “patch board”
on the computer console. Plug-in “patch cords” are used
to connect the components as specified by the equations
to be solved. The conveniently measurable computer
voltages will then vary so that the records of their values
or behavior constitute solutions of the given problem.
Although the analog computer utilizes electronic
components and electrical circuit characteristics in its
operation, it is not essential that the analog computer
user have an extensive knowledge of electrical circuits.
In fact, experience has shown that any good engineer
can become proficient in the use of the analog computer
in a relatively short time; often in a matter of a few
weeks. With the analog technique it is only necessary
to translate the problem equations into a series of con-
nections between the standard computer components.
These connections are shown on a computer diagram
which results from programming the problem. For sim-
plification in programming, the computer components
are represented by symbols, characteristic of the mathe-
matical functions performed by the component, which
can be thought of as mathematical building blocks. The
resulting computer diagram becomes in effect a detailed
signal flow diagram of the problem being solved. The
analog computer symbols in most common use are shown
in Table 1.3
Analog Solution of Differential Equations
The type of problems best adapted to solution on an
analog computer are those involving systems of simul-
taneous differential equations, linear or non-linear, with
constant or varying coefficients. Fortunately, the com-
plexity of problem set-up is increased only slightly for
non-linear problems and problems involving non-
constant coefficients. Problems other than those which
belong in the category of ordinary differential equations
can also be satisfactorily solved with an analog computer.
The solution of differential equations with the analog
computer is based upon the mathematical technique of
repeated integration. The use of the method of succes-
sive integration is illustrated by the following sample
problem. Consider a spring-loaded pneumatically oper-
ated diaphragm motor used for operating control valves.
Oldenbourg and Sartariusthave shown that the equation
of this arrangement is
da? d
Moo + Dt Kx = Fit) (1)
Solving the equation for the highest derivative
d?x F(t) D dx Kx (2)
dec ~~ M M dt“ M
Equation (2) states that if voltages proportional to
F(t)M, (D/M)/(dx/dt), and Kx/M, are summed
the resulting voltage will be proportional to the second
derivative or valve travel. Integrating this voltage gives
the first derivative of valve travel. This second voltage
when integrated yields the valve travel x. Thus the
problem variables represented by computer voltages re-
quired for forming the second derivative of valve travel
have now been developed. These voltages are multiplied
by the system parameters and scaling constants and then
summed and integrated in the first integrator. The
complete computer diagram for solving this equation
is shown in Figure 1.
These simple programming techniques, although
here applied to a relatively simple system, are the same
as those used in programming more complex systems.
The ease in programming is evident. Each equation or
physical component may be treated separately, hence,
more complicated mathematical operations can be per-
formed through combinations of a limited number of
simple operations. Thus the analog computer can, in
effect, simulate the operation of large physical systems
since it combines and simultaneously solves the many
equations representing its behavior.
Applications To Industrial Processes
Because of the nature of their operation d-c analog
computers are particularly well suited for engineering
Application
Bulletin No.
EQUATION:
COMPUTER CIRCUIT:
— + KX = F(t)
]
+F(t) fa.
\wy
|
FIGURE |.
analysis since they lend themselves readily to the solu-
tion of the mathematical expressions describing the per-
formance of physical systems. A realistic appraisal of
analog computation techniques should point out, how-
ever, that the analog computer must be regarded only
as a tool for the solution of engineering problems. Since
the mechanics” of their use are relatively straight forward
and easily visualized any discussion of computer applica-
tions must deal primarily with derivation of equations
used to describe physical systems. This is a difficult sub-
ject to accommodate since the techniques of analysis of
physical systems encompass the entire field of engineer-
ing practices and fundamentals. Consequently, this dis-
cussion will primarily attempt to point out areas where
analog computers have been used with success in the
design of industrial processes and control systems.
‘Their use in solving engineering problems in the
processing industries will generally fall into two areas
of application: (1) Control System Design, and (2)
Applied Research and Processing Design.
Control System Design
Perhaps the most extensive and fruitful applications
have been in the field of automatic control engineering.
ANALOG SIMULATION OF A PNEUMATIC CONTROL VALVE MOTOR
D-c analog computing elements lend themselves naturally
to the representation of feedback loops analogous to
those used in control systems, and the analog nature of
the computer input and output data permits one to intro-
duce components of actual systems into the feedback
loops for system tests.
In the design of process control systems one of .the
questions to decide is which of the many types of control
systems is likely to give satisfactory operation. Ideally
each control system should be designed as a complete
unit to produce the required quality of contro] at the
least initial and running costs. In the recent past this
has not been done because the potential economic advan-
tages of such a design either could not or was not evalu-
ated. It has been normal practice for the designer of the
plant or process to call for certain standard types of con-
trol equipment, taking the advice of the instrument
manufacturer on their suitability for a given application,
and to depend on the flexibility of such equipment to
permit its adjustment to meet the plant requirements.
It has become increasingly apparent to many indus-
trial organizations® that this approach does not always
result in satisfactory performance of the process. They
Electronic Associates Inc.
THEORETICAL EQUATION: ,
P(t) = Ket —\ |[€-at
c
COMPUTER CIRCUIT:
SET POINT
+100
—MEASURED
VARIABLE
RESET
FIGURE 2. ANALOG SIMULATION OF A THEORETICAL PROPORTIONAL PLUS RESET CONTROLLER
VALVE CHARACTERISTICS:
% MAXIMUM F LOW
% VALVE TRAVEL
COMPUTER CIRCUIT:
+ VALVE TRAVEL o-
FIGURE 3: ANALOG SIMULATION OF CONTROL VALVE CHARACTERISTICS.
Application Bulletin No.
TRANSFER FUNCTION:
COMPUTER CIRCUIT:
+1) oO
K
t
ip) =
T, (TPH TP +N)
FIGURE 4. ANALOG SIMULATION OF JACKETED THERMOCOUPLE
are beginning to realize that the mere fact that a system
is working is not sufficient criterion for qualification as
a suitable control system. Consequently, control systems
for process requiring speed and precision of control are
ptesently being designed with the aid of the so-called
system engineering techniques. These techniques, based
on the theory of feedback control, offer a systematic pro-
cedure for a consideration of the dynamic behavior of
all components in the system, including the process. The
study of the dynamic behavior of a closed loop necessi-
tates the proper defining of the dynamic characteristics
of all the elements constituting the loop. Since the study
of the performance of physical devices involves a con-
sideration of their energy transfer characteristics, the
definition of the operation of control system components
normally results in some form of differential equation.
Broadly speaking, therefore, what is required is a
mathematical formulation of the measuring system, the
control valve, the controller, and the process. The first
three are rather easily definable since their mechanism,
although complex in many cases, can be usually repre-
sented by well defined physical fundamentals. Conse-
quently, the mathematical descriptions of instrumenta-
tion devices have become almost standardized and the
usual practice in the analog simulation of process control
systems is to use standard analog computer diagrams
with parameters or constants depending upon the par-
ticular manufacturer. In Figure 1 the analog-circuit for
a pneumatic control valve operator was presented. Fig-
ures 2, 3, and 4 show the computer simulation for a
theoretical proportional plus reset controller, a valve
body, and jacketed thermocouple.
The process because of its complexity and general
lack of basic data regarding its energy transfer mechan-
isms is generally much more difficult to define mathe-
matically. "The quantitative treatment of chemical pro-
cesses is complicated because heat, momentum, and mass
transfer frequently occur simultaneously with chemical
reactions. This coupled with the complexity of analysing
and correlating rate data for chemical reactions has hin-
dered the development of mathematical expressions for
defining chemical operations. Recent progress in the
science of chemical kinetics plus better experimental
techniques have led to significant progress in interpreting
processes where chemical reactions are accompanied by
physical transfer operations. This increased ability to de-
fine chemical process behavior has encouraged the use
of electronic computers which has in turn further in-
creased the understanding of process operation.
Initial attempts to define the controlability of indus-
trial processes were based on the time lag properties of
physical systems. Perhaps the first published attempt at
understanding the’ dynamics of processes was in a paper
by Ivanoff* in 1934, followed in 1936, by a joint effort
by Callendar,* Hartree, and Porter who investigated the
effects of time lag in a control system. The next step was
the application of the Servomechanism techniques to
process control systems discussed in 1950 by Rutherford.®
In 1952, McMahon! and Ackley pointed out those pro-
cess characteristics which affect automatic control. Since
that time much work has been done in the application of
these techniques to process control, and gradually a better
understanding of the dynamic behavior of a wide variety
of plants is being obtained.
These early papers, although doing much to help in
an appreciation of the basic energy transfer mechanisms
of industrial processes, primarily dealt with simple
processes on a linear basis. Despite the simplicity of this
treatment of processes equations difficult to solve manual-
ly were obtained when they were combined with instru-
mentation devices into a closed loop system. Hence, the
services of a computer were required. Medkeff and
Matthews! discuss the use of analog computers in solv-
ing these simplified process control problems.
Electronic
Associates Ine.
Such control investigations were based on the as-
sumption that the process lags are caused by capacitance,
resistance, and dead time effects. Although the visualiza-
tion and the sizing of these effects were based on a con-
sideration of the transfer of energy, there was a growing
realization that the energy mechanisms of physical
processes should and could be considered on a more
fundamental basis. Such an approach utilizes the basic
laws of physics, chemistry and mathematics. Mathematics
serve as the tool by means of which knowledge of physi-
cal and chemical principles can be applied.
Batke, Franks and James!” describe the results of a
more fundamental approach to the control of a large
chemical reactor. Their approach was to divide the
reactor into a number of physical zones. Heat and ma-
terial balances were then made for each zone resulting in
a series of differential equations, which when solved
simultaneously gave the performance of the reactor.
Although this approach is largely intuitive, it does
have a theoretical basis. The theoretical approach would
be to write the equalibrium equations with respect to
the proper space dimensions. The application of finite
difference techniques to the resulting partial differential
equation produces exactly the same equations as solved
by the authors.
As an illustration of this approach consider the trans-
fer of heat by conduction in a heat exchanger. Howe’
discussed the solution of heat conduction on the analog
computer by the application of finite difference tech-
niques.
EQUATION:
Mp Cp
g dt g
n
dTp =A
To be theoretically accurate, heat flow, like the flow
of electric charge, must be represented by a system which
has parameters regarded as distributed throughout the
body. For this reason “lumped” parameters are usually
assumed by dividing the heat transfer surface into sec-
tions of uniform temperature. The mass in each section
is assumed to be confined to the boundries of the section
while all increases or decreases in temperature are con-
sidered to take place between the sections of uniform
temperature. The accuracy of the analysis naturally de-
pends upon the number of sections considered.
The mathematical justification for this approach
is based upon the theory of equations of finite differences.
Equations obtained from the application of the theory
of finite differences to the general equation for the heat
flow by conduction are essentially those arrived at by
making a heat balance for each of the geometric sections
of the heat exchange equipment. A heat balance equation
for each section is a simple statement of the fact that
RATE OF HEAT STORED IN SECTION = HEAT
FLOW IN—HEAT FLOW OUT (3)
Since rates of heat flow are involved in the heat
balance, this equation usually takes the form of a simple
differential equation. The resulting equation and the
computer circuit required for solving it are shown in
Figure 5. The output of the computer circuit will be a
d-c voltage proportional to the average temperature in
the nth section providing the correct scale factors are
applied.
- nio+n non
= (T, Tp )+ 205C5(T, Tp )
COMPUTER CIRCUIT:
FIGURE 5. ANALOG SIMULATION
OF THE nth secTION OF A HEAT EXCHANGER.
Application Bulletin No.
EQUATION:
ttt Xntt= Vn Yn + WXw
COMPUTER CIRCUIT:
tha Xn sy
FIGURE 6. ANALOG SOLUTION OF MATERIAL BALANCE FOR BOTTOM PLATE OF A DISTILLATION
Since the equation was developed from a basic energy
balance it can, with some modifications, be applied to a
variety of heat transfer problems. Should the process
being analysed possess some form of chemical reaction,
then additional mass balance equations for each section
must be included. The conversion of mass by chemical
reaction must of necessity be included. By application of
these techniques exceedingly complex chemical processes
can be analysed on a dynamical basis.
Unfortunately, for the engineers, the analysis of most
physical systems, if carried out on a rigorius basis, re-
sults in some form of partial differential equation. The
problem of solving such equations is not simple, no
matter the method used. Solutions are usually restricted
to simpler cases in two or possibly three dimensions.
Fortunately, many engineering problems can be solved
with these restrictions. The analog computer can be used
to solve these simplified equations. For the two dimen-
sional case, with linear or non-linear equations, the results
are quite satisfactory with a reasonable amount of equip-
ment. However, for three or more dimensions, the
amount of equipment required becomes excessive and
the problem is seldom attempted.
Again the accuracies obtained depend upon the num-
ber of sections used and hence the amount of equipment
required. Fisher't shows that the accuracy of the finite
difference method can be expressed as a function of the
order of the difference and the number of sections.
For a second order difference
error = (Ax)?
COLUMN
For a fourth order difference
etror —= (Ax)*
Thus for a second order difference ten sections will
result in a 1% error in the method. For a fourth order
difference five sections will result in an error of 0.16%.
One encouraging factor is that a higher order difference
requires the same number of operational amplifiers as a
second order difference. However, more coefficient poten-
tiometers are required but they are relatively inexpensive
computer components.
What results can be obtained from an investigation
of process control systems with the analog computer?
Typical questions quickly answered are:
1. Will the proposed control system do a good job of
control?
2. What is optimum control?
3. What are the controller settings which will give
optimum control?
4. What is the best start-up procedure?
5. Are the pre-construction system specifications
adequate?
The question naturally arises as to how this approach
can improve control system design.
1. Different control schemes can be tried quickly and
easily.
2. The simulated control system can be disturbed
without upsetting production.
3. Safety limits can be evaluated without danger.
4, Efficient start-up procedure can be worked out.
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5. Process and instrumentation parameters can be
quickly changed.
Applied Research and Process Design
Because of the success of analog computers in solving
control problems for the aircraft industry, it was gen-
erally thought that the use of the analog machine in
the processing industries would be for control studies.
Although the analog has indeed proven valuable for
control studies, there is a definite trend toward more
of a preponderence of applications in the design of
processes.
The design of chemical processes requires a knowl-
edge of those parameters affecting the process and the
method of operation of the process. This information
along with the required production of product constitutes
the design conditions. They are the variables that must be
chosen by the engineer before the design can be carried
out. The optimum design is that which will be the most
economical, ie. that which will require the lowest total
cost per unit of product. Included in the total cost will be
initial design and construction expenditures and all
operating costs. The pattern for the ideal approach to
process design is apparent: first, the calculations should
be carried out for a number of design conditions which
are likely to result in low total costs and second, the
optimum conditions are chosen from the result of these
calculations. In the past this approach to design has not
been possible because of the complexity of the problem,
time involved, and cost. The rapid solution of design
equations by electronic computers allows the engineer
to at least approach this ideal in design practice.
Many calculations required for the design of chemical
and petroleum processes are based on trial and error
solutions. Classic examples are those which occur in
distillation calculations where many heat and material
balances have to be made to obtain a design. The elec-
trical rebalance feature of the analog computer can be
used to advantage for these types of calculations without
having to program a convergence to a solution. Franks
and O'Brien'” utilize this principle in performing steady-
state design calculations for a multi-component, non-
ideal distillation column. Consider the material balance
equations for the bottom of a distillation column:
Li +1 Xn + ae Vinym + Wxy (4)
Franks and O'Brien have developed the computer
circuit shown in Figure 6 for solving this equation.
The principle of electrical rebalance performs the
material balance as follows; assume that the output of
the integrator driving the servo multiplier is E, volts.
The integrator will continue to integrate until the net
sum of the inputs is reduced to zero and the computer
comes to balance with a constant output voltage. There-
fore,
Ln4t E, —_ Vin¥n —Wxy — O (5)
A comparison with the material balance equation
shows that E, is now equivalent to the liquid composition
x. Thus by supplying voltages proportional to the pro-
ducts Vinyu, Wx, and the value L, this circuit will
always rebalance to give the value of liquid composition
which satisfies the material balance equation. Since the
design of many types of processes require the calculation
of equalibrium conditions, analog circuits such as this
can be extremely useful in solving chemical design prob-
lems.
Investigating reaction mechanisms are another group
of problems which can be solved with the analog com-
puter. The investigation of reaction kinetics for various
proposed mechanisms is largely a trial and error pro-
cedure since a promising rate equation is assumed, and
then the design calculations are carried out to see whether
or not the predicted conversions agree with experimental
data. By repetition of this procedure an equation for
the rate can be found.
As an example consider the following example of a
first order reaction carried out adiabetically. Typical
equations for this reaction are:
dc
= ke (6)
dt
If k is given by the usual Arrhenius Equation, the
equation for the rate of change of the reaction rate can
be shown to be
dk kE dT
—=;|—|; — (7)
dt RT- dt
This equation can be solved with the computer cir-
cuit shown in Figure 7. By changing the parameters in
the reaction, the reaction curve obtained on the compu-
ter can be matched to experimental data. When the data
is matched, the proper mechanism has been obtained.
Should several reactions be taking place they are simul-
taneously represented by similar circuits. Usually a mass
and heat balance is made to obtain the reaction tempera-
ture and the product produced. The net effect is to design
a chemical experiment electronically to give a simulated
pilot plant. Since full scale operating conditions are
easily scaled into the computer as pilot plant data, the
computer simulation is usually based on the actual
process. This does not mean to imply that a pilot plant is
no longer required. The hope is, however, that with the
analog computer the use of pilot plants in process design
can be put on a mote rational basis.
Analog computers are also well suited to the solu-
tion of catalytic reactor design problems. Programming
simplicity is maintained even when complex reactions
are considered for steady state or transient cases. Wehner
and Wilhelm!’ have discussed an isothermal reactor in-
volving axial diffusion and flow with a first order
Application Bulletin No.
EQUATION:
COMPUTER CIRCUIT:
FIGURE 7, ANALOG SIMULATION FOR A FIRST ORDER CHEMICAL REACTION.
EQUATION:
COMPUTER CIRCUIT:
NOTE:
X=|(-Z
FIGURE 8. ANALOG SIMULATION OF SIMPLE MODEL OF A CATALYTIC REACTOR
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reaction. The computer circuit for this simple model
is shown in Figure 8. Note that the circuit is set up so
that the integration is carried out in the opposite direc-
tion; ie., from the back of the reactor to the front. The
conclusions reached by the authors dictate this change
in variable if an analog computer solution is to be
obtained. This basic model may be expanded to more
complicated investigations with small changes in cir-
cuitry. These are; (a) higher order reactions, (b)
adiabatic reactor, (c) nonsteady state operation, and
(d) consideration of radical and axial diffusion.
Because of the similarity of the basic equations de-
scribing chemical reactions, there appears to be a wide
variety of chemical design problems amiable to solution
by the analog computer. A partial list of possible appli-
cations are:
Classical Kinetics Cooler Condenser
Particle Fluid Transfer Drying
Transfer in the Flowing Liquid Distillation
Catalysis Absorption
Solvent Extraction
Zone Melting
Fluidized Bed Reactors
Adiabatic Absorption
Ion Exchange
Conclusions
This discussion has attempted to point out the types
of processing problems which can be solved by Analog
Simulation. The use -of the analog computer in the
processing industries has increased tremendously during
the past year and indications are that this trend will
continue. Perhaps the most important reason for this
increase is the enthusiastic acceptance of the analog
computer as the “engineers tool” by those who have
performed calculations with the machine. The simplicity
of programming has allowed the engineer with the prob-
lem to conduct the investigation himself. Rapid changes
in programs, which were unforseen until problem solu-
tions were obtained, are easily made during the course
of the investigation and can be guided by the engineer
who knows the problem. The net result is that the
engineer develops a “feeling” for the problem which
helps him produce better results and often at a cost of
several magnitudes less than that of other forms of
computation.
Experience has shown that analog computers are
highly capable tools for solving engineering problems
associated with the design and control of industrial
processes. Their use effectivel” expands the utility of the
engineer. Yet they cannot think for him nor will they
automatically solve his problems. In fact, considerable
effort must be expanded before solutions to problems are
obtained. Because of this, the use of a computer in
engineering work generally does not replace any men
but rather increases the amount of effective work that
can be done by the same staff. One important result is
that problems and methods are set up for computer
solution that would never be considered wichout the
machine. Another advantage and one that is often over-
looked is that the techniques associated with the use of
a computer allows problems to be classified which in
itself encourages a broader investigation of possible
solutions to problems. This cannot help but result in
improved engineering effort.
Nomenclature
M = Mass of moving parts including diaphragm
plates, Ib/FT-sec °.
D == The pneumatic damping constant, Ib/FT-sec.
K = Spring constant, lb/FT.
x == The travel of the Valve stem, FT.
F — The force of the actuating air pressure, lb.
P, = Controller output pressure, PSI.
é = Controller error, dimensionless.
K. == Controller gain, dimensionless.
Tr = Reset time constant, seconds.
Ty = Thermocouple output temperature, "F.
T; = Thermocouple input temperature, °F.
Ki = Thermocouple proportionality constant, di-
mensionless.
71 == Thermocouple well time constant, seconds.
Ts = Thermocouple time constant, seconds.
l = Number of sections used to represent heat
exchanger.
Te == Temperature of vapor in nth section, °F.
Ty, == Temperature of product in nth section, °F.
T,* = Average temperature of product in nth sec-
tion, °F.
t = Time in seconds.
@, = Product mass flow rate, lbs/sec.
M, = Product mass, Ibs.
Cc, = Product specific heat, Btu/Ib-°F.
U = Overall heat transfer coefficient, Bru/sec.-
FT?-°F,
A = Heat transfer area, FT?.
Lisi = Liquified flow from plate above, Ib. mol/hr.
Xn+1 = Liquid composition on plate above in mol
fractions.
Vn = Vapor flow from bottom plate, Ib. mol/hr.
Yu == Vapor composition on bottom plate in mol
fractions.
Ww = Bottoms product take off rate, lb. mol/hr.
Xy = Bottoms composition in mol fraction.
k = Reaction rate constant.
Cc == Concentration of reactant.
E = Activation energy.
R, == Gas constant
P.. = Peclet number (uL/D), dimensionless.
R = Reaction number (kL/u), dimensionless.
f = Fraction of a reactant remaining, dimension-
less.
Zz == Normalized distance variable, dimensiontess.
u = Flow velocity, length/time.
L = Reactor length, length.
References
1. PASCHKIS, V.; and H. D. BAKER; “A Method for
Determining Unsteady-state Heat Transfer by means
of an Electrical Analogy,’ Trans. ASME, 1942.
2. SHANNON, C. E.; “Mathematical Theory of the
Differential Analyzer’; J. Math and Phys., 20:337,
1941.
Application
Bulletin No.
»
“Symbols for PACE General Purpose Analog Com-
Puter,” Simulation Bulletin No. 7; Electronic Asso-
ciates, Inc.
OLDENBOURG, R. C.; and H. SARTORIS; “The
Dynamics of Automatic Control,” Text Published
by ASME February, 1958.
JOHNSON, C. L., “Analog Computer Techniques,”
McGraw - Hill Book Company, Inc., New York,
N. Y., 1956.
JACKSON, WARREN, JR. “Sohio Gives Analog
Computers A Trial,” The Oil and Gas Journal, July,
1957.
IVANOFF, A., “Theoretical Foundations of the Au-
tomatic Regulation of Temperature” Journal Insc.
Fuel, 1934.
CALLENDER, A. HARTREE, D.R.; and’ PORTER.
A. “Time Lag in a Control System,” Phil. Trans..
1936,
RUTHERFORD, CL; “Practical Application of
Frequency Response Analysis to Automatic Process
Control,” Proc. Instrument Mechanical Engrs., 1950.
10. McMAHON, J.B.; and R.A. ACKLEY: “The Process
11,
12.
13.
14.
15.
16.
Characteristics Which Affect Automatic Control’;
ASME Paper No. 52-SA-3.
MEDKEFF, R.J.; and H. MATTHEWS; “Solving
Process Control Problems by Analog Computer”;
Instruments and Automation, October, 1954.
BATKE, T.L.; FRANKS, R.G.; and EW. JAMES;
“Analog Computer Simulation of a Chemical Re-
actor,” ISA Paper No. 56-7-2.
HOWE, R.M., “Application of Difference Tech-
niques to Heat Flow Problems Using the Electronic
Differential Analyzer,’ Univ. of Michigan, May,
1954.
FISHER, M.E., “Higher Order Differences in The
Analog Solution of Partial Differential Equations,”
Journal of Assoc. Comp. Mach., Vol. 3, No. 4, P.
325, October, 1956.
FRANKS, R.G.; and N.G. O'BRIEN; “Application
of General Purpose Analog Computer to Steady
State Distillation Calculations,’ A.I.Ch.E. Seattle
Meeting. June, 1957.
WHENER, J.F.; and R.H. WILHEIM; “Chemical
Engineering Science,’ 6 89 (1956).
Electronic
Associates
Inc.
TABLE I
SYMBOLS FOR PACE GENERAL PURPOSE ANALOG COMPUTER
NAME SYMBOL FUNCTION DESCRIPTION
HIGH
AMPLIFIER 8
G>I0
u V, Vo =-(V, +10Va —5Vz) AMPLIFIER
SUMMER +V2 0-———— fe) fe) { 2 3
—Vv3 0 MULTIPLE INPUT
+v,o-——5] |
INTEGRATOR Vo Vo= -ftsy -Vp) dt AMPLIFIER
=Vp o-——_ MULTIPLE INPUT
=
COEFFICIENT V van V Vo= KV; MANUALLY SET
POTENTIOMETER 1° ay, o 0 O<K<| POTENTIOME TER
+Vo Viv
SERVO + | Ne
SERVO Va z+ SERVO DRIVEN
MULTIPLIER +ILo y, POTENTIOMETER
100 Vo HIGH GAIN
DIVISION Yo* + Dee
CIRCUIT Hoo SERVO DRIVEN
| POTENTIOMETER
ViVe
ELECTRONIC Voz ——35 ELECTRONIC
MULTIPLIER +Vig MULTIPLIER
V y
2 + V2 /N2 Vo
SERVO
+ {LO )
SERVO FUNCTION
GENERATOR Vv
ARBITRARY
DIODE Vio-_——|_ pF Vo FUNCTIONS
FUNCTION V
GENERATOR 0
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