Handbook of Analog Computation
HANDBOOK OF
ANALOG
COMPUTATION
HANDBOOK OF
ANALOG
COMPUTATION
© ELECTRONIC ASSOCIATES. INC. 1967
•
PRINTED IN U.S.A.
•
PUBL. NO. 00800.0001-3
•
JULY 1967
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Computation Center
HANDBOOK OF ANALOG COMPUTATION
Prepared by
The Education and Training Department
Edited by
Alan Carlson, George Hannauer,
Thomas Carey and Peter J. Holsberg
SECOND EDITION
Electronic Associates, Inc.
Princeton, New Jersey
~ Electronic Associates, Inc., 1967 -- All Rights Reserved
These notes, or any part thereof, may not be copied or
reproduced in any form without written permission of
Electronic Associates, Inc.
PREFACE
These notes on Analog Simulation have been developed from the experience
gained by the Education and Training Department of EAI in presenting intensive
short courses in analog computer operation, programming, and applications
for nearly a decade.
The objective of these courses has been to provide scientists and enginee~s
with a working knowledge of the analog computer and its uses. They have proven to be most effective when lectures and demonstrations are supplemented with
laboratory sessions allowing students to put theory into practice. The solution of problems on an analog computer, using effective and efficient programming techniques and check-out procedures, has proven to be invaluable in gaining
familiarity both with the machine and its potential as an engineering tool.
Many of the procedures and techniques described in the notes have been used
and found to be effective in EAI Computation Centers throughout the world.
A course in analog computation utilizing these notes could readily meet the
requirements of an accredited, one semester, 3-credit-hour university course.
A course in differential equations as a prerequisite is desirable.
The wide range of application for the analog computer permits the introduction of actual applications appropriate to courses in all scientific disciplines. The EAI Applications Reference Library is a source of a large number
of such studies describing applications in such areas as electronics, chemical
processing, aerospace engineering, and life sciences.
These notes represent the combined efforts of a large number of people within
the EAI organization. Contribution to the notes and the editing were made by
A. I. Katz, O. Serlin, H. Davidson, and J. J. Kennedy, as well as many others.
Many sections in the notes were derived from material generated by the various
Departments in the Research and Computation Division of EAI.
The editors, in particular, would like to acknowledge the efforts of the secretarial staff, Helen Lynch, Bette Davis, and Ginny Gafgen in helping organize
the typing and production of these notes on a time schedule that was agreed
upon by all as being impossible.
TABLE OF CONTENTS
CHAPTER I ••••• THE ANALOG COMPUTER AND ITS ROLE IN ENGINEERING ANALYSIS. • •. 1
CHAPTER II.oooTHE GENERAL PURPOSE ANALOG COMPUTER . . . . . . . . .
CHAPTER 111
0
••
CHAPTER Vo •••• TECHNIQUES IN FUNCTION GENERATION • .
CHAPTER Vloo •• TRANSFER FUNCTION SIMULATION . . .
0
•
••••
16
ANALOG COMPUTER PROGRAMMING AND CHECKING PROCEDURES . . . . . .84
CHAPTER IV.oo.ANALYSIS OF LINEAR AND NON-LINEAR SYSTEMS
CHAPTER VII.
0
0
CHAPTER IX o•• ANALOG MEMORY . . . . .
0
•
0
••••
•
•
•
0
•
It
• • • • • • •
0
•
0
0
0
•
0
•
0
•
0
•
CHAPTER X••••• PROBLEM PREPARATION PROCEDURE •.
0
••••
0
TRANSPORT DELAY SIMULATION
CHAPTER VIII •• REPETITlVE OPERATION.
0
0
• • • • • • •
0
••
0
0
0
••
0
0
•••
•.
•
•
•
•
•
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0
•
0
••••
•
•
•
•
•
•
•
•
•
•
0
•
0
••
0
• • • • • • • •
0
CHAPTER XIII •• EXAMPLES OF EFFICIENT PROGRAMMING . . .
0
0
0
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•
0
•••
0
~
• • • • • • • • •
0
163
191
220
241
248
260
. 279
•
• • • • •
CHAPTER XIV ••. FACTORS IN PLANNING AND OPERATING AN ANALOG LABORATORY
•••
•
•
•
CHAPTER XII ••• ACCURACY OF ANALOG COMPUTER SOLUTIONS
0
•
•
CHAPTER XI ••• oANALOG COMPUTER SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
Appendix Ao.~.LaPlace Transforms . . •
••
135
289
296
. • . 354
• • • • • • • • • •
366
Appendix B.o •• Transfer Function Circuits.
369
Appendix C.o •. Diode and Relay Circuits.
375
Appendix D•..• Selected Applications Bibliography • • • • • • . . . . . • . . 384
CHAPTER I
THE ANALOG COMPUTER AND ITS ROLE IN ENGINEERING ANALYSIS
A.
Introduction
The role of the electronic, general purpose analog computer in modern-day
industry best can be explained by considering the concept of engineering design. When a design is required, one or more engineers or scientists propose
a system which they feel 'viIi satisfy the design criteria. Design proposals,
however, involve approximations and estimates and there may not be concrete
agreement as to which design is best. Therefore, some form of evaluation of
the proposed system is desirable.
In evaluating proposed systems or designs, one can, in general, select either
of two paths: an experimental program, or an analytical evaluation of the
system. The experimental approach is usually characterized by a minimum of
of analysis, the construction of a prototype of the system, and considerable
"trial-and-error" experimental work. The objectives are to evaluate the experimental data and suggest appropriate modifications which will result eventually
in an optimum or nearly optimum design. The cost and time required for this
experimental approach are normally much greater than those incurred in an
analytical evaluation. In the analytical approach,the task is to derive a set
of equations (a mathematical model) whose solution will describe the behavior
of the system in terms of its geometry, time, and parameters. These solutions
then can be used to obtain operating conditions and parameters which will result in optimum system performance.
Since the derivation of mathematical models frequently requires approximations,
and the results obtained are often based on limited input data, prototype
experimentation usually is required. However, pilot plants designed on the
basis of extensive analytical investigations frequently are near optimum and
require little or no modification. The only experimenta~esults required
are those which validate the mathematical model. Once the model is validated,
additional experimentation can be performed analytically, which results in a
considerable cost reduction compared 'to the experimental approach.
Not all proposed designs, unfortunately, lend themselves to a choice of evaluation programs. If one cannot obtain a mathematical model, there is no recourse
except an experimental program. On the other hand, if the cost of a prototyp~
is prohibitive (e.g. a nuclear reactor),its design is restricted to analysis.
The major considerations in selecting the proper evaluation path are a compromise between cost, time, and objectives.
B.
Ma thema ti ca 1 M()'de 1 s
A system is best described analytically in terms of the causal relationship between
its component parts, such as one would find on a detailed block diagram of the systen
-1-
The analyst then can derive equations for each subsystem, and the set of equations is the mathematical model for the entire system. The individual equations are derived from basic mathematics and physical laws such as the conservation of energy, matter, etc. and, at times, from empirical and semi-empirical
equations such as fluid film resistance in heat transfer. The mathematical
model can be a collection of integral or algebraic equations, although differential equations are most frequently obtained.
Typical examples Jf equations encountered in practical applications are:
1)
Algebraic and Transcendental Equations, e.g. the effect of
temper~ture on physi~a1 properties of materials.
Thus
k = thermal conductivity of a metal = k
Cp = specific heat of a gas
~
2)
o
+ aT
a + bT + cT
2
-A T
= viscosity of a fluid = ~ o e
Ordinary Differential Equations, e.g. the kinetics of a chemical
reaction
A
x
2B
whose mathematical model is
dA
= - k A
dt
1
and
3)
COOLANT
OUT
~
HOT
IN
Partial Differential Equations,e.g. a amcentric pipe,heat
exchanger (Figure I-I)
h
~h
Tw (f,x)
T ( f ,x)
t
•
T2 (ftx)
~
I
64
Figure 1-1.
Simple Heat Exchanger
-2-
HOT
OUT
COOLANT
IN
whose mathematical model is
oT
2
~T2.
~ + V2 ox
+ a 2 (T 2 - Tw)
0
where
T (t,x)
w
wall temperature
Tl (t,x)
coolant temperature
T (t,x)
2
primary (hot) fluid temperature
Two types of models, linear or nonlinear, are possible and are a
measure of the complexity of the system. Simple linear models are "nicer"
since they lend themselves to rapid analytical solutions. Unfortunately, because of the interaction of physical laws, the need for semi-empirical or empirical equations to des~ribe this interaction, and the nature of most physical
systems themselves, the majority of the mathematical models encountered in
practice are nonlinear. This is unfortunate because little is known about the
analytical solutions to nonlinear equations, and those solutions that are
obtained are usually difficult to interpret and evaluate. If a system is
nonlinear, its behavior is a function of its initial conditions, which makes
its analysis even more essential if optimum performance is desired.
C.
Solving Mathematical Models
Solutions of mathematical models can be obtained analytically by classical
methods, numerical methods, or by electronic computation.
Classical solutions of simple models are possible if the model is composed of
ordinary linear and/or partial differential equations and certain classes of
non-linear differential equations. Frequently, this technique can be applied
to limiting cases of complex models if approximations are acceptable. Analytical solutions for nonlinear models are rare, and, hence, variable substitutions
are made to linearize the model as required. Depending upon the model and the
results required of a study, phase-plane techniques may be applicable. Unfortunately, as was previously mentioned, linear systems seldom arise in practice and classical solutions are usually reserved for limiting cases and linearized approximations.
-3-
Numerical solutions involve the transformation of a mathematical model into
a set of algebraic equations by replacing all derivatives in the model with
appropriate algebraic, finite difference approximations. The resultant set
of algebraic equations is then solved simultaneously to affect a solution.
This technique not only is time consuming but may suffer from accuracy, stability and convergence problems.
To illustrate the classical and numerical solutions for a differential equation, consider the problem of a solvent tank (Figure 1-2) which can be filled
by two feed streams (Ql and Q2) in 4 and 5 hours respectively. Two drain
pipes, Dl and D ,
2
Figure 1-2.
Solvent Tank System
can empty the tank in 3md 6 hours respectively. If the tank is half full
and all feed and effluent streams are used will the tank fill, empty, or
reach steady-state? How long will it take?
The mathematical model for the tank is the nonlinear differential equation
~
dt
(1)
where
y
hlh a
(2)
The analytical solution of this equation, which can easily be obtained by
consulting a table of integrals, is,
-4-
Y
I
v;-
J
dy
1:
0.45-y2
d
2f!, l.l
v;:0.45-1-L
c:t
(3)
~o= ~
Yo=~
or,
-~
0.45
+ 2([; --{4;) = t
0.45 -\fY
2
y
0.9 ln
(4)
The obvious difficulty in applying this equation is that y, the level in the
tank, does not appear as an explicit function of time, t. Even though we have
an analytical solution, considerable effort is still required to produce a
useful relation between y and t, say, in the form of a graph. The eventual
height of the solvent in the tank will be the steady-state solution, y , of
e~ation (1) ( obtainedby letting dy/dt equal zero)
s
y
s
= (0.45)2 = 0.203
(5)
The time required to reach this height in theory is infinite; therefore, a
practical value of the steady state time must be obtained graphically from a
plot of y versus t.
Since the time required to attain the equilibrium height also can be obtained
from a numerical solution of equation (U,
let us now consider this method of
solution.
Integrating equation (1) one obtains
t
y
0.45 t
-f Y
1:2
o
dt +y
(6)
o
where the initial value of y, y is 0.500. Recalling that integration is
the area under a curve, Figure £-3, equation (6) can be rewritten in terms
of finite or discrete intervals of time:
n=co
= y
o
+ 0.45 nf::. t
-I
f::.t
(7)
n=l
where
t
= n f::. t
(8)
The accuracy of the solution obtained from this equation depends on the magnitude of the time interval (accuracy increases as f::. t decreases).
-5-
Ij~I!1 ~~~eE UNDER
FUNCTION
OF TIME
•
Y
'IIII~=~
O~
0.450
t,
__
t
1:2
o
t2
ERROR DUE TO
FINITE APPROXIMATION
OF INTEGRAL
----------I!I... t
INTEGRAL
OF THE FUNCTION
--~~~-------~·t
Figure 1-3
Illustrations of Numerical Integration
-()-
Equation (7) is solved in the
at say 0.5 hours.
fol.lm~li ng ,~:rrrrr::" :~:t
dEtra:r :.~L 0.~E'
1)
Compute Y"2 -
2)
Compute
3)
Compute
4)
Compute 0.45 n~t
5)
Compute Yn
6)
Let n -- n + 1 and retut'n to f;tep 1.
n 1
c';::~n
sele;;.:tecl;
(Yo ia known and LS U5~: ~s G startj~3 point)
Results obtained from both the numerical and ..m.ulytical solutio:ns are shown
in Figure 1-4.
0.5
t
o
NUHERICAL RESULTS
o
ANALYTICAL RESULTS
0.4
y:JL
ho
0.3
o
o
0.2 --1----_ _ _ _ _ _-=::::~==~==:il=~dd
4
2
3
4.5
TIME IN HOURS
Figure I-4
Numerical Solution of Solvent Tank Problem
-7-
From the curve shown in Figure 1-4, it is apparent that the tank (Figure 1-2)
will fill and reach equilibrium in approximately 4.5 hours. It should be noted
that an increase in the accuracy of the numerical solution would have required
additional computations and, hence, increased computation time. The same
procedure would have been followed for a smaller increment of time, ~t. The
error of the numerical solution is indicated by comparison to analytical results obtained from equation (4). Computer solutions are best understood
after an explanation of the type and methods of computer operation is presented. However, it is convenient to point out at this time that the numerical solution illustrated above is typical of digital computer solutions and
the itemized instructions are typical of a digital computer flow chart.
If one considers a flow diagram for the solution of equation (6), which is
shown in Figure 1-5, insight to the analog computer solution can be obtained.
It will be shown later that the analog computer is composed of components
which perform the mathematical operations described in Figure 1-5.
f
-
0.45
}:
Oo4s!dt
_.
-
Yo
--
L
--
y (t )
t
Y I /2( t)
..
-
[:t
-h
l/ 2 dt
y"2{t)
Figur e 1-5:
Flow Diagram of Solvent Tank
Solution
V- --
At this point, the justification for using computers can be considered.
In our modern society, machinery of various sorts has relieved human muscle
from a great deal of routine and repetitive operation. In doing so, it has
multiplied the effectiveness of that human muscle both in industry and in the
home.
The computer has performed a similar service for the mind essentially by
mechanizing routine mental processes, leaving the mind free to examine new
problem areas. Studies of the behavior of entire complex systems can ,be
performed with great speed and, consequently, our actual knowledge of complex
systems has increased greatly. Equally important, our capacity for control and
prediction, and for insight into these complex systems also has been extended.
The inf~uence of the computer on our common life, therefore, lies in its contribution, in the broadest sense, to science and technology.
Investigations in science and engineering can be carried out on a scale unheard of only one or two decades ago. Scientific principles and models can
be verified against experimental facts at small cost, without hazard and with
-R-
considerable flexibility. Thus, new areas of scientific knowledge have been
established and will continue to grow as a result of research and development
performed on computers.
D.
Computer History and Characteristics
A computer is a device that is able to receive information (equations, instructions, data, etc.) and process it in a predetermined manner to obtain useable
results.
For example, a human being may be a computer. He can take information in
through his senses, use principles stored in his memory to process or perform
operations on this information in many ways, and produce an answer, perhaps
in the form of an action.
Similarly, a machine may be able to accept information of a suitable form,
receive instructions on how to operate on this information, perform the required operations, and give the answers. Machines may take many forms varying from simple beads on a frame to the incredibly complex, expensive and
highly sophisticated modern machines.
1. Early Computers---The history of computing devices may well extend to the
very beginning of civilization. For our purposes, they can be divided into
two categories (see Figure I-6):
o Mathematical instruments, the more complex of which are known
as analog computers. These are exemplified in simple form by
the slide rule.
o Calculating machines, more often known as digital computers.
These can be represented simply by the desk calculator.
Early forms of digital computations could be considered to exist when man
first started to use his fingers or pebbles for counting.
The earliest known record of analog computation is its use in surveying and
map making for the purpose of taxation (Babylonia, 3800 Be). The earliest
digital machine is probably the Abacus. In its early form, it consisted of
a clay board with grooves in which pebbles were placed. It later appeared
in the form of a wire frame with beads. It is still used extensively in Asia
and the East for remarkably rapid calculations.
The development of computational aids can be traced from these early instruments through the invention of logarithms, slide rules, linkages, analytical
engines, and desk calculators to the large-scale general-purpose machines of
the present day.
The first large-scale general-purpose digital computer was completed at Harvard
in 1944. This machine, the Harvard MBrk I Calculator, was built jointly by
IBM and Harvard, and used electromechanical relays. The Moore School of
Engineering also completed its all-electronic digital computer for the Aberdeen
-9-
1COMPUTERS I
I
DIGITAL
ANALOG
COUNTING DEVICES DISCRETE,STEP BY STEP
SERIAL OPERATION
ANALOGOUS SYSTEM OPERATES IN
PARALLEL ON CONTINUOUS VARIABLES
i
I
l
GENERAL PURPOSE
I
I-'
I
I
SPECIAL PURPOSE
~ (GENERAL
INDIRECT
PURPOSE)
I
I
I
I
I
DIRECT
SPECIAL PURPOSE
I
o
I
I
_I
i
I
J
-.l
i
I
ELECTRICAL
MECHANICAL
ELECTRICAL
MECHANICAL
ELECTRICAL
MECHANICAL
ELECTRICAL
HYDRAULIC
MECHANICAL
ENIAC
UNIVAC
STRETCH
MANIAC
ABACUS
DESK
CALCULATOR
ACCOUNTING
MACHINES
STOCK MARKET
TOTE BOARDS
BANKING
SYSTEMS
AIRLINE TICKET
RESERVATION
SYSTEMS
GASOLINE
PUMP
ODOMETER
ELECTRONIC
SLIDE RULE
LINKAGES
BUSH ANALYSER
PLANIMETER
NORDEN
BOMB SIGHT
RESISTANCE CAPACITANCE
ARRANGEMENT REPRESENTING ELECTRICAL NETWORKS
TOWING
TANKS
SCALE MODELS
WIND TUNNELS
PILOT PLANTS
PNEUMATIC
ACCOUSTICAL
AND OTHER
7090
HARVARD
MKI
ANTI-AIRCRAFT FIRE CONTROL PREDICTORS
GENERAL PURPOSE
ELECTRONIC
ANALOG COMPUTERS
Figure 1-6
R C NETWORKS REPRESENTING HEAT TRANSFER PROBLEMS
Computer Devices
Proving Grounds in 1944. This machine, the ENIAC, ~hich contained 18000
vacuum tubes, now has many direct descendents.
Mechanical integrating devices of the late 19th century were improved on
during World War I, when Hannibal Ford increased the torque output of the
ball-and-disc integrator and used it to make a naval gun fire computer.
This was followed by more experimentation in the 1920's.
At M.I.To, Dr. Vannevar Bush completed the first large-scale mechanical
differential analyzer in 1931. This machine is now installed at Wayne
University in Detroit where it is still being used effectively. At the
present time, there are several large scale mechanical machines in operation. Simultaneous equation solvers and harmonic analyzers of many types
also appeared in the 1930's.
Special computers, in the form of network analyzers for the simulation of
power networks, appeared around 1925. The network analyzer is a passive
element analog. A scale model of the particular network to be studied is made
with resistors, capacitors, etc. The early network analyzers could be used
to investigate only steady state problems; that is, voltage drops along
lines, possible current flow in lines, etc. The most recent network analyzers
can be used to investigate transient conditions during faults on networks or
switching on networks. These may be considered to be true general-purpose
computers.
2. Analog and Digital Computers---In digital computers, numbers are operated upon directly. The basic operation in these machines is counting. This
enables the machine to perform the four fundamental operations of arithmetic,
addition, subtraction, multiplication and division. The basic operation of
any digital computer is similar to that of the abacus where numbers are represented by.beads and the counting of these beads is the basis of addition and
subtraction. In digital computers, all mathematical calculations depend ultimately on counting, whether it be beads, gear teeth, or electrical pulses.
In analog machines, numbers are reoresented by physical quantities whose
magnitude is determined by the magnitude of the number. Mathematical
operations are represented by physical events; that is, the machines do not
count, but perform continuous manipulations equivalent to the mathematical
operation required. The result of these manipulations is another physical
quantity, whose magnitude and behavior represents the solution to the problem.
Probably the most useful example of the analog computer is the slide rule.
Here, to multiply one number by another, the discrete numbers are converted
to logarithms, the logarithms are converted to linear distances on sticks
which, when placed end to end (i.e. added together--the continuous operation
in this case), give another length representing the product of the numbers.
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There exists on the analog a complete analogy between the physical quantities,
events, and the mathematical numbers and manipulations. If many events are
taking place at the same time in the physical world, they will also take place
at the same time, or in parallel, on the machine. An analog device will, consequently, arrive at a result in a shorter time than a digital machine which
must perform all its operations serially.
The precision of a digital machine is theoretically boundless. To increase by
ten the precision of a decimal counting device, it is a matter simply of accomodating one more place (decimal) throughout the equipment. However, to achieve
the same end on the analog, e.g. the slide rule, the length of the slide rule
would have to increase by a factor of ten. This is not always practicable.
Analog devices, are characterized by continuous operations performed in parallel,
as opposed to digita.l machines which are discrete, 'serial devices. The analog
solutions are obtained in a continuous manner since all parts of these devices
operate simultaneously.
3. General Purpose Analog Computers---We have said that in analog devices
numbers are represented by physical quantities. Theoretically, any physical
quantity may be used as long as it can be made to obey those laws necessary
to represent the mathematical relationships involved in the original problem.
Purely electrical relationships, which have the mechanical advantage of no
moving parts and, a high speed of operation, have been found most suitable
for analog devices.
The introduction of the operational amplifier made possible the newest class
of general purpose analog computers using voltages as the 'physical quantity' .
Lovell of Bell Telephone Laboratories is generally credited with the introduction of the operational amplifier during the Second World War. These amplifiers can be divided into two groups, those which op~rate on a-c voltages and
those which operate on d-c voltages. The a-c amplifiers exhibit certain difficulties and do not lend themselves to any direct form of integration.
Therefore,only d-c amplifiers are considered in these notes since they are
most common in comrner~ially available general purpose analog computers.
E.
Industrial Uses of the Analog Computer
As a result of the tremendous competition in industry following World War II,
more economical designs and more thorough evaluations were needed. The concept of a fully automated plant, or system, operating at an economic optimum,
demanded from the engineer a more extensive knowledge of each element, and
its behavior. The engineers, in turn, demanded a more complete analysis of
mechanisms and transport properties from the basic research scientists.
It is important, at this point, to state the range of the computer's usefulness, and to delineate those areas where it is not suited.
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1. Initial Research and Development---The initial work in development
usually takes place in the laboratory where bench-scale studies, thermodynamic
calculations of feasibility and other preliminary calculations are made. Because this stage of the work is so intimately concerned with mechanisms, most
of which are dynamic in nature, the use of the computer is particularly advantageous. Consider, for example, the determination of chemical reaction velocity constants. A series of isothermal batch reactions may be run and data
collected on the compositions of the various components as functions of time.
A kinetic model then is assumed, i.e., the orders of the various reactions
are estimated, and programmed on the computer.
The problem is one of matching the results of the computer with the data from
the test runs. Different redction velocity constants can be tried or different models assumed until a good match is obtained. In this way, a reliable
model of the isothermal chemical kinetics is quickly obtained.
The laboratory work then may be extended to include temperature changes and,
possibly, other types of reactors. The computer is used in each step to
simulate the mechanisms, check the model and the assumptions, and obtain
system
param~~ers for design purposes.
2. Intermediate Development---The use of the analog computer in the prototYPE
stage of development represents a powerful tool for improving the overall
efficiency of the development procedure. By combining the philosophy of model
building with the philosophy of simulation, a complete study of a component
or system can be obtained. The conditions of optimum operation can be determined and quickly evaluated over a wider range of variables than is often
possible with the hardware or plant itself.
Consider, for example, a development program in which a pilot plant is simulated with an analog computer. In order to achieve a meaningful simulation,
certain basic facts must be known, and these are found from preliminary pilot
plant or bench-scale data. Certain heat-transfer coeffici"ents or diffusion
constants might, for example, be determined from specific tests in the pilot
unit. The simulation then is checked against normal operating data obtained
from the plant on the computer where "runs" can be made in a more economical
fashion.
Three areas of study thus are defined. In the initial phase of investigation
it can be seen that the knowledge gained is,perhaps, not inmediately useful
as design data. The second phase is equivalent to normal operation, except
that the computer is added and the model obtained. Finally, the parallel
operation of computer and pilot plant, or prototype,results in a greater
amount of information at a substantial decrease in cost and time, since the
simulated plant runs faster than the actual plant and does not require any
raw materials.
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3. Final Deve10pment---After pilot plant work is completed, final design
calculations are undertaken. Most design calculations are based upon a steadystate type of operation and, hence, are primarily algebraic equations. In
many such cases--for example, mu1ticomponent distillation calculations, heat
exchanger sizes and capacities, vessel specifications, structural rigidity, etc.-complete digital computer programs have been worked out. In such cases, even
though the analog is capable of solution, it is obvious that the digital computer should be used if available.
One of the areas in which the analog computer is particularly applicable is
the choice of pr~cess instrumentation and control. The large amount of work
that has taken place in control engineering recently, in fact, is an excellent
illustration of the fact that, while design may be steady-state, the operation
of a process is always dynamic. With the analog computer model of a process,
the interrelation of its various unit operations can be examined easily, suitable
control systems can be tested, and the proper settings on controllers determined.
In most cases, the control system itself is simulated; in others, the controllers to be used in the plant are connected directly to the computer. The
use of the analog computer in such applications is expanding rapidly, and is
one of the primary means the process engineer has available to improve process
efficiency.
The last step in the development of a process is start-up--often a difficult
and expensive task. If a computer model has been determined, the proper
values of the flow rates and other variables can be tested under different
start-up conditions, and the optimum ones selected. In many cases, an appropriate start-up procedure for a plant can be determined long before the unit
is ready for operation.
4. Post Development Work---After a plant is running satisfactorily, the
computer model can be adjusted to match the particular idiosyncrasies of the
unit.
Further experimentation is than possible with the computer. As with the pilot
plant, the real plant can be tested for different optimum conditions. The
economics of the operation can be investigated under changing values of the
products. The range of operation can be extended to determine some of the
safety precautions to be observed in the plant. Finally, the computer model
can be tested for use of the equipment with different materials, reactions,
etc.,in the event that a changeover ever became necessary.
It is interesting to note that these suggested areas of application are not
aimed at replacing with the analog computer important procedures in the standard process development program. Rather, they supplement the ways and means
by which decisions can be made. Thus, in the pilot plant simulation, it was
necessary to retain the pilot plant as a check on the simulation, but the simulation could be extrapolated outside the range of the actual plant capabilities.
The general program involving the analog computer in the process development
scheme is characterized by the high rate of information exchange between
experiment and simulation. Such a program shows a great improvement over the
usual procedures because at no time does it become necessary for the development program to become "boxed in" by previous studies. The computer provides
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an economical means for the complete evaluation of the investigation, since
these studies are brought into focus, gaps in data are filled and predictions
of major importance are obtainable.
5. Inappropriate Areas for the Analog Computer---As mentioned previously,
steady-state algebraic equations, can be and have been solved on the analog
computer. As a general rule, however, large scale algebraic problems are
better solved with a digital computer.
In general, problems involving high accuracy are not suitable for the analog
computer. Some perturbation schemes have been developed for handling problems
up to five and six places but only certain problems can be solved in this way.
Under ordinary circumstances, the computer is accurate to about 0.1% for small
simulations, and, depending on the type of problem, may range from 0.5% to 1.0%
or more for very large simulations. Most engineering data, however. are not
that accurate. For example, heat transfer coefficients, and reaction velocity
constants and modules of elasticity, are usually in the 5% to 20% range of
accuracy_ Consequently, the computation accuracy is usually not a problem.
Specific Illustrations of Analog Computer Applications
The above discussion has been of e general nature. However, a selected
bibliography of computer applications , categorized by specific industrial
areas, is presented in Appendix D.
F. References
Jackson, A.S.:
Analog Computation, McGraw-Hill Book Company, New York, 1960.
Johnson, C.L.: Analog Computer Techniques, McGraw-Hill Book Company, New
York, 1956.
Korn, G.A. and Korn, T.M.: Electronic Analog Computers, 2nd Edition, McGrawHill Book Company, New York.
Roedel, J.: "An Introduction to Analog Computers", ISA Journal, Volume 1,
No.8, August, 1954.
Rogers, A.E. and Connolly, T.W.: Analog Computation in Engineering Design,
McGraw-Hill Book Company, New York, 1960.
Soroka, W.W.: Analog Methods in Computation and Simulation, McGraw-Hill Book
Company, New York, 1956.
Wass, C.A.: Introduction to Electronic Analogue Computers, McGraw-Hill Book
Company, New York, 1956.
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CHAPTER II
THE GENERAL PURPOSE ANALOG COMPUTER
A.
Introduction
Analog computers have been constructed in a number of forms which, by definition, appeal to the similarity between the laws of nature. For example, co~
sider the analogy between mechanical, electrica