James 0. Osburn University of lowa Iowa City
I
Analog computer instruction
A Plugboard Teaching Aid
The recent development of computers is creating fundamental changes in research and teaching. The use of computers creates new opportunities in all fields of science, including chemistry and chemical engineering. No chemistry teacher can afford to disregard the possibilities for using computers in his own teaching, nor of preparing his students to use computers in research and in industrial work. Computers are classified according to operation as digital or analog. A digital computer can do a wider range of calculations, and has a higher precision than an analog computer. Nevertheless, the use of analog computers is increasing a t a rapid rate (1). The analop: com~uterhas several advantaxes (10): . . Programs are relatively easy to set up. I t is particularly suited for the simulation of physical systems. The program can include information, such as curves, which have not been reduced to mathematical form. Parameter magnitudes can he changed easily to give a series of solutions for different parameter values. This is particularly valuable in preliminary curvefitting. I t can be combined with nou-computing elements, such as control systems. For problems of moderate size, the analog computer is less expensive (6). Teachers will find many opportunities to use analog computers for demonstrations in reaction kinetics, equipment design, and control. Some applications for which computers are well suited are (4,9,12) :
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are as follows: 2 ' =
(2)
-az
z/' = az - by 2' =
(3)
by
(4)
where x is the concentration of substance A, y is the concentration of substance B, z is the concentration of substance C, and the primes represent derivatives with respect to time. The circuit for solving this problem is derived from the basic principles of analog computation. Fundamental to the electronic analog computer is the high-gain dc amplifier, represented by the symbol shown in Figure 1. The gain, or ratio of output voltage to input voltage, is commonly between 20,000 and several million. If the output is not over 100 volts, therefore, the input will be less than 5 millivolts.
This can be t,aken as essentially zero. The importance of this will become apparent in the circuit analysis which follows. Resistors and capacitors are connected to the amplifier to form sub-units which perform addition, multiplication, or integration. The operation of these subunits can be understood from an elementary consideration of the current flow in the sub-unit. For addition and multiplication, a circuit is put t'ogether as shown in Figure 2.
Consecutive and simultaneous reactions Catalytic Reactor design and operation Distillation column dynamics Control system simulation Air pollution studies Figure 2.
A Typical Demonstration
Problems in chemical kinetics are often so complicated that a mathematical solution is hopelessly involved. Such problems can often be set up rather easily on a computer and the solutions demonstrated to a class. The following section illustrates how this would be done for the simple case of two consecutive first order reactions. This example includes a derivation of the basic concepts of analog computer programming. Those interested in pursuing the subject further should consult one of the excellent books on the subject (2-5,s). The eauations for the consecutive first order, irreversible reactions A-B-C (1) 492
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Journal of Chemical Education
Circuit for addition and multiplication.
The current through a resistor is equal to the voltage drop divided by the resistance. Applying this lam to Figure 2, we obtain el
- eo
through R,, the current = R1 e2 - eo through R., the current = R, eo
- e.
through R,, the current = --
R.
No current goes through the amplifier, so the current through R, is equal to the sum of the currents through R, and Rr, or
Xow since eo is essentially zero, we may write
For integration, the circuit is shown in Figure 3,
Figure 3.
SET AT 0
Circuit far oddilion ond integration.
The current through a capacitor is given by the expression C(dE/dt), where C is the capacitance, E the voltage across the capacitor. Equating currents as before, we obtain
For e,
= 0,
%J I
MEGOHM
RESISTOR
I YICROFARAO CAPACITOR Figure 5.
For multiplication by a positive number less than 1, a potentiometer is used as shown in Figure 4. Figvre 4.
Potentiometer.
Circuit diagram and board wiring for x'
of the input terms is y'.
= ox.
By equation (3),
y' = ax
- by.
A voltage representing ax is already available, and -by is obtained from -y by another pot,entiometer. These are added as shown in Figure 6, again using 1 megohm resistors and 1mfd capacitors.
Applying these principles to the reaction problem, we start with equation (2) : The output voltage of an integrating amplifier is arbitrarily assigned the value of the concent,ration x. The voltage a t the input resistor, by equation (6), then represents -xJ. (Note that an amplifier always gives a reversal of sign.) The integrating amplifier, as shown in Figure 5, is an amplifier with a capacitor in parallel and a resistor leading to the input. Leads also must be placed from "initial conditions" terminals to fix the value of x a t zero time, when the computat,ion begins. By equation (2), Assuming a to be less than 1, one obtains ax from x by using a potentiometer. The condition that -x' be equal to ax is obtained by connecting the points representing these two quantities, as shown in Figure 5. This figure shows the wiring diagram in conventional symbols, together with a sketch of the wiring on a computer plugboard, which will be described in a subsequent section. Use of a 1 microfarad capacitor resistor reauires that time be expressed and 1 megohm in seconds. The circuit for y is designed in a similar fashion. Addition and integration are performed by the same amplifier. Let the output equal -y, whence the sum
SET A T b
49 w
+I00
I MEGOHM
%-' Figure 6.
I
RESISTOR
MICnuFbI~bD CbPbCITOR
Circuit diogrom ond board wiring for y' = ax
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Finally, the simulation of z is performed by letting the output of a third amplifier be z, the input - z l , and noting that The voltage representing -by is already available. This part of the circuit is shown in Figure 7.
Heathkit computer using the plugboard which is described in the next section. The initial conditions are 100 for x, zero for y and z; the values of the coefficients are a = 0.4, b = 0.3. The equations were also solved by calculus, and a few calculated points plotted to show the validity of the computer solution. The plugboard set up for solving this problem is shown in Figure 9.
Figure 9.
-
\d I
MEGOHM
RESISTOR
I MICROFARAD CAPACITOR
Plugboard programmed for rolution of kinetics ,,roblem.
The computer can be useful even for this simple example in showing t,he effects of changing coefficieuts and initial conditions. These can be changed by resetting potentiometers to different values, and new solutions obtained very easily. The computer can also be used for reversible reactions, reactions of higher order (11), several reactions in series and compet,ing react,ions where a mat,hematiral analysis ~ o u l dhe impractical.
Figure 7. Circuit d i a g r m and boord wiring for r' = by.
The Auxiliary Plugboard
The solution begins when the three switches which bypass the three amplifiers are opened simultaneously. The solution is in the form of three time-voltage curves, which are the analogues of the time-concentration curves in the physical sit,uat,ion. Figure 8 shows the curves which were obtained on a
T I ~ E Figure 8. Computer $olulion for simultaneous flmt order reaction% Computer, 0 Cdcvlakd.
494
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J6urnol of Chemical Educofion
After watching a demonstration on a computer, many students will want to learn how to operate the computer themselves. In view of the growing use of computers in industry, the student can profit by learning how to use a computer while in school. Many feel that this is a proper part of the undergraduate curriculum in chemistry and chemical engineering (7). For teaching computer use, laboratory instrnction and practice is very desirable. Although a high precision computer is beyond the budgets of many college departments, a less expensive model is quite satisfactory for instruction. These computers of lower accuracy are quite suitable for teaching computer programming and operation. Computers can be obtained for as little as $200 in kit form, and several companies offer small computers for less than $5000. For information about available computers, the reader is referred to a recent list compiled by Williams (9). A satisfactory recorder will cost about $500. Thus, investment in a computer should offer no serious problem. However, even with an inexpensive computer, the instructional cost per pupil is apt to be excessive. Space a t the computer is limited, and a student tends to use a lot of time in setting up a problem. This sets a severe limit to the size of a class, and the instructor's time is taken by a very few students. One solution has been to give lectures on computer operation, then let each student reserve time on the
computer and work out his problems by himself. This has not worked well, as students usually need some help and supervision. We have found that the use of auxiliary plugboards has provided a very satisfactory solution to the problem of student use. Essentially, the auxiliary plugboard consists of a number of terminal jacks arranged to resemble a section of the computer plugboard. When the board is plugged into the computer, the iacks are connected in parallel to similar jacks on the main plugboard. The board shown in Figure 10 was developed for use with a Heathkit computer; it could easily be adapted to other computers. A commercial model has been introduced by the Donnrr Co. for use with their Model 3500 computer. The base of the auxiliary board shown in Figure 10 is a 3 X 8 X 12 in. aluminum chassis. Colored jacks for four operational amplifiers are arranged in imitation of the Heathkit board. In addition, plugs for reference voltages, ground potential, and three sets of initial conditions are included. Each jack or set of jacks is wired to an 18-connectoroutlet. The plng which mates with this outlet makes connection with the corresponding jacks on the main plugboard. "Initial condit,ionsl' terminals are wired to initial conditions voltage sources through relay switches. Four 100 K ohm carbon potentiometers are wired as shown on the sehematic painted on the board; one side of a potentiometer is connected to ground, the other side and the
and connectors. Each capacitor or resistor is attnrhed to a General Radio double jack, which can be inserted into the appropriate plugs on the board. After a lecture on the principles involved in the solution of a problem, the student proceeds to set up the program on his board. When he is ready, he plugs the board into the main computer with the assistance of the instructor. The solution is recorded in a short time. For some problems, the solution is a voltage read on a voltmeter. For problems in which there are differential equations, the solution is a voltage us. time trace recorded by a recording voltmeter, or a trace appearing on the face of an oscilloscope. By using the inexpensive plugboards, several students can be working on their programs a t the same time, with the help of only one instructor. Each student uses the main computer only a relatively short time. More complicated problems, such as second order equations, require function multipliers or function generators. Commercial units are fairly expensive, but units which have enough accuracy for student laboratory use can be assembled for a small fraction of the cost of a commercial unit, and will serve for instructional purposes. (The author will gladly answer requests for information as to details of such units and of the construction of plugboards.) Acknowledgment The National Science Foundation supported the development of the auxiliary plugboards through their Program for Development of Teaching Aids. This support is gratefully acknowledged. Literature Cited (1) C and EN Special Report, Chem. Eng. N e m , 39, No. 6, 118126 (Feb. 6, 1961). (2) JACKSON, A. S., "Analog Computation," MeGrem.-Hill Book Co., Inc., New York (1960). (3) JOHNRON, C. L., "Analog Computer Techniques," MeGrawHill Book Co.., Ine.., New York (1956). (4) KARPLUS, W. S., "Analog Simulation," MeGran.-Hill Book Co., Inc., New York (1958). (5) KORN,G. A,, AND T. M. KORN,"Electronic Analog Com-
~.
puters," 2nd ed., McGraw-Hill Book Co., Inc., New York
Figure 10. Auxiliary plugboard
center tap connected to plugs which are located near the amplifier plugs. The potent,iometers were calibrated in place. In the computer laboratory, each student gets an auxiliary hoard and a supply of capacitors, resistors,
(1956). (5) PINK,J. F.,Petrol. Refiner 38, 21.520 (March 1959). (7) ROGERS, A. E., J . Eng. Ed. 50, No. 2, 107-112 (Nov. 1959).
(8) WARFIELD, T. N., "Electmni~Analog Computers," Prentice-Hall, Inr., Englewood Cliffs, N . J. (1959). (9) WILLIAMS, T.J., AND V. A. LAUHER,Chem. Eng. Progr., 56, 71-5 (Feb. 1960). (10) WILLIAMS, T. J., Ind. Eng. Chem., 50, 1631-5 (1958). (11) WILLIAMS, T. J., Chem. Eng. 67, No. 7 , 139-144 (Apr. 4. 1960). (12) WOODS,F. A., Ind. Eng. Chem., 50, 1627-30 (1958).
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