Joe D. Cunning ond George Burnet, Jr. Iowa Stote Univers~ty Ames
Octave Levenspiel Illinois Institute of Technology
Chicago
I
II
Data Correlation Experiment
The correlation of data usiug statistical techniques is brcoming increasingly impcrtant. Many industrial processes and laboratory experiments rrquire careful data correlation in order that maximum
7 Figure 1.
lndustrialoperation~rimulator.
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~xsultscan be obtained. Graduating chemists and chemical engineers should have at least a rudimentary understanding of the techniques of data correlation. In thc Chemical Engineering Department at the Illinois Institute of Technology and later a t Iowa Statc University a device that has heen called an Industrial Operations Simulator \\-as designed to provide an inexpensive, yet unique, method of obtaining suitable data for teaching students with very little statistical background the techniques of data correlation. The Industrial Operations Simulator is an electror~ic device based on the total current draw of simple triode circuits plus a motor operated random error switch. Solutions have been built into the simulator for the equation:
where c is a normally distributed random error with mean zero and standard deviation u. The magnitude of the error term may he set to be either large or small, thus enablimg a wide or narrow scattering of the data points. The x values can be varied continuously from 0 to 10. The values of b, c, and d can each be set at one of two values, thus enabling a variety of equations to be solved. The Circuit
The Industrial Operations Simulator (Fig. 1 ) circuit is a simple one based on the total current draw of a triode circuit with a 24 position motor driven switch to provide the random error. The output (y) is read on a large 0-50 ma meter. The circuit is shown in Figure 2. The list of component parts is shown in the Parts List. The magnitude of the values for b, c, and d are set by the switches S 3 , S 4 , and 8 5 . The magnitude of the resistors R33 through R38 set the plate voltage that results in values for b of either -0.31 or -1.46; for c, +0.50 or +1.00; for d, -0.44 or -0.78. Each may also be set a t zero. The values of XI, x2,and x8 are set using the potentiometers R3, R4, and R5 to set the grid bias voltage of the 6J5 triodes. The values of the grid bias voltage vary from 0 to -12 v dc which is well within the linear region for a 6J5. The negative values for b and d are set by grounding the R3 and R5 potentiometer such that the grid bias voltage changes from -12 to 0 v dc as xland x3 are increased from 0 t o 10. The random error circuit is shown in detail in Figure 3. This circuit consists of a 24 position rotary switch operated by a 4 rpm motor. When switch S2 is pushed, the motor stops with the particular resistor connected from the 300+v dc to ground. The current through this resistor is added into the total current draw of the simulator and provides the error term. This error, controlled by switch 87, may be either large or small. The error circuit may be turned off using switch SG.
The random error circuit is designed to approximate the Gaussian distribution as follows:
The value of h was selected to be 0.1, for ease of calculation. With this choice for h the value z = 7.03 corresponds to one standard deviation. This was chosen to be 1 ma for the small random error. Appropriate resistors (R8 to R32) were then selected such that the distribution of the error approximated the Gaussian distribution in 24 equal-area increments. The large random error was made to correspond to 3 ma and the same procedure was followed. With this selection the maximum contribution of the error was 12 ma in which case the limit of the 50 ma meter would not be exceeded. The mean current through the random error circuit was set a t 6 ma, one-half of the maximum. The values of R6 and R7 were calculated so that the mean current would be the same for either a large or a small error. A counter records the number of readings made, and a red light goes on when S2 is pressed to indicate the Simulator is operating. When switch 5 2 is not pressed, the meter is shorted out for protection of the meter. When switch 8 2 is depressed, the power to the motor is cut off, the random error circuit is connected, the counter is activated, the red light goes on, and the meter bypass is disconnected. The Experiment
A variety of experiments can be conducted, with the Simulator. One of these is outlined below. The experiment may be presented as a simulation of a practical situation, or may be extented and amplified as suits the individual instructor. Parts List for Figures 1 and 2
T Bl,B2
Transformer, Stancor PC8419 Drake type 105 post light (one red, one white)
VI, v 2 V3, V4, V5 L Counter Motor Meter F
GX~TUK'
6J5 Tube 80 henry, 75 ma choke Lafayette type F553 80 in-ox, type RSM, 4 RPM 0-50 ma, Trlplett 420 PL 1 amp fuse
SPST (toggle) 3PDT (lever) SPDT (center off toggle) SPDT (slide switch) lOmfd - 450 v electrolytic 20mfd - 50 v electrolytic 158K, 'I1 W 20K, 'IzW 1.0 megohm pot.
Figure 2.
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Basis drcuil.
Journal of Chemical Education
TO switch positions I b a l o r l
All p0.i110n, 01.
Iooblnp 10.0rd
12
motor
Figure 3. Random error circuit.
Part I . The Method of Least Squares. The purpose of this experiment is to help the student to become familiar with the method of least squares for correlating data involving more than one independent variable. It also will introduce the student to the idea of experimental design, in particular to illustrate the power and efficiency of the factorial experiment in studying a number of factors simultaneously. The experiment also can serve to familiarize the student with the desk calculator and/or high speed digital computer. The student is asked with a fixed number of experiments, say 8 or 16, to find the constants of best fit for t,he equation by the proper selection of dial positions for XI, xz (and and by recording the corresponding meter readings (9,). ,., ,Part 2. Linear Regwssion. I n this exercise, the student is asked to show how the "t"-test (or F-test) is used to find whether a correlation between variables exists and la verify or reject proposed relationships between variables. xz)
Only one dial, perhaps XI, is varied and its positions and corresponding y values are recorded. Since only linear relationships are considered here the student first must evaluate the best value of b in the equation
He is then asked whether it is reasonable to conclude from the data that b # 0, in other words, that a relationship between XI and y exists. This is then extended to test whether hypothetical relationship, b = bl is consistent with the data. Attention is again focused on the idea of efficient experimental design by letting the student discover for himself that observations should not be taken a t equal intervals of XI but should be divided evenly among the two extremes of x,. The Simulator can also be used to flustrate and teach more advanced topics in correlation, regression, quality control, and optimization. It is felt that this device is a useful and versatile tool to help introduce the student to the basic ideas of efficient experimental design and proper analysis of data. We would like to acknowledge the contribution of V. M. Putrius who constructed the pilot model of the Simulator at Illinois Institute of Technology. Bibliography DAVIS,D., "Nomograhy and Ernpit.ical Equations," The Waverly Press, Baltimore, Maryland, 1955, chap. 3. LEVENSPIEL, 0 , WEINSTE~N, N. J., A N D LI,J. C. It., Ind. Eny. Chem. 48, 324 (1956). OSTLE,B., "Stati~ticsin Itesearel~."Iows Stato University I'ress, Ames, 1954, chap, 6. SNEDECOR. G. W.. " S t a t i ~ t i dMetlurds." 5th ed.. Iuwa State
lied
,
.
VOLX,W., Statistics for Engineers," McGriw-Hill Buok Cu. Ine., h'ew Yark, 1958, ehap. 8.
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