THE INDUSTRIAL CHYMIST Graphical Calculus B. J . Luberof, Ph. D.
I like gadgets, particularly those that do a job simply by clever use of some rather fundamental principle. Here are a couple that I’ve found useful from time to time in handling both mathematical and shop problems. There’s a “differentiator” and a n “integrator.” If at a point you know the slope of the perpendicular to a curve then you also know the slope of the curvei.e., its first derivative or tangent, at that point. This follows since the slope of the curve is the reciprocal of the normal to that curve. Estimating the tangent from the normal is far more precise than any direct method I know and that’s just what the first two gadgets do. Why bother? Well, the most frequent use I’ve found is in If estimating the rate of things. some variable-e.g., concentrations
DR. LUBEROFF has been working at the interface between chemistry and engineering since receiving his Ph.D. from Columbia in 7953. He has been associated with American Cyanamid and Staufer Chemical Co., and is presently Manager of Process Research for The Lummus Company, Bloomfield, N . J . His publications which appear in the scient& and patent literature, both here and abroad, number several dozen. They are characterized by the diversity of fields they cover: from pesticide residue analysis through homogeneous catalysis to full process flow sheets. 4
or pressure-is plotted us. time, the slope at a point is the instantaneous rate of change a t that point. This datum itself is useful in, for example, estimating residence time in a continuous kettle from batch rate data. However, if one tabulates slopes of a concentration-time curve, i.e., (dc/ dt) us. concentration (c) and then plots them up on log-log paper, one has a dandy handle on kinetics. This follows since such a plot represents the equation log
dc -
dt
=
n log
c
+ log k
or taking antilogs dc - = kcn dt With an estimate of n, one can then proceed analytically. “Other graphical applications will be obvious to those skilled in the art.” Now, how does one erect the desired perpendiculars. Both gadgets are ridiculously simple. I n one, you just lay a length of glass rod on the curve-roughly perpendicular to itand then turn it until the image of I he curve seen through the rod shows no discontinuities with the curve. T h a t is, until the curve appears to pass smoothly through the rod. The rod is then perpendicular to the curve. Any of the glass rods found in the lab will do fine. The second method has seemed somewhat more precise. A mirror is held on the curve in a plane roughly perpendicular to the plane of the curve. Look into the mirror and rotate it (still perpendicular to the plane of the curve) until the curve appears to
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
pass smoothly through the mirror. The silvered surface is then perpendicular to the curve. Since most mirrors are back-silvered, draw the perpendicular along the back of the mirror. Ladies’ purse mirrors work fine, but beware the double-faced kind that’s silvered in the middle. Any other plane, glossy surface also works-microscope slides, even a knife blade. Obviously, both methods can be used to draw perpendiculars in other useful places. One that comes to mind is locating the center of a circle. The perpendicular bisectors of any two chords intersect in the center of a circle. This, then, is a handy way to reconstruct a broken circular part from a piece of it. Our other gadget is a kind of planimeter which can easily and “reversibly” be made from the large compass of most drafting sets. SVith it, one can measure area to *2yG which is a t least as good as counting boxes on graph paper and is a lot handier than the “cut it out and weigh it” technique for graphical integration. I’ve never known anyone to swipe this kind of planimeter either. This planimeter is made by replacing the pencil of the compass by a small, curved blade which lies in the plane of the compass-hence the name, Hatchet Planimeter. A curved Xacto knife blade is best, but a washer of about 1-in. diameter, which has been file-sharpened, works too. Means for clamping it where the lead usually goes is usually obvious. A fold of paper will tighten things up, if necessary. Place a startingpoznt near the center of gravity of the curve to be in-
LETT ER S
QUESTION: HOW can YOU
leach bigger tonnages at lower oDerating costs? tegrated and draw a line from it to the curve. Lock the compass legs apart a distance about equal to a “diagonal” of,{ the figure to be measured. Then hold the compass perpendicular to the plane of the curve with the tip on the starting point and the blade somewhere near the middle of the plane. Mark its starting point. Now holding the tip loosely, slowly trace the line out to the curve, go around the curve, along appropriate axes, and then return to the starting point. The “hatchet” will follow dutifully, but since it can only rotate or travel parallel to itself it will trace an odd path. Mark its stopping point. T h e area traversed is the product of the span of the compass and the distance between the initial and final positions of the hatchet. T o assure that the compass span hasn’t changed during the traverse, it is wise to compare the distance between the starting point of the tip and the initial and final blade positions. Obviously, one should try this gadget on some known areas to get the hang of it-circles and squares. If the distance traversed by the blade is too short for precise measurement, shorten up on the span of the compass. If the blade’s journey causes it to leave the paper, start the blade in a different spot and/or traverse in the opposite direction. Although it’s fairly obvious how the optical differentiators work, I have never understood the integrator. If any reader has a n explanation, do send it along. We would also like to have knowledge of your favorite gadgets, problems, or whatever else will interest Industrial Chymists.
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Units Again Sir: This is in response to your invitation for comments on the article by Dr. Adrian Klinkenberg, “American Engineering System of Units and Its Dimensional Constants g,,” [IND. ENG. CHEM.,61 (4), 53-59 (April 1969)l. We are chagrined to have made the errors in units which he noted in ref. 41. T h e reason (but not excuse) for our carelessness was the knowledge that all of our work was actually done and expressed in dimensionless terms. T h e problem of units is vexing to both undergraduate and graduate students, and I have done considerable experimentation to discover what is most successful in teaching. I have concluded that it is best to stick to the M L T system in my presentations and thus avoid the distraction of the extra terms g c and J . However they must regularly utilize data in terms of Btu’s, calories, and pounds force. Hence I must also teach them to convert to the M L T system from the F M L T and F M L T Q systems. I a m appreciative to Dr. Klinkenberg for an enjoyable and constructive article. Stuart W. Churchill
The Carl V. S. Patterson Professor of Chemical Engineerzng University of Pennsylvania Phila de& hia, Pa. 19 I04
THE FRENCH OIL MILL MACHINERY CO. Piqua, Ohio 4 5 3 5 6 . A/C 513-773-3420
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