T H E J O U R N A L OF I N D C S T R I A L
58
THE GAUGING OF STORAGE TANKS BY R
L OGDEN
Received December 10, 1915
One of t h e problems which often confronts t h e chemical engineer is t h e accurate gauging of storage tanks. When these are merely of upright cylindrical shape, t h e solution is very simple, b u t when t h e cylinder has dished ends a n d lies on its side, i t becomes a rather complicated problem, as there are two variables t o be considered, t h a t is, t h e cylinder a n d t h e spherical segments at t h e ends. It is t h e purpose of this article t o present a method of solution which, while rather easy of application, yet yields accurate results. The usual method of constructing these tanks is to make t h e diameter of t h e cylinder t h e radius of t h e sphere of which t h e end is a segment. The depth of t h e segment can better be calculated t h a n measured; if d equals depth of dish, R = radius of sphere a n d Y = radius of cylinder, then d = R - d R z - r z : in this case R = z r because of t h e method of tank construction. The arc of t h e dished end being only one-sixth of a circle, it may, for all practical purposes, be considered as t h e curve of a parabola, inasmuch as t h e curve of a circle a n d of a parabola for this distance very nearly coincide, and t h e spherical segment may be considered as a paraboloid. T h a t this assumption yields sufficiently accurate results will be shown by a special example later on. The volume of t h e cylinder may be estimated for each inch b y integral calculus, but a far simpler method consists in t h e use of geometry and trigonometry. The first inch is calculated by finding t h e area of t h e sector whose segment has a depth of one inch a n d subtracting the area of t h e enclosed triangle. This multiplied by t h e length of t h e cylinder gives t h e volume of t h e first inch of the cylinder
d$---F-
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I n Fig. I , I)' = /z base of angle x r - I = area triangle. Sine i/z 0 : i/z base triangle : : I : T
tri-
Having found t h e angle 0 we may easily compute the area of the sector, since angle 0 : 360' : : area sector : area circle. By subtracting t h e area of t h e triangle from t h a t of t h e sector we obtain t h e area of the segment. This same method may be applied for every inch a n d , of course, yields a n absolute value, but it involves far less work and is sufficiently accurate for all purposes t o find t h e length of t h e medial line of each one inch segment of t h e circle. This may be done by t h e ordinary methods of geometry. Thus t o find t h e area of the second segment, t h a t is, t h a t lying between t h e first a n d second inch, we have: -~ --
I/Z
medial line
=
~
x'rz - ( r - I ~ / z ) ~
This result doubled gives t h e length of t h e medial line which in linear measure very closely approximates t o the area of the one inch section in square inches. I t rarely happens t h a t t h e average inside diameter is expressed in even inches. I n this case it is neces-
sary t o consider 1 he inch section at t h e center as composed of two parts divided by the diameter; calculate each separately and add t h e t w o together The second half of t h e circular section is computed in t h e same manner inntil one comes t o t h e top segment, which is calculated as shown for t h e first inch. By adding together t h e values of each one inch section successively, t h e areas of t h e circular segments are found for each successive inch of depth. These areas multiplied by t h e length of t h e cylinder give t h e volumes of t h e cylinder for each inch of depth. The volumes of t h e spherical segments at t h e ends are computed for each inch in somewhat t h e same manner, assuming t h e segments are those of a paraboloid instead of a sphere. T h e area of the medial plane of a one inch section of a paraboloid in square inches very closely expresses t h e volume of t h e section in cubic inches. The area of a parabola is approximately the product of t h e base by two-thirds of t h e altitude. The base line of the medial plane of each one inch section has already been ascertained as it is identical with t h e medial line of each one inch section of t h e cylinder. If t h e altitudes of these planes are known it is a simple matter t o compute t h e areas of t h e medial planes and consequently t h e volumes of t h e one inch sections.
I n Fig. z t h e lines 0 B, E H, F I, represent t h e medial planes of each one inch segment, supposing t h e segment t o be continued t o form a sphere. The radius 0 B, as stated, is nearly always t h e diameter of t h e cylinder A C. The depth of t h e dished end, D B, having been found by t h e formula d = R . \ I F $ , this is subtracted from t h e radius 0 B t o find t h e length of D 0. This value D 0, common t o all t h e planes, added t o t h e height of each one inch segment, gives t h e radius of each circular medial plane. . Fig. 3 represents a side view of such a plane passing through 0 B, H E, I F. The radius 0 A or 0 B is dcD5qxD3. 0 D is common t o all t h e planes and A D is one-half t h e length of each medial line found previously in the cylindrical section. 0 B - D 0 = B D which is t h e height of t h e segment A B C. By now assuming t h a t A B C is a parabola, A C X 2/aB D = area A B C, and expresses very closely t h e volume in cubic inches of t h e one inch section of which A B C is t h e medial plane. By adding together successively t h e volumes of these one inch sections on t h e two ends and adding t o t h e corresponding volumes of t h e cylindrical part
T H E J O LTRLVAL O F I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y
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$ 9.19 25.89 47.21 72.17 ion. 15 130.73 163.59 198.45 235.09 273.33 313.01 353.99 396.13 439.29 483.37 528.25 573,85 620.05 666.77 713.91 761.39 809.13 857.03 905.03
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we obtain t h e total volumes of t h e t a n k for each SUCcessive inch of depth. T h e volume of t h e first inch of t h e spherical segments is almost negligible and may be found by assuming it t o be a pyramid a n d calculating in t h e usual way. T h e foregoing remarks are set forth in a tabulated form of calculation which i t might be found convenient to adopt. In this case t h e inside diameter of t h e cylinder is 48 inches a n d t h e length 1 2 0 inches. The depth of dish by calculation is 6 43 inches. The inside diameter, being in even inches, it is necessary only t o calculate t h e volumes of each section for onehalf t h e diameter and obtain t h e others by simple subtraction of each segment from t h e total volume of t h e cylinder.
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41.83 42.40 42.92 43.40 43.86 44.29 44.70 45.07 45.43 45.76 46.06 46.34 46.61 46.84 47.05 47.24 47.41 47.56 47.68 47.79 47.87 47.94 47.98 48.00
0.26 0.83 1.35 1.83 2.29 2.72 3.13 3.50 3.86 4.19 4.49 4.77 5.04 5.27 5.48 5.67 5.84 5 99 6 I1 6.22 6.30 6.37 6.41 6.43
1749.22 1797.78 1841.70 1883.81 1923.78 1961.64 1998.00 2031.86 2063.68 2093.63 2121.69 2147.90 2722.00 2933.76 2133.82 2311.61 2477.90 2261.67 2271.75 2283.60 2291 65 2297.84 2301.66 2304.06
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1106 3128 5724 8780 12223 16004 20084 24430 29016 33819 38818 43998 49337 54819 60430 66155 71982 77895 83882 89930 96027 102161 108318 114490
14 25 38 53 69 87 106 126 146 168 191 214 237 262 286 312 337 363 389 416 442 469 496
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I t will be noted in t h e above summary t h a t t h e total volume of t h e t a n k by this method of calculation is: 2 X 114,490 = 228,980 cu. in. The -.absolute volume of the cylindrical part is, /,@I or 217,150 cu. in. T h e absolute volume of the dished ends is, 2(1/’2xR2h I / e x h S ) = 11914 cu. in. Total absolute volume. . . . . . . . , . . . . . . . . 229,064 cubic inches. Total calculated volume. , . . , . . . , , , , . . . , 228,980 cubic inches.
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84 cubic inches.
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This difference of 84 cubic inches, or about onethird of a gallon in the total, between t h e calculated and t h e absolute becomes correspondingly less for t h e smaller gauges and is negligible for all practical purposes. 133 MILTONAVENUE,RAHWAY, N. J.
THE UNIVERSITIES AND THE INDUSTRIES ~~
~~
Addresses before the New York Section of the AMERICAN CIIEMICAL .%s~IF~TY in joint session with the New York Section of the SOCIBTYOF CIIEXICAL INDUSTRY and the AMERICAN ELECTROCHEMICAL SOCIETY, Chemists’ Club, December 10, 19 15
UNIVERSITIES AND INDUSTRIES B y RICHARDC. MACLAURIN President of Massachusetts Institute of Technology
The title of this address may give rise to some comment as to the place of the Massachusetts Institute of Technology in the general series of discussions that are being carried on by this Society. It can hardly be necessary, however, to say to an audience such as this that the fact that the M. I. T. is not called a university is merely an accident of names and does not touch the substance. For if you examine the matter a t all closely you will see that in everything but name what is sometimes unhappily called “the Boston Tech” is a university. The institution of the university, like much else that is potent in our midst to-day, has roots that go deep into the past and when you look back to the early days of the university you see clearly what was the fundamental idea that underlay that medieval name. The idea was not, as has sometimes been supposed, without the slightest support from history, that a university is a place where all subjects are studied. Fundamentally, i t was a place to which men came from all parts to study. The
medieval phrase was “university of persons” or “university of nations” and the first test of a university was whether it drew its students from a limited region or from many lands or many sections of one country. The Massachusetts Institute of Technology is as national and international in its scope as any of the so-called universities. It has long drawn men in large numbers from all states in the Union and has a very large proportion of foreign students to-day, nearly twice as many, I am told, as any other institution of higher learning in the country. The other tests were two: jijirst, the character of the subjects taughtwere they elementary or did they represent what were called the higher faculties; and second the character of the teacherswere they men of high standing, the practical test coming to be were they graduates of a recognized university. In the course of ages, a fourth test has been added, one that has long been inherent in the idea of a university, the test of investigationwhether the institution concerns itself merely with imparting knowledge or whether it endeavors to extend its bounds. I need not speak to you of the distinction of the three hundred men who form the instructing staff of the Institute of Technology
