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T H E J O U R N A L OF I N D U S T R I A L A X D ENGINEERING C H E M I S T R Y
Vol. 8, No.
j
LABORATORY AND PLANT GAUGING OF STORAGE TANKS-METHOD OF ACCURATELY AND RAPIDLY DETERMINING THE VOLUME CONTENT OF MATERIAL IN HORIZONTAL CYLINDRICAL TANKS WITH BUMPED HEADS By
K. B. HOWELL
Received January 19, 1916
M a n y horizontal cylindrical t a n k s find employment in industrial operations of t h e present day. Some of these serve for storage or transportation of liquid materials, others for purposes of distillation, still others for purposes of admixture or agitation of liquid materials. In co-nnection with these tanks, there often arise occasions where i t becomes extremely desirable t o know the quantity of liquid material which is contained in them. There have been published from time t o time various formulas charts and mathematical tables which aim TO calculate t h e volume of material contained in these tanks f r o m a knowledge of the vertical height of t h e material in t h e t a n k a n d t h e dimensions of t h e t a n k These formulas and methods of calculation. however, are prac-
tically without exception based on the assumption t h a t t h e tank is a true cylinder. They, therefore, become a p plicable with accuracy only t o those cases u-here t h e t a n k or still has flat heads. In t h e majority of cases met with in practice, however, t h e mechanical advantages t o be gained have required t h a t the heads of the tanks be bumped. T o such tanks it is impossible t o apply the aforementioned method of calculation without the introduction of a considerable error. I n t h e case of t h e average 8,000-gal. t a n k car, t h e volume content of t h e t w o bumped heads is about 2 7 7 gals., or about 3.5 per cent of t h e total contents of t h e car. Since, therefore, the percentage content contained in the heads varies from 0 , when the t a n k is empty. t o 3 j per cent, when it is one-half full. i t becomes necessary for
us in many cases t o employ methods of measurement which take account of this varying volume content. Probably t h e only relatively accurate method is b y a direct empirical calibration of t h e t a n k filling the same xi-ith measured quantities of water. This method is quite slow and laborious. I t has, therefore, seemed t o t h e writer advisable t h a t a method of calculation be worked out for t h e determination of the volume content of t h e liquid material contained in a horizontal t a n k containing bumped heads which shall eliminate t h e necessity of a laborious calibration of the t a n k and a t the same time permit the calculation t o be of a reasonable degree of accuracy. I n other words. the method of calculation should possess the following features : 1-Simplicity of Calculation 2-General Applicability 3--Accuracy Commensurate with the Possible Accuracy of Measurement
Expressing the proposition in a somewhat more mathematical form, our problem then resolves itself
F~CTOQ FIG. I-FOR DETERMINATION O F LIQUIDCONTENTS O F n U 4 l P G D T A N K SI N HORIZONTAL POSITION Liquid Contents of Tanks (Exclusive of Bumped Heads) is equal t o Factor X Diameter2 X Length of Tank in ft.
into a determination of a method of calculating t h e volume of the figure formed b y the sides and heads of a “bumped” t a n k and a horizontal plane a t any height, H , from the bottom of t h e t a n k . This volume we have represented by V . It may be considered as consisting of two parts. the volume V6 contained in t h e true cylindrical portion of the t a n k which is exclusive of the bumps, a n d the volume It’, contained in each bumped head. The relatioliship between these 1-dues is expressed b y the following equation: I’ Vc + 2 v-b
T H E JOCRNAL OF I N D U S T R I A L A N D ENGINEERING CHEMISTRY
&fay, 1916
T/, This volume is t h e segment of a t r u e cylinder. I t s volume can. therefore, be readily determined by dfly of t h e methods, charts or tables recognized in general engineering practice a n d referred t o previously i n t h i s article. Fig. I is a graphical adaptation of one of these tables from Kent’s ‘‘ Mechanical Engineers’ Handbook,” page 1 2 1 . It is used in connection with t h e following formula: METHOD OF DETERNINING
where
V c = 7.48 X ( F . No. 1) X DZL = length of tank in f t . = diameter of tank in ft. 7.48 = conversion factor for cu. f t . t o gals. F. N o . 1 = factor obtained from Fig. I
L D
F. KO. I is obtained as follows: Express t h e height H as a fraction of t h e diameter D. Locate t h e resultant value on t h e vertical axis of t h e chart.
From
43 I
of a complex figure formed b y t h e surface of a sphere,
a fixed vertical plane a n d a horizontal plane a t varying height, H , from t h e lowest point, L , of intersection of t h e vertical plane a n d sphere. This may be otherwise expressed as t h e volume of a portion of a spherical segment cut off b y a plane at vertical height, H , from t h e point L. The development of a n y general formula or method of determining this volume is rendered simpler a n d of somewhat more general application by reason of t h e very general practice of t a n k design by which t h e radius of t h e bump of t h e t a n k head is made equal t o t h e diameter of t h e t a n k . While it is quite possible t o design a n d determine upon a method of calculation which applies t o a different radius of “bump” yet t h e
V
s
+ % S
0 .+ V
s”
Height
H
Expressed as
cf
Fractror,
Diameter FIG I1
D.
this point, move horizontally across t h e chart until intersection with t h e curve occurs. F r o m t h e point of intersection, drop down t o t h e vertical axis. T h e point of intersection with t h e latter represents t h e desired factor F. S o . I . M E T H O D O P D E T E R M I N I N G T/(, This proposition is considerably more complex. It consists essentially in t h e determination of t h e volume
FIG. 111-FOR DETERMINATION O F LIQEIDCOXTEXTS OF BUMPED T A N K SIN HORIZONTAL POSITION
almost universal acceptance of t h e above formula has led us t o confine our proposition t o t h e above condition. I n our subsequent calculations, therefore. we shall t a k e advantage of this condition which results in making t h e diameter of t h e base of our segment equal t o t h e radius of our sphere. Expressed in terms of mathematical symbols, our problem is as follows:
T H E JOCRiYAL OF 1,VDCSTRIAL AKD’E-VGISEERISG C H E M I S T R Y
43 2
WETHOD O F C A L C U L A T I O X
Given: 2 D
SPHERES : Diameter
=
SPHERICaL
Diameter = D ; Height = D -
SEGMENT :
-D -J?t3
CIRCULAR AREALEFC : Diameter = D CIRCULAR SEGMENT C E F : Base C E = B ; Height = H CIRCULAR AREAG E C K : Diameter = D CIRCULAR SEGMENT G E C : Base = C E - B; Height = H ’ ; Area = d SECTION OF SPHERICAL SEGMENT GELC
Required: Volume V b o i Section GELC
This volume may be expressed by the following expression :
T o make this expression capable of integration the value must be expressed as a function of H . The expression for 21 in terms of H contains the inverse sine and is, therefore. difficult of ready mathematical integration. I t has, therefore, seemed more advisable t o plot this function graphically on Fig. I1 and perform the integration graphically. This method of calculation consists essentially of four parts: (1)-Determination of A for given values of H (2)--Plotting these values on Fig. I1 (?)-Planimetering of Fig. I1 t o determine 1-alues of I-b for given \ d u e s of H (4)---Plotting these values on Fig. I11 ( I ) D E T E R M I X A T I O N OF A--A is the area of a segment of height H‘ of a circle of diameter D’. I t s ‘value may be determined from the tables of Kent if the r a l u e of D’ and H‘,/D‘ are known. These values are obtained as follows:
Expressing H’ in terms of fractions of D‘, we obtain numerical ~ a l u e for s H‘, D‘. Employing these values and using t h e tables in Kent for t h e determination of the area of a segment, we obtain the values shown inTable1. TABLEI Factor
II 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0
D
D D 0 D
D
0 D D
H’jD’ 0,0270 0.0505 0.0707 0.0879 0.1022 0.1138 0.1227 0.1290 0,1327 0.1339
No. 1
0.00246 0.00603 0.00964 0.01297 0.01583 0.01854 0.02020 0.02156 0.02240 0 02265
TABLEI1 A 0.00782 0 % 0.02026 0 ; 0.03287 0 0.04721 D 2 0.05936 D i 0.07119 D 0.0i898 0 2 0,08538 D z 0.08938 D ? 0.09060 D 2
H 0.05 D 0.10 D 0.15 D 0.20 D 0.25 D 0.30 D 0.35 0 0.40 D 0 4.5 D
vb
0.00017 0 3 0.00085 D 3 0.00221 0 3 0.00420 D 3 0.00687 D3 0.01048 0 3 0.01386 0 3 0.01805 0 3 0.02234 0 3 0 5 0 D 0.02697 Da
Factor No. 3 0.00017 0.00085 0.00221 0,00420 0.00687 0.01048 0.01386 0.01805 0.02234 0.02697
I
O F FIG. Ir--The values of A are t h e n plotted on Fig. I1 against the corresponding values of H expressed as fractions of the diameter. The scale of this chart is as follows: (2)
PREPARATIOS
Vertical-I in. =C= 0.02 D ? Horizontal-1 in. === 0.05 U 1 sq. in. = 0.001 D3
(3) DETERXISATIOS O F T’b-The area of the portion of Fig. I1 beneath the curve and between the origin and any
I7ol. 8. No.
j
given height, H , represents the desired volume I.;] as cut off b y the plane a t height H. These areas have been very carefully planjmetered, the values of determined for varying \-slues of H and t h e results tabulated in Table 11. (4) P R E P A R A T I O X O F FIG. 111-The values V b a r e expressed as decimal fractions of the cube of the diameter of the t a n k . The decimal portion of these expressions we have plotted on Fig. 111. against t h e values of H expressed as a fraction of t h e diameter L). By reference t o this chart, then, we are enabled t o obtain for a n y height, H . expressed as a fraction of D , the necessary decimal fraction with which t o multiply the cube of t h e diameter of t h e t a n k in order t o obtain the volume contained in the bump. This \-olume can be expressed in terms of gallons as follows: where
T’b = 7.48 X (F. S o . 3) X = diameter of tank in it. I .48 = conversion factor (cu. it. t o gals.) I;. S o . 3 = factor obtained from Fig. I11
p
0 3
in a manner exactly similar t o the manner in which F. KO.I is obtained from Fig. I . THE GESER.4L
EXPRESSIOS
.Is previously stated, the total ~ o l u m eis expressed by the formula: 1’ = r L 2 T’,i K e h a r e then the following formula for the desired partial volume 1~’ of a bumped headed t a n k ‘
+
‘i’ = 7.48 X (F. S o . 1) X 0 2 X L -L 2 X 7.48 X (F.No. 3) X D3 where D =diameter of t h e tank in it. & = l e n g t h of the tank in it. j.48 = coni-ersion factor (cu. i t . t o gals.) F. S o . 1 and F. KO.3 are factors obtained from Fig. I and Fig. 111 as preriously described
The use of this formula not only permits the determination of the gallon contents of liquid material contained in bumped tanks from a knowledge of t h e dimensions of t h e tanks and t h e height of the liquid, h u t also permits the calibration of these tanks by a simple mathematical calculation avoiding t h e laborious method of filling vi.ith known quantities of liquid. ;It the same time this method affords a n accuracy r.5 great as is generally desired in manufacturing operation. DISCL-SSIOK OF D E G R E E O F A C C U R A C Y
The method of calculation employed is theoretically accurate. The degree of accuracy is limited only by the size scale on which t h e graphical results are plotted and the number of numerical figures carried in the mathematical calculation. I n so far. however. as the actual mechanical construction of tanks and stills 7-aries necessarily t o some extent from the design. it is hardly of a n y advantage t o employ a method of calculation of a n accuracy greater t h a n one yielding a maximum error less t h a n 0.1per cent of the total content o€ the t a n k . To obtain this desired degree of accuracy requires t h a t the maximum possible error in determinating 17b and I’, should he less t h a n 0 . I per cent of T * . Since TTb is seldom greater t h a n j per cent of T’, this requires t h a t t h e accuracy of F. S o . 3 and t h e corresponding Figs. I1 and I11 shall be such as t o permit an error not over z per cent. I n plotting these charts and in
& l a y , 1916
T H E J O C R Y A L OF I N D C S T R I A L A N D ENGINEERIXG CHEMISTRY
all calculations, a considerably greater accuracy was maintained. I n t h e case of Vc, however, a certain a m o u n t of difficulty arises. Vc represents approximately 9 j t o I O O per cent of V . T o maintain t h e desired final accuracy, requires t h a t F. No. I be obtained with an accuracy corresponding t o a maximum error of approximately onet e n t h of one per cent, a n d consequently t h a t Fig. I be plotted on a scale capable of reading t o this degree of accuracy. While such a scale is quite possible, it is quite inconvenient a n d t h e writer would recommend in cases where extreme accuracy is desired t h a t t h e value F. No. I be determined from t h e table for “Determination of Areas of Circular Segments” given in Kent’s ‘‘ hTechanica1 Engineering Handbook,” page I z I. The factor under t h e heading “Area” represents t h e desired value F. S o . I a n d is given t o very great accuracy. I n general, however, t h e above Fig. I plotted o n scale 0.1t o t h e inch is sufficiently accurate for most manufacturing calculations. 17
T H E BARRETTCOMPAXY NEW YORK
BATTERY PLACE,
THE UNIT OF VISCOSITY MEASUREMENT By P B R K B R
c. MCILHINEY
Received March 1, 1916
The scientific world expresses t h e results of measurement of viscosity in terms of absolute viscosity of which t h e units are directly related to t h e fundamental units of mass, length, a n d time. The practical world speaks of Saybolt seconds, Engler numbers, etc. The absolute C. G. S. unit of viscosity is a relatively large one so t h a t water a n d similar liquids have absolute viscosities which are inconveniently small numbers, a n d furthermore, without giving it a name it is impracticable t o use such a unit in commercial testing. M a n y people have t h e idea t h a t absolute viscosities cannot be determined except when t h e viscosity is deduced from t h e absolute dimensions of t h e instrument and t h a t t h e practical instruments are not adapted t o determine absolute results. The fact is t h a t there are many practical methods by which viscosity may be measured, some of t h e m better t h a n others, all of t h e m subject t o disadvantages which differ according t.o t h e circumstances of their use, but all of t h e m giving results capable of being translated into t h e common language of absolute viscosity. The s t u d y of this important physical property of liquids has been seriously hampered b y t h e lack of a n y kind of uniformity in its measurement. T h e principal use which is made of viscosity measurements to-day is in t h e case of lubricating oils, b u t their use is not more widely extended because workers in different countries with different kinds of instruments think t h e y are unable t o obtain anything b y their work but results of merely personal interest. Few of t h e m know how t o translate t h e results which they obtain into results comparable with those obtained with another instrument. This s t a t e of affairs would be radically improved if there were some unit of measurement of viscosity which was generally intelligible a n d in which t h e results of a n y determination with any instrument might
433
be expressed. The absolute unit as already mentioned is inconveniently large a n d i t has no name. T h e suggestion has been made b y Deeley a n d Parr’ t h a t t h e unit of viscosity expressed in C. G. S. units should be called t h e “poise” in honor of Poiseuille, b u t t h e suggestion has not been adopted generally a n d it is customary t o simply speak of t h e “absolute” viscosity of a liquid. If t h e “poise” is adopted as t h e name of t h e absolute unit, i t has been suggested t h a t we might use t h e decimal multiples a n d submultiples of this c p = 0.01 9unit, a n d t h a t t h e n t h e centipoise-1 would be almost exactly t h e viscosity of water a t 2 0 ’ C. or 68” F. T h u s for all practical purposes in t h e lubricating oil business, i t would be sufficiently near t h e t r u t h t o say t h a t the viscosity expressed in centipoises is t h e specific viscosity, t h a t is, t h e viscosity as compared with water a t 2 0 ’ C. or 68” F. as a standard liquid. There are, as is well known, a Gariety of instruments with which t h e viscosity of liquids generally m a y be determined. There are three of these instruments which are in commercial use largely for t h e examination of lubricating oils, namely, t h e Saybolt Universal, t h e Engler, a n d t h e Redwood. All three of these instruments are capable, as shown b y t h e work done a t t h e Bureau of Standards in of determining t h e viscosity of oils with t h e accuracy usually required in present-day industrial testing. In these instruments, t h e number of seconds required for a given a m o u n t of oil t o flow through a small t u b e or orifice in t h e instrument is measured. Tables have been prepared by t h e use of which the t r u e viscosity may be calculated from t h e number of seconds required for a n y one of these three instruments. If these tables could be brought into general use, a determination of t h e number of seconds required b y the Saybolt Universal, Engler, a n d Redwood instruments would be reported in terms of centipoises. If the centipoise is used as a u n i t , t h e figure obtained will be 1.0042 for water a n d a larger number for all oils, a n d t h e number will represent t h e t r u e relation between t h e viscosity of the oil examined a n d t h a t of water with sufficient accuracy for commercial purposes, a n d also t h e t r u e or absolute viscosity. I n scientific work, t h e use either of absolute viscosity expressed in C. G. S. units or o f - t h e centipoise for convenience, will enable t h e workers in all t h e different fields in which viscosity may be determined, t o express their results in a universal language a n d the result will certainly be t h a t t h e use of viscosity as a valuable physical property will no longer be confined practically t o oils b u t will be extended t o many other lines of work. If, for example, t h e instrument recently described b y MacMichae13 should prove serviceable in many fields, investigations in viscosity can be carried on with i t upon materials for which i t is adapted a n d t h e results readily compared with those “The Viscosity of Glacier Ice.” Phil. M a g . , [6] a6 (1913). 8 5 . Dr. C. W. Waidner, “Conversion Tables for Saybolt Universal, Engler, and Redwood Viscosimeters,” Proc. A n . SOL.Test. A!fal.. 15 (1915). 1
2
284. 3
R. F. MacMichael, “ A New Direct-Reading Viscosimeter,” THIS
JOURNAL,
7 (19151, 961.
