INDUSTRIAL AND ENGINEERING CHEMISTRY
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sion was determined n-ith the General Electric recording spectroTABLE 11. EFFBCTOF PLASTICIZER ON PHYSICAL PROPERTIES photometer. OF F I L ~ I S 15% D 2 . o
Test Elmendorf tear Fold endurance,, cycles Light transmission througl 3-mil film, 4000 A. 5000 A.
7000 a
A.
D.P.
-
20% D.P.
100 88.000
850
77.0
69.0 81.5 89.0
86.0 89.5
25% D.P.
62.5
78.0 86.5
dibutyl phthalate
Table I1 shows the effect of plasticizer on other physical properties of films. Block temperature was determined by folding a strip of cast film lengthwise upon itself, applying a load of one pound per square inch upon the film, and placing the sample in a forceddraft hot air oven for one hour. The temperature at which the film adhered to itself so that marring or defacing resulted from pulling the halves apart is considered the block temperature. Heat-seal temperature was determined on the continuous-belt Saran sealer (15 feet per minute), inm-hich the sample to be sealed is carried between two continuous belts through a heated zone and a cooling zone. The heat-seal temperature is taken as that point which produces a seal stronger than the film surrounding it. Elmendorff tear tests were run on the standard tester of that name by the recommended technique. Fold endurance was determined a t 25’ C. and 50y0relative humidity in the Tinius Olsen model 31633 by the recommended technique. Light transmis-
USE APPLICATIONS
The Saran coating latices h a w been found useful in a broad range of applications. As coatings on paper, cloth, leather, plastics, and foils they provide heat-sealable finishes of excellent appearance, resistant t o water, greases, oils, and a wide variety of chemicals, and cxhibit low rates of vapor transmission. Latexcoated paper was until recently being used for packaging foodstuffs and small parts for the armed forces. I t is now available for civilian applications. As base resins for water paints these latices appear promising. Among the outstanding characteristics for this application are the rapid drying rate of successive coats and the resistance to scrubbing after a one-hour drying period. This is of particular interest to bakeries, laundries, and brevieries where the rapid application of a maintenance paint having good resistance to water is paramount. The finish may be applied by either brushing or spraying. Free films and tapes cast from the latices are of interest in the packaging field. Excellent adhesives ranging from rigid to flexible have been prepared for paper, cloth, leather, conveyor belts, belt drives, etc., and have shovn high strength and durability in use. Other uses include binders for materials such as cork, clays, mica, yarn floes, etc., in insulation, wallboard, and floor coverings. Because of the recent introduction of this type of latex and allocations which restricted use t o the war effort, the wide and varied fields of application are still unexploited. Development work is continuing to provide improved formulations for specific applications.
Construction of Nornographs with “YP erbolic Coordinates WALTER HERBERT BURROWS State Engineering Experiment Station, Georgia School of Technology, Atlanta, G a .
E
ARLY treatises on nomography, such as the works of d’Ocagne (5) and Lipka (S), were largely investigations of the equation forms arising from various arrangements of scales and index lines, with applications of these forms to t h e solution of engineering formulas. There was much duplication, and the same formula might fall into a number of different forms. The tendency among more recent authors, such as Sllcoclr and Jones (1) and Navis ( I ) , has been to devote primary interest t o the single case of three scales cut by one index line, with the various modifications and extensions possible within this form. The defining equation for nomographs ef this form is the equation of the straight line (index line) through points on the three scales. I n determinant notation this equation is yu
xu xo
f/w
xw
=
0
(1)
1
where x is the Cartesian abscissa and the y is the Cartesian ordinate of the scale points employed in constructing the U , V , and W scales. If all z values are constants, the three scales are straight and parallel. If any y = 0, the scale lies on the base line of the chart. If any y is a function of the corresponding z,
the scale may be curved. Thus, a wide variety of equation forms may be accommodated by this general nomographic form. When vie add the device of “variable Constants”, giving rise to “network scales”, thc form becomcs very versatile. The difficulties in constructing a nomograph to represent a given formula are not primarily those of converting the formula to the form of Equation 1 but of constructing the scales in such a manner as to yield a well-balanced and accurate chart. Two factors are involved in this step-the moduli of the scales and the angle between the coordinate axes. The latter factor has never presented any difficulty, since oblique coordinates are as easy to use as rectangular coordinates. On the other hand, it is not possible to change the modulus of any scale without simultaneously changing both the modulus and the position of a t least one other scale. Xomographs constructed without due consideration to the best relative positions and moduli of the scales are frequently impractical from the standpoint of ease of reading and interpolation, as indicated in Figure 5. I t is necessary, therefore, to devise means of varying the moduli and posltions of the scales. Earlier texts employed formulas relating these factors; later
INDUSTRIAL AND ENGINEERING CHEMISTRY
May, 1946
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473
I 0
.I
2 .3
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P
.5
O
Figure 2.
I
5 IO
Location of Ordinates
- -
A n ordinatem i located on a line drawn from point Z 0 to point on the Y-axis. The abacisma scale has a value of r 1.
IO 20 00
W
Figure 1. Scales for Locating Abscissas From the ranges of the variables, a suitable value of factor r is determined. The abscissa scale corresponding to this value of r is used. Scales for multi lea of these r values are obtained by multiplying the scale points the mame factor: that is, a scale for r 20 would be formed from the scale for r 2 by multiplying each scale point by 10.
gy
-
-
q
texts introduce a "matrix of transformation". Either method is laborious and none too satisfactory. I n reality, both are attempts to circumvent the rigid confinement of Cartesian coordinates, in which all ordinates have a fixed modulus. The purpose of this paper is to demonstrate that the defining equation may be stated in terms of a system of coordinates in which the moduli of vertical axes can be altered a t will; the transformation can thus be made upon the coordinate system rather than upon the defining equation. The same defining equation will, then, serve for constructing nornographs in which the scales have any possible positions and moduli with respect to one an-, other. It is necessary merely to select the base line in advance, much as the angle of the coordinates is selected in advance. Methods of making this selection from the ranges of variables are demonstrated in the construction of three nomographs. HYPERBOLIC SCALE
T h e problem of adjusting the moduli and positions of the scales of a nomogra'ph, in order to increase accuracy and legibility, is treated from the standpoint of the coordinate system rather than the defining equation of the nomograph. A coordinate system is described such that variations in the value of a single factor, r, will produce the desired variations in the positions and moduli of the scales. The general defining equation for nomographs is derived and shown to be independent of r. Thus the scale arrangements can be altered without changing the original equation of the nomograph. The mechanics of constructing nomographs on this coordinate system and the method of selecting the desired value of r are illustrated by the construction of three nomographs.
The scale referred to in this paper as the hyperbolic scale is a line segment of unit length divided into subsegments of length s = p / ( p r ) , where p i s a variable and r is a constant. Figure 1 shows hyperbolic scales for several values of r . The subsegments are marked with values of p. Scales for multiples of these T values are obtained by multiplying both r and the scale points (values of p ) by the same factor. A scale for r = 20 would be formed from the scale for r = 2 by multiplying each scale point by 10. For, if the multiplying factor is n, the function plotted is s = n p / ( n p nr) = p / ( p r). This method of producing multiples extends the range of r from 0 to a,and the scales of Figure 1 are thus sufficient for all nomographs. I
+
+
+
HYPERBOLIC COORDINATE SYSTEM
PRINCIPAL AXES. The hyperbolic coordinate system consists of three principal axes, X , Y,and 2.
414
INDUSTRIAL AND ENGINEERING CHEMISTRY
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Y
0
0 ,
2
5
Vol. 38, No. 5
SIGNIPICAXCE OF T . The above poiiits \\-ere located on a chart whose base line was the p-scale for T = 1. I n Figure 3 the same points are located on ordinates \\hose r = 0.1, 1, and 10. The reaulting shift in position of thc points relative to one another and to the nxcs iq ample dc,monstration of the flrxibility of thcie coordinates. RELATION TO C I R T L 6 1 1 N C'oORDIS4TbS. If a Set Of cart&arl cwordinates are superimposed upon hyperbolic coordinatrs so that the two X-axes coincide, with n: = 1 at the origin of the %axis, and the two Y-axes coincide, then a point (z,ui on the cartesian system becomes the point (p, q ) on the hyperbolic s~stcm This point lies on the line from 2 = 0 (.z = 1) to the point, ?/ = q . That is, its cartesian intucepte are N = I , y = 4. From thr intercept form of the equation of t h e straight linc.,
0 But Thc.refore
0 I
G
0 P
.i i
.5 Figure 3.
I
1
1
.5
I
I
Significance of r
The points of Figure 2 are located for three separate values of r . The shift in position of the points relative to one another and to the aws illustrates the flexihilit, achieved by changing factor r .
The X-axis is a line sepriieiit subdivided hyperbolically by onc' of the scales of Figure 1 or a multiple thereof. Point p = 0 is called the "origin" and point p = m, the "terminus" of the Xaxis. This axis may bc, either perpendicular or oblique to the other axes. The Y-axis is a vertical line whicali passes through the origin of the X-axis. I t s subdivisions are linear with a modulus, m y . The Z-axis is a vertical line u-hich passes t,hrouph the terminus of the X-axis. Its subdivisions are linear and in t,he snme sense as the Y-axis. Its modulus is r times that of the Y-axis (where is the cvnstant of t,he scxlr: subdividing the X-axis). That is, mZ = r7iiP-. The subdivisions, linear and hyperbolic, of these axes girt? propert,ies of the axes, not necessarily of functions plotted on these axes, just as Cartesian coordinates are fundamentally linear, although scales of any function may be plotted on such axes. LOCATIOK OF .4ascrssas. Abscissas of points on the liyperbolic system are given in terms of p rather than 2. Points having the abscissa, 2, lie on the vertical through point p = 2, on the X-axis. L o c a ~ r o sOF ORDIXATES. Ordinates are given in ternis of q and are determined by lines from the origin of the Z-axis to the values of p on the Y-axis. Points having the ordinate, 3, lie on the line from 2 = 0 to Y = 3. Figure 2 shoTw t h e location of the following poink: P 0.50 I .00
9 3.5 6.0
P 0.50 1.m
v 7.0 10.00
The abscissas are first located wit1i rcs1)cc.t to tila scale on the Xaxis, and a, light vertical line is drawn in o n cinch. Then L: straight edge, pivoted a t 1; = 0, is laid t,o the values of q on the Y-scale. The points of intersection with corresponding verticals are the required points.
= v/(p
+
(4)
T)
Eyuwtions 3 and 4 give the hyperbolic coordinates of any point, (2,y) on cartesian coordinates and will be used for deriving thv equation of the straight line from Equation 1. EQUATIOX OF STRAIGHT LIKE. Equation 1 represents in cnrtesian coordinates the straight line passing through points (zu, ?it,),(z,,y v ) , and ( x w , yTd). The equation in hyperbolic coordinates is obtained by substitutjng in Equation 1 the values of z and y from Equations 3 and 4. However, there are several caws to be considered, and each will be derived by a different set O F substitutions. . of thr hyperCASEI. NOPOI ST^ LIE ON - 1 ~ ~ sSubstitution bolic equivalents of z a,nd y in Equation 1 give,?:
wlh iQuTl(Pi) rlwr/ (pW ~
++ + r)
Pul(Pu
T) 1.)
p,i(p, pw/(pV,
-tT ) -t1.)
+ r)
1:
11
=
0
(5)
1i
Equation ,5 is transformed by the following steps: (1) subtracting the second column from the third to obtain a new third columri, ( 2 ) factoring T from the first and third columns, and (3) factoring l/(pu r ) from the first, r o ~ l, / ( p T ) from the second roJv, a n d l / ( p w T) from the third row (if no point lies on t,he Z-axis, ;ill o f t,hcse p valuw arc finite). I rcsult. of thcse transformations Equation 5 bccornt~a:
+
+
+
C.\SE; 11. OXE POINT LIES os X-.~XIY.If the point (xu,4%) lies 011 the X-axis, then
r,
= Pu!(Pu =
%I
+ r)
0
Substitution of these values lor t,ho first-ran. elcirieiits of Equation ,i,followed by the steps outlined above, gives the equation:
INDUSTRIAL AND ENGINEERING CHEMISTRY
May, 1946
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3, it indicates points on the Z-axis. The l a t h is the only case in which the third-column element may differ from 1; in that case 1 must occur in the second column. DEFININQEQUATIONOF NOMOGRAPH. As previously explained, the defining equation of a nomograph consisting of scales for three variables and a straight index line is the equation of the straight line through three points. In Cartesian coordinates this is, for the general case, Equation 1. In hyperbolic coordinates the defining equation is, for the general case, Equation 7 ; for specific cases, Equation 8, 9, or 10, or modifications of them. Three modifications of primary interest are: SCALES AND A CURVED SCALE CASEV. Two VERTICAL
’OO!
50
CASEVI. THREE PARALLEL SCALES
lh
Y
0
z
k 1
:I
0
= 0
(9)
CASE VII. 2-TYPENOMOGRAPH, Two PARALLEL SCALES AND ONETRANSVERSE SCALE
Figure 4.
Construction of Nomograph for the Ohm Law,
E = I X R
Without loss of generality,
m y
may be set equal to unity, and if
xv = 0
I!
1’ 11 = 0 01
0
p”
(10)
These various forms are illustrated by the following examples. In each case the X-axis has been placed oblique to the Y- and Z-axes in order to increase the symmetry of the nomograph or to render i t more compact. The axes are not shown in the figures but the descrbtion of the construction indicates their positions.
then yv = y
’
These substitutions are made in place of the second row of Equation 5 . By the above transformation, the resulting equation reduces to
CASEIV. ONEPOINT LIES ON Z-AXIS. If the point (xw,yw)lies on the Z-axis, its ordinate is y,
=
or yW =
zmz TZ
=
rzmy
( m y = 1)
Also, since the Z-axis intercepts the X-axis at its terminus, XW
= 1
Substitution of these values in place of the third
