Construction of Circular Nomographs with Hyperbolic Coordinates W-4LTER HERBERT BURROWS State Engineering Experiment Station, Georgia Institute of Technology, Atlanta, Gu.
Hyperbolic coordinates greatly simplify the construction of nomographs because of the ease with which the positions and moduli of the nomographic scales may be altered t o accommodate the ranges of the variables represented. Upon application to circular nomographs, this simplicity is enhanced by the fact that hyperbolic coordinates and circular nomographs are both based on a hyperbolic abscissa scale. This scale, with a range from zero to infinity, provides a nomographic form which, in certain cases, is superior t o the usual logarithmic- and Z-type nomographs for multiplication.
A'""
T IOUS article (1) has described a coordinate system for use in- the construction of nomographs. The value of this coordinate system in nomograph construction lies in the fact t h a t the coordinate system may be easily modified in producing a well balanced, accurate nomograph, whereas in the use of Cartesian coordinates it becomes necessary t o modify the defining equation of the nomograph t o accomplish this purpose. For nomographs 1
of complicated formulas, modification of the defining equation is a tedious task. hIodification of the coordinate system, however, involves only a change in the value of a constant of the system; the constant is a simple number, often an integer. This paper shows t h a t the circular nomograph of Douglass and Adams ( 3 ) is easily constructed by the author's method; the hyperbolic coordinate system adds to the simplicity of the construction by the nature of its abscissa scale. Figure 1 shows the method of locating points on a hyperbolic coordinate system whose constant is T = 5 . This constant has a dual effect: It determines the distribution of points with given abscissas on the X-axis, the corresponding Cartesian abscissa being 5 = p / ( p T ) ; and it determines the modulus of the 2axis relative to that of the Y-axis, because m, = 7mY. Figure 2 shows the effect of varying the value of r; the hyperbolic constant in this case is 0.5. In both figures the three points lie on a straight line. In the previous paper ( 1) it was shown that the defining equntion of any nomograph with a straight index line has the form
+
2
Y
Figure 1.
Location of Points on Hyperbolic Coordinate System
Figure 2.
158
Effect of Varying r
INDUSTRIAL AND ENGINEERING CHEMISTRY
January 1951
3
4
I l l
159
1.V
5
Inside Diomrbr Ift.)
l l l , l , A
B
I
I
2
10.
I
4
I
I
,
6
IO
,
20
Velocity (fps.)
3
4
5
Wl
Figure 3.
Circular Nomograph for the Product,
UI =
VI
x
w,
*5
Figure 4.
1.0
Circular Nomograph for Scobey’s Formula for Wood-Stave Pipes V
where the p’s and q’s are the hyperbolic abscissas and ordinates, respectively, of points on the U-, V-, and W-scales, as indicated by the subscripts. CIRCULAR KOMOGRAPIiS
If the q’s of the defining equation have the values q. = d p ; and qw = Equation 1becomes
6, 0
dG I-dG
m
*
pu
p”::
:I
qu = 0,
entire range from 0 t o 0 0 . d4tthe same time, it is possible t o select a value of r t h a t places any desired portion of this range in the middle of the scale where readings may be most accurately made. SCOBEY FORMULA
1
(2)
-O
T h e resulting nomograph has its U-scale on the X-axis, and its Vand W-scales are semiellipses. For greater ease of construction, the modulus of the Y-axis may be taken l / l / q t i m e s the length of the X-axis, in which case the ellipse becomes a circle. There may be cases, however, in which an elliptical figure would provide greater legibility. T h e simplicity of this nomograph, in either form, lies in the fact that the V- and W-scales lie on a fixed curve; thus, it is unnecessary t o locate both the abscissa and the ordinate of any point. Either the abscissa or the ordinate fixes the position of the point on the curve. A straight index line placed across this nomograph makes pessible solutions of the defining Equation 2, which upon expansion becomes p , = l/=. Obviously, the p’s may represent any functions of U , 8,and W , and the above device may apply t o a large variety of formulas. If the values of these p’s are p , =
l/cp*
=8 1,and p , = W1, where U1, VI, and W1 are functions of U , V , and W , respectively (Figure 3), the nomograph provides solutions of the relationship
u, = v1 x WI
= 1.65DO.46HO.665
(3)
This nomograph, in certain cases, is superior to the usual nomograph for products constructed of three logarithmic scales (straight and parallel) because the hyperbolic scales include the
For comparison, Figure 4 shows this method applied t o a formula which was used as an illustration in the previous paper (1)-the Scobey ( 2 ) formula for velocity of a liquid in woodstave pipes, V = 1.65D065SH0.555. I n the defining Equation 2, p , is set equal to 2/V/1.65, and p v = and p, = Ho.555. In plotting these scales it is unnecessary to locate any of the ordinates, because all scale points lie either on the X-axis or on the circle. The circle is first drawn, the X-axis with the desired value of T (in this example r = 1 ) is laid on the diameter, and the abscissas of points on the three scales are marked, completing the nomograph. Here the ranges of D and H are of the same order of magnitude. Cases arise in which the elements of the product are of different orders of magnitude, and these may be treated by t h e use of modifying factors. If, for instance, the ranges of Ut, Ti,, and W1 a r e U I = 1 t o 10, 81= 0.01 to 0.1, and W1 = 10 t o 100, it would be
4%
desirable t o set pu = p , = 30Vl, and p , = W1/30. By these substitutions, Equation 2 still becomes VI = VI X W , (the factors 30 and 1/30 cancel). LITERATURE CLTKD
Burrows, W. H., IND.ENG.CHEM., 38, 472 (1946). (2) Davis, C. V., “Handbook of Applied Hydraulics,” 1st ed., p. 9, New York, McGraw-Hill Book Co., 1942. (3) Doudaas, R. D., and Adams, D. P., “Elements of Nomography,’’ p. 147 ff., New York, McGraw-Hill Book Co., 1947.
(1)
RECEIVED june 9, 1950,
