Nomograph for Formulas Containing Fractional Exponents W.HERBERT BCRROWS State Engineering Experiment Station, Georgia Srhool of Terhnology, .4tlanta, G a .
T h e use of fractional exponents in the derivation of empirical formulas is widespread in engineering fields, especially chemical engineering, hydraulics, air conditioning, etc. Frequently empirical formulas are encountered which require the simultaneous irlrolution of terms to fractional powers and multiplication or di\ision of several such terms. These terms are cumbersome to handle by ordinary means, such as the log-log slide rule. A general nomograph, derived by blcblillen, gives a method of simultaneous involution and multiplication of such terms. A new nomograph has been derived which accomplishes the same results and which generally requires fewer operations; hence manipulation of the chart is simpler. A t the same time basic changes have been made which lead to greater accuracy of reading and interpolation of the scales and to a wider range of products. The method is illustrated by the solution of typical problems.
to the same possible errors of manipulation which afflict all nomographs. Hig!i accuracy of readings will be of more credit to the manipulator than t o the chart. I t best the readings may be considered satisfactory fur engineering calculations, where errors of 2 to 5y0 are allowable. The method is t o be considered, in short, primarily as a time saver. CONSTRUCTION OF CHART FOR EXPOSEKTS.The positions of the vertical lines for exponents are determined as shown in Figure 2 . Linear scales are laid off on the parallel extreme scales, using moduli in the same ratio as that t o be used in laying off the logarithmic scales later; in the examples shown, this ratio is 2 to 1. A line is drawn betn-een the zero points on the two scales and a straight edge (dotted lines) is laid from - 1 on the T-xale to successive points on the P-scale. The points of interesection of the straight edge with the 0-0 line mark the abscissas of the horizontal scale, the values of these points having the same numerical value as the corresponding points on the P-scale. This construction, as shown, gives rise to a scale which reads from right t o left, a factor which facilitates the use of the chart by right-handed persons; such persons will generally operate the pivot with their right hands and thus leave t,he T-scale unobstructed. Left-handed persons will doubtless find a reversed chart easier to use for the same reason. LOGARITHMIC SCALES.The T- and P-scales are logarithmic. The moduli of these scales must be in the same ratio as the linear scales used t o mark off the vertical lines for exponents. Those portions of the P-scale (lo2 to 104 and lo-* to lying along the top and bottom edges of the chart are laid off by extending the P-scale two cycles beyond the upper and lower extremes of the chart and projecting these cycles onto the horizontal position by a straight edge laid through point 1 on the Tscale. Except for evolution, this portion of the scale may be used only in the reading of final products, not for initial terms. I n other words, since i t is formed by projection through the point T = 1, it is valid only when the straight edge lies through t h a t point. vERTIC.4L
CHART for the evaluation of fractional powers of terms and for the direct accumulation of a product of odd-power terms was previously prepared by 1IcYIillen~. The usefulness of such a chart in chemical engineering calculations, where the log-log slide rule is somewhat more cumbersome, was well presented in the article accompanying that chart. The present chart differs from the original in two respects: First, tlie ratio of the moduli of the T - and P-scales, I' = 2 , is used as compared to I' = 1 in the original chart. This provides for greater accuracy in the reading of the individual terms, wider range for the products, and a more even spread of the vertical lines of the exponent scale. Second, the revised mode of operation described provides additional range for the produet scale by projection of the P-scale, extended, onto horizontal lines along t o l o 4 is the top and bottom of the chart. -1range from bhov-n, although the same principle could be applied t o extend the range from IO-'* t o 10'2 if it n-ere advisable. The principal deviation from the original i, in the mode of operation, which will be described in detail later. In the original method the simultaneous involution and multiplication of power terms required two straight edges or tn-o operations for cuch term (in certain special caseb, three operations for each two terms), and the product TYas accumulated term by term on the P-scale. I n the present method one position or operation of the straight edge is required for each term, and no product need be shown on the P-scale until the final reading. The master chart, from which the reduction slion-n in Figure 1 n-as made, is 16 X 20 inches, exclusive of scoring3 and numerals. Consequently, eacil cycle of the 7'-bralo is 10 inches in length, and each cycle of t,he P-scale is 5 inches in length. It does not follox that the scales have the accuracy of slide rule scales of corresponding length, since the latter are machine-engraved whereas the chart is entirely hand-ruled. Aforeover, the chart i. subject 1
LISM
USE OF CHART
The method of construction of 1x1OLKTIOS A X D EYOLPTION. the scale of exponents shows that the segments of the 5"- and Pscale intersected by t \ Y o transversals crossing a t point e of the exponent scale are in the relationship, -4 = eB, where A and B are linear segments of the P- and T-scales, respectively. If A and B are logarithmic segments, then log A = e log B , or A = B e . Rule 1 . T o raise the number, n, to the eth poTver, connect 1 on the P-scale TYith n on the 5"-scale. Place a pivot a t the inter,v.xtion of the straight edge n-ith the vertical line for e , pivot the straight edge t o 1 on the 2'-scale and read the answer, ,ne, on the P-scale. The example given (Figure 3) shows t h a t 2.6'.'j = 5 , 3 5 . Evolution proceeds in an exactly reverse fashion, as in the same
SIcl\Iillen, E. L., IKD. ENG.CHEM.,30, 71 (1938).
586
JU.I:-,
1946
581
INDUSTRIAL AND ENGINEERING CHEMISTRY
80 60
50 40 30
20
j: -5
.-
in L5
Figure 1.
Chart for Formulas Invol\ ing Fractional Powers
t
INDUSTRIAL AND ENGINEERING CHEMISTRY
588
-. I
E-SCALE
‘I: L-6
-5
Figure 2.
Construction of Scale for Exponents
3Ioduli of T-and P-scales are i n a ratio of 2 to 1.
1 7.5 example, v 5 . 3 5 = 2.6. I n both cases, as in the cases to follow, terms are located on the Tscale,except initial terms in multiplication; their powers and products are located on the P-scale. ~IULTIPLICATIOS ~ N D DIVISIOX. The method of lIclIillen for simple multiplication is shown in Figure 4, path 1. The number 2.5 is raised t o the first power, as in ordinary involution giving the point 2.5 on the P-scale. The number 4.2 is then raised to the first porver, but the initial point for this operation is taken as the final point of the previous operation (P = 2 . 3 , so t h a t the resulting reading on the P-scale is the product 2.5 X 4.2 = 10.5. Taking this as the initial point, a third member, 8.6, is then raised t o the first power, giving the product 2.5 X 4.2 X 8.6 = 90.3. This could be repeated for any number of terms, the result being like t h a t shown by path 1 of Figure 4 except t h a t it could not extend beyond the limits of the vertical portion of the P-scale. Division is accomplished by pivoting in the reverse direction. This is, of course, a simple application of the addition and subtraction of logarithms to obtain a product; hence, no proof seems necessary beyond the example given. X simpler and more direct method is t h a t diown by path 2. Instead of projecting each term back onto the P-scale and measuring all units from t h a t scale, the pivot is merely moved one additional unit to the left for each additional term, and the final product is obtained by projecting onto the P-scale from t h e final
Figure 1. Figure 3. IiiroluLion and Etolution, S h o w i n g a JIethod of Obtaining Odd Powers or Roots of Any N u m b e r
Vol. 38, No. 6
T w o RIethods of JIultiplication
Path 1 shows the method of RlcXIillen, in which a product is accumulated by adding the logarithms of the terms on the P-scale. Path 2 shows that the same product is obtained by accumulating the product in the “field” of exponentu and projecting from the final position hack onto the P-scale. 0 indicates position of pivot.
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
lune, 1946
position of the pivot. The product so obtained may fall anyv here on the P-scale. Division is accomplished by moving the pivot t o the right instead of t o the left, for each term in the divisor, as this aniouiits t o multiplying by that term raised t o the -1 poxer.
,d
589
Rule 5. For terms n i t h powers other than oiit, move the pivot a number of spaces t o the left or right corresponding t o the power to which the term is t o be raiwd. T h e product i i obtained as before. For example (Figure G ) , t o evaluate the, product,
C'
,starting with 2.35 on the P-scale and 0 . 1 on the T-scale, move the pivot 0.15 t o the left. Pivot to 0.18 and again move the pivot 0.15 to the left (i.e., to 0.3). Pivot to 7.6 and move the pivot a n additional 3.32 to tlie left (i.e., t o 3.62). Pivot t o 1.725 and move the pivot 1.25 to the right (i.e., to 2.37). Pivot t o 1 and read the product on the P-scale
"'\.
//
,/'
r
Figure 5 .
E s a m p l e iii 3Iultiplicatioti and DiTision
Figure 1 is graphic proof of the identity of the products ob-
. tained by the tn-o methods, and proof of the validity of t h e secorid
method vi11 be based on that. I t is obvious from Figure 4 that tlie initial setting on tlie P-scale need not be 1, but may be t h e first tcmn of t h e product (that is, the first step in path 1 might t)e omitted). This reduces by one the nuinher of settings on the T- d e , arid leaves t o the method one setting for eacli term antl one for t h e product. Rule 2 . To find the product of a series of ternic, 11, X n2 X x s X . . ., connect ?L1 on the P-wale with 712 on the T-.vxk antl place the pivot a t vertical line 1. Pivot to ring on the T-ecale and niove the pivot t o vertical line 2. Proceed in this fashion for subyequent terms, moving the pivot onc i.iiiit t o the left with each term. For division, move the pivot one unit t o the riglit for each divisor. When the pivot has been located for the last term, pivot to 1 on the T-scale and read the product o n tlie P-scale. For example (Figure 5), t o obtain the product, 17.5 X 0.22 X 0.655 X 4.30 0.31
=
31,8
start with 17.5 on the P-scale and 0.22 on thc T-scale. Pivot a t vertical line 1 t o 0.655 on the T->cale. Pivot a t vertical line 2 to 4.3 on the T-scale. Pivot at wrtical line 3 to 0.34 on the Tscale antl, *ince this term is in the denominator, move the pivot back one space t o vertical line 2. Pivot on this point to 1 on the T-scale and read the product on the P-scale. SIMCLTANEOLZ ~,ICLTIPLICATION A N D I s v o t u ~ ~ o x The - . motion of t h e pivot b y units t o the left or right was for terms v-hosct powers were 1 or - 1,rc.pwtivrly.
/
/'
,/" Figure 6.
Siniultaneoue \Iultiplic~ationand l t i r o l i r tion
A-OTES. If, during the course of an opcration, t l w pivcir move; too far t o the right or left, i t may be brought back into range t)? multiplication by any poircr of onc. This incans that, n-it11 the straight edge laid through point 1 ?n the T-wale, thc Fivot may he moved into any desired po>ition without affecting tlie previou,. terms or t h e final product. T o evaluate n', There n lies outside tlic range of the T-scale, factor n into tlvo terms by simultaneouc multiplication and ciivision by a convenient poivcr of 10. For example, to evahiaie 1680°,i0, write this term a.5 1.G80°.S0 X (103)0.60= 1.6800~0x 102.40and obtain the product of theae T K O terms, as previou-ly shown. If several such terms occur in the tame expression, it is advisable for the sake of time and accuritcy t o combine the factor9 of 10 before multiplication. For example,
+
.
