ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT tion in which the diffusion constant is 0.16 X 10” square cm. per second show excellent agreement with parabolic velocity distribution down to values of y of 0.002 inch. In addition to the deviation caused by diffusion, it has been suggested that the observed deviations may be inherent in the flow system or in the instrumentation; however, changes in both of these have had no effect on the data. This observation, together with the data shown in Figure 7, has led to the conclusion that molecular diffusion is the only significant factor.
greatly expedited the work, and the assistance is gratefully acknowledged. The authors also wish to acknowledge the support of the Beckman Instrument Co., which made a spectrophotometer available in the later stages of the work. Nomenclature
b
= constant in Beer’s law
C = concentration of dye Co = initial concentration of dye d
Conclusion
The dye displacement technique has been successfully applied to the measurement of the velocity distribution of a fluid in laminar flow in a smooth circular tube. Velocity measurements have been made by this technique to within 0.002 inch of the wall of a ljAnch-diameter tube. These measurements are in a,greement with the parabolic velocity distribution predicted by the theory of laminar flow. In this technique, wall-adjacent velocity measurements are limited by molecular diffusion of the dye. Measurements extremely close to the tube wall may be made if molecular diffusion is sufficiently reduced. Acknowledgmen!
The experimental work reported in this paper is closely related to a project being carried out by Wright Air Development Center a t Iiorth Carolina State College. Cooperation with this project
= = = T =
t to
T, To
= =
= uo = z = y =
v
diameter of tube time time for a point moving with velocity Z J to ~ travel distance z relative transmittance transmittance a t concentration C transmittance a t C = 0 velocity of any point in tube velocity at center of tube distance along axis of tube radial distance measured from wall of tube
References
(1) Furth, R., and Ullman, E., ,RoZZoid-Z., 41,307, (1927). ( 2 ) Robinson, C., and Hartley. G. S.,Proc. Roy. Soc. 134 A, 20 (1931). (3) Seneca], V. E., and Rothfus, R. R., Chem. Eng. P r o g ~ . ,49, 533, (1953). RIPCEIVED for review hIsy 26, 195-1. ACCEPTEDAugust 20, 1954. Presented before the Division of Industrial and Engineering Chemistry at the 125th kIeeting of the h 1 E R I C A N CHEMICAL SOCIETY, Kansas City, hl0.
Direct Construetion Nomonranhs V
.
W. H. BURROWS Engineering Experiment Station, Georgia lnstifufe o f Technology, Atlanfa, Ga.
A
LTHOUGH a formula may be available for use as a defining
equation for the construction of a nomograph, it happens occasionally that the defining equation is not readily manipulated to produce the form of nomograph desired. It is then desirable to prepare from the formula a table of values which may in turn be used for the construction of the nomograph. Some freedom of choice may be exercised in the selection of one of the scales, or in the spacing of the scales. The consequences of making a particular selection are best illustrated in symmetrical nomographq of which the logarithmic mean temperature difference nomograph is a typical example. I n the case of unsymmetrical tables, the selection of a suitable scaIe function for the dependent variable is critical, since it vastly affects the ultimate shape and arrangement of the scales of the nomograph. Consequently, it is necessary to have not only a general method for the construction of nomographs from data tables, but also a means of selecting the critical scale function. logarithmic Mean Temperature Difference I s Example of Development for Symmetrical Case
An excellent example of a formula which yields an undesirable form of defining equation and for which it is desirahle to construct a nomograph by other means is that for the logarithmic mean temperature difference January 1955
M.T.D. =
AT2
- Ti
In ATg - h
where AT1 and AT2 are the temperature changes experienced by two fluids as they pass simultaneously through a heat exchanger, and M.T.D. is the logarithmic mean temperature difference. This formula takes the determinant form ( 1 )
The nomograph constructed on the basis of this defining equation is shown in Figure 1. A modified form of this nomograph is given by Helves and Seward ( 4 ) . The ATl and A T z scales are identical, making readings of M.T.D. difficult and uncertain for cases in which AT1 and AT, have nearly the same value. Aside from this, however, the unsymmetrical form of this nomograph makes adjustment of scale moduli and positions especially unwieldy Consequently, the construction of an accurate nomograph by the use of this defining equation is very difficult. A much more desirable form would be one in which the AT, and AT2 scales were separate, with the M.T.D. scale between them. However, conversion of Equation 1 into a defining equation for such a form involves determining the nature of defining scale functions, such as f1 in Equation 2.
INDUSTRIAL AND ENGINEERING CHEMISTRY
33
ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT
Although this is the form of the defining equation for Figure 3, no general deductive process is available for determining the nature off,, and it is usually a matter of chance that such a conversion can be effected inductively.
35-
--1 4 -
-
30 -
/ I
i
-
25-
3
--
IC3
i
Yo-;
20-7
1 -
83.-
i *
~
70s6C
,
5-
_1
7 10-
0
-
L 5 --4 Figure I . Nomograph for Logarithmic Mean Temperature Difference Constructed from Defining Equation
Figure 2. Construction of Symmetrical Nomograph from Data by Successive Subdivision
the scale is to be subdivided by graphical interpolation. In Figure 2 this method is illustrated for the logarithmic mean ternperature chart with parallel scales. Table I shows the values of the ordinates for all three scales for the points of subdivision obtained by this method of cross plotting. Figure 3 is the n o m e graph, vi ith scales completed by graphical interpolation. B siniilar nomograph appears in a publication of the Girdler Corp. ( 3 ) . The defining equation (Cartesian coordinates) of this noino-
Whether a nomograph of the type dpsired is compatible with the formula is generally indicated by examination of the formula. I n the case of the logarithmic mean t'einperature difference, the folloxing aspects indicate the form of nomograph which might be constructed: 1. In the formula A T l and AT2 may be iiit,erchaiiged without altering the value of X1.T.D. This indicates the possibility of a symmetrical arrangement with scales for AT1 and A T 2 on opposite sides of the M.T.D. scale. 2 . For the case AT1 = A T 2 , the value of 1I.T.D. is equal to that of AT1 and AT%. Consequent,ly, if the symmetrical forin can be constructed, ordinates on t'he three scales will be equal for equal values of AT1, A T P ,and M.T.D.
Method for Parallel Scales. In attempting construct,ion of a symmetrical nomograph of t'his type, the obvious procedure ia as follows (Figure 2 ) : 1. Locate suitable points, A , A', B , B', for t,he extremes of the exterior scales. 2 . Cross plot, I, between opposite points to obtain a point on the middle scale and compute the value of that point. 3. Draw a horizontal line, 11, t,hrough this point intersecting the exterior scales and compute the values of these points C, C', on the exterior scales. 4. Cross plot, 111, between these points and the terminal points of the scales t o obtain two more point's on the middle scale. Compute the values of these points. 5 . Draw horizontal lines,. IV, , through - these points intersectingthe exterior scales. Repeat this process of subdivision until enough points have been located that the remainder of each scale mav bv " be plotted _ graphical interpolation.
I n actual practice the table of values (Table I ) of the points obtained by this erose plot are calculated in advance from the formula and need not be inserted a t the plotted points until
34
i
Figure 3. Nomograph for Logarithmic Mean Temperature Differenee Constructed from Data
INDUSTRIAL AND ENGINEERING CHEMISTRY
VOl. 47, No. 1
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Table I. Ordinates Obtained b y Successive Cross Plotting of M.T.D. between AT1 and AT2, Extremes 10 to 100 Cross Plot Number 1 2 100.00)......
4
3
. . . . . . . . . . . . . . . 90.22 281.13..
> 6 4 . 8 3 ..
21.53. \
I
.
......
%lo 17.95
14.95..
Method for Special Scales. Cases arise in which it is desirable that the M.T.D. scale be linear or logarithmic so that the M.T.D. nomograph may be combined with other scales in the construction of a more complex nomograph. I n this case the middle scale is first laid o f f ,and a point, P I , is selected for one of the termini of an exterior scale. Directly opposite this point (specifically, on a line perpendicular to the M.T.D. scale and equally distant from the M.T.D. scale) a point, Pz, i s placed for the corresponding terminus of the other exterior scale. Ordinates corresponding t o the values of Table I are located on the M.T.D. scale and horizontal lines A , B, and C are drawn to establish the corresponding ordinates of points on the exterior scales (Figure 4). Abscissas of those points are located by diagonal lines, I, 11, 111, corresponding to the method of cross plotting by which the values of the points in Table I were derived. Figure 5 shovs a nomograph having a linear M.T.D. scale constructed in this manner. Similar nomogrrtphs have previously
quired. A suitable point of projection, P, is employed, and points on the three scales having equal ordinates are located on lines passing through this point of projection, rather than on horizontal lines. Instead of equidistance from the M.T.D. scale as measured
Figure 5. Nomograph for Logarithmic Mean Temperature Difference with Linear M.T.D. Scale
Figure 6. Nomograph for logarithmic Mean Temperature Difference with Logarithmic M.T.D. Scale
January 1955
Figure 4. Determining Abscissas for Scales with Predetermined Ordinates
INDUSTRIAL AND ENGINEERING CHEMISTRY
35
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT I
’
HYPERBOLIC REFERENCE SCALE
Figure 7.
’
I
e
I l l
3 4 5
-,
hand, with well selected scale functions it is usually possible to construct a useful nomograph, even for a table presenting seemingly great irregularities. The follox-ing discussion presents a iiiet,hod for converting asymmetrical data tables into nomographic form and some feasible steps leading to a proper selection of the most critical scale functions. Arbitrary Features. The features of the nomograph that may be selected at will, subject to certain reservations, are a scale for the dependent variable (Scale I), usually taken as one of the extreme scales, and the extreme points of one of the other scales, usually the scale (Scale 111opposite to the dependent variable scale. Figure 8 shows this stage in the construction of a nomograph. As construction of the nomograph proceeds, it is often necessary t o revise the features originally selected. The required revisions are usually clearly indicated by the shapes of the scales resulting from cross plot,ting betn-een the original points of the nomograph. Cross Plotting. By using the points originally !aid out, some points of Scale IT1 may be located by cross plotting, as shown in Figure 9. A sufficient number of such points should be located to determine whether the resulting scale will be sufficiently
t
Spacing Scales and Subdivision
c
for “Symmetrical” Nomograph of Unequal Scale Moduli
on a linear scale, “equidistance” on an hyperbolic scale ( 1 ) is substituted. Figure 7 shows the method applied to an 1I.T.D. nomograph with parallel scales.
CF
Method for General Case Includes Means of Selecting Critical Scale Function
I n the symmetrical case a great deal of freedom could be exercised in the choice of a scale for the M.T.D.; the scale function could be varied to accommodate various demands other than that of providing a nomograph having certain degrees of accuracy and compatibility of its scales. Most data tables, however, are unsymmetrical and leave far less choice in the selection of scale functions. A poorly selected function for one of the scales may result in a nomograph that is so distorted as to be virtudly illegible over portions of certain of its scales. On the other
t
\
I_ ~
I
//
-
--i
~
Figure 8. Arbitrary Features of Genera I Nornogra ph Constructed from Data
IF t-
1... .
I
\
~. .
*-i
~...
Figure
36
9.
Cross Plotting Points for Middle Scale
Figure l O. Determining Positions of Intermediate Points on Exterior Scale
Figure 11. Modifying Shape of Dependent Variable Scale
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vel. 47, No. 1
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT that should be considered M ith regard to range and distribution of points are y = 9,y = e=, y = log 2, y = x / ( z T ) , y = l/z, 1~ = ura br c, y = sin o, y = tan L, etc. It is desirable to be familiar with the distribution of points on scales representing the above functions over the entire range of x from - m to 0 toltofm. If one of these functions gives a scale that approaches, but does not reach, the required standards, it may be modified by the method shown in Figure 12. The scale is laid out on the abscisss, of coordinate paper, and the proposed extremes of the modified scale are marked off on the ordinate. Corresponding extremes of the two scales determine points in the plane, as shown. Approximate positions of additional ordinates are marked off to provide the desired distribution of points. These ordinates, with the corresponding abscissas from the original scale, also determine points in the plane. A smooth curve is then drawn through the approximate positions of these points; it is much more important that the curve be Rmooth and regular than that it pass exactly through the points. The points of the original scale are then projected upward to the curve and across to the ordinate scale, thus producing the modified scale. Variations in Plotting. Variations may be made in the plotting method, producing inversions of the scale positions. One variation is the substitution of the second independent variable for the one first plotted opposite the dependent variable scale. The other variation is transposition of the terminal points of that scale. I n most nomographs these variations contribute little t o the appearance or legibility of the nomograph. However, such modifications should be considered if it is desirable that certain scale points lie in predetermined positions or if the length of a scale is out of proportion to the degree of accuracy required of the variable it represents. Use in Analysis of Data. These methods may be applied to other formulas for which a suitable defining equation may not be obtained, or to empirical tables for which no formula has been derived. In the latter case, the nomograph scales may be analyzed to obtain general expressions for the functions defining their ordinates and abscissas, using standard methods of analysis. The values may then be inserted in a skeleton defining equation which, upon expansion, give,s rise t o an empirical formula for the relationships inherent in the table and nomograph.
+ +
O F “6”fiNO “ 3 ”
ORIGINAL
SCALE
(LOGARITHMIC1
Figure 12. Modifying Distribution of Points on Dependent Variable Scale accurate and well proportioned before proceeding t o the next step; these points should be sufficient, also, t o locate the remaining points of Scale 11. The intermediate points of Scale I1 are located by a similar manner of cross plotting, as shown in Figure 10. Portions of Scale I11 which could not be plotted from the terminal points of Scale IS are now plotted. At times, it develops that the scale for the dependent variable, w originally laid out, fails to yield a satisfactory cross plot over its entire range. I n such cases, it is usually possible to remedy the defect by a shift of points on that scale as the cross plotting proceeds toward the extremities of the scale. The shift is made by methods similar to those employed in cross plotting Scale 111, so that the points on the revised scale remain consistent with Scales I1 and 111, as shown in Figure 11. The result is a curved scale; i t is through this device that empirical nomographs with three curved scales sometimes make their appearance. Selection of Scale for Dependent Variable. I n the selection of a scale for the dependent variable, three points deserve primary consideratjion--the range of the scale must be such as to include all the values of the dependent variable given in the table; the subdivision of the scale must be such as to provide the degree of accuracy required in each of these values; and the scale must be free of discontinuities foreign to the dependent variable. A good starting point in the selection of the scale is one of the common functions, or a function known t o be generic with the variable to be represented. Some of the common functions
January 1955
+
literature Cited
(1) Burrows, W. H., IND. ENG.CHEM.,38, 472 (19.46). (2) Chemical Engineering Catalog, p. 99, Reinhold, New York, 1942-
43. (3) Girdler Corp., Louisville, Ky., “The Votator,” p. 20. (4) Hewes, L. I., and Seward, H. L., “Design of Diagrams for Engineering Formulas,” p. 63, McGraw-Hill Book Co., New
York, 1923. ( 5 ) Sherwood, T. K., and Reed, C. E., “Applied Mathematics for
Chemical Engineers,” p. 342, McGraw-Hill Book Co., New York, 1939. RBCEIVED for review November 30, 1953. ACCEPTEDAugust 23, 1954. Presented before the Division of Industrial and Engineering Chemistry at the 124th Meeting of the AMBRICANC H E M r C A L SOCIETY, Chioago, Iu.
INDUSTRIAL AND ENGINEERING CHEMISTRY
37
