August 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY k 8 - I1n
t
ACKNOWLEDGMENT
(a-a,)
- at
where a is composition of starting material, ae the composition a t equilibrium, and at the quantity converted after time, t, in hours. Since there was considerable drift in the IC values, the results are not exact but are of the right order of magnitude. These values are shown in Table 111. I
CONCLUSION
From the experimental data it is postulated that the hexanes isomerize in a stepwise manner: n-Hexane 2-Methylpentane
1823
11
11
3-Methylpentane
2,3-Dimethylbutane
11
2,2-Dimethylbutane The methylpentanes are rapidly interconverted; this prohibits deducin whether only one or both of the methylpentanes are intermegiates in the transformation of n-hexane to 2,3-dimethylbutane. The rate-determining step in the formation of 2,2dimethylbutane from n-hexane is the slow isomerization of 2,3dimethylbutane t o 2,2-dimethylbutane. I n the isomerization of n-hexane t o 2,3-dimethylbutane, the rate-determining step is the isomerization of n-hexane to the methylpentanes.
The authors are indebted to associates, especially B. H. Shoemaker, H. M. Grubb, and A. P. Lien, who contributed their advice and services. LITERATURE CITED (1) Egloff, Hu!!~, and Komarewsky, “Isomerization of Pure Hydrocarbons, New York, Reinhold Publishing Co., 1942. (2) Evering, B. L., and d’Ouville, E. L., J . Am. Chem. Soe., 71, 440 (1949). (3) Evering, B. L., d’ouville, E. L.,Lien, A. P., and Waugh, R. C . ,
Petroleum Division Preprints, 111th Meeting, AMERICAN CREMICAL SOCIETY, Atlantic City, N. J. , (4) Frankenburg, Komarewsky, and Rideal, “Advances in Catalysis,” Vol. 1, p. 251, New York, Academic Press Inc , 1948. (5) Leighton, P. A , , and Heldman, J. D., J. Am. Chem. Soc., 65, 2276 (1943). (6) Oblad, A. G., and Gorin, M. H., IND. ENG.CHEM., 38, 822 (1946). (7) Powell, T. M., and Reid, E. B., J. Am. Chem. Soc., 67, 1020 (1945). (8) Schneider, V., and Frolich, Per K., IND. ENG.CHEM.,23, 1405 (1931). (9) Wackher, R. C., and Pines, H., J . Am. Chem. Soc., 68, 1642 (1946). RECEIVED Maroh 1948. Presented before the Division of Petroleum Chemat the 113th Meeting of the AMERICAN CHEMICAL SOCIETY,Chicago, Ill.
istry
Construction of Three-
Dimensional Nomographs 1
We Ha BURROWS
State Engineering Experiment Station, Georgia Institute of Technology, Atlanta, Ga.
Use of three-dimensional nomographs for engineering formulas has not been tested in practical application, because nomographs of this type have been virtually nonexistent. This article presents comparatively simple methods for construction of three-dimensional nomographs. Although they present a convenient means of handling formulas with a proportionately larger number of variables than those handled by the more familiar two-dimensional nomographs, three-dimensional nomographs are more difficult to construct and to read. Nevertheless, in addition to the role they play as aids in computation, they provide a method of graphic representation which, as an aid to visualization, is difficult to achieve by three-dhensional graphs or two-dimensional nomographs.
T
HE construction of three-dimensional nomographs for certain types of formulas was described by d’Ocagne (6) as
early as 1921. Virtually no application has been made of the methods which he presented ; the reasons are apparently twofold: first, the theoretical discussion provided no simple means of adjusting the moduli of the several scales to accommodate variations in the ranges of the variables; and secondly, the method was applicable .only t o polynomials. This article presents a simple theoretical approach t o the construction of three-dimensional nomographs for all formulas capable of representation in a nomograph with a planar index surface. The hyperbolic coordinate system on which the construction is made may be modified to accommodate desirable variations in moduli and positions of the scales of the nomograph.
THREE-DIMENSIONAL HYPERBOLIC COORDINATES
XY-PLANE. The XY-plane is identical with the plane hyperbolic coordinate system described previously ( 2 ) . Both principal axes are hyperbolic-i.e., they are of finite length-with points on the axes defined in terms of p’s for which the corresponding distance of the point from the origia is p / ( p r ) ; r is an arbitrary constant. The hyperbolic constants of the two principal axes, rz and ru, are independent. Points in the plane are located with respect to points p , and p u on these axes by means of coordinate lines drawn to these points from the termini of the Y- and X-axes, I*espectively(Figure 1). Points with infinite abscisqa and ordinate are located on a n auxiliary Z-axis passing through the termini of the X- and Yaxes, its origin on the X-axis and its terminus on the Y-axis. Points on thi8 axis have the values of p;; the hyperbolic constant is r;, where r,- = r y / r a . VERTICAL AXES. The principal vertical axis is the Z-axis on the - origin of the X- and Y-axes. Auxiliary vertical axw are the Y-axis on the terminus of the X-axis, and the X-axis on the terminus of the Y-axis (Figure 2). These axes are linear; their moduli have the relationships, m; = rzmr and m; = rumz. Points on theRe axes have the values p;, q;, and q;, respectively. VERTICALPLANES.Thc vertical axes and the axes of the X Y plane together determine three vertical planes; the X Z plane, the YZ-plane, and the Zy-plane. Each of these planes is identical with the semihyperbolic planar coordinate system previously described ( I ) , consisting of a single hyperbolic axis a n d two linear axes. Points in these planes are located in the manner shown in Figure 3: abscissas, by vertical coordinate lines; and
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
1824
Vol. 43, No. 8
x, = L
100 x i , P
+2 I-' (100 - L )
where X~ is the mole fraction of component i in residual liquid (range 0 to 1); xii, mole fraction of component i in original liquid (0 t o 1): L, total moles of liquid (0 to 100j; P,, vapor pressure of component i (0 to 5000 mm.); and P , total pressure (0 to 7500 mm.). The formula may be expressed in determinant form
I
\
0
\
x
PX
Figure 1. llIethod of Locating Points on the Plane Hyperbolic Coordinate System
-c
X
0
igure 2.
The characteristics of the rows in this determinant indicate the following positions of the nomographic scales: 2% on the Y-axis,, zf, .. on the Z-axis, L on the XY-plane a t p , = 1, and P i / P on t,he Z-axis. From the ranges of these variables the desirable moduli of the vertical axes and the hyperbolic constants of the horizontal axes may be derived: m; = 10 inches, m, = 10 inches, r; = 1: whence, r2 = m;-/m, = 1, rU = rz/r; = 1, and m; = m L ~= , 10. The nomograph so constructed is shown in Figure 5 ; the base of the figure is on t'he XZ-plane. RAYLEIGH EQUATIOX.A somewhat more involved example is the Rayleigh equation ( 7 ) for biliary mixture.:
Semihyperbolic Solid Coordinates
ordinates, by coordinate lines drawn through the terminus of the hyperbolic axis. POIXTS IN SPACE. Points not lying on the axes or abovementioned planes are located with respect to their X , Y, and Z coordinates ; the corresponding coordinate planes lie on the 3-, p-,and Z-axes, respectively (Figure 4). DEFINING EQUATION FOR NOhIOGRAPHS
GEKERAL FORX O F DEFISISG EQUATIOS.The relationship of any four points (PZ,, PU,, G,), PA^, P,,, Y& (pzg, q z 8 ) , and (p,,, p u p ,qs,) lying on a plane is given by the expression
9y8.
Any formula in four variables which can be expressed in this form, with the functions of any one variable confined to a single row of the determinant, can be represented as a three-dimensional nomograph. The elements of that determinant corresponding to the p's and 9's of Equation 1 determine the values of the dimensions of the nornographic scales for the four correRponding functions. Such a zero-valued determinant is called the defining equation of the nomograph. SPECIFICTYPES. Equation 1 is the form taken by nomographs in which none of the scales lie on the axes or axial planes of the coordinate system. Scales lying on axes or planrs are indicated by modified r o w in the determinant, as follows: 0 y. qI
p, 0 p,
pr 0 p, q. 0
9;
0 0
q; q;
1 0
0
p;
p, p, 0 1 p, 0 0 0 1 1
where L1 is the moles (or pounds) of original charge ( 2 5 to 250); Lz, mole8 (or pounds) of'rcsidual charge (25 to 250); 21, mol(> fraction (or weight fraction) of more volatile component in original charge (0.01 to 1.0); zg,mole fraction (or weight, fraction)
2
I
1 for scales lying on the SY-plane 11 for scales lying on the XZ-plane 1 1 for scales lying on the YZ-plane 0 ' for scales lying on the Z g p l a n e 1 I for scales lying on the X-axis 1 1 for scales lying on the Y-axis 1 1 for scales lying on the Z-axis 0 I for scales lying on the z-axis 0 1 for scales lying on the Y-axis 0 I for scales lying on the Z-axis APPLICATIONS
As a firat example MOLEFRACTIOSS IN R E S ~ ~ ULIQUIDS. AL this method is applied t o the equations for mole fractions of components in residual liquid ( 6 ) :
Method of Locating Points the Plane Semihyperbolic Coordinate System
Figure 3. 011
INDUSTRIAL AND ENGINEERING CHEMISTRY
August 1951
(Xirn
+20.89.1)XL -C -A
- 16F
(Atm
1 + 9.l)hhrn 1 0 0
1825
XhmAh:
+ 9.1 0 1 0
1 0
=o
0 1
-X Figure 4. Method of Locating Points in Semihyperbolic Solid Coordinates
of more volatile component in residual charge (0.01 to 1.0); and 01, relative volatility (0 t o 10). From the ranges of LI and h, In ( h / L 2 )= 0 to 2.303. The determinant form of this equation is
l
o
The rows of the determinant indicate for the nomographic scales the following positions: L I / L Ion the Z-axiu, x2 on the YZ-plane, zl. on the plane p, = 1, and 01 on the 3-axis. The moduli and hyperbolic constants derived from the ranges of these variables = 1. are m, = 1, m; = 0.5, m,- = 0.5, rz; = 0.5, T $ = 0.5, and Figure 6 Rhows this nomograph with the XY-plane as its base. CASTLESEQUATION.In general, polynomials may be satisfactorily represented by planar nomographs, as shown by Burrows
Figure 6. Three-Dimensional Nomograph for the Rayleigh Equation
Figure 5. Three-Dimensional Nomograph for Mole Fractions in Residual Liquids
Figure 7.
Three-Dimensional Nomograph for the Castles Equation
1826
INDUSTRIAL AND ENGINEERING CHEMISTRY
rz = 0.1,r y = 0.002, and r; = 0.02. The nomograph constructed on these values is shown in Figure 7. METHODSOF READIXG.Reading three-dimensional nomographs is not so simple as reading planar nomographs, since constructing a plane through three given points presents more of a problem than drawing a straight line through two points. Nevertheless, several feasible methods present themselves, one of which is the method suggested b y d’Ocagne ( 5 ) . The nomograph is constructed in the form of a device having a peephole on one of the scales, through xhich the user sights t o a line drawn between given points on two other scales. The point a t which t h a t line appears t o intersect the fourth scale is the desired solution. The author prefers to use two or three contacting straight edges, such as rulers or stretched wires. These are much simpler to construct and mag be applied to any type of three-dimensional nomograph. In most applications, computation is the prime purpose of nomographs. However, they are also useful for the graphical
Vol. 43. No. 8
representation of functionally relt~ted parameters, serving as aids to visualization of involved relationships. For this latter purpose the method of reading need not be precise. Hence, the three-dimensional nomograph is peculiarlg well adapted to that purpose and presents a means of demonstrating relationships among four related parameters vihich is not easily achiwed with three-dimensional graphs. LITERATURE CITED
(1) Burrows, W. H., ISD. ENG.CHEX.,38, 472 (1946). (2) Ibid.,43, 1193 (1961). (3) Burrows, W.H., J . Eng. Education, 36, 361 (1946). (4)Castles, W., Jr., J . Aeronaut. Sci., 12, 477 (1945). (5) d’Ocagne, M., “Trait6 de Nomographie,” 1st ed., p. 348, Paris, Gauthier-Villars, 1921. (6) Perry, John H., “Chemical Engineers’ Handbook,” 2nd ed., p. 1383, Xew York, McGram-Hill Book Go., 1941. (7) I b i d . , p. 1396. RECEIVED March 9, 1951.
High Pressure Metal-to-Glass Fitting P. C. DAVIS, T. L. GORE, AND FRED 16URL4TA University of Kansas, Lawrence, Kan. During a study of high pressure vapor-liquid equilibria of hydrocarbon mixtures, it became necessary to make a pressure-tight closure between a glass capillary tubing and high pressure steel tubing which was capable of withstanding pressures up to 2000 pounds per square inch under considerable vibration. A satisfactory closure was developed, utilizing end thrust of glass tubing against a rubber O-ring. This closure can be assembled by hand to seal against pressures in excess of 3000 pounds per square inch under considerable vibration and is easily constructed and assembled. Modifications of the closure described would provide effective high pressure seals against a variety of brittle materials which can be formed by heating and/or grinding. These closures could be quicldy dismantled and reassembled and would be more resistant to vibration than soldered or welded fittings.
I
N CONNECTION with a study of vapor-liquid equilibria of
hydrocarbon mixtures, i t became necessary to make a pressure-tight closure, capable of containing pressures up to 2000 pounds per square inch, between a glass capillary tubing and high pressure steel tubing. In order to keep to a minimum the amount of the fluid contained within the closure, it was necessary to minimize the volume within the fitting itself. It was first attempted t o use the usual closure in which the glass is plated with copper over platinum, and the plated portion is soldered into a metal fitting t o which the high pressure steel tubing can be connected. The seal thus obtained was pressure-tight but proved t o be too fragile to resist vibration. This weakness
5/,“
was attributed to localized stresses set up in the glass during the soldering which could not be relieved by annealing because of the low melting point of solder. Also considerable skill is required in the plating technique t o effect a pressure-tight seal. For these reasons other types of closures were investigated. Onnes and Braak ( 1 ) used a closure in which glass tubing was connected to a metal socket with shellac, but this type of seal was unsuitable. Figure 1 illustrates a closure which eliminates the soldered joint and which has proved satisfactory. It has successfully contained pressures up to 3000 pounds per square inch and has withstood considerable vibration. The pressure a t which the seal failed could not be determined because of the lower bursting pressure of the glass capillary. The seal is effected by conipressing a rubber O-ring gasket, which is supported by a groove in the metal male fitting, against the ground end of the glass capillary. The glass tubing is upset t o form a shoulder against which a metal support ring can bear and this metal ring is held in the female fitting by means of a set screw. The female fitting is cut away on
I
C Set Serow
6/64m
x 3 ’ o d by 0.110”ld.
A?
‘/16” eraphite A9beltoB washer
0.10‘ad by ‘/64” id. Drive Fit
Figure 1. Detail of High Pressure Metal-to-Glass Connection Scale 1 inch
=
1 inch
