1036
T H E JOURNAL OF INDUSTRIAL AND ENGINEERING CHEMIXTRY
CoNvERsIoNS-Read lower denomination or greater number on vertical, and higher denomination or lesser number on slant over conversion factor (Fig. 17); thus, days on vertical and years on slant through intersection of vertical and radial conversion factor 365. 2 ROOTSAND PowERs-Square and cube roots and powers read directly from square and cube curves. Either radial or slant is factor in square roots and powers. Slant only is cube root (Figs. 7 to 12). Higher Powers--5th power: read to square curve twice and then to root line. 6th power: to cube and then to square curve (Fig. 13). 7th power: to cube curve, then to square curve, and then to root line. 8th power: to cube curve, to square curve, and to root line twice. ALGEBRA Resolve fractions as in multiplication and division of fractions and resolve equations as in proportion.
Vol. 14, No. 11
GEOMETRY CIRCLES,SPHERES-Read circumference on vertical and diameter on slant over radial T line (Fig. 15). AREAOF CIRCLE-Radial radius to square curve, then up vertical and down radial to slant T line: answer on vertical from T (Fig. 14). TRIGONOMETRY RIGHTTRIANGLES-vertical is lesser side, radial is greater side, slant is angle. Read B angles if base is involved as given or desired quantity, H angles if hypotenuse is involved, H complementary angles if both base and hypotenuse are involved, Hypotenuse always the greater side. Base greater than altitude under 45', and lesser over 45' (Figs. 20 to 24). I OBLIQUETRIANGLES-All angles and their opposite sides intersect on the same radial; angles under H are read. If side and opposite angle are not given, break into right triangles (Figs. 25 and 26).
A Simple Graphical Method of Calculating t h e Number of Plates Required for a Distilling Column' By W. H. Rodebush2 CHEMISTRY
DEPARTMENT, UNIVSR9ITY
N HIS recent book on distillation,x Clark S. Robinson
I
treats the theory of a continuously operated column by a method developed by W. K. Lewis. This method appears somewhat involved and is strictly applicable to a column with an infinite number of plates. Now the rectification in an ordinary continuously operated column with a limited number of plates is a step-by-step process from plate to plate, and the following method of treatment, while not essentially different, appears simpler and more exact. Let the rate of flow be expressed in mols per unit time and the composition by the mol per cent of the more volatile constituent. Following the notation of Robinson, let Vb be the rate of flow of the vapor of composition Y b upward from plate b, R b the rate of reflux of liquid of composition 2 b downward from the same plate, and V, be the output of composition yc from the top of the column per unit time. From an inspection of Fig. 1 the conditions to be satisfied for a steady operation of the column are readily set down. The &st condition is that the liquid leaving any plate, as b, for the plate below, is in equilibrium with the vapor leaving this same plate for the plate above. This is, of course, a limiting condition reached only with a column which is functioning perfectly, but it may be approached in a column properly designed and operated, and deviations from this condition may be corrected by allowing a constant factor for imperfect performance. This condition may be expressed mathematically by the equation Yb
- "cb = 2
(1)
where E is the difference between vapor and liquid composition when the two phases are in equilibrium. In Fig. 2, 1 is plotted as ordinate against r , the liquid composition for ben1 Received
April 24, 1922. Associate Professor of Physical Chemistry. a "Elements of Fractional Distillation," McGraw-Hill Book Co., 1922.
OF I L L I N O I S , U R B A N A , ILLINOIS
zene and toluene. The data used are those obtained by Rosanoff .4 This diagram is fundamental in calculating the conditions necessary for the separation of two liquids, and, where the data are not to be found in the literature, they may be obtained in a few hours in the laboratory with sufficient accuracy by distilling in an ordinary distilling flask small portions successively from a large amount of liquid whose composition varies slowly, and determining the composition of distillates and residues. In addition, the two following equations given by Robiison hold for any cross section of the column RS s in Fig. 1.
Solving these equations,
Suppose the vapor entering the column above plate 0 has the composition yo. For an infinite amount of reflux or zero output, the last term in Equation 4 becomes zero and we have xb = yo. This value is obtained geometrically (Fig. 2) by measuring off with a pair of dividers the distance 1, along x from 2,. By repeating this process of laying off ,a along x from xn until vG is reached, we get the minimum number of plates for infinite reflux, each measurement corresponding to a plate. Of more practical significance is the limiting ratio of output to reflux possible when the number of plates is infinite. If the number of plates is very large, the composition of liquid varies but slightly from plate to plate and in the limit za = x b . Hence Equation 4 becomes:
f
4J. Am. Chem. SOL.,86 (19141, 1999.
THE JOURNAL OF INDUSTRIAL AND ENGINEERING CHEMISTRY
Nov., 1922
given by or
1 at -
the point x = 78 per cent, where the
uc-v
xc
angle 2/n vcxn is a minimum. Here Owing to the approximate validity of Trouton’s rule for heat of vaporization, the vapor and liquid flows will not vary appreciably from plate to plate throughout the column for liquids whose boiling points are not far apart.6 A constant reflux throughout the column being assumed, the actual ratio Vc/R will be fixed by the plate in the column 1 where -is a mini-
k
d
Yc
-Y
mum. That is to say, the output of any constituent per unit time cannot be greater than the net amount of that constituent which rises past any plate in the same time. FIG. 1 Yc
-
Y
will be a mini-
mum a t the point on the diagram (Fig. 3) where the angie yycx is a minimum.6 The limiting ratio Vc/R having been determined by this method for an infinite number of plates, the number of plates actually required in a working column may be calculated graphically by the following modification of the method described above.
1037
Assume now a working ratio
R
=
1
- (approximately). 3
5=1 7‘
Proceed as above to
R
find the point x = ya = 46.5 per cent. Now from Equation
5, x b = ya
VC x b is evidently to the left of the - (y, - yo) E,
point x (= y,) by the amount (yc
- ya) Ev c.
distance from yc to the point x = y., the distance (1
(uc - y),
Then
xb
is the
will lie a t
+ 2) (yL - Y a ) from yc or r ( p - Y a ) . 8
Draw a line ycz .at such an angle with the x axis that its 8 secant is - Then with a pair of dividers lay off a distance 7’ from yc on the x axis equal to the distance ycxlwhere z l is the intersection of ycz with the perpendicular to the x axis a t x = ya. Then xb will be a t the point located. The proof is obvious by geometry. The process is then repeated, starting with 2/b the composition ‘of vapor in equilibrium with 2 3 , and continued until y, is reached. Each construction corresponds to a plate in
FIG 3 - E T A A N O L - w A T B R
FIO.2-BENZENE-TOLUENB
In Fig. 3 we have the vapor-liquid composition diagram for ethanol and water according to the data given by W. K. Lewis.’ Let the vapor entering the column be of composition &I = 46.5 per cent, the output yc = 86 per cent. Then the limiting ratio of
v-‘ for an R
infinite number of plates is
Thib generalization can only be applied satisfactorily to those liquids which Hildebrand has called normal liquids. [ J . Am. Chem SOC.,87 (1915), 970.1 Water and ethanol differ markedly from the normal liquids but their values for the molal heats of vaporization do not differ from each other enough to invalidate the above approximations for alcohol-water mixtures. W. A. Peters, Jr., has suggested expressing the composition in units of equal heat of vaporization instead of mols. This would solve the problem quite satisfactorily except in the case of a substance such as acetic acid, which has an abnormal vapor density that varies with temperature and pressure. The heat of vaporization is likewise a variable quantity and the cornposition would have to be expressed in a variable unit. I ‘It can be shown mathematically that Y y is a minimum when yc
1 Y,--Y+L
Hence 1
1
- + is the tangent of the angle y y c x .
.-
is a minimum; but yc d
y;q is a minimum when the angle yycx is a minimum.
THISJOURNAL,
l a (1920).496.
the column. Approximately 15 plates are indicated by this method as necessary to produce 86 per cent alcohol* from 46.5 per cent with a ratio of output to reflux of 1 to 7. The foregoing applies only to the upper portion of the column above the entering vapor. For the lower portion of the column similar equations may be derived. If liquid is being drawn off from the bottom of the column a t a rate vo = va - Vc of composition xo = V a Y a - YcVcs then if
VO
r and s be successive plates proceeding downward in the
column, ys =
xr
+ (ys - xo) vo
(7)
-(x, -
(8)
or ys
-
xo=
xo)
1 - - vo R
If R is assumed to be constant throughout the whole column, the number of plates in the lower portion may be determined graphically on the vapor composition diagram by a method precisely similar to that employed for the upper portion. 8It should be remembered that the compositions are in mol per cent. 86 mol per cent composition is equivalent to 94 per cent by weight.
