I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
December, 1946
A = 1.50 f 1.10 = 2.60 B 1.50 - 1.10 0.40 By any of the usual chart (6) or slide rule (2) methods available for obtaining the log mean average, Equation 27 becomes Atd
=
2.60
- 0.40 =
and At,,, = 58.8’ F. as before. From the true mean tempertiture difference the outside tube area P L is now readily obtained from Equation 1 :
and since the perimeter P i s
21r = 0.525 ft. 12
1271
I n cases where D is equal to or greater than 3 times E as read from Figure 4,D gives directly the true mean temperature difference per unit change of tube side fluid temperature, with an error of 4% or less. Analysis shows that no change is required in either equations or nomenclature if the tube side material is the heating medium, with the cooling medium in the shell, nor are changes required in cases where the tubes are provided with fins, aa in Figure 6 . The concept of equivalent greater and lesser temperature differences appears t o have a general application t o heat transfcr problems where arbitrary arrangements of cocurrent and countercurrent flow of the heating and cooling media are specified, provided that each fluid may be considered as mixed (of uniform temperature) at any cross section of its flow. A preliminary investigation indicates that i t is applicable with benefit of simplicity to all such cases for which mean temperature difference equations are readily available-namely, the 1-2 and 2-4 arrangements of shell and tube passes in multipass heat exchangers as given by Underwood ( 7 ) , the 1-4 arrangement of Yendall ( I ) , and the unbalanced pass arrangement of Gardner (3).
the required length is ACKNOWLEDGMENT
“ 1
L
=
3 = 13.5 ft. 0.525
The writer wishes to acknowledge the helpful suggestions of
If the effect of the interchange of heat between tube side streams were neglected-that is, r = 0-the log mean of the terminal temperature differences in countercurrent flow woula apply, giving a mean temperature difference of 72” F., or over 22% greater than the true value.
J. A. Davies, J. J. Gabor, and L. P. Gaucher of The Texas Company, Particular acknowledgment is due to K. A. Gardner, of the Griscom-Russell Company, who reviewed the manuscript and kindly made available his own uncompleted article on the subject, in which the same basic equations are derived. LITERATURE CITED
DISCUSSION
Equation 26 and Figure 5 show that, other things being equal, smaller values of E give larger meaq temperature differences. The flow arrangements of cases I1 and 111 are thus superior to those of cases I and IV in this respect, as can be seen by comparing Equations 30 and 31. It is therefore desirable to have countercurrent flow between the shell side fluid and that in the annular space.
(1) Bowman, Mueller, and Nagle, Trans. Am. SOC.Mech. Engrs., 62, 283 (1940). (2) Dalton, T. N., IND. ENG.CHEM., 30,1081 (1938). (3) Gardner, K.A., Ibid., 33, 1215 (1941).
(4) Gel’perin, N. I., Khim. Mashinostroenie, 4, 1 (1939). (5) Sherwood and Reed, “Applied Mathematics in Chemical Engineering’’, p. 94, New York, McGraw-Hill Book Co., Inc., 1939. (6) Tubular Exchanger Mfr. Assoc., Inc., “Standards of Tubular Exchanger Manufacturers Association”, p. 19 (1941). (7) Underwood, A. J. V., J . Inst. Petroleum Tech., 20, 145 (1934).
Calculation of Number of Theoretical
Plates for Rectifying Column A. E. STOPPEL University of Minnesota, Minneapolis, Minn.
By
extending the equilibrium curve and operating line of the typical McCabe-Thiele distillation diagram beyond their usual limits, the operating line will be found to cut the equilibrium curve at two.points, once on the chart proper, and again somewhere outside the chart. The number of theoretical plates required to operate the column may be expressed as a simple power function of these two points of intersection.
T
HE problem of the rapid calculation of the number of theoretical plates required to effect a given separation by distillation of a two-component system was studied by a number of investigators (1-4,7 , 8) who have approached the problem in a variety of ways. Some of the proposed methods make simplifying assumptions and must be regarded as approximate when
used over wide ranges of operating variables. Others are empirical in nature and involve the use of charts. A few are exact for infinite reflux but do not hold a t lower reflux ratios. A method which is exact for any reflux ratio, purity of overhead product or waste, or condition of feed, but which assumes constancy of the relative volatility, was proposed by Smoker ( 7 ) . More recently Harbert (6) developed equations designed to “jump over” a number of plates in a single step. The method described here leads to an expression algebraically identical to the equation proposed by Smoker ( 7 ) . By proper choice of constants picked from a modified McCabe-Thiele chart, the equation may be expressed rather simply and in a form resembling the expression obtained by Fenske (3) for conditions a t total reflux. Figure 1 is an expansion of the familiar McCabe-Thiele (6) diagram. Within square OPQR are represented equilibrium curve
INDUSTRIAL A N D ENGINEERING ,CHEMISTRY
1272
ODQ and two operating lines A K and BD. Extension of the equilibrium curve and rectifying line to the right to the point of intersection a t I permits the construction of rectangle H I J K , within which are represented the operating conditions of the section of the column above the feed plate. The rectifying line is the diagonal of this rectangle; this sug-
(YZ - YO)
(ZE
(22
(YE
- ZO)
- 22)
- YZ)
Vol. 38, No. 12 = E
E
ZOYE
(5 )
A sinular expression may be written for each plate above the feed plate. The product of all these expressions leads to the series
In Figure 1 point (21, y2) falls on the diagonal of the rectangle. I t is obvious from the geometrical construction that area ( Z E a) (yz - yo) = area ( Y E - ~ 2 )(21 - 20). Also ~ Z E- 2 2 ) (y3 yo) = (ZZ - 20) ( Y E - ya), etc. Cancellation of these equalities from Equation 6 givcs
n-hich is a general equation for calculating the number of plates above the feed plate. For conditions below the feed plate the equilibrium curve is extended belo\\-the origin t o permit intersection Kith the stripping line at, points (z;, y ' ) and (z;, uh). A rectangle is constructed with these points on the diagonal. The derivation of the equation for the number of plates belox the feed plate is entirely analogous to that given and results in the equation
xby /
Figure 1. itIodified McCabe-Thiele Diagram for
Distillation Column
Points ( 2 0 , yo) and ( Z B ,Y E ) , the intersections of the operating line and the equilibrium curve, may be locat,ed by solving simultaneously the equation of the operating line, y = mz c, and the equilibrium relation, y(1 - s)/z(l - v) = a, obtaining: m(a - 1 ) 2 2 (m - a ac - c ) ~ c = 0 and ( a - 1 ) ~ 2 ( m - a - ac c)y ac = 0. The solution of these quadratics for zo,Z E , yo, and Y E is obviously somen-hat involved and leads to cumbersome expressions. However, the products Z O Z E and YOYE are useful and are simply stated: yoys = a c / ( a - 1) and ZOZE = c/m(a - 1). Dividing these equations and rearranging gives ynuE/axoxE = m, and, since m = ( Y E - yo)/(xa - xo),
The total number of plates required for the entire column is the sum of n1and n2,where n2 includes the still pot as one of the plates in the stripping section. Equations 7 and 8 are algebraically identical to those proposed by Smoker ( 7 ) , although the similarit,y is by no means obvious. When this derivation is applied to conditions a t total reflux, the equation reduces t o y l ( 1 - z , , ) / z v ( l - yi) = on, as given by Fenske (3).
+
+ +
(YE
+
-
+
+
+
?/O)/(zE
- 20)
=
(1)
YOl/E/a%ZE
Consider the arc cut by a straight line through points and (xl,yl). From analogy to Equation 1this leads t o (y1 -
y0)/(21
- Zo)
=
(YE
- yl)/(ZE
-
21)
=
(2)
Ylyo/azlzo
and, for the straight line through points
(21,VI)
ylyE/axlzE
(50,YO)
and
(Xi
- Yo) - Zo)
(ZE
- zl)
(YE
-
Yl)
=
2//E ZOYS
X
y
yl zw
a m c
= mole fraction of more volatile component in liquid = mole fraction of more volatile component in vapor = mole fraction of more volatile component in over= = = =
head product mole fraction of more volatile component in waste relative volatility slope of rectifying line y-intercept of rectifying line on z,y diagram )Then
x=o
zo, yo = lower point of intersection of rectifying line and
equilibrium curve upper point of intersection of rectifying line and equilibrium curve z;,yb and z;,y i = lower and upper points of intersection of stripping line and equilibrium curve, respectively x9, yg = point of intersection of rectifying and stripping lines = equilibrium value of y when z = z9 &' = number of theoretical plates for rectifying section nl = number of theoretical plates for stripping section nz ZE, Y E =
( Z E ,Y E ) ,
(3)
Dividing Equation 2 by Equation 3 gives (!!I
NOAIESC LATUKB
(4)
which represents ratios of slopes from the top plate t o opposite corners of the rectangle. 4 ratio around point (z2,y2) results in the expression
LITERATURE CITED
(1) Dodge, B. F., andHuffman, J. R., IND. ENG.CHEM.,29, 1434-6 (1937). (2) Faasen, J. W., Ibid., 36,248-52 (1944). (3) Fenske, M. R., Ibid., 24,482-5 (1932). (4) Gilliland, E. R., Ibid., 32, 1220-3 (1940). (5) Harbert, W.D., Ibid., 37, 1162-7 (1946). (6) McCabe, W.L., and Thiele, E. W., I b i d . , 17, 605-11 (1925). (7) STmoker,E. H., Trans. A m . Inst. Cham. Engrs., 34,165-72 (1938). (8) Lnderwood, A . J. V., Trans. I n s t . Chem. Engrs. (London), 10, 112-52 (1932).
