Rectification of Binary
Plate Efficiency In a of Bubble Cap bubbleColumns cap - column
Mixtures
the liquid o n each plate is i n c o m p l e t e l y mixed , with the result that the plate efficiency, as ordinarily defined, becomes higher than the local “Murphree” W. K. LEWIS, JR. efficiency of vapor enrichment. With the aid Massachusetts Institute of Technology, of certain simplifying assumptions, equations and Cambridge,Mass. plots for binary mixtures are obtained which show the plate efficiencies resulting from various conditions of flow. In addition, a relation is derived between the plate efficiency and the number of plates required to N THE design of bubble-cap columns it effect a given has been customary to assume that the the vapor always rises in B vapor rising through the liquid completely separ a direction perpendicular to the mixed the latter. Kirschbaum (5) and Gadwa (4, however, tion. liquid streamlines.) Of t h e found considerable liquid concentration gradients across total vapor rising through the plates only a few inches in diameter, even at superficial velociplate, call dV the number of moles per unit time rising through ties (Gadwa) of 6 feet per second. In view of this result it is that part of the plate in which the composition of the liquid clear that, in columns of commercial size, mixing of the liquid increases from xn to xn dx,. The usual assumption is mads cannot be complete and is probably very small. Peters (IO) that V and 0 are constant. Then by a material balance we
+
recently emphasized the possibility of securing large increases in vapor enrichment per plate-i. e., higher over-all plate efficiency, EO,by proper control of the flow in such cases. If the liquid on the plate in question is perfectly mixed, it will have a t all points on the plate the same composition as that of the liquid leaving the plate. If, on the other hand, mixing is incomplete, then only the liquid in the neighborhood of the down pipe will be as lean as the liquid leaving the plate, the liquid on all other parts of the plate being richer than this. Hence, the vapor enrichment will be greater in the latter case. It is the purpose of this paper to develop the necessary modification of the conventional computation technic of the McCabe-Thiele method (7)for rectification of binary mixtures, and to present certain important conclusions from the calculations. To make computation possible, it will henceforth be assumed that the liquid flows across the plate with no mixing whatever. The only published methods of calculation are those of Brewster (2) and Kirschbaum (6),both of whom assume the liquid on a given plate to be divided into a number of pools in series, with perfect mixing of the liquid in each pool. Brewster’s assumptions are unsound while Kirschbaum’s method, though correct, is far too tedious for practical use. Fortunately the method of the calculus, properly manipulated, offers a readier solution. Consider a rectangular plate, sketched in Figure 1. (The shape of the plate is actually immaterial, provided the quantity of vapor rising through the liquid between any two pairs of streamlines on the plate is always proportional to the volume of liquid flowing between those streamlines, and provided
obtain: ( ~ n-
yn-1)dV = 0 d x n
For convenience, replace dV by dW 0 dx,
yn - y n - l = - = V dW
d V / V . Thus,
=
dx
It is evident that W varies from 1 to 0 as the liquid travels across the entire plate. From this point on, it will be assumed that over the range of the plate in question the equilibrium curve, y* vs. 2, is linear but does not necessarily pass b, and dy*/dx = K . through the origin-i. e., y* = Kx This is legitimate since we are primarily interested in the regions where each plate represents but a small step in the McCabe-Thiele diagram. Then it is readily seen, placing K / R = A, that;
+
Or, employing the definition of Murphree local efficiency (8), E = % - !in-1 we obtain: u*n
-
b - 1 ,
(3)
Equation 3 will now be applied to three specific cases. For simplicity, E will henceforth be assumed constant, although this assumption is not made in Equation 3. 3919
INDUSTRIAL AND ENGINEERING CHEMISTRY
400
Case
I
In case I the vapors enter the plate completely mixed. Here is a constant, denoted by Yn-l. By integration of Equation 3 from W = 0 to W = W ,
The resulting values are given in Table I.
VOL. 28, NO. 4
equilibrium relation, y* vs. 2 , by the same straight line, Knzn b,, which is employed for plate n. The operating line is also assumed straight in this range, having the slope R,. The range must be sufficiently large so that the conditions on any plate below this range have negligible effect on either plate n or on plate n - 1. I n general this required range is only two or three plates; furthermore, the case of importance is where each plate represents only a small step on the x-y diagram. Hence the above assumptions are completely justified except for the first plate or two above the still and for the first plate or two above the point of feed (the plate just above the still falls under case I). Granting the assumptions stated in the previous paragraph, the y vs. W curves for plates n and n - 1 must have the same shape, differing only in size and in position on the diagram. I n other words, if we write zn = yn - (yn)o-i. e., zn = 0 when W = 0-then 2, = (YZ,,.-~ for any given W (where a! is a constant). Upon substituting the relations zn = yn - (yn)o, zn-l = yn-l - (ynyl)O,and zn = a!znwl in Equation 3 andrearranging, we obtain : yYn =
+
where kn =
(y,)o
TABLE I. VALUESO F Eo FOR THREECASES En
7
E
X = 0
7
X = 1.0
X = 1.5
X = 2.0
0.22 0.49 0.82 1.23 1.72
0.23 0.55 0.97 1.55 2.32
0.25 0.61 1.16 1.98 3.19
0.22 0.21 0.50 0.45 0.86 0.72 1.33 1.03 2.00 1.43 Case I11 0.21 0.22 0.44 0.49 0.70 0.81 0.97 1.16 1 21 1.50
0.23 0.56 1.03 1.73 2.79
0.25 0.63 1.25 2.26 3.92
A = 0.5
Case I
0.20 0.40 0.60
0.21 0 44 0.70 0.98 1.30
0.20 0.40 0.60 0.80
0.80 1.00
1.00
The result of integration between W = 0 and W = W is: In
Case I1
0.20 0.40 0.60 0.80 1.00
0.20 0.40 0.60 0.80 1.00
0.20 0.40
0.20 0.40 0.60 0.80 1.00
0.60 0.80 1.00
0.23 0 54 0.94 1.41 1.90
-l)~,-i
(CY
0.25
[IC,
But
I%, =
- (zn-1)11
I1
In case I1 the vapors rise from plate to plate without mixing and the liquid flows in the same direction on all plates (Figure 2 ) . It will be assumed in cases I1 and I11 that the vapor leaving a given plate a t any one point rises directly to the corresponding point on the plate above. By assuming the vapor rising from the still to be of uniform composition, we could, in theory, work up the column, applying Equation 3 successively to each plate. I n practice, however, the integrals become hopelessly complicated after the first two or three plates. However, a good approximation can be obtained as follows: It is assumed that in a range consisting of a sufficient number of plates immediately below plate n, which is the plate in question, we can represent the
- E l [ k n - ~ - (Zn
=
-
( ~ n - l ) ~=
1- E a: [kn
Arrows Represent Liquid Flow
Arrows Represent Liquid Flow
FIG. I
I
-
(zn-1)11
The following equation will be proved later: Eo =
CY-1
Equations 7 and 8 enable us to obtain the value of Eo vs. A. The results are given in Table I.
I11
In case I11 the vapors rise from plate to plate without mixing, and the liquid flows in opposite directions on alternate plates (Figure 3). Granting the same assumptions made in solving case 11, we see as before that the y vs. W curves for plates n and n - 1 must have the same shape. I n this case, however, because of the alternation of direction of liquid flow on successive plates, the y vs. W curve for plate n must be the mirror image (except in size) of the corresponding curve for platen - 1. That is, zn ~ n - 1
FIG. 2
-2)11
(7)
where 'To Plate Below
=
Since, in general, a! # 1 - E , thenIc, = (zn-&. Substituting this last relation in Equation 6 when W = 1 and rearranging algebraically, X=(;+-J)lna: 1
Case From Plate Above,
(6)
a:+-E-1
c r L 1 and zn = c r ~ , - ~ ;therefore kn
Case
+ kn = XE(a: - 1)W
kn
From the definitions of E, z, and k plus the relation (y*,), (y*nll)l which holds in this case, we can obtain:
0.60
1.10 1.73 2.47
- (Yn-l)o
= yn
-
yn-1
(zn)~ (yn)o
a(Zn-1)l-W
- (Un-111
After substituting in Equation 3 and rearranging, we obtain:
APRIL, 1936
INDUSTRIAL AND ENGINEERING CHEMISTRY
kc vs.
When CY > 1, we avoid imaginary numbers by employing the equivalent form (obtained from t,he mathematical relation COS-' B = - 0 cosh-' B):
E CASE
&.=
40 1
I
n
where CY > 1 As in case 11,we combine Equations 13 and 14 with Equation 8 [Eo= ( a - l)/(x - 1)1 to obtain the curve of EOvs. A. The results are given in Table I. When = 1, the equations derived for cases I, 11,and I11 (2 - E)E reduce to EO= eE - 1, EO= 2 2E - E' and Eo = 2(E2/3 - E 1; respectively.' Taking the ordinate as Eo/E,these three equations are plotted as curves a , b, and c, respectively, in Figure 4.
+
Over-All Efficiency
+
[ c Y ( z ~ ) w . ( z ~ ) I - - w aknl
wherek,
= (yn)0 -
X
E" (%)w
= -
+
1-E
E
dz,
(vn-l)~
This equation can be more readily solved if the origin is shifted from W = 0 to W = 1/2. (For the solution of this equation the writer is indebted to Douglass, 3.) Denoting the new abscissa by x, we obtain:
Equation 9 can be solved by integration in series as shown in any book on differential equations (9), with the final result:
OS.
Number of Plates
The practical effect of an increase in the over-all efficiency Eo, whether due to nonmixing of liquid on the plates or to any other cause, is measured by the fractional decrease in the number of plates secured, not b y the absolute value of EO. The effect of EOupon the number of plates will now be shown. Figure 5 represents a portion of the McCabe and Thiele diagram, A H being the operating line and BG the equilibrium line, which will b e a s s u m e d straight between B and G. As above, the slopes of the equilibrium and operating lines in the and R,involved range respectively. are denoted ABC repreby K
4
sents o n e p e r f e c t s t e p (i. e., Eo = 1) while A D E F H r e p r e sents two steps for which Eo = Eo. By definition Eo = AD/AB = EFIEG; then
FIG.
5
EK=AB-AD=
e=4
E2(1 -
R
but
where a = an arbitrary constant
=
EG
CY')
- (1 - E)*
AD/DE
=
=
AD/BK and K
EK + K G = 6
=
(& - 1 + % ) A D
a = EF/AD = EG(Eo)/AD = 1
(This solution can be verified by substitutJing it back in (i. e., W = 0), zn = 0. Equation 9.) But a t z = - ' / z Eliminating a from Equation 10 by this relation and then (i. e., W = l),we obevaluating Equation 10 €or n: = tain :
or
+ Eo(&- 1)
a-1
Eo
Let A D = dl, EF =
KG/BK
__.
x-
1
(8)
etc., and let AB = al; then dz = dim U~EOCY
d2,
Similarly, da = d z a , etc., assuming that the equilibrium line remains straight for a number of plates. Hence for the mth plate, d, alEo~~m-1 From the definitions of E , X,-I, k,, and kn--l, plus the rela= (Y*~-])o which holds in this case, it follows that tion But k , = a kn-l and k, - (z,-l)O = [l - E ] [kn-l (zn-1)o]. ( z , ) ~ = a ( z n - l ) O ; therefore,
+
If T denotes the total vertical rise on the diagram for plates 1 to m, inclusive, then T = dl + d z ...... +dm
+
=
Substituting this relation in Equation 11 and reducing:
or, since by trigonometry, cos 2A = (1 - tan2A)/(1
where
CY
