December, 1934
INDUSTRIAL AND ENGINEERING CHEMISTRY
These results are shown graphically in Figure 1, in which
the logarithms of the concentration of salt in the outflowing wash water are plotted as ordinates against the times as abscissas. As is to be expected from Equation 5, the graph for the period of diffusion washing in each experiment is a straight line. From Equation 5 we can derive the equation: (h
- ti)/(logioCi - lcg,oC,)
= 2.303
L/kF
(7)
in which tl and tz are any two instants during the period of diffusion washing and C1and C2 are the corresponding concentrations of the outflowing wash water. Therefore from the negative reciprocals of the slopes of the graphs as plotted, the thickness of the cake, and the rate of flow of the wash
1333
water in each experiment it is possible to calculate the value of k for the experiment. The results are as follows: Run No.
1 2 3 4 1.1 1.1 0.5 0.5 0.001225 0.00145 0.000793 0.001345 6.4 5.4 22.8 13.3 140.5 142 146.5 146.8
L
F 2.303 L / k F
k
These values of k are constant to within the limit of error of the experiment. These results demonstrate the validity of the equation developed above and indicate that, with a cake of thickness comparable to that used in filtration in leaf and plate presses, the value of k is practically independent of the rate of flow and of the thickness of the cake. RECEIVXID June 27, 1934. The experimental result8 were obtained by the clam in chemical engineering laboratory a t Cornell Univeraity.
Effect of Entrainment on Plate Efficiency in Rectification F. H. RHODES,Cornell University, Ithaca, N. Y.
Equations are developed to express the eflect of a known amount of entrainment, at known rejlux ratio, on the eficiency of a fractionating plate in a distilling column.
V
ARIOUS methods have been suggested for calculating the number of perfect plates required in a rectifying column to give, with a known reflux ratio, a product of specified purity from a charge of known composition. Of these, the method proposed by McCabe and Thiele (1) is most commonly employed. I n the actual design of columns of the bubble-plate type, allowance must be made for the fact that the individual plates are not perfectly efficient. The liquid and the vapor are not usually brought into such intimate contact with each other that true equilibrium can be established in the short time for which any particular portion of vapor and of liquid are simultaneously on the same plate; drops of liquid may be entrained in the rising vapor and carried upward with it to contaminate the liquid on the plates higher in the column. The effect of failure to attain equilibrium between liquid and vapor has been discussed by Murphree (2) who has proposed a method of correcting design calculations for the lack of perfect efficiency caused by this failure. The effect of entrainment in decreasing plate efficiency is commonly recognized, but no equation has been developed to correlate quantitatively the amount of entrainment and its effect upon plate efficiency.
ITSEFFECT ON EFFICIENCY
CORRELATION O F E X T R b I N M E S T AND
V , = vapor passing upward from any plate, n, in unit time, moles R = reflux fed at the top of the column in unit time, moles E = liquid entrained in one mole of vapor from any plate, moles yn = molar fraction of the more volatile component, A , in the vapor from late n zn = molar fraction of in the liquid on plate n zn+>= molar fraction of A on plate n + 1, immediately above plate n ye = molar fraction of A in the product The following assumptions are made:
We have the following equations: Zn Yn - fun (1) where fy, = difference between the molar fraction of A in the vapor rising from plate n and the liquid on platen Vn+EVn = total molar quantity of material passing upward from plate n in unit time R = 1 EV, = total molar quantity passing out, in unit time, from that portion of the column above plate n Vn EVn = R 1 EVn or Vn = R 1 (2)
+
+ +
+
+
Equating the moles of A that enter and leave, in unit time, from that section of the column above plate n, V n y n + E V x n = Rzn+l + EVnxn + ye + 1)Yn + E ( R + 1 ) =~Rxn+l + E ( R + + y e(3) +I
(R
PLATE
In this discussion, the following symbols are used:
1
(1) E is constant throughout the column. (2) The unit of time is the period required to distill one mole of withdrawn product. (The term “withdrawn product,” a8 used here, designates that portion of the total distillate from the top of the column that is removed as finished material and is not returned to the column to provide reflux.) (3) The vapor leaving each plate is in perfect equilibrium with the liquid on that plate. (4) The molar rates of vapor and of reflux are constant throughout the rectifying section of the column.
l)Zn+l
From Equations 3 and 1,
+ 1)Yn + E ( R + 1)Yn - E ( R + 1)fYn = [R + E ( R + 1 ) 1 ~ + + 1 = [ ( R + 1)(1 + E ) / ( R + E ( R 41))l~n+ E ( R + 1)) - ye/ [R + E ( R + 1)I (4) [E ( R +
(R
~c
~ n + i
lf~n
or
Xn+1
=
Aym - Bfym
- CY,
(5)
When E = 0, this reduces to: ~ n + l=
[(R
+ 1)/RIyn - yo/R
which is the equation of McCabe and Thiele. As R approaches infinity, coefficient A in Equation 5 approaches 1 as a limiting value, coefficient B approaches E/
INDUSTRIAL AND ENGIKEERING
1334
Vol. 26. No. 12
CHEMISTRY
(1+ E ) , and coefficient C approaches 0. Therefore, at infinite reflux ratio, when all the distillate is turned back into the top of the column, Equation 5 becomes:
is so drawn that the vertical distance between a point on the operating line and the corresponding point on the actual curve bears the ratio k to the vertical distance between the point on the operating line and the corresponding point on the ZWI = Y n - E/(1 El.f~n (6) equilibrium curve. From Equations 1 and 6, In any distillation the net plate efficiency is the product of the efficiency as determined by the intimacy of contact ~ n + l= Zn + f ~ n- E/(1 E)f~n between liquid and vapor, assuming no entrainment (i. e., or sn+1 - ~n = 1/(1 E)fyn (7) the Murphree effiWith no entrainment the change in the composition of the ciency) and the ef- A liquid as we pass from plate n to plate n 1 would be fun. ficiency a s d e t e r Therefore the effect of an amount of entrainment equal to E , m i n e d b y t h e in distillations made with an infinite reflux, is to reduce the amount of entrainplate efficiency from 1 to 1/(1 E ) . A similar equation has ment, assuming perfect contact between been proposed by Underwood ( 3 ) . In the graphic method suggested by McCabe and Thiele l i q u i d and vapor. for determining the number of plates required in a rectifying Within rather wide column, this effect of entrainment in decreasing plate effi- l i m i t r a t l e a s t , ciency may be provided for by plotting an operating line the Murphree effiwhich ,lies between the vapor composition line and the theo- ciency, k , a n d t h e retical enrichment line, and which is so placed that the hori- amount of entrainzontal distance between any point on the vapor composition m e n t , E, are concurve and the corresponding point on the operating line has stant as long as the B x the ratio 1/(1+ E ) to the horizontal distance between the total rate of flow of FIGURE1. EFFECTOF 50 PER CENT same point on the vapor composition curve and the corre- vapor is constant. If we make two disENTRAINMENT AT INFINITE REFLUX sponding point on the theoretical enrichment line (Figure 1). tillations with the RATIO In calculating the number of plates in the column, this operatX molar fraction of A in liquid same total rate of Y molar fraction of A In vapor ing line is used instead of the theoretical enrichment line. B , gnl P = vapor-composition-curve flow of vapor but A general equation for the effect of enrichment on plate B , zn, Zn+i P = operating line B , M ,P -’ enrichment line efficiency, with any finite reflux ratio, may be developed. with two different reflux ratios, R1and By substituting in Equation 4 the value yn = xn fun, R,, we should obtain two different values, A , and A,, for the ~ n + l= [(R RE E 1 ) / ( R RE E)lsw net plate efficiencies. For these two values the following [(R RE E 1 - RE - E ) / ( R RE E)lfyn equations would apply: Y J ( R RE E) (sn+1 - 2 Iaotual = [ ~ n ( R l ) . f ~ n - ycl/(R RE E)
+ +
+
+
+
--
+ + + + ++ + + +
++
+ +++ + + +
If there were no entrainment, we should have: X”+1
Or
(%+I
=
(R
- Zn)theoretioal
+ I ) / R Y ~- y c / R
=
[Zn
f ( R f l).fVn
Letting the ratio A1/A2 = A and solving for E , E
- y,l/R
=
R&(A
- l)/[R1(1+ Rz) - RzA(1 + R1)1
(8)
If in one of the distillations the reflux ratio is infinite, this Therefore the ratio of the actual change in composition be- equation becomes: tween plate n l and plate n, which would be the plate efficiency, would be R / [ R E ( R l)]. In a diagram of the McCabe and Thiele form, this effect of entrainment may be represented by plotting an operating line which is so placed that the horizontal distance between Therefore it is possia point on the vapor-composition curve and the correspondble, by measuring ing point on the operating line has the ratio R / [ R E(R l)] the plate efficiencies to the horizontal distance between the point on the vaporin two distillations composition curve and the corresponding point on the theomade with the same retical enrichment line (Figure 2). charge and the same total rate of flow of CONDITIOKS WHEN VAPORIs KOTIN PERFECT EQUILIBRIUM vapor but with difWITH LIQUID ferent reflux ratios, to calculate the apIn the foregoing discussion we have assumed that the vapor proximate amount rising from each plate is in perfect equilibrium with the liquid of entrainment and on that plate. As has been pointed out by Murphree and FIGURE2. EFFECT^ OF 50 PER CENT the effect of this enothers, this is not usually the case. The equation developed ENTRAINMENT AT 1:l REFLUX RATIO trainment upon the above can readiIy be modified to apply even when the vapor X = molar fraction of A in liquid efficiencies of the is not in perfect equilibrium with the liquid. In such cases, Y = molar fraction of A in vapor B C , yn, P vapor-composition curve plates under the conwe substitute for fyn the term f’yn, in which f’yn = k(fyn), c,’Zn, zn+l D operating line d i t io n s prevailing where k is the “Murphree efficiency”4. e., the ratio of the C,z c = edrichment line within the column. actual difference between the composition of the vapor and In a series of distillations made with a constant reflux ratio the composition of the liquid to the theoretical difference in composition that should have been attained if the vapor were and with perfect contact between liquid and vapor on each in perfect equilibrium with the liquid. In plotting the dis- plate but with different amounts of entrainment, the plate tillation diagram we use, instead of the equilibrium vapor- efficiency should decrease progressively with increase in the composition curve, an actual vapor-composition curve which amount of entrainment. For example, if we were to make a
+
+
+
D
+
+
--
December, 1934
series of distillations with a reflux ratio of 1:l and perfect equilibrium between the liquid and the vapor on each plate but with different amounts of entrainment, the variation in plate efficiency with change in the amount of entrainment would be as follows : PLATE
ENTRAINMEIT EFFICIESCY L 0 0.1 0.2
0.82
PLATE ENTRAINXENT EFFICIEICY 0.5
0.5
1.0
0.33
0.71
If, in a series of distillations made with perfect contact between liquid and vapor the amount of entrainment is kept constant and the reflux ratio is varied, the plate efficiencies should decrease with decrease in the reflux ratio. For example, in a series of distillations made with 50 per cent entrainment, the change in plate efficiency Tvith change in the reflux ratio would be as follows: REFLUX RATIO Infinite
4:l
CHEMISTRY
INDUSTRIAL AND ENGINEERING
PL.4TE
EFFICIENCY 0.66 0.62
REFLCX RATIO 1:l 5:l
PLATE
EFFICIEXCY 0.5 0.4
It is eT-ident that, in any discussion of plate efficiencies in rectifying columns operating under such conditions that a considerable amount of entrainment occurs; we must take into consideration the fact that the efficiency is a function of the reflux ratio as well as of various other factors. The plate efficiencies of various types of plates under various conditions have usually been determined by experiments made with infinite reflux ratio, since under such conditions the experimental work is considerably simplified. Results so obtained may not, however, be applicable in the design of columns that are to operate a t finite reflux ratios. APPLICATION TO EXHAUSTING SECTIONOF A C o ~ u m -FOR CONTINUOUS DISTILLATION The foregoing discussion applied directly only to a rectifying column used in batch distillation or to the rectifying section of a continuous still. Analogous equations may be derived to apply to the exhausting section of a column for continuous distillation. In the derivation of these equations it is assumed that the unit of time is the time required to distill one mole of withdrawn product from the top of the rectifying column, that the feed liquid enters a t its boiling point, and that the molar quantity of vapor rising from each plate to the next higher plate is constant throughout the column. The following addition symbols are used: Rj = feed admitted in unit time, moles R, = waste discharged in unit time, moles R, = reflux from any plate, m, in the exhausting section in unit time, moles ym = molar fraction of the more volatile component in the vapor from plate m
zm
=
V,
= =
zw
1335
molar fraction of the more volatile component in the liquid on plate m vapor rising from plate m in unit time, moles molar fraction of the more volatile component in the waste
Since for the continuous still as a whole the input of material is represented by RJ and the output by R, l,
+
R, = R ,
+1
(10) Since all of the vapor reappears either as reflux or as product,
V,
=
R,
+ 1 = Rj + R,
- Rw
The total amount of the more volatile component entering that portion of the column below plate ( m 1) is (R, E V , R)zm+ the total amount of the more volatile component leaving this section is V m y m EV,zm Rd,; therefore,
+
+
PILES, SANTA
+
+
+
+ EJ'mzn + R w zto (12) If we substitute for Rf its value (R, + l ) , for X m its value (ym--fym), and for Vm its value (R, + l), we have (Rn
+ EVm + R / ) z m + l
When E
=
=
V m ~ m
0, this reduces to
zm+1
= Vm?/rn/Rm+I
+ Rdu/Rm+i
(14)
which is the equation developed by McCabe and Thiele (1). The change in composition of the liquid between plate m and plate m 1 is - z m or zm+l- y m f u m . From this relationship and from Equation 13,
+
+
(zm+l
(1
= [Rw(zm +(%( R n++l)f~mI/ I)f~ml/[(Rn + 1) +-E+) +E(Rn R,I = [Rw(zW + 1)l (15) 2,) 2,)
zm)aotual
[%+I
When E = 0, this becomes
Dividing Equation 15 by 16 gives:
From this equation it is possible to calculate the effect of a given amount of entrainment, E, upon the plate efficiency in the exhausting section of a continuous still. LITERATURE CITED (1) McCabe and Thiele, IND. ENQ.CHEM.,17, 605 (1925). (2) Murphree, Ibid., 17,747 (1925). (3) Underwood, Trans. Inst. Chem. Eng. (London), 10,140 (1932).
RECBIVED July 7, 1934.
Courtesu, U. S. Bureau of Public Roads
TREATED CONCRETE
(11)
MARIARIVER BRIDGENEAR SANTAMARIA,CALIF.
