ENGINEERING AND PROCESS DEVELOPMENT
(BATCHWISE FRACTIONAL LIQUID EXTRACTION)
Calculation of Deviation from Steady State EDWARD G. SCHEIBEL Hoffmann-la Roche, h e . , Nutley, N . J .
s
INCE the author (6) first proposed a simple empirical equa-
tion for calculating the deviation of a batchwise countercurrent fractional liquid extraction operation from steady-state conditions, considerable additional work has been done to obtain a more accurate expression, Peppard and Peppard (4) have observed that the equation deviates considerably from the true value a t large numbers of cycles and stages and have proposed two other empirical equations which are accurate over a wider range. Auer and Gardner (I), using a set of coordinates similar to that of Stene ( 7 ) to identify the stages, developed a set of equations which could be solved to give the deviation from steady state. However, like Stene, they concluded that the solution of the equations for a nonsymmetrical extraction pattern is too tedious for convenient application. Satsangee (5),using the numbering system originally employed by Craig (S),developed the relationship between the successive term8 of the rigorous equation expressing the variation of the product streams with time for a symmetrical extraction pattern, but the solution of his equations for large numbers of stages also became increasingly complex. Using the latter numbering system, an equation has now been developed to give a direct determination of all the terms of the equation representing the variation of the product quantities a t each cycle. This equation, originally developed by Stene (7), and used in subsequent publications (4-6') is
Pt = PN[Pl
+ Pzpp + P d P d * + . . . +
Pt(pdt-11
At a constant value of iV, representing the feed stage for the heavy solvent a t steady state, the above expression a t t cycles becomes
---\ 4--16 6-
N+M-IzTOTAL STAGES
1'
c
F
,/'
1M :-
(1)
The present author (6) observed that this equation approached a geometric series as t increased and developed a simple expression based on the geometric series, the sum of which equalled the steady-state quantity. Peppard and Peppard (4) observed that the first few turns deviated so much from the geometric series for large numbers of stages that their effect was to produce a large error in the deviation a t large numbers of cycles. They proposed a different ratio for the geometric series, which gave accurate values a t a higher range of values. However, recognizing the limitations of this equation, they developed a second empirical equation which could be employed over the range of smaller deviations from steady state with reliable results. Like all empirical equations, extrapolation outside of the range investigated is uncertain.
t :M+N
Figure 1 shows the numbering system for the stages as used in the derivation of the equations in this paper and for the continuing symmetric pattern. Craig (3) has shown that the quantity in any stage is given as
Figure 1.
Extraction Pattern
This relationship holds only for the continuing pattern, so that when the pattern is cut off a t N , the amount in any cycle after the first has been found to be equal to =
Equations Are Applicable to:Both Symmetric and Nonsymmetric Systems
January 1954
,
rJO
Q'
nsl-
[( t -( Nl)!+( N2t +- t3)!- 2)! -
The second term results from not returning the necessary quantity a t the N 1 stage to the second cycle. The resulting pa& tern may be considered as the sum of the original pattern and a superimposed negative feed pattern to correct for the amounts lacking in the fresh solvent introduced along the line at N . The above relationship holds only when the pattern on the right con-
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
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ENGINEERING AND PROCESS DEVELOPMENT tinues to infinity. HopT-ever, the number of stages in a practical pattern is limited in this figure to M. Thus, the shortage of negative feed at stages below 1M will show up in the quantity along N after M cycles. The expression for this shortage can be derived from the general expression deduced for Equation 4,
Lf -
(n
=
Qnli
+.
2t I)! ( n
- 3)! -~
+ t - 2)! -
where t now identifies the stages in a vertical line at n. Thus, the coefficient of the additional quantity of the component, which is missing at A' after M cycles, due to the withdrawal of product along M is (1
tion of @ for a certain number of cycles, the ratio necessary to approach the steady-state quantities can be compared with the ratio of thc last two values. Thus, after t cycles, it may be shown that if P , is the fraction of the component in the light stream product a t steady state
+ -
(V 2t~ _3)! ____ I)!(S t nf - 2 ) !
- .lI -
and the value of r from this equation may be compared with the ratio, @t+ I/&, to ascertain whether additional cycles must be calculated before assuming a constant ratio. The value of P , can be calculated from the relationship of Bartels and Kleiman ( 2 ) for steady-state extraction,
+ +
+ 2 cycles the deficiency is equal to ( N + 21 - 3 ) ! - hl - l)!(A- + t + M - 2 ) ( N + 2t - 3)! ] ( t - 31 - 2 ) ! ( X + t + M - l)!
which for a symmetrical pattern when N = M becomes
and for M [(t
E'Y
P, = -
ES
pn+t-lql-l
Similarly, Then this negative feed pattern with the apex a t L+r reaches M , some of the negative feed is withdrawn so that after 1cI S cyclas the adjustment will be positive. All these opera-4' cycle intervals to infinity and the general tions repeat at M equation for the coefficient, @ t , in Equation 1 is given as
+
+
,=- 'If1 -+1 .?'
t = t--2
--
,
=
c i = O
c
M+ N
B i ( 2 BJ! -
i = O
A!
(B, - 1)!(A
- R, + I)!
-
+1
The deviation from steady state a t any cycle t has been previously defined (6) as
When Ai # M, the product cycle t will not be represented by the same horizontal line for both sides of the extraction pattern. Thus, the product streams will not approach steady state a t the same rate as has been recognized for the symmetrical pattern (6). If the quantity of the component in the heavy solvent is desired, the values of iV and M , as well as p and q, must be interchanged in the equations. Since p and q always occur as a product in the series expansion, only the numerical values of @ will be affected. Similarly, when two components are involved with different values of E, each will approach steady state a t a different rate. Summary
where i a n d j have only integer values of 0, 1,2, etc., not exceeding the upper limit. If the upper limit is below 0, no terms of the series are included. Also,
A = S + 2 t - 3 B,
=
t
B, = t
- 1 - i(3.I + A-) - M - 1 - j(hf + Ai)
..ls a sample calculation, the 21st cycle at .V for a system in which S = 4 and A9 = 6 can be evaluated by calculating the upper limits for i and j in the four summations as 2.0, 1.9, 1.4, and 1.3 in sequence. Then, by the following tabulation of the values of B, and B , a
B,
0
20 10 0
1 2
j 0 1
A, 14 4
the expresiiion for Bt becomes 43! 43! 43! 43! _ _ - _43! _ 14! 29! 4! 39! + 13!30!+
Calculation will show that the first few terms of all series are the most important, and the subsequent terms may be neglected. Thus the values of p approach a constant ratio; and, af pr evalua-
44
INDUSTRIAL
I
Equations are presented for the direct calculation of the approach to steady-state conditions for symmetrical and nonsymmetrical fractional liquid extraction. The equations involve an expanding series of terms as the number of cycles increases; however, for slide rule calculation, generally only the first few turns of each series need be evaluated until the approach to a geometric series has been recognized. The constant value for the ratio of the coefficients may be compared with the value calculated to make the total of the remaining geometric series equal to the necessary amount to give the total product quantitv required by the steady-state calculations. Nomenclature
+
= .Y 2f - 3 B; = function in summation = t - 1 - z ( M S) Bj = function in summation = t - M - 1 - j ( M S) concentration in light phase D = distribution coefficient = concentration in heavy phase LD E = extraction factor = H H = quantity of heavy phase to each stage per cycle i = general term of summation j = general term of summation L , = quantity of light phase to each stage per cycle m = general term for stages to right of feed stage 111 = stages between feed stage and heavy product drawoff stream, including- feed stage n = general term for stages to the left of feed stage
A = function in summation
N D E N G I N E E R I N G CHEMISTRY
+
+
Vol. 46, No. 1
-
ENGINEERING AND PROCESS DEVELOPMENT A- =: stages between feed stage and light product drawoff stream, A' including feed stage (total stages in system = 111' 1) r = ratio between coefficients of terms of geometric series E p =: fraction of component in light phase in each stage = E+1 P = fraction of feed appearing in light product stream q =: fraction of component in heavy phase in each stage =
+
-
Subscripts
n,J,T , and t identify stages of exwaction pattern t and s refer, respectively, to any cycle t and to steady-state conditions
~
1
E+1 Q
=
&'
=.
t 8 ,
= = =
6
total amount of component in any stage amount in product stream for given number of stages at any cycle cycle number starting a t first product drawoff coefficient of terms of expanding series deviation from steady state expressed as fraction
Literature Cited
(1) Auer, P.L.,and Gardner, C. S.,IND.EXG.CHEM.,46, 39 (1954). (2) Bartels, C. R., and Kleiman, G., C h a . Eng. Progr., 45, 589 (1949). (3) Craig, L. C.,J. Bid. Chem., 155, 519 (1944). (4) Peppard, D. F.,and Peppard, 31..4.,IND.ENG.CHEM., 46, 34 (1954). (5) Satsangee, P.P., D.Ch.E. dissertation, Polytechnic Institute of Brooklyn, June 1952. (6) Soheibel, E. G., IND. ENG.CHEH.,43, 242 (1951). (7) Stene, S., Arhiv. Kemi,Mineral. Geol., 18H,No. 18 (1944). RECEIVED for review May 29, 1953.
ACCEPTEDOctober 31, 1953.
Torpidity of Stirred Phlegmatic 1iquids K. C. D. HICKMAN' Research laboraiories, Eastman Kodak Co., Rochester, N. Y.
W
ORK presented recently ( 5 ) showed that a binary liquid
mixture, 2-ethyl hexyl phthalate plus 2-ethyl hexyl sebac a b , evaporating freely into high vacuum, can evolve vapors of varying composition, depending on whether the surface is expanding or contracting. It should follow that when a partly submerged paddle is moved in a liquid evaporating in the micronmillimeter pressure range, different quantities and compositions of vapor should be emitted on the fore-and-aft sides of the paddle. Stationary Probe Measures Rate of Evaporation of Moving liquids
A partly immersed paddle can be rotated as a radial arm from a shaft held over a pool of liquid in a vacuum container, and probes for the vapQr can be fastened to either side of the paddle. I t is more practical to employ stationary probes on one side of a fived paddle and to move the liquid toward and away from the probe side by means of a completely submerged stirrer. Figures l , a and I,b give details of apparatus in which this has been done. Figure 2 shows the attachments for purifying the liquid during use. Thus, d is an inverted bell jar, 20 cm. in outside diameter, which serves as container, boiler, and part condenser of the liquid Closure is made a t the top by a steel plate, B, fitted with a large stainless steel cold finger, C. The plate rests on a neoprene gasket, D. Vacuum is applied by the usual pumps and traps at the side arm, E . Side tubes F and G serve for overflow and return of experimental liquids. The liquids are heated by a grid work of Nichroine wire stretched between two sides of a tubular glass support, H , which, in turn, is suspended through the stainless steel tube, J. The stationary stirrer-barrier, K , is made of glass and is attached radially to a glass ring, 4 cm. in diameter, which serves to occupy and dispose of any liquid vortex that might otherwise form a t the center of the pool. The ring and barrier are attached to the grid support, H . The rotating stirrer, L, is constructed from four crescent-shaped arms of aluminum fastened to an oblong permanent magnet which, in turn, is rotated by the external horseshoe magnet and motor, M . The experiments involved only measurements of rate of evaporation. The probe for determining emission of vapor consists of a small fan and a tachometer, N , perfected and described by Trevoy and Torpey (5). The probe is fitted with a thermocouple with its tip just beneath the surface of the liquid. Wires from 1
Present address, 136 Pelhitm Rd., Rochester 10, N Y
January 1954
the probe, thermocouple, a i d heater are brought through rubber stoppers in holes in the flange, B. This procedure se,emed justified for a first test of the stirrer hypothesis. It was soon found that little condensate reached this region of the jar, and contaminants coming from the rubber were removed by the purification train. The pressure of residual gas in the apparatus wa8 measured by the Pirani gage, P , and the duplicate ionization gages, Q. The motor of the external stirrer could be operated a t eight speeds toward and away from the barrier. The motion imparted to the liquid was don-er than the stirrer and varied along the radius of rotation. Average surface speeds, estimated by the passing of floating particles, are given in Table I.
Table 1. Stirrer Setting 1
2
3 4
5 6
7 8
Stirrer Setting and Solution Speed R.P.N. 4.4 8.4 15 22 34 46 62
82
Approx. Speed of Surface, C m . ( S L Torpid Working
'4 6 8 12 16
0.9 1.7 3 4 4
7 0
12
16
The thermocouple was composed of S o . 30 copper and constantan wires fused to a small ball, 0.3 mm. in diameter a t the tip. Compared with the elaborate, hair-thin thermal probes that have been reported in other work on evaporating surfaces, this thermocouple was crude. The justification for itscontinued use has been the uniformity of the temperature readings near the liquid surface and the reproducibility with which emission of vapor corresponds with recorded temperature. Also, the variations in emission caused by passage from torpid to working surface would correspond with changes in a single kind of surface of 2' to 40' C., which are tens or hundreds of times greater than the likely errors of the thermocouple. The temperature gradient in the first few molecular layers is not considered in this paper. The temperatures measured experimentally are those of the upper quarter million molecular
INDUSTRIAL AND ENGINEERING CHEMISTRY
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