Effect of Residual Coke on Behavior of Cracking Catalyst A. C. WHITAKER AND A. D. KINZER Gulf Research 6% Development Co., Pittsburgh, P a .
A
BOUT 19 years ago the first commercial catalytic cracking unit (Houdry fixed-bed) went on stream. Since then catalytic cracking has had a tremendous growth and much technical effort has been expended in learning how t o use cracking catalysts most effectively. The fundamental purpose in testing cracking catalysts is t o predict t h e manner in which they will behave when used in commercial units. Those aspects of catalyst behavior which are of greatest importance are activity, or extent of conversion which the catalyst produces; selectivity, or relationship of desirable t o undesirable products of cracking; and stability, or resistance of catalyst t o aging. The most desirable method of catalyst testing mould be employment of test conditions t h a t duplicate those used in commercial plants. I n the case of fixed-bed operations the catalyst test method is merely a scaled-down version of the commercial operation; i t is also applicable t o a moving-bed process which employs fairly long catalyst residence times and low ratios of catalyst t o oil. I n the case of the Fluid operation, however, short catalyst residence times and high ratios of catalyst t o oil are used commercially and these conditions are practically impossible t o duplicate in a fixed-bed bench scale unit. Test procedures for fluid catalyst which call for long contact times and low ratios of catalyst t o oil have been found entirely misleading in predicting the behavior of certain catalyst types. These procedures are particularly inadequate for predicting selectivity, which is of everincreasing importance as heavier gas oils of high sulfur, nickel, and vanadium contents are charged t o the Fluid units. A method of using fixed-bed test results, which is more reliable than prior methods in evaluating fluid catalysts, employs a static bed of powdered catalyst and two runs of different reaction periods. As both conversion and carbon on catalyst are linear functions of the duration of cracking when plotted on log-log paper, the experimental data may be extrapolated t o give activity and selectivity values a t the high catalyst-oil ratios of commercial conditions. This method cannot b y itself evaluate the third catalyst attribute of stability, but is an important adjunct t o aging methods used for this purpose. NEED FOR IMPROVED TEST
The gas oils charged t o the catalytic cracking processes each day in the United States total over 3,000,000 barrels and are expected t o increase steadily. A portion of this growing capacity has been and will be heavy vacuum distillate oils which contain harmful trace metals such as nickel and vanadium. These metals have proved t o be severe catalyst poisons. This situation emphasizes t h e need for adequate equipment and techniques for the evaluation of the selectivity of both new and equilibrium catalyst with maximum confidence. It has been long recognized that the behavior of a catalyst in a commercial fluid catalytic cracking unit and its behavior in bench scale testing units are somewhat different. T h e results obtained in bench scale testing units are at present used on a relative basis. This in many cases has led to faulty conclusions regarding the performance of a catalyst. An example of this type of misinterpretation is shown in Table I, which gives bench scale and pilot unit data pertaining t o synthetic and natural equilibrium catalysts.
The bench scale testing d a t a (Table I ) could be interpreted to indicate that the commercial performance of these two catalysts would be identical. However, the behavior of these catalysts was shown t o be considerably different in pilot unit and commercial operation. For example, the natural catalyst produced about 30% more coke a t the same conversion. On the other hand, it had the higher effective activity, as i t was necessary t o increase the space velocity over this catalyst by 367' in order t o maintain equal conversions. This basic weakness in tests using low catalyst-oil ratios was again brought out with the introduction of SR catalyst (Table V). The lack of agreement between such laboratory results and the commercial performance of a catalyst is due t o basic differences in the operating conditions, However, it has not been a simple matter to eliminate the discrepancies of bench scale catalyst testing, particularly those pertaining t o catalyst residence time and catalyst-oil ratio. I n a single-cycle, bench scale unit a catalyst-oil ratio of 10, commonly used commercially, --ould involve catalyst beds t h a t are inordinately large in relation t o the oil charges. Such a condition is usually attended by a high incidence of unsatisfactory material balances.
Table I.
Comparison of Synthetic and Natural Catalysts
Catalyst Type Bench-scale d a t a Activity (Kellogg, 2 hours) Carbon factor Pilot unit d a t a Charge stock Recycle, vol. % fresh feed Space velocity (total feed), wt./hr./wt. Conversion, vol. % '
Coke,w t . %
Synthetic 28.6 1.16
Natural (Grade D) 27.9 1.10
28' API Mid-Continent gas oil 45 45 3.3 60.3 6.3
4.5 60.6 8.3
DEVELOPMENT
I n view of the difficulties encountered in duplicating commercial conditions on a test basis, it was decided t h a t a catalyst-oil ratio of 10, for example, might be more feasibly approached by extrapolation of experimental data obtained a t easily established laboratory conditions. It was reported by Voorhies (6) t h a t when carbon yield is plotted as a function of reaction period on loglog paper, a straight line is obtained. H e also derived correlations which defined mathematically the interdependence among feed rate, conversion, and length of time between regenerations. These relationships were corroborated by Shankland and Schmitkons (a) and extended t o include catalyst activity as a function of reaction period. On the basis of these findings, i t should be possible, with the use of logarithmic graph paper, t o define the carbon-time and conversion-time relationships of a n y given catalyst by two bench scale cracking runs of different durations. The selection of a method for the measurement, with suitable accuracy, of the activity and selectivity of fluid cracking catalysts involved much careful consideration. There appeared t o be three principal types from which t o choose: fluidized fixed bed (4),static bed of powder ( 5 ) , and static bed of pellets (1-3). Each had advantages and shortcomings. The fluidized fixed bed was tentatively discarded because efficiency of fluidization was sensitive t o particle size distribution of the catalyst sample, 2153
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and it was felt that the removal of the finer fractions, particularly from equilibrium samples, would be undesirable. From the standpoint of simplicity, the static bed of pellets seemed very attractive, provided the effects of particle size and pilling pressure could be neglected. I n order to evaluate these effects, an unused Houdry 5-36 catalyst was tested as full-sized pellets and after being ground and sized to various mesh ranges. The portions recovered as powder were combined and pilled in a Stokes rotary machine using 5% poly(viny1 alcJhol)(Grade 52-22) as lubricant. The pills were again ground, sized, and tezted as before, after regeneration at 1050" F. for 3 h-.;rs. The equipment and test method used for this comparison are described below.
The testing data given in Table I1 indicate that as particle size decreases, both conversion and selectivity improve, and the pilling pressure affected both activity and selectivity adversely beyond the particle size effect. Moreover, the effects of pelleting appear t o be permanent, since the repilled material, after being ground and sized to 16 to 30 mesh, was inferior t o the original cast pellets ground to the same size. The total effect of pilling a fluid catalyst may be approximated by comparing the 16- to 30mesh fraction which was prepared from the original pellets with the full-sized laboratory pellets. I n view of these results, the idea of evaluating fluid catalysts in pill form was abandoned, and it was decided to adopt a test involving a static bed of powder.
Vol. 47,No. 10
inches and an inside diameter of 1 inch. The catalyst section is about 8 inches high and is located slightly below the middle of the tube. Above and below the catalyst section dead men have been installed t o minimize thermal cracking. The reactor thermocouple extends no more than 1 inch into the catalyst bed, as i t was found that deeper penetration promoted channeling of the oil vapors. The reactor tube is surrounded by a furnace consisting of five heating zones. The temperature in the third, or central, zone, which corresponds t o the catalyst section, is controlled automatically. The charge. a 35" API mid-continent gas oil. is forced into t h e reactor b y a ' positive displacement p u k p . The cracking run involves passing the gas oil over 73.8 grams of catalyst (50 grams for pelleted catalysts, owing t o limited reactor volume) at 920" F. and a weight space velocity of 2.0 for a preselected period of time. Sample weight is based on condition of catalyst following t h e standard pretreatment of 3 hours a t 1050" F. in a gentle current of dry air. The cracking run is followed immediately by a nitrogen purge of known volume, which is collected with the unit gas. The synthetic crude is transferred directly to a Vigreux column and distilled t o a 410" F. end point. Conversion is expressed as 100 minus weight per cent gas oil, corrected to 100% material balance. A composite sample of the gas is analyzed for gravity and hydrogen content, both of which are corrected to account for the known volume of nitrogen present. For the carbon determination, the entire sample, whether powder or pellets, is thoroughly pulverized with a mortar and pestle, and a representative sample is burned in a Dietert 2-minute carbon determinator. All cracking runs involving test catalysts in powder form are accompanied b y a pressure drop, the magnitude of which depends largely on the particle size distribution of the catalyst and, to some extent, on catalyst activity. I n testing fluid catalysts, therefore, facilities are required for correcting the data, notably conversion and carbon yield, to atmospheric pressure. The correction factors were obtained by using, as a standard, a synthetic silicaalumina catalyst through which there was essentially no pressure drop, and operating over a range of pressures imposed by a valve a t the outlet. With this arrangement, pressure is uniform throughout the bed and effective pressure is equal to gage pressure. I n testing a n unknown catalyst of ordinary particle Pize distribution, no preseure is imposed a t the outlet. There is a pressure gradient through the bed and the effective cracking pressure, in this case, is some value between gage pressure and the outlet pressure, which is a t 1 atm. It has been found t h a t the effective pressure is equivalent to 0.65 X gage pressure ( 5 ) .
APPARATUS AND PROCEDURE
The apparatus (Figure 1) and procedure were patterned after those described by Shankland and Schmitkons ( 5 ) . The reactor tube is a stainless steel pipe having a length of 48
Table 11. Effect of Pilling and of Particle Size on Cracking Characteristics of Houdry s-36 Catalyst Catalyst As reed. Whole pills 8-16-mesh 16-30-mesh Repilled Whole pills 8-16-mesh 16-30-mesh
Process Period, Min.
Conversion, W t . yo
Carbon, Wt. yo
Carbon Factor
60 20 60 20 60 20
39.1 48.7 41.2 81.8 41.9 52.2
1.39 2.66 1.31 2.38 1.28 2.46
1.32 1.42 1.07 1.10 1.00 1.11
60 20 60 20 60 20
37.2 46.5 38.4 49.1 40 3 50.2
1.41 2.40 1.23 2.28 1.40 2.50
1.55 1.44 1.23 1.20 1.22 1.25
Y
P
CONVERSION, WT. 9/.
Figure 2. Effect of pressure on conversion at 20-minute catalyst residence time
L
I n Figures 2 and 3, conversion and carbon on feed, respectively, are plotted as a function of pressure a t several activity levels (20-minute reaction period). I n Figure 3, absolute pressure was used so t h a t the origin provided an additional point in constructing the curves. Figure 4 is a plot of carbon on feed as a function of conversion, from which carbon factors for test catalysts are
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calculated. Similar curves were developed for the 60-minute reaction period, and, in addition, a carbon-conversion curve based on extrapolated 3-minute data. The adequacy of the pressure calibration employed is demonstrated in Table 111, which includes data for four fractions of a single catalyst. The fractions, obtained by air elutriation, varied in average particle size and provided a rather wide range of pressures, The samples were assumed t o be equivalent in cracking characteristics, as each was composed of 10% fresh synthetic silica-alumina and 90% of the same material which had been thermally inactivated. The reproducibility of the method is demonstrated by the data in Table IV, which contains the results of seven consecutive 20-minute cracking runs over an equilibrium catalyst during a 4-month period.
Table 111. Correction of Conversion and Carbon Yield to Atmospheric Pressure Conversion, Lb./Sq. Inch Wt. % Corr. Actual Effectivea Exptl. 32.0 31.3 1.9 1.2 32.4 7.3 36.0 11.2 37.0 32.3 17.0 11.1 37.0 31.0 26.9 17.5 a Actuallb./sq. inch X 0 . 6 5 .
Carbon, Wt. Vo Exptl. Corr. 0.73 0.69 0.98 0.74 1.06 0.72 1.15 0.68
Carbon Factor 0.94 0.96 0.93 0.94
~~~~~~
Table IV.
Reproduoibility of Test Method
Run X o . 1 2 3 4 5 6 7 Catalyst Equilibrium SR Conversion,at.% 3 2 . 8 31.9 32.0 32.6 32.3 32.9 32.5 1.23 1.21 1.21 1.23 Carbon factor 1.21 1.27 1.14 Standard deviation Conversion 0.377 Carbon factor 0.039
Two catalysts, a n equilibrium fluid Filtrol SR and an equilibrium Filtrol TCC, were chosen for the investigation. Cracking runs of 3-, 5-, lo-, 20-, 40-, and 60-minute duration were made over each catalyst a t the conditions previously stated. At the 10-minute process period, it was necessary to combine the synthetic crudes from two runs to provide sufficient liquid for accurate distillation. For the 5-minute data, the combined liquid products from three runs were required. No attempt was made t o determine conversion in the 3-minute runs, because of the increasing difficulties attending the operation as the process period is shortened. These runs, therefore, merely served t o extend the range of the investigation in regard t o carbon yield. DISCUSSION OF RESULTS
The cracking data for the powder and the pellets are presented in Tables V and VI. The conversion and carbon on catalyst are plotted as a function of process period in Figure 5 . The series pertaining t o the pellets was performed before i t was realized that maximum accuracy in carbon determinations required thorough powdering of the sample. Thus, the deviations in the 40- and 60-minute points, particularly, are larger than would be normally expected. It is apparent that on log-log paper these relationships are linear, and each can therefore be completely defined by two points. The significance of the log-log plots is t h a t by making two runs a t feasible process periods-e.g., 60 and 20 minutes-both conversion and carbon on catalyst can be extrapolated t o a process period of 3 minutes, which, a t a weight space velocity of 2.0, corresponds to a catalyst-oil ratio of 10. UTILITY OF METHOD
At the time SR catalyst (sulfur-resistant activated clay) was introduced t o the market, this laboratory employed a fluidized fixed-bed procedure ( 4 ) with a process period of 2 hours for the bench scale evaluation of fluid catalyst. The test conditions involved a temperature of 850" F. and a weight space velocity of
0
w
E
"
2
I
o
5
4
3
6
7
a
COKE, W T . %
a
Figure 3. Effect of pressure on coke yield at 20-minute catalyst residence time
0.6. B y this test, a fresh laboratory sample of SR gave a conversion of 33 t o 35 volume % and a carbon factor of 1.8, compared with a 66% conversion and a carbon factor of 1.0 for the ground silica-alumina standard. On the basis of these results SR did not appear t o be a desirable catalyst. However, CAT-A ( I , 2 ) data on pelleted SR indicated t h a t the performance of the catalyst improved with increasing catalyst-oil ratio. This was particularly true in regard t o selectivity. For example, a typical commercial shipment of SR catalyst gave carbon factors of 3.24 and 2.22 in the 60- and 20-minute runs, respectively. The calculated carbon factor for a 3-minute run, however, based on extrapolated conversion and carbon yield, was 1.27. It was then apparent t h a t SR catalyst could not be accurately evaluated except a t catalyst-oil ratios approaching those in commercial operations. EFFECT OF RESIDUAL COKE
The preceding discussion has been concerned entirely with the test method, and no attention has been given t o the preparation of the sample. For all fresh catalysts and equilibrium catalysts containing negligible quantities of contaminants, the universally accepted stabilization by heat treatment would probably be adequate t o obtain reliable comparisons. However, the increasing use of heavy gas oils of high sulfur and metal contents has
Cracking Data for Equilibrium Fluid SR Catalyst
Table V.
Process Period, Min.
a
No. of Runs Averaged
Conversion, Wt. %"
Carbon
on Catalvst,
Wt.
70"
Corrected to atmospheric pressure.
Table VI.
Cracking Data for Equilibrium Filtrol TCC Catalyst
Process Period, hlin. 3 5
10 20 40 60
No. of Runs Averaged 2 4 2 2 2 2
Conversion, Wt. %
...
49.2 43.7 36.9 31.2 28.9
Carbon on Catalyst,
Wt. % 0.409 0.536 0.663 0.905 1.155 1,570
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Vol. 47, No. 10
The most striking example of the effect of residual coke was encountered when, during a Fluid pilot unit program, improper storage of a feed stock resulted in a marked increase in the zinc content of the circulating catalyst. -4sharp rise in carbon factor, as determined b y the 2-hour fluidized fixed-bed test, accompanied the contamination, although the pilot unit yields failed t o indicate the presence of catalyst poisoning. These data are shown in Table VIII, which also includes extrapolation results for both the completely regenerated catalyst and that containing about 0.1% residual carbon. It is apparent that t h e coke deposit in both pilot and bench scale units completely masked the nonselective tendencies of the zinc, which are so much in evidence in the results obtained with clean catalyst. It is considered significant t h a t the extrapolated data on the coke-bearing sample and t h e pilot unit data lead t o the same conclusions-i.e., t h a t the catalyst activity was unaffected by the zinc accumulation and the selectivity, if anything, was slightly improved. There is undoubtedly a threshold concentration of metal beyond which t h e effects of contamination become manifest. Evidence for this lies in the experience of refiners with vanadium poisoning. It seems likely also t h a t the deactivating, or masking, capacity of t h e residual coke would vary from metal t o metal. CONVERSION, WT.X.
Figure 4. Yield of coke as function of conversion
Table VIII.
Downflow t e a t unit data for Diakel catalyst a t 20minute catalyst residence time
increased the complexity of bench scale catalyst evaluation. For example, under certain conditions, the performance of a cracking catalyst may be affected t o a marked degree by the presence of residual coke. As catalysts in use are never entirely free of coke, it would seem desirable t o test the activity and selectivity on pilot or commercial unit samples as they are obtained from the regenerator-i.e., without burning them clean. I n regard t o fresh catalysts, a coke layer resembling commercial residual coke in its effects may be simulated b y making a very short run in a fluidized fixed-bed unit t o lay down a coke deposit of 0.6 t o O.8yOby weight of the catalyst. The catalyst sample is then removed t o a treating unit and heated a t 900" F. for 15 minutes under vacuum (about 5 mm. of mercury). Table VI1 contains data relating t o the steam and hydrogen sulfide deactivation of a sample of fresh SR catalyst in the absence and presence of a layer of simulated residual coke.
Combined Effects of Extrapolatioin and Residual Coke
Data Test unit (%hour test) Conversion, vol. Yo Carbon factor Pilot unit Convereion, vol. % Debutsnized gasoline vol. 7, Gas (Cr and lighter), % CokejO.9 carbon, 0.1 hzdrogen), wt. Test unit (Extrapolated) Completely regenerated Conversion, wt. % Carbon factor With approx. 0.17, residual coke Conversion wt. 70 Carbon factor
at.
a
Before Zinc Accumulation
After Zinc Accumulation
19.6 1.52
18.4 2.54
43.1 33.2 9.7 4.9
43.0 33.5 9'7 4.6
49.0 0.84
39 .O 1.85
45.0 0.81
45.0 0.65
Yo
3-minute values obtained by extrapolating 60- a n d 20-minuterdata.
60 50 40
30 25
Table VII.
Effect of Residual Coke on Steam and Hydrogen Sulfide Stability
Treatment of Sample Conversion, wt. % 60 minutes 20 minutes 3 minutes6 Carbon factor 60 minutes 20 minutes 3 minutesb a Steam, l l O O o F., inch, 2 hours. b Extrapolated.
Fresh 28.4 38.4 65.0
Steam and Coked (0.7% C ) , HzSa Steam and H2S5 23.2 31.2 53.0
23.4 31.5 53.0
Coked, Steam and Hisa, Regenerated 27.3 36.3 60.0
2.56 2.14 2.91 3.24 1.61 1.75 2.02 2.22 0 . 8 6 1.30 1.21 1.27 then HzS, 1000" F., 5 Ib./sq. 15 lb./sq. inch, 8 hours;
,
-- EQUIL. FlLTROL SR
1.5 I.o
0.8 0.6 0.4
0.3
2
4
6
810
20
40
60 80
T I M E , MINUTES
Figure 5. Conversion and carbon on catalyst as a function of cracking period
The salient features of these results are: (1) The coke layer affords appreciable protection against the deactivating influence of the steam and hydrogen sulfide treatment (compare third and fifth columns); (2) the 3-minute carbon factors in the second, third, and fifth columns are substantially equivalent, indicating t h a t SR catalyst has excellent sulfur stability; and (3) the low extrapolated carbon factor in the fourth column is caused not by protection b u t b y the masking effect of the coke. This is brought out b y the rise in extrapolated carbon factor following regeneration.
CONCLUSIONS
Both carbon yield and conversion in bench scale catalyst evaluation yield straight lines when plotted against cracking duration on log-log paper. This linear relationship permits, by extrapolation, the estimation of cracking characteristics at short residence times and high catalyst-oil ratios, conditions which are difficult t o obtain with satisfactory precision in bench scale equipment. Complete regeneration of test catalysts may result
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INDUSTRIAL AND ENGINEERING CHEMISTRY
in bench scale data which indicate serious contamination, whereas in actual operation the concentration of contaminants does not exceed that which may be rendered ineffective b y the ever-present deposit of residual coke. LITERATURE CITED
(1) Alexander. Proc. Am. Petroleum Inst.. 27 (111).51 (1947). (2j Alexander and Shimp, Natl. Petroleum ’lVews, 36 ( 3 i ) , R-537 (1
944).
2157
(3) Berkhimer, Macwa, and Leum, Proc. Am. Petroleum. Inst., 27
(111),90 (1947). (4) Diakel Corp., “Physical, Chemical, and Catalytic Testing of Diakel Powdered Cracking Catalyst,” June 7, 1943. (5) Shankland and Schmitkons, Proc. Am. Petroleum Inst.,27 (III), 57 (1947). (6) Voorhies, IND. ENG.CHEM.,37, 318 (1945). RECEIVED for review October 15, 1954. ACCEPTED April 21, 1955. Division of Petroleum Chemistry, 126th Meeting ACS, New York, N . Y., 1954.
Fractional Precipitation or Crystallization- Systems EFFICIENCY OF FRACTIONATION E. F. JOY1
AND JOHN H. PAYNE, J R . ~ Mound Laboratory, Monsanto Chemical Co., Miamisburg, Ohio
T
HE precipitation of barium-radium chromate mixtures from homogeneous solution is a suitable method for concentrating radium because of the high distribution coefficient obtained and the ease with which the fraction precipitated can be controlled (6). Concentrating the radium present as a microcomponent requires repeated fractional precipitation in a systematic scheme. This paper deals with the method of selecting the most efficient fractionation scheme for a binary mixture of this general type. T o determine the “efficiency” of a fractionation step some quantitative measure of the separation achieved must be designated as a basis on which the “efficiency” is t o be calculated. Consider the separation of B from a mixture with A where B is present in small concentrations. When the mixture is split into two fractions in which B is enriched in one fraction and depleted in the other, the amount of separation which has been achieved is related t o the amount of B recovered in the enriched fraction and to the degree of enrichment which it has undergone. At the same time, the amount of B which is now in the depleted fraction must be discounted, taking into account the degree of depletion. The problem is one of balancing these two opposing factors t o obtain a maximum separation. This problem has been solved by a thermodynamic approach involving the entropy of mixing. The entropy change for one fractionation step was found t o be a function of the product of the yield of B and logarithm of the enrichment of B, summed up for the enriched and depleted fractions. The conditions for obtaining a maximum negative entropy change corresponding t o maximum separation have been determined, The “efficiency” of separation of a fractionation step is designated as the ratio of the entropy change for the fractionation step t o the maximum obtainable entropy change for the system being considered. The entropy change depends only on the initial and final states of the system and is independent of the mechanism of the separation. The “efficiency” of separation as used is concerned only with measuring the amount of separation and is not concerned with the means of making the separation. I n a continued fractionation, the mixing of fractions of differing composition results in an increase of entropy and this loss in efficiency of separation should be avoided. I t is important, therefore, t o have a point in the scheme where the composition of the starting material is duplicated so t h a t fresh additions can be made without loss of efficiency. Previous workers ( I , 7) have stated this need and have solved the problem for specific cases in a n Present address, J. T. Baker Chemical Co., Phillipsburg, N. J. a Prevent address, Monsanto Chemical Co., Research Dept., Inorganic Chemicals Division, Dayton, Ohio. 1
empirical way. A more systematic approach t o the general case has been made and the mathematical relationships for obtaining repeating compositions have been developed. The efficiencies of fractionation of several workable systems were calculated from their entropy of separation and the one chosen was the simplest method for separating radium from barium. THEORY OF SYSTEMS WITH REPEATING FRACTIONS
A typical triangular fractionation scheme (8) is shown in Figure 1. Individual fractions are represented by the circles. The fractions are designated by a horizontal row number and a diagonal column letter, Consider the separation of two components, A and B, by fractional crystallization or fractional precipitation. If a constant fraction z of Component A is precipitated a t each operation and fractions are combined in the usual triangular fractionation scheme, the fraction of the original A which mill appear a t any point in the scheme is given by the terms of the binomial expansion [2
+ (1 - 211” = 1
(1)
as shown in Figure 1. I n similar fashion for Component B , where a different but constant fraction y is precipitated a t each step, the distribution of B in the various fractions will be given by the expansion [Y
+ (1 -
!/)In
=
1
(2)
Fractionation schemes are simplified when fractions of repeating composition recur a t regular intervals. The first fraction which can have the same composition as the original mixture is fraction 2b. I n this case every fraction will have the same composition as the fraction vertically above it in the scheme. The composition of fraction 2b will be the same as the original if the fraction of A which reaches that point in the scheme is the same as the fraction of B which also reaches that point, Le.,
241
- 2) = 2y(l - y)
(3)
This equation is satisfied by two solutions 2 = y
x = l - y
(4)
The first solution, z = y, is a trivial solution. Although the concentration of the 2b fraction repeats the original concentra-
