Dehvdration of Aqueous Methyl Ethvl Ketone -
J
J
J
EQUILIBRIUM DATA FOR EXTRACTIVE DISTILLATION AND SOLVENT EXTRACTION MORRIS NEWMAN, CURTIS B. HAYWORTHl, AND ROBERT E. TREYBAL New York University, New York 53, N. Y .
The generalizations of Ewell, Harrison, and Berg (3) are shown t o be useful i n the preliminary selection of solvents for concentrating aqueous methyl ethyl ketone solutions by extractive distillation or solvent extraction. Vapor-liquid equilibria for the binary systems ketonebutyl Cellosolve and butyl Cellosolve-water, as well as the ternary system ketone-water-butyl Cellosolve are presen ted and correlated. Ternary liquid equilibria are presented for the systems ketone-water-trichloroethylene, -1,1,2 trichloroethane, and -chlorobenzene. All the solv e n t s are shown t o be useful for the suggested processes.
A
QUEOUS methyl ethyl ketone solutions are relatively unusual in that, although at atmospheric boiling tem-
peratures there is a region of incomplete miscibility extending from 18 t o 85% ketone, the minimum boiling azeotrope falls outside the solubility limits and consequently forms one liquid phase on condensation. As a result the usual two-tower distillation process for separation of partially miscible liquids involving decantation of the condensed azeotrope cannot be used with this system. Furthermore, Othmer and Benenati (6) have shown that this situation pertains over a t least a moderate pressure range, so that in all probability reasonable variations in total pressure of a distillation would not make such a separation scheme feasible. There are several other methods available for separating such mixtures. In the case of methyl ethyl ketone-water solutions, a salting-out procedure involving solid sodium chloride may be used (IO). Meissner and Stokes ( 4 ) have suggested the use of calcium chloride and other brines for salting-out processes, with a liquid extraction type of flow sheet. The water might be removed by azeotropic distillation with the use of a so-called entrainer, or extractive distillation with an added high-boiling solvent is a possibility. The latter has been proposed by Pierotti and Dunn ( 8 ) , with polyhydric alcohols such as ethylene glycol and other solvents as third components. Liquid solvent extraction of the methyl ethyl ketone with organic solvents would be a further possibility. Of all these methods, extractive distillation and solvent extraction are perhaps the most attractive technically, for reasons of cost of operation, corrosion of equipment, and simplicity of the process itself, provided desirable third components can be found. EXTRACTIVE DISTILLATION
Benedict and Rubin ( I ) have discussed in detail the factors governing the choice of solvent for extractive distillation. I n addition to practical considerations of cost, toxicity, and availability, perhaps the most important are volatility, capacity, separability of solvent from nonsolvent, and selectivity. The more volatile the solvent, the more solvent is found in the Present address, General Chemical Company, Edgewater, N. J.
vapor phase and the poorer is the enrichment in distillation column. The capacity of the solvent refers to the amount of nonsolvent which will be tolerated in the liquid phase without phase separation. Good selectivity on the part of the solvent for one component often leads t o immiscibility with the other component, and thus it is necessary t o balance selectivity against capacity in order to obtain optimum operating conditions. If the solvent is sufficiently nonvolatile with respect t o the nonsolvents, separation should be easy. If, because of extreme deviations from ideality, an azeotrope between solvent and one of the nonsolvent components forms then special means of separation such as decantation of the condensed azeotrope may be required. The selectivity of a solvent may be rated by the sepaartion which it produces between the two nonsolvent components a t any given concentration, and it is on the determination of this property that most attention is focused. I n searching for a solvent of adequate selectivity several generalizations are available; perhaps the most useful is that of Ewell, Harrifon, and Berg (9) based on hydrogen bonding. These authors have classified compounds into five groups, and on the basis of this classification, qualitative conclusions can be drawn as t o the result oi mixing together liquids of different groups. Methyl ethyl ketone is a class 111 compound in this classification, whereas water falls in class I. The solvent selected should have’a region of miscibility with both components, and t o be effective it must possess a different degree of selectivity with each of the components. It is clear that class IV and V compounds (such as hydrocarbons and chlorinated hydrocarbons) do not have compatibility with water, and thus likely solvents must be chosen from classes I , 11, and 111. I n general, when any combination of compounds in class I, 11, or I11 is made with those of class I or 111, hydrogen bonds are both formed and broken; this results in complex agglomerations usually with positive deviations from Raoult’s law. As a preliminary investigation of possible solvents in accordance with this reasoning, experiments were made with ethylene glycol, furfuryl alcohol, diethylacetamide, ethylenetriamine, and 2-butoxyethanol (butyl Cellosolve).
ETHYLENE GLYCOL. Dworzak and Herrmann ( 2 ) report a condensation between methyl ethyl ketone and ethylene glycol with formation of water requiring the presence of sodium sulfate or other alkali salts. Mixtures of the two were made in the absence of salts and held at boiling temperatures. After a short time, a flaky precipitate was noticed and the liquid separated into two phases on cooling. Such reaction eliminated the possibility of using ethylene glycol as a separatiq agent, since effective solvents must be chemically inert to the mixture to be separated. FURFURYL ALCOHOL.This material is polymerized by mineral acids a t room temperature to yield a resin (9). I n the absence of acid it was successfully refluxed for several hours with little change in color. Attempts therefore were made to obtain vapor-liquid equilibrium data for the system methyl ethyl ketonefurfuryl alcohol, but at high concentrations of the alcohol, evidence of reaction was clear and the material was abandoned. DIETHYLACETAMIDE. This solvent was refluxed successfully by itself for several hours without evidence of decomposition,
2039
2040
Vola 41, No. 8
INDUSTRIAL AND ENGINEERING CHEMISTRY but in the prese n c e of m e t h y l ethy1 k e t o n e , r e a c t i o n or decomposition o c curred r a p i d l y . No additional work was done with it. ETHYLENETRI-
AMINE. Thiv subs t a n c e showed evidence of considerable decomposition after only 10 minutes of refluxing and was discarded as a possibility. 0 02 04 06 08 10 BCTYLCELLOMOLE FRACTION METHYL ETHYL KETONE SOLVE. Repeated X IN LIQUID, Y IN VAPOR. and extended boilings with methyl Figure 1. Vapor-Liquid Equilibria, ethyl ketone and Methyl Ethyl Ketone-Butyl water gave no inCellosolve at 760 Mm. Hg dication of reaction or decomposition of this ~ o l v e i i t xud , as its physical properties seemed favorable, further investigation of its usefulness as an extractive distillation agent was made.
BUTYL CELLOSOLVE A S THIRD COMPONENT
In the determination of the vapor-liquid equilibria herein reported, the Othmer-type equilibrium still ( 5 ) , fitted with a barostat to control the pressure a t 760 A 0.5 mm. of mercury, was used. Analyses of solutions were made by use of the rcfractive index (by Abbe refractometer), the densitv (by pycnometer), or both, as the situation required. The methyl ethyl ketone mas fractionated in a 6-foot distillation column packed with 0 2A-inch Raschig rings. The central portion of the distillate had a refractive index ny = 1.3762, specific giavity 0.8019 (d;:), and a normal boiling point of 79.7 C. as compared &th accepted valurs of 1.3761, 0.8020, and 79.6 for these properties, respectively ( I O ) . The butyl Cellosolve, similarly fractionated, had a boiling range of 170" to 171" C., and a, specific gravity 0.9003 id;;) a3 comp~redrvith t h e puhlished 0.899 (13). Vapor-liquid equilibrium data tor the system methyl ethyl ketone-butyl Cellosolve are presented in Table 'I and Figure 1. those for butyl Cellosolve-water in Table I1 and Figure 2 The data for the system methyl ethyl ketone-15 ater were recently published (6) and were not determined. To calculate the activity corfirirnts for the systems studied, it was necessary to have at hand adequate vapor p r e s s u r e data for the pure compounds ovei the temperature range of the investigation. In the case of butyl Cellosolve, f o r which vapor pressures below 1 atmosphere were required, a 0 0.2 04 06 08 Lo Washburn-Read MOLE FRACTION B U T Y L CELLOSOLVE apparatus ( 1 4 ) X IN LIPUID, Y I N VAPOR was used and Figure 2. Vapor-Liquid Equilibria, proved to be Butyl Cellosolve-Water at 760 Mm. e n t i r e l y satisHg I
TABLE I. VAPOR-LIQUID EQUILIBRIA FOR THE SYSTEM &lETHYL ETHYL KETONE-BUTYL CELLOSOLVE AT 760 MM.Hg ~
~
C.
~
~
~l l o l~e Praction ~ ~ Ketone t , Liquid Vapor-
171.2 160.6 148.3 143.8 131.6 127.0 107.1 93.0 87.0 83.0 79.7
0 0,0292 0.0675 0,0880 0.1430 0,1692 0.3145 0.5170 0.6645 0.8070 I . 0000
TABLE IT.
VAPOR-LIQUID
,
Activity Coeffioient ~ ~ , Butyl Ketone Cellosolvr
0 0.2865 0.5625 0 . G430 0.7528 0.8105 0.9140 0.9635 0.9795 0.9585 I 000 ~
EQUILIBRIA FOR
..
1.00
1,55 1.77 1.60 I .48 I .48 1.41 1.27 1.18 1.10 1.00
THE
0.86
0.91 0.89 1 05 0 92 1.07 1.15 1.21 1.42
SYSTEM BUTYL
CELLOSOLVE-WATER AT 760 Mix. Hg Mole Fraction Butyl Cellosolve ___~_ Liquid Vapor
I'emperatnre
6.
0 0.0386 0.4008 0.4835 0.7238 0.7655 0,7700 0,7440 0.9020 1.0000
100.0 9 8 . 8 (IS) 101.8 102.3 115.2 120.4 122.8 125.2 150,2 171.2
0
0.0386 0.0475 0.0535 0.1155 0.1415 0.1530 0.1720 0.4490 1,0000
Activity Coefficient Butyl Cellosolve
Water 1.00 1,04 1 49 1.66 1.93 1.83 1.72 1.37 1.19
IL1:i 1.26 1.15 0.99 0.94 0.93 1.00 0.91 1.00
..
TABLE111. VAPOR PRESSURE OF BUTYLCELLOSOLVE AND METHYLETHYL KETONE 1 emiperaturt.,
r ,
0
c.
Vapor Pressure, Mm. Hg Methyl ethyl ketone ____.._I_,
Butyl Cellosolve 83 116 239 414
106.1 113.4 133.1 149.1 107.1 111.1 116.0 118.2
.,
.) ..
1560 1695 1905 2050
factory. D a t a for the ketone were required nearl2- t o 8 atmospheres for the purposes of this study, and such extreme extrapolation of the available information ( I O ) proved unsatisfactory, However, after first making cherk determinations with pure water.
0 000
8000 6000
4000
?OOO
8000
800 600
40 0
2 2
Figure 3.
24 25 26 27 RECIPROCAL TEMPERATURE, l / * K X IO'
2.3
2.8
zoo
' 3
Vapor Pressure of Butyl Cellosolve and Methyl Ethyl Ketone
INDUSTRIAL A N D E N G I N E E R I N G C H E M I S T R Y
September 1949
2041
TABLE IV. VAPOR-LIQUID EQUILIBRIA IN THE TERNARY SYSTEM METHYLETHYL KETONE-BUTYL CBLLOSOLVE-WATER AT 760 MM. Hg Temp.,
c.
141.6 157.4 142.7 148.2 90.1 98.2 139.6 117.8
n
Mole Fraction in Liquid Butyl Water Ketone Cellosolve
Mole Fraction in Vapor Butyl Water Ketone Cellosolve
0,114 0.053 0.100 0.043 0.291 0.335 0.017 0.087
0.491 0.329 0.391 0.177 0.466 0.572 0.347 0.431
0.016 0.031 0.049 0.095 0.279 0.148 0.069 0.145
0.870 0.916 0.851 0.862 0.430 0.517 0,914 0.768
0.160 0.193 0.315 0.445 0.508 0.371 0.375 0.443
Water Exptl. Calod.
1.15 1.09 1.01 0.92 2.24 1.82 5.78 2.71
0.349 0.478 0,294 0.378 0.027 0.057 0.278 0.126
it was found possible to make measurements using a Reid vaporpressure bomb of the type used for gasoline inspections, and while admittedly the results are not precise, they are believed to be sufficiently accurate for these purposes. The data are presented in Table I11 and Figure 3. Activity coefficients were calculated for the two binary systems herein reported and are plotted in Figures 4 and 5. The curves plotted are the best representations that could be obtained for these data using the more simple of the various integrated forms of the Gibbs-Duhem equation as presented by Wohl(16). Using Wohl's notation, the curves for the methyl ethyl ketone-butyl Cellosolve binary are two-suffix van Laar equations with Az-3 = 0.58 and As-2 = 1.03; those for the butyl Cellosolvewater binary = are four-suffix Margules equations with AI-8 = 0.475, 2.22, and D = 0.558. The data for methyl ethyl ketone-water (6) were fitted with the two-suffix van Laar equations using AI-* = 1.82 and Az-1 = 3.68. These are not shown graphically here. While it is possible to estimate ternary vaporliquid equilibria from binary data alone (15), it was thought preferable t o obtain a t least several det e r m i n a t i o n s of
1.66 1.31 1.31 0.93 1.67 1.73 0.95 0.79
Activity Coefficients Ketone Exptl. Calcd. 0.65 2.26
Butyl Cellosolve Exptl. Calod.
0.99 1.01 1.02 1.00 0.76 0.88 1.01 1.03
0.99 0.74 0.78 0.84 1.09 1.36 0.78 0.92
1.11 0.91 1.36 1.41 0.87 1.50 1.29
1.03 1.42 0.94 1.36 1.63 1.30 1.17
TABLEV, SPECIFIC GRAVITYAND REFRACTIV~ INDEX FOR THE TERNARY SYSTEMMETHYL ETHYLKETONE-WATERBUTYLCELLOSOLVE
Cellosolve/Ketone, Mole Ratio
Mole % ' Ketone
3.995
20.02 15.76 11.78 7.67 41.91 35.62 25.30 4.49 1.76 60.77 47.19 35.73 20.15 4.73 74.02 58.02 40.16 24.59 7.48 96.25 72.96
1.386
0.645
0.351
0,0381
Refractive Index, 22" C.
Specific,, Gravity, das
1.4128 1.4128 1.4099 1.3990 1.4059 1,4052 1.4030 1 3838 1.3519 1.3983 1.3983 1,3970 1.3869 1.3582 1.3918 1,3921 1.3918 1.3939 1.3620 1.3803 1 3801
equilibrium compositions of the ternary system methyl ethyl ketone-butyl Cellosolve-water (Table IV). The best representation of these data found possible was given by the four-suffix Margules equation using the binary constants given above (with D = 0 for both water-methyl ethyl ketone and ketone-butyl Cellosolve) and three ternary constants, C1 = 10.1, Cp = 0.6, and C) = 14.4. A comparison of experimental activity coeffiMETHYL ETHYL KETONE
1
I I I 0.2 0.4 0.6 0.8 MOLE FRACTION B U T Y L CELLOSOLVE IN
0.81
1.0
0
5
LIQUID.
---REFR.
INDEX.
X * AZEOTROPE.
Figure 4. Activity Coefficients for the System Butyl Cellosolve-Water at 760 M m . Hg
3
% w
I
I
I
I BUTYL CELLOSOLVE
0 0.2 0.4 0.6 0.8 MOLE FRACTION M E T H Y L ETHYL K E T O N E IN L I W O
0.SS
10
Figure 5. Activity Coefficients for the System Methyl Ethyl KetoneButyl Cellosolve at 760 Mm. Hg
0.w
0.95
a91
0.9 3
WATER
Figure 6.
BUTYL CELLOSOLVE
The Ternary System Methyl Ethyl Ketone-Water-Butyl Cellosolve
Specific gravities dig; refractive indexes
nv; and liquid equilibria ( 2 5 O
C.)
2042
INDUSTRIAL AND ENGINEERING CHEMISTRY
cients and those calculated using this equation is presented in Table IV. For certain of the data the agreement is poor, but a good deal of the discrepancy ib due to difficulty in analysis of the ternary mixtures. I n the binary systems analysis by specific gravity or refractive index was simple. I n the ternary system both properties must be used, and as Figure 6 and Table V s h o w , a t low water concentrations these properties tend t o follow somewhat parallel c u r v e s , making a n a1y 6 is uncertain. Even small discrepancies 111 composition below mole fraction 0.1 cause Figure 7. Effect of Addition of large charWs in Butyl Cellosolve as a n Extractive the activity coDistillation Solvent on Vapor-Liquid efficient, on the Equilibria i n the System Methyl llhole the agreeEthjl Ketone-Water ment is good. With the help of the four-suffix bIargules equations for the ternary system, it was possible to calculate the effect of various percentages of solvent on the vapor-liquid equilibria for the methyl ethyl ketone-water solutions in the manner indicated in Figure 7. Somewhat larger percentages of solvent are required to remove the azeotropic effect than are customarily used but a separation is possible. Furthermore, although the butyl Cellosolve-water solutions which would result from using this solvent in an extractive distilhtion do form a binary azeotrope, the condensed azeotrope can be separated into two liquid phases of different composition. This can be accomplished by cooling the condensate to a temperature between the upper and lower critical solution temperature, thus making the process feasible. I n order to provide more complete data for the system, the ternary liquid equilibria a t 25 C. were determined by the usual means (11) and are shown in Table V I and Figure 6 .
Vol. 41, No. 9
TABLE VI. TERNARYLIQCID EQUILIBRIA IN THE SYSTEM ETHYL KETONE-BUTYL CELLOSOLVE-WATER AT 25 C. METHYL Solubility Curve, Weight % Ketone Butyl Cellosolre 90.0 74.2 71.0 55 2 43.1 38.36 25.6 24.4
0 6.02 7.33 12.4 11.1 9.55 1.60 0
Tie Lines Water-Rich Layer, W t . % Ketone-Rich Layer, Wt. Yo Butyl Butyl Ketone Water Cellosolve Ketone Water Cellosolve 28.7 29.5 33.7 29.9 25.7
67.3 66.0 59.2 65.1 72.4
4.0 4.5 7.1
5.0
l.Q
EQ.1 oF .1 49.7 56.5 74.4
29.7 31.7 38.1 31.4 18,s
11.2 12.2 12.2 12.1
5.8
Figure 9. The Ternary System Methyl Ethyl IletoneWater-1,1,2-Trichloroethane a t 25' C.
METHYL ETHYL KETONE
WATER
Figure 8.
TRiCHLOROETHY LENE
The Ternary System Methyl Ethyl KetoneWater-Trichloroethylene at 25' C.
Figure 10. The Ternary System Methyl Ethyl KetoneWater-Chlorobenzene at 25' C.
2043
INDUSTRIAL AND ENGINEERING CHEMISTRY
September 1949
S O L V m T EXTRACTION
TABLEVII. LIMITINGSOLUBILITY DATA FOR THE SYSTEMSMETHYLETHYL KETONE-WATER-CHLORINATED HYDROCARBONS AT 25 C . Ketone 6.84
18.01 29.13 43.81 58.90 75.40 90.0 24.40 22.00 15.04 8.08
Weight % Triohloroethylene 93.10 81.82 70.60 55.56 39.90 21.09
0
...
0 0
0.16 0.23 0.40 0.40
...
dz5 1.3747 1.2676 1.1739 1.0696 0,9763 0,8953 0.8305 0.9620 0.9665 0.9715 0.9870
.... ....
Ketone 14.65 29.97 44.38 58.38 71.80 83.30 90.0 24.40 20.07 15.11 8.06 2.83 0
Weight % Trichloroethane 84.82 69.20 54.14 39.05 23.91 9.30 0
0 0.10 0.14 0.25 0.37 0.44
d:' 1.2764 1.1535 1.0554 0.9749 0.9069 0,8531 0.8305 0.9620 0.9704 0.9774 0.9872 0.9441
....
Ketone 15.23 30.14 44.49 58.52 71.95 84.08 40.0 24.40 20.01 15.10 8.12 0
...
-
I n the search for suitable solvents for solvent extraction of the ketone from Weight % methyl ethyl ketone-water solutions, Chlorobenzene dz6 again the generalizations of Ewell, 84.46 1.0406 Harrison, and Berg (3') are of consider69.36 0.9903 able value. Solutions of the ketone 54.38 0.9446 39.34 0.9043 (class 111) and water (class I ) result 24.28 0.8681 9.28 0.8406 in positive deviations from Raoult's 0 0.8305 law. Solutions of ketone and class IV 0.9620 0 0.07 0.9688 solvents (certain chlorinated hydrocar0.06 0.9764 0.14 0.9853 bons) always show negative deviations 0.40 .... from Raoult's law. This combination ,.. .... of escaping tendencies should make the chlorinated hydrocarbons excellent extraction solvents ( l a ) provided they boil sufficiently higher than the ketone to make for good separation of the extract solutions. Ternary liquid equilibria were determined at 25' C. for two such class IV solvents, 1,1,2-trichloroethane and trichloroethylene, as well as chlorobenzene which has been shown useful for acetone extraction ( 7 ) . The usual procedures were followed ( l l ) , with specific gravity as a means of analysis. Data are presented in Tables VI1 and VIII, and Figures 8, 9, 10, and 11. In the triangular figures the crosses represent over-all compositions of mixtures used to determine tie lines. Inspection of the distribution curves of Figure 11 shows that all of the solvents are excellent for extracting the ketone from water solutions. Furthermore all of these solvents are sufficiently high-boiling and water-insoluble that separation of the extract solutions is simple ( 7 ) . To determine which of the processes would be the better, a complete economic study of the two would be required; cost data for such studies were not completely available. A limited study of the equilibrium data, however, indicates that for recovery of the ketone from dilute water solutions solvent extraction with one of the proposed solvents would probably be more favorable than extractive distillation with butyl Cellosolve. The information herein presented permits detailed comparison if adequate cost data are also available. LITERATURE CITED
NEIGHT PERCENT METHYL ETHYL KETONE IN WATER LAYER
Figure 11. Distribution of Methyl Ethyl Ketone between Water and Chlorinated Hydrocarbons at 25" C.
TABLEVIII. TIE-LINE DATAFOR 1-
THE SYSTEMSMETHYL ETHYL KETONE-WATER-CHLORINATED HYDROCARBONS AT 25" C.
Water-Rich PhaEie, Weight % Ketone Solvent dz5
Solvent-Rich Phase, Weight % Ketone Solvent di6
0.18 0.22 0.32 0.36 0.38
Trichloroethylene as Solvent 81.90 13.10 0.9687 25.31 71.65 0,9760 58.10 41.68 0.9831 0.9863 43.70 66.00 0.9886 22.10 77.72
0.8606 0,9120 0.9801 1.0710 1.2330
18.15 12.78 9.23 6.00 2.83 1.02
0.11 0.16 0.23 0.30 0.37 0.41
1,1,2-Triohloroethene as Solvent 75.00 19.92 0,9733 58.62 38.65 0,9804 44.38 54.14 0.9853 0.9902 31.20 67.80 82.58 16.90 0.9942 0,9966 5.58 94.42
0.8904 0.9720 1.0553 1.1425 1.2554 1.3626
18.10 13.10 9.90 7.65 5.52 3.64
0.05
20.71 16.10 11.03 8.50 6.75
0.08
0.12 0.16 0.21 0.28
Chlorobenzene 0,9726 0,9796 0,9841 0.9872 0.9902 0.9928
as Solvent 75.52 58.58 43.68 29.65 17.40 8.58
20.60 39.28 55.15 69.95 82.15 91.18
(1) Benedict, M., and Rubin, L. C., Trans. Am. Inst. Chem. Engrs., 41, 353 (1945). (2) Dworzak and Herrmann, Monatsh., 52, 83 (1929). ENG.CHEM.. (3) Ewell, R. H., Harrison, J. M., and Berg, L., IND. 36,871 (1944). (4) Meissner, H. P., and Stokes, C. A., Ibid., p. 816. (5) Othmer, D. F., Ibid.,20, 743 (1928) ; 35, 614 (1943). (6) Othmer, D. F., and Benenati, R. F., Ibid., 37,299 (1945). (7) Othmer, D. F., and Ratcliffe, R. L . , I b i d . , 35,798 (1943). (8) Pierotti, G. J., and Dunn, C. L., British Patent 563,164 (April 22, 1943). (9) Quaker Oats Company, Chicago, Ill., Furfural (1946). (10) Shell Chemical Company, San Francisco, Calif., Methyl Ethyl Ketone (1938). (11) Smith, J. C., IND. ENG.CHEM.,34,234 (1942). (12) Treybal, It. E., Ibid., 36,875 (1944). (13) Union Carbide and Carbon Corp., New York, N. Y., Cellosolve and Carbitol Solvents (1947).(14) Washburn, E. W., and Read, J. W., J. Am. Chem. S O ~ .41, , 729 (1919) *
(15) Wohl, K., Trans. Am. Inst. Chem. Engrs., 42, 215 (1946). RECEIVED September 2,1948.
0.8590 0.9041 0.9474 0.99221,0324 1.0649
h
t
A
