838
INDUSTRIAL AND ENGINEERING CHEMISTRY
penetration. This demonstrates that plasticity in these highly sheared greases cannot be attributed to the mechanical interference of the fibers with each other since such a pronounced change in length/diameter ratio would have required a comparable change in penetration. It is considerably more reasonable to relate the plasticity of these greases, containing such fibers as those shown in Figure 2C, to the ability of the fibers or particles to attract each other. It is possible that special conditions developed during this work to obscure the direct effects of a changing length/diameter ratio on the consistency of the samples, but it is definitely indicated that particle interaction plays a major role in the phenomenon of plasticity in greases, Whatever the exact nature of the interactive forces may be, there are other indications of their existence and importance, As Dean has suggested (6) syneresis or the contraction of a gel may be due to the forces operating between fibers. Likewise (‘setting up” or rebodying in a grease that has been sheared may be due to the re-establishment of many disrupted bonds. The different properties that have been produced in greases by varying the composition or chain length of the fatty acid portion of soaps (6, 7), may well be resolvable as quantitative differences in the interactive forces. Analogously, Chawalow (4)has shown that in sodium soaps the Van Der Waals forces acting between adjacent hydrocarbon chains and the forces prevailing in the ionic layers directly affect the crystal forms that the soaps assume. He has shown how variations in chain length produce variations in crystal form when formed under the same conditions. It is likely that more complete researches along these lines will result in sufficient understanding of grease gels so that it will be possible to make quantitative predictions of their physical properties from a fuller knowledge of the average soap fiber or particle. This knowledge, of course, would include the nature and strength of interparticle attraction and the size and geometry of the particle; it would, of necessity, presuppose a standard state of particle distribution such that variations in secondary structure and particle orientation would not distort the prediction. I n a strict sense the observations made here should be restricted to the MIL-G-3278 greases since the test work was performed only on this group, but is likely that they will apply as well to
Vol. 47,No. 4
many other classes. It is also recognized that there may be deviations in other classes of greases whose plastic nature differs sharply from that of the greases studied. CQNCLUSIONS
As a result of the test work performed, the following conclusions are made for the MIL-G-3278 greases and extended to other types: 1. Progressively larger amounts of shear or increasing shear rates applied to a grease do not necessarily result in a progressively lower consistency. The manner in which shear stress is applied must be considered as well as its magnitude and duration. 2. Mechanical breakdown of grease consistency could not be directly related to a lowering of the length/diameter ratio of soap fibers. Other factors, apparently, influence the breakdown of consistency. 3. A greater recognition must be given t o particle interaction as a basic factor contributing t o the plastic nature of greases. The role of mechanical interference of the particles to movement should be re-evaluated. ACKNOWLEDGMENT
The author wishes to express his appreciation for the assistance given him by E. R. Lamson who developed the ball bearing shear apparatus and who cooperated in evaluating the soap fiber dimensions. LITERATURE CITED
Bondi, A. A., Cravath, A. VI.,Moore, R. J., and Peterson, W. H.,Inst. Spokesman, 13,No. 12 (1950). Brown, J. A., Hudson, C. N., Loring, L. D., Ibid., 15, No. 11 (1952). Brbwning, G. V., Ibid.,14, No. 1, 10 (1950). Chawalow, M.L. E., J . Phys. Chem., 57,354-8 (1953). Dean, W. X., Inst. Spokesman, 16, No. 12 (1953). Dean, W. K., presented at annual meeting of National Lubricating Grease Institute, Chicago, Ill., Oct. 30, 1950. Aileyer, H. C., Inst. Spokesman, 16, No. 12 (1953). Moore, R. J., and Cravath, A. M., IND. EN^. CHEM.,43,No. 12 (1951). Vold, 14.J., Inst. Spokesman, 16, No. 8 (1952). Vold, R. D., Coffer, H. F., and Baker, R. F., Ibid., 15, No. 10 (1952). RECEIVEDfor review December 21, 1953.
ACCEPTEDDecember 3, 1964. The opinions expressed herein are those of the author and do not necessarily represent the views of the Naval Air Experimental Station or the Department of the Navy.
Adjustment of Vapor-Liquid Equilibrium Data D. B. BROUGHTON AND C. S. BREARLEY Universal Oil Products Co., 30 Algonquin Road, Des Plaines, I l l .
A
-
LTHOUGH methods are available for testing the thermodynamic consistency of equilibrium data, no methods have been proposed for refining data when they do not meet such tests. Since many of the systems reported in the literature fall in this class, such a method is highly desirable to permit maximum utilization of available data. A rational method of making such adjustments is based on an analysis of the systematic errors likely to be encountered in operation of the usual type of equilibrium still. T h e method has been applied successfully on a number of binary systems to bring into agreement the discrepant sets of data from different sources on the same system. One set of thermodynamically inconsistent ternary data, adjusted by the proposed technique, gave results in good agreement with consistent data on the component binaries as reported from other sources.
EQUILIBRIUM STILL CHARACTERISTICS
Most of the reported data have been obtained in recirculating stills of the types developed b y Othmer and Scatchard, with various modifications, Systematic errors may be introduced by: 1. Rectification, resulting from partial condensation of vapor on the still walls 2. Partial ‘Lflashing”of the condensate returped to the still, with failure of this flashed vaDor to come to equilibrium wlth the bulk of the still contents 3. Entrainment
Items 1 and 2 would tend to give an overhead vapor too rich in the more volatile component. It appears reasonable t o express this effect analytically by assuming that the still comprises the equivalent of s theoretical contacts a t total reflux. The cor-
INDUSTRIAL AND ENGINEERING CHEMISTRY
April 1955
rect and observed relative volatilities are then related to the still factor, s, by (Yc
=
ff,dl.d.
(1)
Items 1 and 2 would give values of s greater than unity. The quantitative effect of entrainment is closely represented by the same function, a t least up to 10 mole % of entrained liquid in the vapor, and leads to values of s below unity. The extent of the error introduced from these causes increases with increasing relative volatility of the components. At an aaeotropic composition, for example, none of these deficiencies would alter the vapor composition from its true value. If the still is operated in the same manner for each equilibrium measurement, it might be expected that the value of s would remain substantially constant, a t least for one system. Examples that follow show that this is true, a t least to the extent required for utility of the proposed method. Adjustment of the reported data then consists essentially of finding a value of s that gives thermodynamically consistent data. Errors in temperature and pressure have substantially no effect on the thermodynamic test or on the method of adjustment. I n fact, isobaric temperature variation can be calculated from the adjusted composition data. Analytical errors are not considered.
Thermodynamic Test. The Gibbs-Duhem equation is usually considered a criterion of thermodynamic consistency, although it is strictly applicable only a t constant temperature and pressure. Since the effect of pressure on activity coefficients is usually small, it is precise for isothermal data. It is also a good approximation for isobaric data ( 4 ) and is improved by the additional assumption that the product T log y is independent of temperature (8). By substituting T log y for log y in Dodge's form of the Gibbs-Duhem equation ( 4 ) ,it becomes
Define a function, r, as
r
= XI
T log
y1
Once a proper curve of T log Zzl versus xz has been established, actual activity coefficients can be calculated from the following relation, derived by integration of Equation 5 and substitution of values of T from Equation 3:
( T log ylLt
=E2".
+ x2T log
y2
log Zzldzz
- xz (2' log ZZI),,
(8)
An accurate curve of isobaric boiling point or of isothermal pressure versus composition can then be constructed if desired. An alternate method of testing data without extrapolation consists of using Equation 8 integrated between limits within the range of the data. However, evaluation of the left-hand side of the equation requires knowledge of the temperature and pressure a t the limits of integration. The accuracy of reported temperatures is in general not sufficient for this method to be useful. Temperature errors could result from fractionation or from superheating of the vapor space by the jacket heaters. Empirical Equations for T log 2 2 , versus x2. Use of Equation 6 for testing data requires extrapolation to x2 = 0 and xz = 1. If the data do not extend reasonably close to the pure component regions, a method of extrapolation is desirable. Let the correct or thermodynamically consistent value, ( T log ZZ,),, be represented by a power function in x2
( T log
B I S A R Y SYSTEMS
839
= a
+ bxz + cxz + ex; +
(9)
The constant a can be eliminated by applying Equation 6, from which
Use of only the first term is equivalent to the symmetrical forms of the Van Laar or Margules-type equations. Use of the first two terms is equivalent to the two-term Margules-type equation, of the first three to the three-term Margules, etc. If the second-degree equation is assumed adequate (which appears true in most cases), Equation 10 can be rewritten in terms of A12and Azl, where Alz is the terminal value of T log yLa t x1 = 0 and Azl the terminal value of T log y z a t x2 = 0:
(3)
Differentiating and substituting from Equation 2 dr = T log
y1
+
d z ~ T log
y2
dx2
(4)
Defining ZZI = y z / y ~and noting that dxl = -dxz, the equation 4 becomes
dr = T log Zzl dxz
Adjustment of Data. From Equations 1 and 7 , the following equation can be derived between the correct and observed values of T log ZZ1and the still factor, s:
(5)
Equation 3 indicates that r becomes zero when x2 = 0 or xz = 1. Equation 5 can thus be integrated between x 2 = 0 and x 2 = 1 to give
Data can then be tested by plotting T log ZZ1versus xzand noting whether the area represented by the integral of Equation 6 is zero. Values of ZZlare obtained from the equilibrium data by
Since the ratio of vapor pressures of the pure components, &, is usually insensitive to temperature, reasonably accurate values of ZZ1 can be calculated from composition data, with only a rough approximation of temperature. This is advantageous, since literature data on some systems report pressure only approximately and temperatures not a t all. However, the method does requii-e extrapolation of the curve of T log Zzl versus x 2 to x 2 = 0 andxz = 1.
This can be integrated against x 2 , assuming T log to give
&
constant,
METHOD1: Plot the data as (5" log Z2i)obsvd. versus x2, and draw the best line by sight through the points, extrapolating to x z = 0 and x2 = 1. Measure graphically the term
From Equation 13, calculate the value of s required to give thermodynamic consistency. METHOD2: If the data do not extend sufficiently close to x z = 0 and x 2 = 1 to permit reasonable visual extrapolation, a systematic method of extrapolation is desirable. If, for instance, the second-degree Margules equation is considered adequate, Equations 11 and 12 can be combined to give
Vol. 47, No. 4
INDUSTRIAL AND ENGINEERING CHEMISTRY
840
- Methanol
Water
140
3
120
100
-a
80
e,
2
60
%
2
40
"
20
N
-E
o
-
-
-
N
cI1
0
I-
I- -20 -40 -60 -80 -100
0
.I
P
A
.3
-5
.6
7
.8
0
1.0
.9
.I
.2
.4
.3
.5
=
.8
.9
f.0
Figure 2
Figure 1
1
.7
x2
x2
- ( T log m p 1 ) o b s v d . - T log
,6
A12(222 - 32;) f
Azi(1
- 422 + 32;)
(14) and xz,the con-
From this equation, using observed values of stants Alz, Azl, and s can be evaluated. The value of s cannot be determined for systems having an azeotrope near the middle of the composition range, since the net area under the curve of ( T log ZU), versus 52 then becomes insensitive to the assumed value of s.
Application to System Water Plus Methanol. In all the specific cases hereafter discussed, water is designated as component I, methanol is 2, and ethanol is 3. Data are given by Othmer, Friedland, and Scheibel(7) a t 760 mm. and by Cornell and Montonna ( 3 )a t unspecified pressures between 730 and 750 mm. Both sets are plotted on Figure l as ( T log Zz1)obsvd. versus 22. It can be seen that a considerable discrepancy exists.
the data. However, this discrepancy does not appear to have affected the calculated value of s significantly, as these values are identical with those obtained by method 1. To compare the data after adjustment, each point was corrected by Equation 12, using these values of S. I n Figure 2 the corrected values, ( T log Zzl),, are plotted against z2. Both sets of data now fall on substantially the same curve. The line shown gives a net area of zero. Systems Water Plus Ethanol and Methanol Plus Ethanol. In preparation for a study of the ternary system, water-ethanolmethanol, data on these two binaries were analyzed by method 2. The data on water-ethanol were taken from results of Carey and Lewis ( 2 ) and on methanol-ethanol from Hausbrand (6). The latter data appear to be of low precision but indicate substantial ideality. The constants obtained follow: Water-ethanol:
Methanol-ethanol:
METHOD1: Lines were drawn visually through each set of
data as shown in Figure 1. Extrapolation involves little uncertainty for the upper line but considerable for the lower. Deviation from thermodynamic consistency, as represented by the net areas under the curves, is minor for the upper line but very significant for the lower. From these areas and Equation 13, values of s of 0.99 for the Cornell and Montonna data and 0.92 for the Othmer data were
s = 0.999;
A13
=
114; A 3 1 = 247
s = 0.94; A D = 0; A32 = 0
TERNARY SYSTEMS
Define r analogously
Thermodynamic Test and Adjustment. to the definition for the binary T
= 51 T
log
YI
+
22
T log YZ
+
s
5 3
T log
~3
(15)
Differentiating and utilizing the Gibbs-Duhem relation with the definition of Z dr = T log 2 2 1 dxz
+ T log
231
dx3
(16)
Integrating of the constants obtained were: S
Cornell and Montonna Othmer
*
0.99 0.92
Aiz 128 118
Azi 67 57
T h e Margules equation actually indicates a greater flattening of the T log Z - 2 line at high values of xzthan appears justified by
ra
-
ra =
T log Zzl dxz
+ fT log
231
dx3
(17)
The values of ZZ1and can be obtained from the composition data, as in the binary case, from Equation 7 . If the data are plotted as T log ZZIversus xz with parameter lines of constant 2 3 and as T log versus z3with parameter lines of constant XZ,the
I N D U S T R I A L A N D ENGINEERING CHEMISTRY
April 1955
integrals of Equation 17 corresponding to a given change in composition can be evaluated as the areas under the path curves on these two plots. Plots of this type are shown in Figures 3 and 4 for the water-ethanol-methanol system. The correct and observed values of these integrals are related to the still factor, s, by similar equations as for the binary case:
REQUIREMENT 1: If the composition is varied around any cyclic path, ( A r ) , = 0, and from Equation 17
$-
i +f +
( T log Z Z I )d~~ z
( T log
2 3 1 ) ~dx3
=0
(19)
841
This can be done by a number of metfiods. One possibility is aa follows: Dram lines through the data on Figures 3 and 4 without reference to the criterion of equal enclosed areas. Equation 20 leads to the requirement that the correspondingly numbered areas on the two plots should be equal. Since the average width of these areas is the same, the average vertical distance between the corresponding parameter lines should also be the same. The most simple method of smoothing would then be to take one parameter line on each plot-preferably the ones best supported by the data-as basic. The heights of corresponding areas on both plots would then be averaged and these averaged values used t o reconstruct both smoothed plots. REQUIREMENT 2: From the definition of T , rc is zero for any pure component, and AT)^ must be zero along any path from one pure component to another. This leads to the relation that, across such a path,
However, substitution of Equation 18 in 19, noting that the last term of 18 becomes zero around a cyclic path, indicates that
4
( T log Z 2 l ) o b s v d . dzz
(T log Z 8 i ) o b s v d .
dzs = 0
(20)
This is true regardless of the value of s. I n other words, enclosed ’areas, corresponding to any given cyclical path, on Figures 3 and 4 should be equal, regardless of errors in the data resulting from fractionation or entrainment. If these areas are not equal, other types of error must be present. Since no method is available for dealing with errors not capable of being incorporated into the still factor, the raw data as plotted on Figures 3 and 4 must first be smoothed to satisfy Equation 20.
Values of s can then be obtained from Equation 21 by following paths between each of the pairs of pure components. The validity of using the still constant t o adjust the data would depend on the constancy of s as calculated from different paths. If the plots have been smoothed to give equal enclosed areas, any path between two given pure components will yield the same value for the sum of the integrals of Equation 21. Consequently,
I
275
2 50
225
225
200
200
I75
( T log Z 3 1 ) observed vs. x3
150
I 75 U
2
150
e
2
125
A
100
5
0
8
75
-
50
01
-
50
c
75
0
I-
CI,
-0
IO0
0
W
h
125
W
-0
25
v
25
0 -25
0
- 25
- 50
- 50 - 75
- 75 - 100
-100
-125 -150
-I 25
0
.I
-2
.3
A
.5
-6
.7
,E
.9
1.0
x2 Figure 3
Figure 4
INDUSTRIAL AND ENGINEERING CHEMISTRY
842
250
I50
,
I
200
'(3
140
I
0)
c
, Water
150
I30
Vol. 47,No. 4
- Methanol 1
1
1
-9
10 .
I
L
0
.0
I20
100
h
N"
I IO
100
-o
90
v
I-
"
50 0
-50 -100
80
I
1
0
.I
I .2
.3
.4
.5
-6
.7
.8
70
f
350 ,
60
01
I
300
2 5 0 I-
250
40
=
30
200
0)
c
2 L
20
0
0
IO
-
150 100
h
0 -10
Nm
50
-B
o
I-
-20
Y
-50 -100
-30
-150
-40 0
.I
.3
.2
.4
.6
.5
.8
.7
,9
10 .
x2
0
.I
.2
.3
9
.5
,6
.7
.E
.9
10 .
Figure 5
there are only three independent values of s to be obtained and compared. USE OF MARGULES-TYPE EQUATIONS
The previous method requires extrapolation of the data to at least two pure-component points for evaluation of s. As in the binary case, empirical equations can be used for this extrapolation. For example, Benedict and coworkers ( 1 ) give a ternary form of the two-term Margules equation] involving the constants for the component binaries and one additional ternary constant. A slightly modified form of this equation is
( T log r l j o = ~ z ( 12$(1
-
221)A13
2222$423
221)A12
+ 22123(1 -
- 2Zz23A32
+
- Z I ) A Z+~ -
3. 22122(1 Xl)A31
22223(1
- 221)A123
(22)
Equations for the other activity coefficients are obtained by rotating all subscripts, noting that A123 = A231 = A312. These equations can be combined to give
(T log Zn),
+ ( T log Z,,), =
(22122
- 22;)AIz
- 42122)Azl + - 2Z;)Als (2: + (2: f (2:
(22123
42123)A3~
(22223
1
-(T log s
+ Z;jAa2 + + T log
CY21
+
+
22223)&3
(22168
f 22122 -
CY3i)obsvd
42223)A123
- ( T log
=
+ Tlog
(23)
The latter equality is derived from the relation between the true and observed values of CY and s. Ternary data could be handled by inserting observed values of the variables in Equation 23 and solving for values of the constants and s. This is a more restricted method than the previous, since it assumes that the corrected data should fit the hlargules expression, Application t o System Water-Methanol-Ethanol. Ternary equilibrium composition data a t 740 to 748 mm. and ternary boiling points a t 760 mm. have been reported by Griswold and Dinwiddie ( 6 ) . No binary equilibrium data are reported. Activity coefficients for all components have been calculated. On Figure 5, T log y1 is plotted versus z1with parameter lines of constant X * / ( X ~ x8). The lines a t parameters of zero and one have been drawn on this plot from the thermodynamically consistent data on the binaries obtained from previously mentioned sources. For simplicity, the data points are designated as falling in certain ranges of parameter values, instead of writing actual values on the graph. Analogous plots for the other components are not presented here. Data on many other ternary systems have indicated that, when all the component binaries show positive deviations from Raoult's law as is true here, the ternary data plotted as in Figure 5 fall entirely between the parameter lines of zero and one for the two components of extreme polarity-i.e., water and ethanol. Only the intermediate component has activity coefficients falling below the lower terminal parameter line. I n other words, it is strongly indicated that the data points shown on Figure 5 should all fall between the two lines drawn. The discrepancy cannot be attributed to systematic errors in
+
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
April 1955
temperature, since the data points on the analogous graphs for the other two components fall, in many cases, above the values anticipated from binary data. PRELIMINARY TESTFOR FRACTIONATION: Water is the component of lowest volatility, particularly so at the high water end of the system. Consequently, the low activity coefficients of water might have been caused by fractionation. As a preliminary test, it was assumed that the correct values of T log y1 for all points at z1 > 0.60 could be estimated by linear interpolation between the parameter lines of zero and one on Figure 5. At lower values of xl, uncertainties in the interpolation method invalidate these estimates. For each data point at 21 > 0.6,the value of s was calculated that would be necessary to move the point to the estimated true location. XI
S
0.593 0.610 0.637 0.653 0.670 0.698 0.765 0.786
1.23 1.21 1.30 1.20 1.23 1.23 1.26 1.29
X1
1.20 1.20 1.23 1.20 1.26 1.21 1.23
The relative constancy of these values of s indicates that fractionation might be responsible for the deviations. APPLICATION OF EQUATIONS. It was assumed that Equation 23 could be applied to this system. The binary constants were taken from consistent data from previous sources as Alz = 66.5, Azl = 127.6, A13 = 113.8, 431 = 246.9, A23 = A32 = 0. Observed data points were then inserted in Equation 23, and the constants s and A123 obtained by the method of least squares. A weighting factor of (4/y1 l / y ~ 1/y3 4/21 1/x2 1/23)-l was used to take account of analytical precision. Calculated values of the constants were s = 1.25 and A123 = 106.3. The reasonable agreement between this value of s and that obtained in the preliminary test lends additional support t o the assumption that errors in the data are due to fractionation a t a relatively constant value of s. THERMODYNAMIC ANALYSIS AND ADJUSTMENT.Neither Of the foregoing methods can be considered conclusive, since the first assumes that y1 must fall between its values in the two binary systems and the second assumes applicability of the empirical Margules-type equations. A more general test is provided by thermodynamic analysis. The ternary data were plotted as (2’ log Z2l)obsvd. versus 2 2 and (Tlog .&)obsvd. versus 23. Parameter lines of constant 2 3 and 22 were drawn and smoothed to fulfill the requirement of Equation 20. The smoothed graphs, extrapolated to all boundaries, are shown as Figures 3 and 4. For simplicity, the data points are shown classified by ranges of parameter values rather than identified by actual parameters. The correspondingly numbered areas on the two figures are substantially equal. Values of s were then calculated from Equation 21 along various paths from one pure component to another:
+
+
+
+
+
.
Path 21 21 z2
= 1 to 22 = 1 = 1 to 2 3 = 1 = 1 to 2 3 = 1
methanol-water and ethanol-water, as derived from extrapolation of the ternary data. These are reproduced in Figures 6 and 7 for comparison with consistent data on the binary systems as obtained from other sources. The discrepancies are large. Lines representing the binary data derived from the ternary system, corrected by a still factor of 1.24, are also shown in Figures 6 and 7. Agreement with the binary data is much improved. CONCLUSION
The methods described allow evaluation of the thermodynamic consistency of binary and ternary vapor-liquid equilibrium data and provide a rational means of adjusting inconsistent data, NOMENCLATURE
8
0.806 0.833 0.833 0.874 0.906 0.917 Av.
S
1.24 1.21 1.28 Av. 1 . 2 4
Any path between given pure-component terminals yields the same value of s. The average value of s is in excellent agreement with those values obtained by the previous methods. COMPARISON OF ADJUSTEDTERNARY DATAWITH BINARIES. The upper lines on Figures 3 and 4 represent the smoothed curves of T log 2 versus 2, not adjusted for fractionation, for the binaries
a43
A,, Po
= =
r
=
terminal value of T log yrn at 2%= 1-0 vapor pressure of pure component a t observed temperature Z(2Tlog y) still factor, defined by log s =
*
S
=
T
=
X znln
= = =
amn
=
relative volatility =
a:,
=
P,yP::
y
=
T
=
activity coefficient in liquid phase = “y P Ox total pressure of system
y
temperature, O K. mole fraction in liquid phase mole fraction in vapor phase
CYC
Ym,/Yn
XmY
n
Subscripts = corrected for thermodynamic consistency 1, 2, 3 = individual components (in specific case discussed, water, methanol, and ethanol, respectively)
C
LITERATURE CITED
(1) Benedict, M., Johnson, C. A., Solomon, E., and Rubin, L. C., Trans. Am. I n s t . Chem. Engrs., 41, 371 (1945). (2) Carey, J. S., and Lewis, W.K., IND.ENG.CHEM.,24, 882 (1932). (3) Cornell, L. W., and Montonna, R. E., Ibid., 25, 1331 (1933). (4) Dodge, B. F., “Chemical Engineering Thermodynamics,” p. 558,
McGraw-Hill Book Co., New York, 1944. ( 5 ) Griswold, J., and Dinwiddie, J. A., IND. ENG.CHEM.,34, 1188 ENG.CHEM.,ANAL.ED., 14, 299 (1942). (1942); IND. (6) Hausbrand, E., “Principles and Practice of Industrial Distillation” (translated by E. H. Tripp), 4th ed., p. 168,Chapman & Hall, London, 1925. (7) Othmer. D. F.. Friedland. D.. and Scheibel. E. G.. reoorted in “Distillation Equilibrium Data,” by J. C. Chu, p. Ik6, Reinhold, Kew York, 1950. (8) Chemical Engineers Handbook (J. H. Perry, editor), 3rd ed., p. 531, 1LlcGraw-Hill Book Co., New York, 1950. RECEIVED for review July 6, 1954.
ACCEPTED November 13, 1954.
