Nomenclature
B, C
D G KIcl KKi L, M
= constants of the Antoine vapor pressure equation = sum of fluid densities a t temperature T = Goldhammer factor =
correction factor for the Klein equation
= correction factor for the Kistiakowsky equation
molal latent heat of vaporization molecular weig!nt exponent of the Goldhammer equation n P vapor pressure R gas constant t = temperature T = absolute temperature u = specific volume Av = difference in specific volumes of saturated liquid and saturated vapor V = molal volume VL’ = molal volume of the saturated liquid a t some temperature T’ at or below the normal boiling point p = density p ~ ’ = liquid density corresponding to VL’ p = dipole moment = = = = =
(8) Goldhammer, D. A., Z.physik. Chem. 71, 577 (1910). (9) Griffin, D. N., J . Am. Chem. Sod. 71, 1432 (1949). (10) Haggenmacher, J. E., Znd. Eng. Chem. 40, 436 (1948). (11) Haggenmacher, J. E., J . Am. Chem. SOC. 68, 1633 (1946). (12) Hanson, E. S., Znd. Eng. Chem. 41, 96 (1949). (13) Herz, W., Neukirch, E., Z. physbk. Chem. 104, 433 (1923). (14) Hougen, 0. A,, Watson, K. M., Ragatz, R. A., “Chemical Process Principles-Part 11,” 2nd ed., pp. 574-83, Wiley, New York, 1959. (15) Keenan, J. H., Keyes, F. G., “Thermodynamic Properties of Steam,” Wiley, New York, 1956. (16) Kelley, K. K., U. S. Bur. MinesBull. 383,1935. (17) Kistiakowsky,W., Z.physik. Chem. 107,65 (1923). Chem. Eng. Progr. 45, 675 (1949). (18) Klein, V. .4., (19) Kobe, K. A., Lynn, R. E., Chem. Rev. 52, 117 (1953). (20) Lange, N. A.: ed., “Handbook of Chemistry,” 8th ed., Handbook Publ., Sandusky, Ohio, 1952. (21) Lu, B. C., Chem. Eng. 66,137 (May 4, 1959). (22) Lydersen, A. L., University of Wisconsin Eng. Expt. Sta. Rept. No. 3, 1955. (23) Lydersen, A. L., Greenkorn, R. A., Hougen, A . O., University of TYisconsin EnF. EXD.Sta. ReDt. No. 4. 1955. (24) Nadejdine, A,: Be;bl. Ann. Piysik. 7, 698 (1883). (25) Osborne, D. W., J . Am. Chem. Soc. 64, 169 (1942). (26) Perrv, J. H., ed., “Chemical Engineers’ Handbook,” 3rd ed., McGraw-Hill, New York, 1950. (27) Pitzer, K. S., Givinn, W.D., J . Am. Chem. Soc. 63,3313 (1941). (28) Riedel, L., Z . Elektrochem. 53, 222 (1949). (29) Riedel, L.. Chem.-Zngr.-Tech. 24, 353 (1952). (30) Zbid.: 26, 83, 259 (1954). (31) Rohm & Haas Co., Philadelphia, “The Methylamines.” (32) Rossini, F. D., Pitzer, K. S.,Arnett, R. L., Braun, R. M., Pimentel, G. C., “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” Carnegie Press, Pittsburgh, Pa., 1953. (33) Sama. D. A , , M.S. thesis in chem. eng., Mass. Inst. Technol., Cambridge, Mass., 1955. (34) Scott, D. LV., et al., J . Am. Chem. Soc. 74, 4666 (1952). (35) Sherwood, T. K., Reid, R. C., “The Properties of Gases and Liouids.” DD.8. 18. McGraw-Hill. New York. 1958. (36) Stuli, D. R.iind. Eng.’Chem. 39,157 (1947). (37) Thodos, G., A.Z.CI1.E. J . 1, 165, 168 (1955). (38) Zbid., 2, 508 (1956). (39) Thomson, G. W.. Chem. Reu. 38. 1 (1946). (40) \Vatson, K. M.. Znd. Eng. Chem.‘35,‘398 (1943). ~
Subscripts b = normal boiling point critical point
c
=
g
= saturated vapor = saturated liquid = reduced state-refers
L r 1 2
T,
to the ratio of the given property to the same property a t the critical point = lo\rer temperature level = higher temperature level = 0.85 = reduced temperature equal to 0.85
literature Cited
Colloid Chem. 52, 1060 (1948). (1) Benson. S. \V.. J . P h y . (2) Birch. F.. Phjs. Rez’.41, 641 (1932). (3) Fishtinr. S. H., Chen. Eng. 69, 154 (Sept. 3>1962).
(4) (5j (6) (7)
Gambill. \ V . R.. Zbid.. 64. 262 (Dec. 1957). Zbad.. 66, 181 (June 15, f959).‘ Zbad.. 66, 157 (Julv 13, 1959). Gates. D. S.. Thodos. G., A.Z.Ch.E. J . 6, 50 (March 1960).
RECEIVED for review May 21, 1962 ACCEPTEDDecember 28, 1962
COM M U N I C A T I ON
S E M I MICRODETER M I N A T I O N OF VAPOR- LlQUI D EQUI LI B R I U M A semimicronlethod for the determination of isothermal vapor-liquid equilibrium in multicomponent systems is described. Gas chromatography was used to analyze the phases. Only about 2 ml. of sample is needed to determine one experimental point. The proposed method is advantageous, especially for systematic choice of extraction agents for extractive distillation.
THE experimental
determination of vapor-liquid equilibrium is usually performed in circulating stills ( 3 ) which in the normal arrangement require about 30 to 200 ml. of liquid mixture to yield one experimental point. T h e proposed method is based on the sampling of very small volumes of the vapor phase which is analyzed by gas chromatography and makes possible the determination of equilibrium data on a considerably smaller amount of the liquid mixture (about 2 ml. for one value). ‘The quantity of substance in the sample is proportional to the peak area; therefore, it is
possible to calculate the relative volatility from chromatograms directly by comparing the ratios of peak areas of components in the liquid and vapor phases. I t is not even necessary to reach the equilibrium partial pressures because it is sufficient to measure the concentration ratio of components in order to determine the relative volatility. T h e small amount of the vapor phase sampled makes it possible to obtain a considerably greater number of analytical data while leaving the concentration in the liquid phase unchanged. Using the still described and a system with relative volatility 01 = 4, ten VOL. 2
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155
samplings of the vapor phase cause a concentration change in liquid phase of about 0.2%, which is within the accuracy range of gas chromatographic analysis. T h e liquid phase is analyzed by injecting into the gas chromatograph a sample containing approximately the same number of moles as that of the vapor phase.
is within the accuracy range of gas chromatographic analysis. A comparison of all available data for this s)-stem is shown in Figure 3, where the relative volatility of n-heptane is plotted against the mole fraction of n-heptane in the liquid phase. I t follows from Table I and Figure 3 that the proposed method is useful for rapid determination of isothermal vapor-
Experimental
The conic glass equilibrium vessel (Figure 1) is sealed to the sampling cock. T h e cock-plug is cut out in the middle (the cavity has a volume of about 0.5 ml.). During saturation, the cut is connected with the vessel. The sample of the vapor phase is taken by turning the cock-plug by 180' so that the vapor phase from the cut is transported by the carrier gas into the gas chromatograph for analysis. The temperature of the cock and vessel are maintained constant within 1 0 . 0 5 " C. The liquid phase is stirred by an electromagnetic stirrer, avoiding a contact of the liquid phase lvith the cockplug. The sampling cock is very carefully polished and is not lubricated. Electric heating is used to prevent condensation of vapor phase between the equilibrium vessel and gas chromatograph. T h e gas chromatograph [the model according to Grubner ( 2 ) with thermal conductivity detection] is kept a t a constant temperature about 1' C. above that of the equilibrium vessel with an accuracy of 10.1' C. Composition of the liquid phase is determined by weighing a vial while adding the individual pure components. The solution of known composition from the vial is poured into the equilibrium vessel. Both phases are analyzed 4 to 8 times.
Table 1.
Experimental and Calculated Values for the System n-Heptane-Toluene at 90" C.
-
X
%rp
0,068 0,193 0,278 0.462 0.616 0,721 0,846
1.86 1.67 1.57 1.39
1.27 1.23 1.15
%*lo
1.819 1,690 1.605 1.428 1,298 1.225 1.149
Yexp
Ycala
0.120 0,285 0,377 0.544 0,671 0,761 0.863
0.117 0.288 0.382 0.551 0.676 0.760 0.864 Mean
Yew Ycaic
4-0.003 -0,003 -0.005 -0.007 -0.005 +0.001 -0.001 f0.004
Results
The still was tested with the system n-heptane-toluene at 90" C. An example of chromatograms is presented in Figure 2. Only isobaric data ( 7 , 4-6) have been published for this system, and it was necessary to convert them into isothermal data. The data ( 5 ) were first converted into log which can be considered independent of temperature, especially in systems of components with near boiling points. Relative volatilities were calculated from the values log y,/y2 for authors' x and used to compute the composition of the vapor phase ycnle. These values were compared with values obtained experimentally. The experimental and calculated values are presented in Table I. T h e mean absolute deviation is 0.4 mole yo; the mean relative deviation is l.OYc. T h e experimental results are in good agreement with published data, and the mean deviation
-
Figure 2. mixture
A.
6.
I @ Y
*.-
h
d
Liquid phase Vapor phase I
I
e
d
P
time
Chromatograms of the n-heptane-toluene
v
I
e
data ( 1 )
3
data ( L )
c,
data
o
data
0
authors
(5) (6)
$ 0
1.4-
g
.-P Y d
t
I
-0 Figure 1. A. B.
156
,
,
5crn
Equilibrium vessel
Front view without cock-plug Complete view
l&EC FUNDAMENTALS
1.2-
1
1.00
0.2
0.4
0.6
0.8
1
liquid equilibrium. Its accuracy can be increased with help of a more accurate analyzer such as a mass spectrometer. Acknowledgment
T h e authors wish tc acknowledge the aid and cooperation of Otto Grubner in developing the chromatographic analyzer. and of the glassblow-er, Alexander eernq. foi his excellent technical assistance.
literature Cited
(1) Bromiley, E. C., Quiggle, D., Znd. En,.. Chem. 25, 1136 (1933). (2) Grubner, O., Inst. of Phys. Chemistry, Prague, private communication, 1961. (3) HBla, E., Pick, J., Fried, V., Vilim, O., “Vapour-Liquid Equilibrium,” 1st ed., p. 253, Pergamon Press, London, 1958. (4) Hipkin, H., Myers, H. S., Znd. Eng. Chem. 46, 2524 (1954). (5) Rose, A , Williams, E. T., Zbid., 47, 1528 (1955). (6) Steinhauser, H. H.. LVhite, R. H., Zbid.,41, 2912 (1949). IVAN W I C H T E R L E
EDUARD H . ~ L A
Nomenclature
Institute o j Physical Chemistry Czechoslovak Academy of Sciences Prague, Czechoslovakia
x = mole fraction in liquid 1: = mole fraction in vapor cy = relative volatility y = activity coefficient
RECEIVED for review January 9, 1963 ACCEPTEDFebruary 15, 1963
COM M UN I C A T ION
NATURAL CONVECTION T O I S O T H E R M A L FLAT PLATE W I T H A SPATIALLY N O N U N I F O R M ACCELERATION Natural convective heat transfer a t steady state to a flat plate at constant temperature, with either a laminar or turbulent boundary layer, modest temperature differences, and a parallel acceleration proportional to distance along the surface measured from the leading edge, is analyzed by the integral method and compared with the classical case of uniform acceleration.
ISTEGRAL METHOD has proved to be a most useful approach to the solution of problems in transport phenomena. I n natural convective heat transfer, the approach has been used successfully for a number of problems (3-5)>including the classical problem of the isothermal vertical flat plate a t steady state with modest temperature differences and a laminar boundary layer ( 7 ) or a turbulent boundary layer (2). HOWever, these cases are for a spatially uniform acceleration such as gravity. I n the present article, the problem of a n isothermal flat plate in a fluid environment of nonuniform acceleration is considered. For the particular problem considered here, the acceleration is parallel to the plate and proportional to distance along the plate, measured from the leading edge. 4 physical example of this situation might be the centrifugal acceleration acting on the boundary layer ad.jacent to a cool plate which is inside of. and which is oriented radially outward from. the center of rotation of a spinning satellite in orbit. [The difference between this type of ]physical system and that of a rotating plate (disk), in othencise stationary surroundings, is lcorth noting. The latter case involves interaction between a rotating element and otherwise stationary surroundinqs, \chile in the former (present) case, the entire surroundings rotate a t the same angular velocity as the element.] In this way, the leading edge of the plate is a t the center of rotation. Thus, the acceleration is represented by Equation 1.
THE
a = wzx
The acceleration is presumed not to be so extreme as to cause significant density differences directly. A warm plate so oriented in a spinning satellite \could constitute a different problem for t\co reasons. First, the leading edge would not be a t the center of rotation which represents zero acceleration, but would be a t the position of maximum acceleration instead. Thus, Equation 1 \could be changed to a = u2(L - x ) . Second, the accumulation of heated fluid near the center of rotation might well hinder convection. Any effect of Coriolis forces caused by the spin can be minimized for the present analysis by allowing the cool plate to face in the angular direction opposite to such forces. Any uncertaintl- in the near vicinity of the leading edge can be largely eliminated by allowing the plate to extend through the center of rotation. In this way: the leading “edge” \vi11 not be a n edge in the usual physical sense, but tcill simply be the center of rotation itself. Analysis
Laminar Boundary Layer. The momentum balance and heat balance equations for the integral method applied to a flat plate in laminar natural convection, respectively. are as follows ( 7 ) :
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