Nomenclature
Davis, A. H., Phil. Mag. 47, 972, 1057 (1924). Elata, C., Lehrer, J., Kahanovitz, A,, Israel J. Technol. 4,
D = wire diameter, cm. De = Deborah number, dimensionless
Ernst, W. D., L. T. S. Research Center, Rept. 0-7100/6R-14
g
87 (1966). (1965).
= gravity acceleration, cm./sec.?
h = heat transfer coefficient, watts/sq. cm., H = Pitot tube readinl,,7 cm. I = intensity, amperes
k = conductivity, watts/cm., L = wire length, cm. R, = wire resistance, ohms
OK.
OK.
T, = wire temperature, OK. Tf = fluid temperature, OK. U = velocity, cm./sec. CY
0 Y
p,
= heat diffusivity, sq. cm./sec. = natural time, sec. = kinematic viscosit,y, sq. cm./sec. = wire resistivity, ohm cm.
literature Cited
Acosta, A. J., James, D. F., unpublished report to ONR contractors meeting, 1966. Alfani, F., chemical engineering thesis, University of Naples, 1967.
Astarita, G., Can. J. Chem. Eng. 44, 59 (1966). Astarita, G., IND.ENG.CHEM.FUNDAMENTALS 6, 257 (1967). Astarita, G., Metzner, A . B., Rend. Accad. Lincei VII-46, 74
Gupta, M. K., Metzner, A. B., Hartnett, J. P., Intern. J . Heal Mass Transfer, 10, 1211 (1967). Leathrum, R. A., Report for Ch.E. 342 Laboratory, University of Delaware, 1966. Lindgren, E. R., Chao, J. L., Phys. Fluids 10, 667 (1967). Maerker, J. M., I3.Ch.E. thesis, University of Delaware, 1967. Marrucci, G., Astarita, G., IND.ENG.CHEM.FUNDAMENTALS 6, 470.(1967).
Marrucci, G., Astarita, G., Rend. Accad. Lincei VIII-41, 355 (1966).
Metzner, A. B., personal communication, 1968. Metzner, A. B., Astarita, G., A.I.Ch.E. J . 13, 550 (1967). Meyer, W. A., A.I.Ch.E. J . 12, 522 (1966). Oliver, R. D., Can. J . Chem. Eng. 44, 100 (1966). Piret, E. L., James, W., Stacy, M., Znd. Eng. Chem. 39, 1088 (1947).
Savins, J. C., A.Z.Ch.E. J . 11, 673 (1965). Serth, R. W., Kieser, K. M., A.Z.Ch.E. J., in press, 1968. Sestak, J., Appl. Sci. Res. 17, 650 (1967). Shaver, R. D., Merrill, E. W., A.Z.Ch.E. J. 6, 189 (1959). Smith, K. A., Merrill, E. W. Mickley, H. S., Virk, P. S., Chem. Eng. Sci. 22, 619 (1967). Uebler, A. F., Ph.D. thesis, University of Delaware, 1966. Vurachi, P., chemical engineering thesis, University of Naples, 1967.
White, D. A., J . FIuid Mech. 28, 195 (1967). RECEIVED for review May 27, 1968 ACCEPTED February 3, 1969
(1966).
Astarita, G . , Nicodemo, Id., A.Z.Ch.E. J. 12, 478 (1966).
E X P E R I M E N T A L TECHNIQUE
SEMIMICROMETHOD FOR DETERMINATION OF PARTIAL PRESSURES OF SOLUTIONS IVAN WICHTERLE AND LUDMILA B O U B L ~ K O V ~ Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, Praha-Suchdol, Czechoslovakia
A semimicromethod has been developed for the determination of partial pressures of nonelectrolyte solutions. By suitably constructing the equilibrium still and standardizing the conditions for taking samples of the vapor phase, sufficiently good reproducibility of the results was attained. The function of the instrument was verified b y measurements with pure substances: benzene, hexane, and toluene. The x-y-P dependence for the system hexane-toluene at 70' C. was also measured. macroscopic experimental techniques (HBla et al., 1967) for obtaining: vapor-liquid equilibrium data for multicomponent real systems have several disadvantages. One is that a considerable amount (up to 200 ml.) of the liquid phase is needed to obtain one experimental point, which means that for the experimental determination of the entire equilibrium curve large amounts of pure substances are required. To procure the original substances in such amounts, and purify them to the desired degree, is usually a rather difficult and by no means negligible part of the work. A further disadvantage associated with the large size of the sample is that a rather long time (from 20 minutes u p to several hours) is required for establishing equilibrium be tween the liquid and gaseous phases. For systems containing more than two components, the rapidity with which the composition of the equilibrium phases can be determined with sufficient accuracy can also control the overall rate of the experimental work. NOWN
The static semimicrostill (Wichterle and HBla, 1963) is free of these disadvantages and in combination with a gas chromatograph it is suitable for rapid determination of the concentration dependence of the ratio of activity coefficients or the relative volatility. Although the accuracy of the results is less than that obtainable with classical methods, it is sufficient for engineering purposes. I n the present work we describe a modified semimicrostill which is suitable for direct measurement of the dependence of partial pressures on concentration. The rapidity of measurement is due to the fact that it is not necessary to wait for phase equilibrium to be established, because the ratio of concentrations of the two components in the gaseous phase does not vary with time. This assumption, that rates of vaporization of components (which do not differ sizably) are approximately equal, was experimentally verified by analyzing the vapor phase at different times after injection of the liquid phase. The time needed for determination of one VOL. 8
NO. 3 A U G U S T 1 9 6 9
585
experimental point is therefore practically limited by the time needed for the gas chromatographic analysis; since the sample can be analyzed several times, this makes it possible to increase the accuracy of the determination. The consumption of liquids is about 3 ml. per experimental point, so that, for example, for determining vapor-liquid equilibria over the entire range of concentrations for a ternary system, about 50 ml. of each component is sufficient. Experimental
The most important component of the modified all-glass equilibrium still is the eight-way stopcock, whose function is shown schematically in Figure 1. A sample of the vapor phase which is in the front bore of the plug can be transferred by a stream of carrier gas into the gas chromatograph, where it is analyzed. The carrier gas enters in pofition 1, it is preheated, and when the sampling space, 8, is saturated (position of the plug as A in Figure 1) it passes directly into the gas chromatograph via tube 2. The feed position of the stopcock is shown as B in Figure 1. The inlet tube, 3, is open only when the equilibrium vessel, 6, is being cleaned or when it is filled with fresh liquid (the filling neck is closed by stopper 4). The vessel is always filled under a stream of the carrier gas in order to eliminate the presence of air in the vapor phase. The electromagnetic mixer, 7, stirs the contents of the equilibrium vessel. Both the stopcock and the equilibrium vessel are enclosed in a jacket which can be thermostated. The temperature is measured in the well, 5. The carrier gas (electrolytic hydrogen, flow rate 50 ml. per second) is bubbled through an alkaline pyrogallol solution, then through concentrated HzS04, and is dried by anhydrous Mg(C104)Z. The gas chromatograph with thermal conductivity detection, developed by Grubner (1967), is connected to an integrator which directly records the integral curves. The chromatographic column is 90 cm. long and 4 mm. in diameter, and is filled with 5% of Apiezon L coated on earthenware; its temperature is maintained at 85’ C. When all measurements are carried out under the same conditions, it is not necessary to wait for equilibrium to be established. In preliminary experiments it was found, however, that the size of the vapor phase is influenced by numerous fac-
tors. The first condition for obtaining reproducible results is the perfectly stabilized operation of the chromatograph-Le., constant flow rate of the carrier gas, constant temperature, and constant heating current in the detector cells. The amount of liquid in the equilibrium vessel must also be chosen carefully: When it is too large, the liquid may climb u p the vessel wall to the stopcock, and when it is too small, it may be splashed by the agitator. We therefore tested liquid samples from 1 to 7 ml. and found that 3-ml. samples are the most satisfactory. The effect of the time of saturation of the feed space by the vapor phase was tested by varying it from 3 to 30 minutes without finding any significant influence. We therefore chose a saturation time of 5 minutes, which for the substances under consideration is also sufficient for completing the analyses of the preceding sample and obtaining the chromatographic record. The effect of the initial period-i.e., the time between the injection of the sample and the first analysis-was also investigated by varying it over the interval 1 to 30 minutes. This factor is not significant, and in agreement with the preceding consideration, we chose an interval of 5 minutes. The effect of the time of transfer of the sample into the chromatograph on the perfect flushing of the feed space was investigated for times from 1 to 30 seconds. For short transfer times the sample is too small, and for long transfer times “tailing” appears on the chromatogram. A period of 5 seconds was chosen for the transfer of the vapor sample. All these preliminary investigations were carried out with benzene. Materials
Benzene was prepared from an analytical grade product (Lachema) by extracting it with concentrated sulfuric acid and redistilling it on a 40 T P column. Toluene was prepared from an analytical grade product (Lachema) by distilling it twice on a 40 T P column. Hexane was prepared by the Wurtz reaction from propyl bromide and sodium; the reaction product was distilled on a 40 T P column. Physical properties are [corresponding data according to Dreisbach (1955, 1959) are given in parenthe.ses] :
I
I
I
I
I
log h
- 2.2 2.1
Q
III
2.0
I
I I
I
19
1.8 A
B
1.7
9 , , , . 5 crn. I .6 Figure 1. 1.
2. 3.
4. 5.
6.
7.
a.
A. B. 506
Schematic view of equilibrium still
Inlet of carrier gas Outlet to gas chromatograph Inlet of carrier gas during filling Filling neck with stopper Thermometer well Equilibrium vessel Stirrer Sampling space Position of plug during saturation Position of plug during sampling of vapor phase
I&EC FUNDAMENTALS
1.5
Figure 2. Dependence of height h of integral curve on temperature (amount of benzene)
---
Dependence of benzene vapor pressure, P, on temperature
Benzene d$' = 0.87943 (0.87901), n B 1.5010 (1.5011) Hexane d P = 0.65953 (0.65937), n z 1.3748 (1.37486) Toluene d P = 0.86660 (0.86694), n 2 1.4963 (1.4969)
y = Pl/(Pl
All substances were tested chromatographically; no impurities were detected. Results
Pure Components. Under the standardized conditions we first measured the dependence of the height of the integral curve on the size of the vapor sample. Since at a given temperature the mass of the sample is proportional to the vapor pressure, the form of the temperature dependence of the height of the integral curve must be the same as the temperature dependence of the vapor pressure-Le.,
log h = A
- B / ( t + 230)
Table 1.
Vapor-liquid Equilibrium Data of Hexane-Toluene System at 70' C.
x1
Yl
P
0.000 0.088 0.150 0.210 0.266 0,370 0.457 0.550 0.622 0.810 1,000
0.000 0,277 0,403 0,523 0.604 0,686 0.752 0,808 0,841 0.920 1,000
203.9 270 311 348 394 45 1 504 585 627 733 791.4
The experimentally determined equilibrium composition and pressures are summarized in Table I, which also gives the calculated values and the deviations in the determination of the vapor phase composition which was correlated by a Margules equation of the third order with constants A H = 0,166 and A21 = 0.063. The dependence of the partial pressures on the composition is plotted in Figure 3. For this system only isobaric data for 760 mm. of Hg have been published (Sieg, 1950)
(11
where h is the height of the integral curve, t is temperature, and A and B are constants. These dependences were measured over the temperature range 20' to 85' C. for all the pure components used, and for each temperature 10 to 12 analyses were performed. As an example, the calibration curve of benzene is plotted in Figure 2, in which the temperature dependence of the vapor pressure is also plotted for comparison. The constants of Equation 1 were evaluated by the method of least squares and the calculated and experimental values were compared: The height of the integral curve was determined with a mean error of 1.5 mm. (maximum height 280 mm.), which corresponds to a mean deviation of 2%. Hexane-Toluene System. Samples of the liquid phase were prepared by successively weighing out the pure components ; the samples were transferred into the equilibrium vessel by a syringe. Samples of the vapor phase were taken in the standardized manner described above. By calculation it could be estimated that for a mixture of relative volatility cy = 5, the removal of 10 vapor phase samples will change the composition of the liquid phase by 0.3 mole %, and that it is therefore not necessary to make a new analysis of the liquid phase on the completion of the experiment. The measurements for this system were carried out under isothermal conditions a t 70.0' C. The partial pressure of the component can be calculated in the following manner. I n the binary system, to each height of the integral curve there corresponds according to the calibration curve (Figure 2) a certain temperature for which the corresponding pressure of the pure component can be found. This pressure equals the partial pressure, PI,of component i in the system investigated a t the temperature of measurement (in our case 70' '2.). Assuming ideal behavior of the vapor phase we can then calculate its composition, y, from the relation
y l calcd 0.000 0.306 0.433 0,523 0.590 0.688 0.754 0.814 0.854 0.940 1.000
Mean
700
-
600
-
500
-
U 0
-
0
.2
.L
.3
.5
.6
.7
.8
1.0
.9
Figure 3. P-x diagram of hexane-toluene system a t
1.t
I
I
I
I
0
SlEG (1950)
0
AUTHORS
70" C.
Y
.I
s .4
.2
0.000
-0,014 0.002 0.002 0.006 0,013 0.020 0.000 0.013
.1
mole fraction of hexane
AY 1 0.000
0.026 0.030
+ Pz)
I
0
Figure 4.
70" C.
I
.2
I
I
I
.6 .8 x.... mole fraction of hexane .L
1 I 1.0
x-y diagram of hexane-toluene system a t
VOL. 6
NO. 3
AUGUST 1969
587
and it was therefore necessary to recalculate them a t least approximately to isothermal data. Assuming that the dependence of the ratio of activity coefficients on the composition does not vary much with temperature, we can calculate from this dependence the relative volatilities and consequently also the composition of the vapor phase a t the desired temperature. The dependence of the ratio of activity coefficients on the composition was correlated on a computer by using statistical weights (Wichterle, 1966). The experimental and calculated results are plotted as an x-y relationship in Figure 4.
the dependence of the height of the integral curve on the mass of the vapor sample. The results obtained were utilized for determination of vapor-liquid equilibrium for the system hexane-toluene. The equilibrium data were correlated by a third-order Margules equation, the mean deviation in the composition of the vapor phase being 1.3 mole %. literature Cited
Dreisbach, R. R., Aduan. Chem. Ser., No. 15, 11 (1955). Dreisbach, R. R., Advan. Chem. Ser., No. 22, 19 (1959). Grubner, O., Czech. Patent 121,672 (1967). Hbla, E., Pick, J., Fried, V., Villm, O., “Vapour-Liquid Equilibrium,” Pergamon Press, Oxford, 1967. Sieg, L., Chem.-Zng. Tech. 22, 322 (1950). Wichterle, I., Collection Czech. Chem. Commun. 31, 3821 (1966). 2, 155 Wichterle, I., Hbla, E., IND.ENC.CHEM.FUNDAMENTALS (1963).
Discussion
The semimicromethod for determining the variables which characterize vapor-liquid equilibrium is rapid and in combination with gas chromatographic analysis it gives good results. By standardizing the conditions for sampling the vapor phase, the analyses were made reproducible to 2.8%. I n the present work we measured over the range of interest
RECEIVED for review January 18, 1968 ACCEPTEDNovember 20, 1968
COMMUNICAT IONS PREDICTION OF POLYSTYRENE MELT TENSILE BEHAVIOR
-
A reasonably accurate prediction of tensile viscosity from dynamic and simple shearing viscosities is made for a polystyrene melt using a llnetwork rupture’’ theory of rheological behavior.
RELATIVELYfew measurements have been made of the
extensional (or Trouton) viscosity, p t , of polymer melts and solutions, but it has been found (Ballman, 1965) that pt is often much greater than ps, the shear viscosity, at equal strain rates. The extensional viscosity is defined for a circular rodlike specimen as the tensile stress divided by the rate of extension GI = au/az, where z is measured along the rod center line. The velocity field is assumed to be v = (Gz, -Gy/2, -Gz/2), the material being incompressible. It is readily shown that for a Newtonian fluid (Trouton, 1906) (1) Figure 1 (circles) shows experimental measurements of pt and pe for a polystyrene melt described previously (Ballman, 1965). The simple shearing rate (du/dy) is denoted by y. The high value of pt/pe when compared at y = G is obvious and here we attempt to predict p t ( G ) given p 8 ( y ) and the dynamic small-strain viscosity, Pd ( w ) . The dynamic viscosity, Pd, is a function of the frequency, w (rad./sec.), of the applied small sinusoidal shear. Suppose that the velocity vector v = (ydeht, 0, 0 ) in a small strain motion, and that the z - y shear stress component is written ?eiot where i is a complex number. Pd is defined as the real part of the complex viscosity, p* ( w ) , where Pt = 3P8
.3 = p*ci (2 1 For a Newtonian fluid ps = Pd = p*. Pd for the polystyrene sample was measured on a Weissenberg rheogoniometer and is shown in Figure 1 (circles). T o predict p t we use a “network-rupture” type of constitutive relation (Tanner and Simmons, 1967) which is a simple form of BKZ fluid. This gives the well-known expression for Fd in terms of n discrete relaxation times, A,: an(1
Pd(w) = n
588
I&EC
+
FUNDAMENTALS
(3 1
10’
ioe
-
\
.-0
a 10’
:10‘ 0 0
> IO5
-
’X
EXPT.
-X- FITTED CURVES --- PREDICTED, B = 1 2
FREQUENCY OR
STRAIN RATE
\
kw \
w(rad/sec.) y,G (seCi)
Figure 1. Measured and predicted properties of polystyrene melt
where a, are constants with the dimensions of viscosity. Choosing X,+1= A,/ -v%,an iterative numerical procedure with smoothing (Tanner, 1968) was used to assign the a,. The longest relaxation time was taken to be 505 seconds and 13 terms were used in Equation 3. The crosses in Figure 1 are points recalculated with the fitting values of G. Choosing a value B of the rupture strain parameter the shear viscosity,
