EXPERIMENTAL TECHNIQUE
VISCOMETER FOR MEASUREMENT
OF
VISCOSITIES IN THE RANGE lo8T O 10" POISES, USING 0.1 ML. OF SUBSTANCE W.
H . COGILL, JOSEPH
D U N L O P , A N D D. N . R O Y A L
Untuerrzty of Ncw South Wales, Kenrm@m, Ncw South Wales, Awfrolra
The viscometer is c a p a b l e of measuring the viscosities of small approximately cylindrical samples.
The
accuracy achieved is +20%. HE viscosities of weathered bitumens frequently exceed 108 Tpoises, and their measurement by means of apparatus such as the Couette viscometer is time-consuming. I t is difficult to obtain a true sample, as traces of recovery agents such as xylol remain after extraction by the normal methods. Following
Kiihn and Rigden (2) small quantities were scraped directly from weathered bituminous concrete aggregate and rolled approximately to cylinders. Kiihn and Rigden's apparatus was adapted for use with an interferometer. Bearings for the plunger were dispensed with, as Teflon bearings resisted a static thrust on the plunger of 200 grams, and steel shims were employed (Figure 1). These had sufficient lateral stability to allow 5-kg. weights to be used on the platform. The cord connecting the (optional) counterbalance weight for the plunger introduces some hysteresis to the system, although this was not measurable by the methods employed. The Heavens (7) spring lever was used to adjust the interferometer plates. Plots of deformation against time indicated that temperature differentials of 10' C. between the center and outside of the cylinder were resolved in these highly temperature-sensitive substances within 2 minutes. Using a cylinder of substance 0.16 cm. high and 500 grams on the platform, the viscosity of a substance having a value of 1.8 X loQpoises could be determined within 20% 5 minutes after commencing readings. Literature Cited
Figure 1.
Viscometer
A. E.
Steel shims. C. Looding plotform. D . Interferometer plater. Spring lever for adjusting lower interferometer plote. F. Specimen Idummyl. G. Counterbalance.
(1) Heavens, 0. S., J . Sci. Imtr. 27, 172 (1950). (2) Kiihn, S. H., Rigden, P. J., Proc. A r m . Asphalt Paning Technol. 28, 372-84 (1959).
R e c ~ r v ~for o review January 14, 1966 ACCEPTED May 12, 1966
SECOND VlRlAL COEFFICIENTS OF ORGANIC VAPORS FROM D ENSlTY BALANCE MEASU REMENTS S.
F. D I 2 1 0 , M . M . A B B O T + , '
D A N I E L ZIBELLO,* AND H. C. VAN NESS
Chemical E@mring Deparrmenf, Renmloer Polyrechnic Imtitutc, Troy, N . Y . HE
virial equation of state has become the standard form to
pressures the series may be truncated to two terms,
(1)
experimental compressibility data to Equation 1. From this equation one obtains the total differential of E :
'
Present address, Essd Research and Engineering, Florham Park, N. . I . Present address, Enjay Chemical Co., NEWYork, N. Y.
...
Z=l+Bp+
I - 22 ~~
)
- + -dP- -
dp
P( " B = z -( 1 .L)[ VOL. 5
P
"I
T
NO. 4 N O V E M B E R 1 9 6 6
569
A vapor density balance may be operated to allow the determination of vapor densities, and hence of second virial coefficients, without reference to the properties of any other material. The only measurements necessary are of temperature, pressure, and mass. Sample data taken with a prototype balance are presented by way of illustration.
I n the lowdensity or low-pressure region, say for Z between 0.95 and 0.99, the factor Z / ( Z - 1) varies from about 20 to 100. Thus determination of B to an accuracy of 2 or 3y0 from low-pressure data requires knowledge of the variables p , P, and T to within several hundredths of 1%. This is not too difficult to achieve for P and T above pressures of 250 mm. of H g and temperatures of 300’ K., but very accurate density determinations are difficult to accomplish in the low-density range. Consider an apparatus consisting of a very precise analytical balance enclosed in a vacuum-tight chamber immersed in a thermostated bath. The chamber is equipped so that it may be evacuated and then filled to a given pressure with the vapor to be studied. A float of mass, m f , is suspended from one end of the balance beam, and a counterweight of mass, me, and approximately the same surface area as the float (to minimize errors caused by adsorption) is suspended from the opposite end. The mass m, is adjusted to be only slightly greater than m,. An experimental run is begun by evacuating the chamber and nulling the balance. Because of slight dissymmetries in the balance and the difference between mc and mj, a “zero mass,” mo, given by
mo
=
m,
- mf
must be added to the float side to achieve null. The vapor of interest is now introduced into the chamber and allowed to come to thermal equilibrium, and such weights are added to the float side of the balance as are necessary to bring the balance back to null. Application of some simple principles of mechanics to the nulled balance yields the expression :
m = pMVE
(2)
Here, m is the mass of the added weights minus the zero mass, mo, and V Ris the effective float volume given by: VE
=
vf - vc + E
where E is a correction for any differences in the volumes ot the balance beam on each side of the fulcrum. If the beam were entireIy symmetrical, E would be zero. Corrections can also be made for the volumes of the added weights, but these are generally insignificant. Thus, from Equation 2, vapor densities could be calculated from buoyant mass determinations if the molecular weight of the vapor, M , and the effective float volume, V x , were accurately known. A single value of p , together with the corresponding pressure and temperature, would suffice to determine B from Equation 2. However, independent determination of V E would require calibration of the apparatus with a reference gas of precisely known volumetric properties, thus making all subsequent density determinations dependent not only upon the precision of the apparatus but upon the accuracy of such calibrations as well. Also, since the contribution to error in the calculated density due to an error in the molecular weight (for want of absolute purity) is of the same magnitude as the error in the molecular weight, M must be known to within several hundredths of 1%. Finally, the determination of a second virial coefficient from measurements a t a single pressure allows no opportunity for statistically averaging out random errors. 570
l&EC FUNDAMENTALS
By the method described, B is obtained from a series of buoyant mass determinations made at constant temperature and successively lower pressures, and in such a way that M V E need not be determined independently. From Equation 2, the compressibility factor of a vapor a t conditions p , P, T , is PMVE Z = P / p R T = -(3)
mRT
Substitution of Equations 2 and 3 into the virial expansion carried to three terms gives
Equation 4 is rigorous and in principle permits calculation of virial coefficients higher than the second. However, this requires extremely accurate data over a considerable pressure range. For low pressures the virial equation can be truncated after the second term : (5) Thus a plot of P l m us. m should yield a straight line with R T / M V E as the intercept and B R T / ( M V E ) zas the slope. B can then be calculated from the slope, intercept, and absolute temperature. The product M V E eriters only indirectly into the calculations through the slope and intercept terms, and may be evaluated from the intercept and the absolute temperature. A similar treatment results if one starts with the virial expansion in pressure rather than density. The advantages of the procedure described are evident. A second virial coefficient is determined from a series of mass measurements made a t various pressures of the vapor, but all at the same temperature and all from a single filling of the apparatus. The vapor may be injected into the apparatus either as a gas or as a liquid. Its amount need not be known. Thus the only measurements necessary are of temperature, pressure, and mass, all of which can be measured directly and to whatever precision one wishes to take the trouble to attain. The prototype vapor density apparatus constructed to operate as discussed above was a modified Ainsworth semimicro analytical balance, enclosed in an aluminum chamber of approximately 20-liter capacity. Buoyant masses were determined with NBS Class S weights, and the pressure measurements were made with a Texas Instruments Co. fused quartz precision pressure gage. The pressure range employed was from 250 to 750 mm. of Hg. Results from data taken for n-hexane, shown in Figure 1, are well correlated by the smoothing curve proposed by McGlashan and Potter (3) as a result of their exceptionally careful and thorough investigation of n-alkanes. Values of B were obtained from the linear relation of Equation 5 . Fitting quadratic equations to the data, as suggested by Equation 4, did not significantly improve the correlation. This procedure would be appropriate with data of sufficient accuracy over a wider pressure range. All data necessary for the determination of a virial coefficient are easily taken in a day. This method is therefore
-800
, b'
h
W J
0 ,' -+IOOCC
0
Acknowledgment
/ '
z
& I
probably quartz, and a torsion suspension would undoubtedly be superior to the knife-edge suspension used.
The authors gratefully acknowledge the financial support provided during the course of this work by NSF Grant GP2199 and by the Lummus Co. Thanks also go to the Dow Chemical Co. for its generous donation of heat transfer fluid.
-1,200
\
0 0 v
m
Nomenclature
-1,600
B, C I
40
I
I
I
80
60
I
P
= = = =
100
H
=
p
= = = = =
m M
T ("C) Figure 1 .
T V
Second virial coefficients for n-hexane
VE 2
This work ( I , 2); Srnoothin,g curve of McGlarhan and Potter (3)
capable of establishing second virial coefficients as quickly as an) technique no\\ available, and operation of the balance is entirely routine. Although the protot)pe balance ( 7 , 2) worked well with nonpolar chemicals such as argon and hexane, it did not produce nearly bo satisractory results with more chemically active reagents such as alcohols and nitriles. The problem undoubtedly had to do \I ith the fact that a commercial balance as converted to serve i n this work, and it included a variety of materials of construction. The most suitable material is
virial coefficients in density expansion mass molecular weight pressure gas constant molar density absolute temperature volume effective float volume compressibility factor = P/pRT
SUBSCRIPTS 0 = zero pressure c = counterweight f = float literature Cited ( 1 ) Abbott, Ivl. M., Ph.D. thesis, Rensselaer Polytechnic Institute,
1965. (2) Di Zio, S. F., Ph.D. thesis, Rensselaer Polytechnic Institute, 1964. (3) McGlashan, M. L., Potter, D. J. B., PYOC. Roy. Soc. 267A, 478 (1962).
RECEIVED for review September 8, 1965 ACCEPTEDAugust 10, 1966
COM M UN I CAT1ONS
SUFFICIENCY CONDITIONS I N CONSTRAINED VAR I A T 1 0 RI S Sufficiency conditions for an optimum of a continuous differentiable function to exist are derived for the case when equality or inequality restrictions are present.
optimization of problems subject to limitation by equality restrictions has been treated using the Jacobian solution (constrained variation) a.s proposed by Edelbaum (2). This treatment, while very clear, omits any discussion of sufficiency conditions. Since these Corm an essential aspect of any optimization study, and because it does not appear generally possible to investigate the character of a stationary point for this type of problem by other methods, no discussion of optimization can be considered complete Tvithout an investigation of these sufficiency conditions. .4dequate sufficiency conditions for the restricted problem do not appear to have been previously developed but can be derived easily as follows from the concepts of constrained variation as noted by Wilde (4)in a recent review article. Let us define the problem as the determination of THE
Optimum { ? ( X I , x 2 , ubject to them restrictions (m< n) g r ( x ~ ,x z
... x,)
... x n ) )
= 0; k = 1, 2
... m
(1)
(2) In the analysis to follow it is assumed that both y and g k are twice differentiable, continuous functions. I t is convenient and proper to visualize varying one of the variables, say x,+
while holding x+, I , xm+ 2: x,+ 4 . . .Y, fixed and at the same time permitting XI, x2 . , . x, to change so that the n restrictions are satisfied. Based upon this concept, the objective function, v , depends only on the n - m variables, x+, 1, xm+ . . . x,, so that a Taylor series expansion about a point (XI*,.YZ* . . . x,*) can be written as
3' = y *
+ Ay + AZy -+ 2!
where
and
h , E (xj
- xI*)
f o r j = rn
+ 1, .
,
. n
Here a constrained derivative has been defined as
(&l= VOL. 5
NO. 4
NOVEMBER 1 9 6 6
571
