B E H A V I O R OF V E L O C I T Y P R O B E S IN V I S C O E L A S T I C DILUTE POLYMER SOLUTIONS GlANNl ASTARITA AND LUlGl NICODEMO Istituto di Principi di Ingegneria Chimica, University of Naples, Naples, Italy The response of hot wire probes and Pitot tubes in dilute polymer solutions was studied experimentally. The hot wires are much less sensitive than in Newtonian liquids, and their use as velocity probes is probably impossible. Pitot tubes give abnormally low readings, particularly at small diameters; the results are reproducible.
THE measurement of
local velocities in dilute polymer solutions is of great interest in the interpretation of the macroscopically anomalous flow behavior of such liquids. In particular, turbulent flow behavior has been discussed on the basis of velocity distribution data (Elata et al., 1966; Meyer, 1966; White, 1967). Such data are subject to considerable doubt, because of anomalous response of usual velocity probes, such as Pitot tubes and hot wires, in polymer solutions. Anomalies of velocity probes have been discussed theoretically (Metzner and Astarita, 1967; Savins, 1965) and observed experimentally (Acosta and James, 1966; Astarita and Nicodemo, 1966; Leathrum, 1966; Maerker, 1967; Serth and Kieser, 1968; Smith et al., 1967). The present work presents some representative results of an experimental investigation of the response of hot wire probes and of Pitot tubes in dilute polymer solutions. In contrast with previously published data, many of the results reported here were obtained under conditions where the local velocity a t the probe was perfectly known: thus, the observed anomalies are to be attributed entirely to probe behavior, and no influence of anomalous velocity distributions needs to be considered. Analysis of Hot Wire Response
Five sets of data on hot wire (or hot film) response in dilute polymer solutions are available (Acosta and James, 1966; Leathrum, 1966; Lindgren and Chao, 1967; Serth and Kieser, 1968; Smith et al., 1967). These results are not consistent with each other, and in one case (Smith et al., 1967) even internal consistency mas not achieved. Published experimental results have been obtained by holding the probe stationary in a stream of liquid. This method gives rise to two problems. The first difficulty is that, while the average velocity of the liquid stream is known (from the total flow rate), the local velocity a t the probe, to which the probe is sensitive, needs to be inferred from an evaluation of the velocity profile. The latter is generally not known, and indeed its knowledge is just the kind of information which the probe should hopefully provide once it is perfected. A vicious circle is therefore encountered, which can be resolved only by inverting the method-Le., by running the probe a t some mechanically controlled speed through a quiescent liquid. The second difficulty is that the probe needs to be very small, to avoid the possibility of appreciably nonconstant velocities along the probe itself. Small probes are more liable to end effects, which are likely to be abnormally large in 582
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FUNDAMENTALS
polymer solutions. This difficulty is again avoided by running the probe through the liquid, because in this case large probes can be used. Consider a hot wire probe. If a current, I , circulates through the wire, an energy balance on the wire yields:
12Rw= nDLh(T, - T j )
(1 )
For Newtonian fluids, and in the absence of appreciable end effects, the heat transfer coefficient can be calculated from the equation : h=
k
(D v U ) O " ( ~ r
- 0.95 D
Equation 2 was obtained from a re-evaluation of the data of Piret et al. (1947) and Davis (1924). Comparison of Equations 1 and 2 shows that the hot wire response-Le., the current, I , needed to keep the wire temperature, T,, constant-is proportional to the 0.2 power of velocity. Though the measured quantity is electrical, and can thus be measured with good accuracy, even in Newtonian fluids the sensitivity of the hot wire probe is rather low. Published analyses of flow around submerged objects of viscoelastic liquids (Marrucci and Astarita, 1966; Metzner and Astarita, 1967) suggest that, at large values of the Deborah number, abnormally thick boundary layers may develop, leading to a marked decrease of probe sensitivity. The Deborah number for the hot wire is defined as:
De = 8 U / D 8 being the "natural time" of the liquid (Astarita, 1966; Astarita and Metzner, 1966). Usual values of D for hot wire probes are of the order of cm.; values of 8 for dilute polymer solutions are of the order of second (Oliver, 1966). Thus, velocities of the order of 1 cm. per second correspond to Deborah numbers of the order of unity: Hot wire probes are expected to exhibit a very low sensitivity over the whole range of velocities of practical interest. The situation discussed above may be different in the case of conical probes. Previous experimental results (Leathrum, 1966) and theoretical analyses (Metzner and Astarita, 1967) suggest that conical hot film probes may be more efficient sensors than hot wires in dilute polymer solutions. In fact, the sensor is located on such probes some distance downstream of the leading edge, and thus the conspicuous kinematic anomalies predicted to occur a t the tip (Marrucci and Astarita, 1967) may have less influence on the heat transfer characteristics of the probe. Furthermore, conical probes
are in principle easier to analyze in terms of boundary layer theory, because there is no contribution from the wake as in the case of wires. Analyses of this type have been attempted (Maerker, 1967; Metzner, 1968). Analysis of Pitot Tube Response
The response of Pitot tubes in dilute polymer solutions differs from that in ordinary Newtonian liquids for two reasons: The prevailing normal stress pattern influences the Pitot tube reading (Astarita and Nicodemo, 1966; Metzner and Astarita, 1967; Savins, 1965); the Pitot tube itself alters the local flow pattern; this effect is more important in dilute polymer solutions, where again an abnormally thick boundary layer may develop. Independent experimental evidence (Vurachi, 1967) shows that the stagnation pressure in dilute polymer solutions is lower than in ordinary Newtonian liquids. Pitot tube measurements in flowing polymer solutions have been rather frequent (Astarita and Nicodemo, 1966; Elata et al., 1966; Ernst, 1965; ,Shaver and Merrill, 1959; Smith et aZ., 1967; White, 1967). Whenever a careful mass balance check was made, the integrated Pitot tube curve did not check the independently measured flow rate; nonetheless, some of the apparent-velocity data, (Ernst, 1965) have been used to interpret the mechanism of turbulence in polymer solutions (Meyer, 1966). Again, previously published data have been obtained by holding the Pitot tube stationary in a flowing stream; this gives rise to the same problems as with the hot wire probe. Running a Pitot tube through a stationary liquid has the additional advantage of eliminating the first effect, because no normal stress pattern needs to be considered. Thus, any anomaly of response encountered in this case should be attributed to distortion of the local stagnation flow caused by the Pitot tube itself. Experimental
Running probes a t constant velocity through quiescent liquids requires use of a, tow tank, equipment which, if linear, becomes of prohibitive length if a rather large velocity needs to be held constant for even a few seconds. This difficulty can be overcome by use of a circular tow tank of sufficiently large radius, through which a probe moves a t constant tangential velocity. The curvature of the probe trajectory is negligible a t diameters exceeding a few inches. A circular tow tank, with an inner radius of 50 cm. and an outer radius of 70 cm., was filled to a depth of 5 cm. with the liquid to be tested. A iseries of radial baffles was provided to minimize the possibility of a net tangential flow of the liquid due to the probe motion. Observation of small floating tracer particles has confirmed that the net liquid velocity was generally below 1%, and never exceeded 3% of the probe velocity. The probe was held a t the tip of a rotating arm, which was driven through a series d gears by a variable speed motor fed through a continuous transformer. The velocity of the probe was measured by timing a known number of revolutions. Probe velocities ranging from 1 to 800 cm. per second could be obtained with this mechanical arrangement The hot wire probe used was a 2.0-cm. length of 0.001-inch diameter platinum wire, soldered between two 0.1-cm. diameter platinum wires. The electrical signal was transmitted to the measuring bridge via tungsten-mercury rotating contacts. A high wattage bridge with low temperature coefficient elements was used; the bridge balance was maintained by varying the power input. This assures constant temperature operation of the wire, so that comparison of the
response with accepted heat transfer correlations was simplified. The highest velocity attainable with the hot wire probes was of the order of 250cm. per second, because of frequent mechanical failure of the wire at higher Velocities. Pitot tube probes have been used, with internal diameters ranging from 0.06 to 0.9 cm. An open-end manometer was mounted directly above the tube a t the end of the rotating arm, so that there was no centrifugal force contribution to the Pitot tube reading. Readings were taken from motion pictures of the rotating manometer. The liquids tested were water and several aqueous solutions of ET597 (a high-molecular-weight additive manufactured by the Dow Chemical Co.) at different concentrations. Only representative data are reported here; a more complete set of data is available (Alfani, 1967).
Hot Wire Data. A preliminary series of runs was carried out in water (Figure 1). Substitution of Equation 2 into 1 and rearrangement yield:
Pw
where pw is the resistivity of the wire. Equation 4 cannot be compared directly with the data unless the temperature difference, T, - Ti, is known. The latter was evaluated as follows: First, the power input to the bridge was regulated so that incipient boiling a t the wire was observed. With the fluid temperature a t 20' C., this implies a temperature difference, T, - T,, of 80'C. The power input a t the same velocity was reduced by 30%, and thus, as shown by Equation 4,the new temperature difference could be evaluated a t 40' C. This setting was then used in all subsequent runs. The straight line in Figure 1 through the water data has the slope predicted by Equation 4,and its intercept would correspond to a T, - Tfvalue of 38' C. The remarkably good agreement with the data shows that the equipment, when used with water, has the expected sensitivity and is controlled by the wire-to-fluid heat transfer with no appreciable end effects (the aspect ratio LID of the wire was 800). Data obtained with six polymer solutions of different concentrations are reported in Figure 1. All the data in Figure 1 were obtained with the same wire and with equal settings of the bridge elements, and can thus be directly compared with each other. The marked difference of behavior in water and in dilute polymer solutions is easily observed. Similar data covering a wider range of velocities are reported in Figure 2. Up to the maximum velocities no leveling
o water
o
6 0.OS'x
v 0.25Ox
a
0 O.SO%
0.10%
0.20%
10
-
%as€
-r
-
06
0.5.
Q.4.
0.3
VOL.
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of the curve of the type reported by Acosta and James (1966) was observed. The slope of the I us. U log-log plot for dilute polymer solutions is always lower than in water, being only 0.048 in the case of the 0.370solution (Figure 1 ) . This is so low that the possibility of reliable velocity measurements with hot wire probes is in practice ruled out. The results in Figures 1 and 2 show that the heat transfer coefficient for external flows of dilute polymer solutions is lower and less sensitive to velocity than in Newtonian liquids. The same behavior has recently been reported also for internal flows (Gupta et al., 1967; Marrucci and Astarita, 1967). A comparison with results obtained by other investigators is in order. Table I reports the range of values of velocities covered by different authors. Smith et al. (1967) obtained their calibration by holding the probe stationary a t the center line of a pipe; reported values of U are average velocities, and thus lower than the true local velocities a t the probe by an unknown, and possibly not constant, factor. Two ranges of velocities were investigated; in the “high” range, which exceeds the maximum velocities obtained in this work, a double-valued response was observed, which is left unexplained. Only the “low” range data can be compared with the present ones. Smith et al. (1967) report the ratio of the measured heat transfer coefficient to the heat transfer coefficient in water a t equal average velocity. This ratio decreases from 0.98 a t U = 50 to 0.45 a t U = 150. Considering that the heat transfer coefficient in water has presumably increased by a factor of (150/50)0.4= 1.55, the odd conclusion is drawn that the heat transfer coefficient in the polymer solution actually decreased with increasing velocity. If this result is considered together with the nonreproducibility of the response in the “high” range, and with the uncertainty concerning the true local velocity a t the probe, the data of Smith et al. appear to be too anomalous to allow any reliable conclusions to be drawn. Serth and Kieser (1968) suggest that the level of turbulence of the flowing liquid may be an important independent variable, neglect of which may cause the apparently erratic behavior reported by Smith et al. The data of Acosta and James (1966) have been discussed
Tube U
- U,,, at u = loo
cm./sec.
d log H / d log U
Table II. Pitot Tube Data I.d.,cm. 0.08 0.15 O.d.,cm. 0.12 0.21 Water
0.05% 0.10%
0 35 ?
2.0 2.13 ?
Water 0.05% 0.10%
water
I
(1.3
1
5
2
10
50
20
100
200
U,cm/cec
Figure 2.
Table 1.
Hot wire data at high velocities
Velocity Ranges Investigated Range of U ,
Ref. Acosta and James, 1966 Smith et al., 1967, low Smith et al., 1967, high This work
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Cm./Sec.
0.6-21 50-150 450-850 1.2-250
FUNDAMENTALS
urns=/ Urnin
35 3.0 1.9 208
I
2.0 1.99 2.13
0 12 15
2.0 2.03 1.81
0.90 1.00 2 5 9 2.0 2.00 1.92
in some detail by Metzner and Astarita (1967). At velocities below 5 cm. per second, the response of the probe appears to be lower, and less sensitive to velocity, than in water. This is in perfect agreement with the present results. At velocities above 5 cm. per second the response becomes totally independent of velocity. The breakup velocity is the same, though two wire diameters (0.0021 and 0.0064 inch) were used; this is rather surprising. The behavior a t higher velocities observed by Acosta and James is not well understood. Pitot Tube Data. Data were collected with four Pitot tubes, with inside diameters ranging from 0.08 to 0.90cm. In each case a short horizontal length of tube (about 4 cm.) was soldered to a 0.20-cm. i.d., 0.30-cm. 0.d. tube fixed a t the tip of the rotating arm; this minimized the net flow of liquid in the tank. Pitot tube diameters are reported in Table 11. Data were collected a t values of U ranging from 7 5 to 250 cm. per second. The water data were in every case consistent with the Pitot tube equation: H = V/2g with the exception of the largest Pitot tube, which gave slightly low readings. These were justified by a nonnegligible net flow in the tank, amounting to a mean velocity of about 4% of the Pitot velocity. Data obtained with two polymer solutions (0.05 and 0.1% by weight) were systematically low. The log H us. log U plot was in every case a straight line with a slope slightly less than 2; values of H were as little as 4Oy0of the value predicted by Equation 5. Indicating with UaPpthe apparent velocity obtained from the equation: Uapp
0
0 22 30
0.32 0.40
= d2gH
(6 )
the difference U - Uapp-i.e., the error which would result by using the Pitot tube as an absolute velocity probe-was in some cases as large as 40% of U . Table I1 reports values of U - Uappat U = 100 cm. per second. The Pitot tube readings approached normality with increasing diameter, an effect quantitatively observed by Smith et al. (1967), yet appreciable discrepancies are observed up to a Pitot tube of 1.0-cm. 0.d.-Le., an unacceptably large diameter. In the 0.1% solution, no meaningful results could be obtained with the smallest Pitot tube because of nonreproducibility. The results obtained here are in agreement with independent observations on the reduction of the stagnation pressure in dilute polymer solutions flow (Vurachi, 1967). Anomalous behavior of Pitot tubes has been reported (Astarita and Nicodemo, 1966; Smith et al., 1967). If the assumption is made that the response of a Pitot tube is the same in a shear field as in a nonflowingliquid, curves such as the ones obtained in this work could be used as calibration curves for velocity distribution measurements.
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
= gravity acceleration, cm./sec.? h = heat transfer coefficient, watts/sq. cm., H = Pitot tube readinl,,7 cm. I = intensity, amperes k = conductivity, watts/cm., O K . L = wire length, cm. R, = wire resistance, ohms T, = wire temperature, OK. Tf = fluid temperature, OK. U = velocity, cm./sec. CY = heat diffusivity, sq. cm./sec. 0 = natural time, sec. Y = kinematic viscosit,y, sq. cm./sec. p, = wire resistivity, ohm cm.
87 (1966). (1965).
g
Gupta, M. K., Metzner, A. B., Hartnett, J. P., Intern. J . Heal OK.
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
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 by 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
NOWN
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.
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. In 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
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