1187
VISCOSITYOF WATERAT HIGHPRESSURES
Viscosity of Water at High Pressures and Moderate Temperatures by E. M. Stanley and R. C. Batten' Naval Ship Research and Development Laboratory, Annapolis, Maryland
2 1 40.2
(Received April 29, 1968)
The viscosity of water was measured at pressures from 0 to 1406.2 kg/cm2 gauge and over a temperature range of 2-30' with a rolling ball viscometer. The data obtained show good agreement with most of the previously reported results. Analysis of the viscosity data shows activation energies of viscous flow decreasing in a uniform manner with increasing temperature and pressure and a possible transition phase at 36' and atmospheric pressure, indicated by extrapolated data.
Introduction In the last several years measurements of the viscosity of water under pressure have become important because of the use of these data in studying the structure of water, the processes of ionic conduction and viscous flow, and certain anamolous behaviors of water. Recently new data have appeared in the literature2-6 to supplement the older work of Bridgman,6 Cohen,' Lederer," and Tammann and Rabe;g but correlation of results among the different authors varies and in some cases large discrepancies are noted. Horne and Johnsons and Wonham2 have pointed out that more information is needed to resolve the conflicting data and to establish the correct values for use in other studies.
Experimental Section Apparatus. The instrument used in these experiments is a rolling ball viscometer. Viscosity is determined indirectly by measuring the time it takes a ball to roll between two points in an inclined barrel closed on both ends and filled with the fluid of interest. Knowing the density of the fluid being investigated, PF, the density of the rolling ball, PB, the measured time of roll, t, between the two fixed points, and a previously determined calibration constant, c, the viscosity, 17, can be calculated from the equation
- PF)
(1) The theory and limitations of this type of viscometer for use at atmospheric pressure have been described by Hubbard and Brownloand further considerations for its use a t pressure have been discussed by Horne and Johnson.* The experimental instrumentation is a viscometer which consists of a ball, a barrel, and sensing coils, in a pressure vessel, and associated apparatus (Figure 1 ) . The associated apparatus includes a pressurizing system for generating and measuring pressure in the viscometer; a temperature bath which circulates and regulates the temperature-controlling liquid around the pressure vessel; two quartz crystal thermometers, one a t each end of the pressure vessel for measuring the tempera9 = Ct(PB
ture of the controlling liquid; electronic logic circuitry which measures the roll time interval of the ball between the two sensing coils; and electronic instrumentation for visually displaying the roll times and temperatures in digital form. Details of the viscometer and its associated apparatus have been described previously." Accuracy and Error. As previously indicated, verification is needed of existing experimental data for the viscosity of water as a function of pressure. The discrepancies in currently available data by different investigators are probably caused by instrumentation errors. A detailed study of rolling ball viscometers, in general, and the laboratory experiments (with the instrument used to obtain the results reported here) indicated several possible sources of error in other data taken with the same type of instrument. The primary causes of error are pressure and temperature effects on the ball and the barrel, turbulent flow, and nonlineapity of the instrument throughout the viscosity and temperature ranges used. The rolling ball viscometer employed was carefully designed and calibrated to avoid or minimize these sources of error. Changes in pressure and temperature affect the clearance between the ball and barrel and were eliminated by constructing the ball and barrel of materials whose coefficient of thermal expansion and bulk moduli were nearly the same. Furthermore, the measuring barrel was placed inside the pressure vessel so that the pressure was equal on all sides of the barrel. Changes (1) The opinions expressed in this paper are those of the authors and do not necessarily represent those of the U. 8. Navy or the Naval establishment a t large. (2) J. Wonham, Nature, 215, 1053, (1967). (3) R. A. Horne and I).8. Johnson, J. Phys. Chem., 7 0 , 2182 (1965). (4) K . E. Bett and 5. B. Cappi, Nature, 207, 620 (1965). (5) W. Weber, Z . Angew. Phys., 15, 342 (1963). (6) P. W. Bridgman, Pfoc. A m . Acad. Arts Sei., 61, 57 (1926). (7) R. Cohen, Ann. Phys., 45, 666 (1892). (8) E. L. Lederer, Kollotd-Beth., 34, 270 (1932). (9) G . Tammann and H. Rabe, 2. Anorg. Allg. Chem., 1 6 8 , 7 3 (1927). (IO) R. M. Hubbard and G . G. Brown, Ind. Eng. Chem., Anal. E d . , 15, 212 (1943). (11) E. M. Stanley and R. 0. Batten, Anal. Chem., 40, 1751 (1968). Volume 79,Number 6 Ma# 1959
1188
E. M. STANLEY AND R. C. BATTEN
Table I: Relative Viscosity of Water Pressure, kg /cm2
qp/ql
?P/?1
for Various Temperatures and Pressures
?Pi?l
?P
I?I
7P/?l
?Pl?l
le191
a t 2.233’
a t 5,998O
a t 10.005°
a t 15.020O
a t 19.997”
a t 20.001°
176
0.9761 0.9743
0.9822 0.9821
0,9844 0.9870
0.9905 0.9904
0.9952 0.9935 0.9947
0,9990 0.9977
352
0,9596 0.9575
0.9686 0.9683
0.9755 0.9779
0.9822 0.9822
0.9917 0.9912
0.9990 0,9991
527
0.9475 0,9456
0.9595 0.9601
0,9701 0,9693
0.9769 0.9777
0 9898 0.9887
1 .0000 1.0004
703
0.9403 0.9411
0.9532 0.9533
0.9610 0.9633
0.9750 0.9754
0.9899 0.9886 0.9873
1 .0049 1.0046
879
0.9347 0.9357
0.9500 0.9501
0.9602 0.9622
0.9752 0.9759
0.9919 0.9907
1055
0.9310 0.9322
0.9491 0.9485
0.9607 0.9614
0.9777 0.9788
0.9939 0.9941
1230
0.9305 0.9314
0.9510 0.9495
0.9806 0.9818
0,9999 0 9984
0.9322 0.9339
0.9522 0.9505
0.9832 0.9863
1 0050 1.0059 1.0042
1406
0.9672 0.9726
PRESSURE
QUARTZ CRYSTAL
PRESSURE ACCUMULATOR
TEMPERATURE
VISCOSIMETER/
1.0143 1.0151
I
I
1.0283 1.0315
1406.2 kg/cm* and 0.03% for a temperature of 30”. The barrel length changes less than 0.03% for the same temperature change. With these variations, estimated changes in the roll time, by equations in Hubbard and Brown,l0 are on the order of O.l%, maximum, which is the magnitude of the repeatability of the roll times. Standard calibration techniques1°J 3 were employed
TIMER
WATER JACKET
I
P
(12) The change in diameter, due to pressure, of a thick-walled cylinder open on both ends where pressure is equal on all sides is given by Timoshenko (S. Timoshenko, “Strength of Materials,” D. Van Nostrand Co., Inc., New York, N. Y., 1948) as
Ad
=
-Pd(l - N ) / E
where d is the inside diameter of the cylinder (barrel) a t atmospheric pressure (0.6502 cm), p is Poisson‘s ratio, P is the gauge pressure, E is the modulus of elasticity of the cylinder, and the minus sign indicates a decrease in diameter. The change in diameter of a solid ball due to pressure can be calculated from the rearranged compressability equation Ad = P d j B K
I Figure 1, Block diagram of viscometer and associated apparatus.
in the radius of ball and barrel, and therefore the clearance between them, can be calculated,12 but because the clearance between the ball and barrel is small, 0.0203 cm, exact calculations of the change are difficult. Calculations using equations of ref 12 indicate that the average clearance between the ball and barrel changes less than 0.65% for a maximum pressure of The Journal of Phyeical Chemiatry
where d is the diameter of the ball (0.6299cm), P is the gauge pressure, and K is the bulk modulus. The change in diameter of the ball and barrel with temperature cain be expressed by the equation Ad = d a A T
where d is the diameter of the ball or barrel, a is the coefflcient Of linear expansion, and T is the temperature. The change in length A2 of the barrel due to temperature expansion is shown by a similar equation A1 =ZaAT (13) W.W.Langston and V. Orr, “Calibration of the Ruska RollingBall Viscosimeter,” United Gas Corp. Intraofflce Report No. 22-52, Jan 8, 1952.
1189
VISCOSITY OF WATERAT HIGHPRESSURES
SlANLIV L BAllEN 2.Z.C
SlANLel h BATlEN A HORNE h JOHNSON 'J ZHUZE @I 01 5.C
6.2%
BRIDGMAN O'C
6 HORNE 6
PRESSURE, k g h * Figure 2.
JOHNSON 2 S C
PRESSURE, kolcmr
PRESSURE, k ~ b a
Comparison of experimental values of relative viscosity at 2.2,6, and 10"with literature values.
I
c
(with distilled water, 10 and 22% glycerol, and 92% ethyl alcohol solutions) to ensure that the instrument was linear in the viscosity range of interest. A Stanton-type diagram10 was used to assure that all data reported were in the laminar region of flow. The excellent repeatability and overall accuracy of the instrument is a result of careful design, meticulous cleaning and filling techniques, and rigorous operational procedures, all of which have been previously described." Roll times for all experiments range from 19 to 35 sec. Average relative standard deviations vary from 0.05 to 0.2% with a mean of 0.1%. Temperatures were measured to f0.015' and pressures to f1.4 kg/cma. The density of the fluid was calculated with an em-
pirical formula derived by Wilson and Bradley" and is accurate to fO.0001 g/cm8. The density of the ball was calculated from a modification of an equation from Horne and Johnsona and is accurate to fO.OO1 g/cm? The absolute accuracy of the calculated viscosities is better than k0.015 oP.
Results The results of the present viscosity measurements are presented in Table I in terms of relative viscosity
(14) W. Wilson and D. Bradley, "Speciflc Volume, Thermal Expansion and Isothermal Compressibility of Sea Water." U. 8. Naval
Ordnance Laboratory Report NOLTR 66-103, June 2, 1966. Volume Y 8 Number 6 Maa, 1969
1190 where the subscript p refers to the values at pressure and the subscript 1 to the values at 1 atm. Each data point shown is an average of a minimum of ten separate time determinations for both v p and 71. Collation of these data with those reported by other investigators is given in Figures 2 and 3. The graphical limits in these illustrations designate the 99% confidence intervals calculated for the present experimental work; that is, the probability is 0.99 that a single value of relative viscosity at a given pressure and temperature will lie between the limits shown.
E. M. STANLEY AND R. C. BATTEN
l
4
O
0
7
Discussion For 2.2" (Figure 2) the present results lie in the middle of all previously reported data. The best agreement is with the data of Horne and J o h n ~ o n , ~ Cohen7 a t lo, and Bett and Cappia4 The Bett and Cappi values were read from a photographic enlargement of their published graph. Good agreement is also shown with Tammann and Rabe,9 as reported by Dorsey,I5 at 0" and below 600 kg/cm2. The poorest correspondence at this temperature is with Ledere? and Bridgman4 whose values are high and low, respectively. The 6" work (Figure 2) shows good agreement with the 5" work of T. P. Zhuze, V. 1. Sergeevich, and A. I. Chestnov as quoted by Richardson, el al.,lB but no agreement with the 6" work of Horne and Johnson.* At 10" (also Figure 2) the accordance with Bett and C a ~ p i Bridgman,6 ,~ and also Zhuze, Sergeevich, and Chestnov, as was reported by Richardson, et al,,le is good. Wonham's2 values, although at 12.65", indicate that if 10" data were available, they would be in good agreement. Lederer8 follows the present experimental data up to 500 kg/cm2, beyond which his values are low. Horne and Johnson3 and Tammann and Rabe,%as reported by Dorsey,16appear to be in some agreement with each other, but their values are high as compared to the present data and those of Bett and CappL4 At 15" (Figure 3) excellent agreement is shown with Cohen' while the values of Horne and Johnson8are high as they were a t 10". Wonham's2 values, although not exactly at 15", again indicate good agreement. Cohen,' Weber,5 and Wonham2 agree very well with the present values a t 20' (Figure 3 ) . Bett and Cappi4 and Zhuze, Sergeevich, and Chestnov, as quoted by Richardson, et U Z . , ~ ~only agree up to 700 kg/cm2 beyond which their values are low, while Horne and Johnsona are again high. Finally, at 30" (Figure 3) excellent accordance is again reached with Bett and Cappi4 and Weber.6 Very poor agreement is shown with the work of Bridgmant6 Tammann and Rabe,g as reported by Dorsey,15 and Lederer,8 whose values are high and agree with those of Zhuze, Sergeevich and Chestnov a t 40" as quoted by Richardson, et ~ 1 . ' ~ The Journal of Physical Chemiatry
TEMPERATURE, 'C Figure 4. Pressure of minimum relative viscosity as a function of temperature.
In summary, the results of the present experiments show good agreement with Bett and C a ~ p i Weber,6 ,~ Wonham,2 Cohen,' and Zhuze, Sergeevich, and Chestnov as quoted by Richardson, et at.,16 for all common temperatures and pressures. As for the remaining authors, Bridgman's and Horne and Johnson's experimental procedures and instrumentation have been examined for possible causes of their variations from the results reported here. (Lederer's work was not experimentally determined, and details of the Tammann and Rabe experimental procedure and instrumentation were not available.) Deviation of Bridgman's results in the high-temperature region may be due to roll timing difficulties and/or variations in the clearance between the ball and barrel due to pressure. Horne and Johnson's experimental data, which were constantly high except for 2.3", may have errors introduced by turbulence and/or leaving the ends of the barrel open during the rolling of the ball, as opposed to closing them, the normal procedure. The minima in the curves of the relative viscosity (Figures 2 and 3) are a function of two processes: (1) the breakup of the hydrogen-bonded water, which causes a decrease in viscosity, and (2) the compaction of the associated and unassociated water molecules, which results in an increase in viscosity. A t the pressure where the relative viscosity is a minimum, the two (15) N. E. Dorsey, "Properties of Ordinary Water-Substance," Reinhold Publishing Corp., New York, N. Y., 1940. (16) J. L. Richardson, P. Bergsteinsson, R. J. Geta, D. L. Peters, and R. W. Sprague, "Sea Water Mass Diffusion Coefficient Studies," Philco Corp. Publication No. U-3021 (Feb 26, 1961), Office of Naval Research Contract No. Nom-4061 (00) I
1191
VISCOSITYOF WATERAT HIGH PRESSURES processes have equal and opposite effects. Figure 4 is a plot of the pressure of minimum viscosity as a function of temperature. It shows that the pressure of minimum relative viscosity decreases with an increase in temperature. If the curve is extrapolated (dashed line), it intersects the zero pressure (atmospheric) axis at about 36". The temperature of 36" agrees well with the temperature of 35" quoted by Franks and Good" where they postulated that at 1 atm water undergoes a change from a quasi-crystalline structure to a suspension of clusters. The present data are plotted with isobars of relative viscosity as a function of temperature (Figure 5). The dashed curve represents the data at 1500 kg/cm2 from a similar graph of Horne and Johnson.a The lack of sharp changes in slope in this graph near 4", such as illustrated by the dashed curve of Horne and Johnson,a indicate that the breakup of the structured regions of water due to temperature is a gradual process and supports the calculations of Nemethy and Scheraga.l* Figure 6 is a graph of the isobars of the calculated activation energies of viscous flow, E,, as a function of
,
5.0
I
,
1
1
1
1
1
1
1
1
I
-
4.8-
4.4 -
-
4.2-
-
4.6
f5
2 4.0, 3.8
-
3.6
-.
3A I l ~ ~ i c ~ ~ r l l l 0 2 4 6 8 10 12 14 I6 18 20 22 24 26 28 $0 TEMPERATURE,'% Figure 6. Isobars of activation energy of viscous flow of water (kcal/mol) as function Of temperature+
temperature. This plot shows a marked difference in features a t low temperatures when compared to the ~ by the results of Horne and J ~ h n s o n ,represented dashed line.lg At low temperatures no sharp increase in activation energies is observed, but a small relative minimum is present in the data taken at low pressure. Some small fluctuations are also observed between 16 and 20" a t high pressure. Although the curves of E , at pressure are not as smooth as E , for atmospheric pressure, data values of the activation energies of viscous flow decrease with increasing pressure and temperature in a uniform manner and tend to follow the shape of the atmospheric pressure curve. At present no explanation can be given for the relative minima of the activation energies at low temperature. However, it should be noted that (1) the data used and calculations of E , are not precise enough (standard deviations of f 0 . 1 kcal/mol) to establish the absolute existence of the minima at the low temperature and (2) the relative minima do not correlate with the decrease in the temperature of maximum density at pressure as quoted by Dorsey.16 Therefore, a more detailed and accurate investigation should be undertaken in this low-temperature region a t pressure.
5
10
15 20 TEMPERATURE,
"C
25
30
Figure 5. Isobars of relative viscosity as a function of pressure.
(17) F. Franks and W. Good, Nature, 210, 85 (1966). (18) G. Nemethy and H. A. Scheraga, J. Chem. Phys., 36, 3382 (1962). (19) The sharp changes in slope of the curye of Horne and Johnson at low temperature for Figures 5 and 6 are caused by the high wp/71 values at 4' and above (see Figures 2 and 3). Volume 78,Number 6 May lQ80
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