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Non-Newtonian viscosity measurements in the intermediate shear rate range with the falling-ball viscometer
Journal of Non -Newtonian Fluid Mechanics, 15 (1984) 61-74 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
61
NON-NEWTONIAN VISCOSITY MEASUREMENTS IN THE INTERMEDIATE SHEAR RATE RANGE WITH THE FALLING-BALL VISCOMETER
Y.I. CHO *, J.P. HARTNETT
and W.Y. LEE
Energy Resources Center, University of Illinois at Chicago, Chicago, Illinois 60680 (U.S.A.) (Received August 2, 1983)
Summary
An attempt to use the falling-ball experiment to measure the non-Newtonian viscosity in the intermediate shear rate range was successfully accomplished by combining the direct experimental observations with a simple analytical model for the average shear stress and shear rate at the surface of a sphere. The viscosity data of aqueous solutions of Carbopol-960, carboxymethyl cellulose, polyethylene oxide and polyacrylamide obtained from the falling-ball viscometer gave good agreement with those from other viscometers, confirming the general applicability of the analytical approach. In the experiments with the highly viscoelastic polyacrylamide solutions the terminal velocity was observed to be dependent on the time interval between the dropping of successive balls. This time-dependent phenomenon was used to determine characteristic diffusion times of the concentrated solutions of polyacrylamide. These values were, in turn, compared with characteristic relaxation times determined by the Powell-Eyring model. The experimental program revealed that the falling-ball viscometer has very limited utility for the measurement of the steady shear viscosity of aqueous polymer solutions.
* Present Address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91103, U.S.A.
62 Introduction
The falling-ball experiment has been widely used to determine the viscosity of Newtonian fluids because of its simple theory and low cost. The use of the falling-ball viscometer for measuring the non-Newtonian viscosity has been of interest to rheologists for some years. However, its application has been limited to very low shear rate ranges where the non-Newtonian behavior can be neglected and the viscosity remains constant at the so-called zero shear viscosity [1,2]. From an engineering point of view, it would be valuable to extend the falling-ball experiment into the intermediate shear rate range where the viscosity of non-Newtonian fluids exhibits considerable variation with shear rate [3]. As noted in several analytical studies which predict the drag coefficients of a slowly moving sphere in a non-Newtonian fluid [4-61, the flow encountered in the falling-ball viscometer is a non-viscometric flow where the shear stress and the shear rate near the wall of the sphere change locally. Therefore, in order to establish the falling-ball experiment as an independent viscometer in the intermediate shear rate range, it is necessary to develop and validate a procedure to average appropriately these locally changing parameters. In brief, the objective of the current study is to develop a simple analytical model to average the locally varying shear stress and shear rate along the wall of a sphere and to experimentally validate the model with some typical non-Newtonian fluids. Average shear stress and shear rate
It is obvious on physical grounds that the average shear stress at the surface of a sphere, 78,ecan be obtained by dividing the viscous drag by the surface area: = 7
ave
(-
JJ
58
sin @>a, dS.
4aR2
wall
0)
Of note is that the contribution of form drag to the calculation of shear stress becomes zero because the direction of the hydrostatic pressure is normal to the surface. Even though there is no exact solution for qe in the case of non-Newtonian fluids, there are several approximate expressions obtained from variational principles in the form of upper and lower bounds [4,7-91. For the upper bound case, Tomita’s stream function modified by Wasserman and Slattery [8] is given by:
63 1c/= -+~R~sin~B[y-~][l
-u”]‘.
(2)
This yields: I-
ave=
(3)
m( v,/R)”
where the power law model was applied: rre = m[+II](“-“‘2d,e.
(4
For the lower bound case, using the trial stress function introduced by Wasserman and Slattery [8], 7r0 = - mArs( v,/R)”
sin 8
(5)
it can be shown that 7ave= $im(v,/R)“.
(6)
The numerical values of u and A in eqns. (3) and (6), respectively, have been reported elsewhere [8,10,11]. Since m in eqns. (3) and (6) can be eliminated, the average shear stress, 7*ve) can be expressed as a function of n only. From the force balance equation, the total drag, F,, can be shown to be: F, = $rR3g( ps - p).
(7)
Also, from the definition of the total drag coefficient, f, which includes both skin drag and form drag it can be shown that F, = 3rrm (2R)2-“vcX,
(~8)
where X is the non-Newtonian correction factor in the total drag coefficient of a slowly moving sphere, i.e., f = 24X/Re,. The numerical values of X have been reported in the literature as a function of the power law index n [8-111. Hence, equating these two:
m=
2% ps- dR”+’ 9vzx
This reduces to the well-known formulation for the determination viscosity of a Newtonian flmd in the falling-ball viscometer: 17=
2 gh 9v,
dR2 *
(9) of the
(94
Using eqn. (9), the average shear stress at the surface of a sphere can be expressed as a function of n only. Approximate expressions for the upper and lower bound average shear stress values are given as a function of n by the following polynomial equations:
64 (Upper bound) 7avc= (0.2827 + 0.8744n + 0.4562n2 - 0.7486~1~) $gR( p, - p),
(LO)
(Lower bound) 7ave= (0.6388 + 0.6418~1 - 0.4344~1~ + 0.1560n3)
$gR( p, - p).
(11)
The average shear rate can be obtained from the following approximation of the power-law model: 7Bve= ~(%,“J
(12)
Using eqns. (10) and (11) together with eqn. (9), it can be shown that (Upper bound) ?W., = (-1.731+
41.28n - 116.0n2 + 123.9n3 - 46.72n4)(
Q/R),
(13)
(Lower bound) Y,,, = ( - 2.482 + 54.35n - 160.1~~2~+ 178.2n3 - 69.04n4)( u,/R).
04)
Therefore, the apparent viscosity q can be expressed as (15)
7) = %W/jlitve’ It is noteworthy that R, ps, p and urnare all measurable parameters. Detemination
of n
To establish the falling-ball experiment as an independent viscometer, the power-law index n must be determined from the falling-ball experiment itself. For most non-Newtonian fluids of interest in engineering applications, n is not a constant. However, if the intermediate shear rate region is divided i&o three or four segments, it can be assumed that n is approximately constant in each division. Since the power-law index, n, does not change over a small range of +, we have d(h T&We) ’ = d(ln Jo,,,) *
(16)
Substituting the above expressions for 78,e and y,,,, it can be shown that over a small segment n =
d(h Rbs - P))
a4 u,/R )) -
(17)
Since both R( p, - p) and u-/R are experimentally measurable quantities, the power-law index n can be determined directly from the falling-ball experiment.
ADAPTOR
/
ELECTRIC HEATER
WATER BATH & C I RCUIATOR u
Fig. 1. Schematic diagram of the falling-ball viscometer.
Experimental study
A schematic diagram of the falling-ball viscometer is shown in Fig. 1. A detailed description of the apparatus may be found elsewhere [ll]. A special adaptor and funnel was designed to drop the individual balls at the center of the cylinder (diameter = 10 cm) filled with the test fluid, a condition which was found to be very important for viscoelastic fluids. Another advantage of this device is that it prevents the attachment of air bubbles to the surface of the balls during the experiment. Spheres with different densities including teflon, nylon, aluminum and stainless steel were chosen to provide experimental data over the desired range of shear rate [11,12]. The diameters of the steel and aluminum spheres ranged from 0.16 to 0.80 cm, while the diameters of the teflon and nylon balls ranged from 0.24 to 0.64 cm. To measure the terminal velocity, a ball was dropped at the center line of the cylinder filled with the test fluid; then the time required for the ball to pass two premarked lines in the lower half of the test section was recorded. The terminal velocity was determined by the distance between the marks and the recorded time. Subsequently, another ball of the same radius and density was dropped in the same manner and the measurement was repeated. The height of the test cylinder (137 cm) was sufficient to ensure the attainment of terminal velocities for all test spheres. To minimize the end effect, the lower 30 cm of the test cylinder were not used in the terminal velocity measurements. For aqueous solutions of the purely viscous non-Newtonian fluid (Carbopol-960) and the weakly or moderately viscoelastic fluids (CMC and polyethylene oxide), the measured terminal velocity of the first ball dropped
66 102
101 0 Gi
,
,,-I
’
,“I
.
..‘-
CARBOPOL 940
90 +0
: :
000 loo I
10-11
4x10-l
100
101 )5,,!sec-l)
lo*
105
Fig. 2. Apparent viscosity vs. average shear rate for aqueous solutions of Carbopol 960 at 23.8”C.
was within one percent of balls of the same radius velocity was corrected for yield the terminal velocity
the terminal velocity measured for all subsequent and density. The average value of the terminal the wall effect using Faxen’s equation [13,14] to at infinity:
08) /[I - 2.104(~~)]~ % = %l.%SS where the values of R/R, in the present study ranged from 0.016 to 0.080. The apparent viscosities of Carbopol-960, CMC and polyethylene oxide were determined using the above-mentioned analytical model in conjunction with the falling-ball measurements. The results presented in Figs. 2-4 are in good agreement with measurements made with the Weissenberg rheogoniometer (WRG, model R-18). The lower bound is best at the lower shear rates
1000 4x 10-2 10-l
loo ~aelsec-ll
lo1
lo*
Fig. 3. Apparent viscosity vs. average shear rate for aqueous solutions of CMC at 24°C.
67
10-11 10-2
10-l
loo
.lO1
YJsec-l)
102
Fig. 4. Apparent viscosity vs. average shear rate for aqueous solutions (WSR-301) at 23°C.
ld polyethylene
oxide
and the upper bound at the higher, when compared with the rheogoniometer data. When the average of these two are used, the viscosity data in the intermediate shear rate range can be obtained from the falling-ball experiment with about 5-20 percent accuracy. It is also of note that the use of the capillary tube viscometer to measure the viscosity in the intermediate shear rate region requires a special device to control the reservoir vacuum pressure, posing some difficulties if the accuracy of the viscosity measurement is to be maintained. Therefore, if one does not have the rotating type rheogoniometer, the falling-ball experiment may be a suitable device to provide engineering estimates of the viscosity in the intermediate shear rate regime. In the course of measuring the apparent viscosity of the aqueous polyethylene oxide solutions with the falling-ball viscometer, a major limitation of the system became apparent. At concentrations of Carbopol-960, CMC and polyethylene oxide below 500 wppm, the terminal velocity of the available balls exceeded the Stokes’ flow limit and it was no longer possible to obtain reliable viscosity measurements. For the (highly) viscoelastic aqueous polyacrylamide solutions, the measured terminal velocity was found to be very sensitive to the time interval between the dropping of successive balls in the cylinder. Therefore, for each polyacrylamide solution, preliminary tests were carried out for each set of balls having the same radius and density to determine the diffusion time of the fluid. The diffusion time (or characteristic time) as used here designates the time interval between successive balls necessary to ensure that the measured terminal velocity of the second and succeeding balls dropped into
68 the viscometer is the same as the terminal velocity of the first ball dropped into the viscometer. For all the polyacrylamide solutions studied a waiting time of approximately thirty minutes between dropping balls was sufficient to ensure that the measured terminal velocity presented the behavior of the undisturbed polymer solution. The resulting average measured terminal velocities were corrected to account for the wall effects using the equation of Cho et al. for (highly) viscoelastic fluids [15]. u, = v,,,
R/R&0.02,
/[LO148 - 0.7375R/R,], %a =&lwa.s
R/R,~0.02.
(19)
Applying these wall corrections together with the above analytical model, the apparent viscosities for polyacrylamide 2000, 3500, 5000 and 10,000 wppm solutions were calculated. e results are shown in Fig. 5, which also presents viscosity data from the Weissenberg rheogoniometer (WRG) and the capillary tube viscometer ( AVS). Again, the lower bound is best at the lower shear rates and the u 7 pe bound at the higher, when compared with data from other @cometers/ In eneral, when the average of the lower and upper bound values are used, th viscosity data for polyacrylamide solutions obtained from the falling-ball e periment gave good agreement with data measured from other viscometer . \ currently available limited the falling-ball The size and density of balls apparatus to concentrations of 2000 wppm and above. Below this concentra-
Fig. 5. Apparent vi 24T.
sity vs. average shear rate for aqueous solutions of polyacrylamide at
69
tion the terminal velocity exceeded the Stokes’ flow limit for all available balls. In order to measure the apparent viscosity of smaller concentrations it is necessary to obtain accurately fabricated spheres with a density slightly higher than that of water. Time-dependent
terminal velocity and characteristic time
The terminal velocity, which depends on the time interval between the dropping of successive balls in the cylinder was studied with aqueous solutions of polyethylene oxide and polyacrylamide. For the polyacrylamide 10000 wppm case, one hundred stainless steel balls of l/8 in, (0.312 cm) I.D. were chosen and a series of tests were conducted in which the time interval between the dropping of the balls was controlled. In the first set of experiments, the time interval between successive balls was held to thirty minutes; in the second set, the time interval was held to fifteen minutes. This process was repeated for various time intervals with the smallest time interval being thirty seconds. The terminal velocity measured for each At, designated as the time-dependent terminal velocity, u,~, is plotted in Fig. 6, demonstrating that the terminal velocity for polyacrylamide 10000 wppm solution in the falling-ball experiment is a function of the time interval between the dropping of the falling-balls. As the time interval was decreased from thirty minutes to thirty seconds, the time-dependent terminal velocity increased by about fifty percent. It is interesting to note that Fig. 6 also shows the existence of an asymptotic value of the terminal velocity for each At. Some preliminary discussions on the time-dependent terminal velocity were given in previous papers [16,17]. The time interval of more than 30 min needed to eliminate the effect of the previous ball is certainly not a measure of the relaxation time in the time tminl
0
10
2030405060708090 time tmin)
Fig. 6. Time-dependent
terminal velocity vs. lag time in the falling-ball experiment.
70 usual sense. Rather, the 30 min are an estimate of the diffusion time required for the disturbed region left by the previous ball to recover its original undisturbed condition. When a ball drops through a viscoelastic fluid medium, it opens or breaks the high molecular weight polymer network matrix locally. As the ball moves down the center line of the cylinder, the solvent (i.e., water) immediately fills the space left by the ball in the center region. Therefore, the fluid in the region of the center lines has a relatively higher concentration of solvent, thus making spheres fall faster in the disturbed condition than in the undisturbed. At the same time, the relatively low concentration of polymer molecules at the tube center results in the diffusion of polymer molecules toward the center line. The existence of an asymptotic value of the terminal velocity at each At suggests that there is an equilibrium between the influx of solvent and the diffusion of polymer molecules for a given value of At. Using the experimental results given in Fig. 6, it is possible to determine the asymptotic value of the apparent viscosity for each time interval, At, and to normalize this asymptotic viscosity with respect to the apparent viscosity determined by the terminal velocity measured in the undisturbed fluid (i.e. the value of u,~ measured at time zero for all runs). The results are shown in Fig. 7 and suggest that the falling ball-viscometer may yield a quantitative estimate of the characteristic diffusion time of the fluid. For example, if it is assumed that the fluid has fully diffused when the apparent viscosity is within 5% of that associated with the original undisturbed state, then the characteristic time of the 10000 wppm polyacrylamide solution is of the order of 19.5 min. The same procedure was used for the 3500 and 5000 wppm polyacrylamide solutions, yielding 3.2 and 10.5 min for the characteristic times as shown in Table 1. It was not possible to measure a characteristic time with the falling-ball viscometer for concentrations of polyacrylamide
SEPARANfAP-2731 10,000~~
0
10
20
30
40
50
60
Atlmm.1
Fig. 7. Normalized vpm.
viscosity vs. time interval for aqueous solution of polyacrylamide
10004
71 TABLE 1 Characteristic (P-E) model
times obtained
Solution
wppm
polyacrylamide (separan AP-273)
Polyethylene (WSR-301)
* Diffusion
from the falling ball viscometer
oxide
2000 3500 5000 10000 5000 10000 20000 30000
time; + Relaxation
(FBV) and Powell-Eyring
x FBV * (min)
P-E Model + (s)
3.2 10.5 19.5 -
8.8 34.5 53.3 133 0.25 3.3 16.4 41.8
time
of 2000 wppm or less. In this range, the characteristic time of the falling-ball experiment is of the order of seconds and cannot be accurately measured. In the case of polyethylene oxide it was not possible to determine a characteristic diffusion time with the falling ball viscometer even at concentrations of 20000 wppm. It is of note that both purely viscous fluids and weakly viscoelastic fluids yielded diffusion times which were too small to be measured (i.e. practically zero) indicating that the diffusion time may be a qualitative measure of viscoelasticity. The characteristic relaxation times of the polyacrylamide and polyethylene oxide solutions were also calculated using the Powell-Eyring model [l&19] along with the viscosity measurements made with the Weissenberg rheogoniometer. These results are given in Table 1. It is of note that the characteristic times calculated from the falling-ball experiment represent a diffusion time, while those from the Powell-Eyring model represent a relaxation time. Physically the diffusion process is much slower than the relaxation process, supporting the data given in Table 1. Since the Powell-Eyring relaxation time for the 30000 wppm polyethylene oxide is greater than that of the 3500 wppm polyacrylamide, it would appear that the falling-ball test should be capable of measuring the characteristic diffusion time for this polyethylene oxide concentration. Unfortunately, the poor visibility of this optically dense solution made it impossible to carry out the necessary observations. These measurements-or lack of them-underscore the limitations of the falling-ball instrument. Further, it should be noted that it is not possible to determine a priori the applicability of the falling-ball viscometer to other highly viscoelastic fluids. For a given polymer, the measurement of the diffusion time following the
72 approach outlined in this section should give some indication. If the diffusion time is in the range of minutes or a few hours the falling-ball viscometer should yield an engineering estimate of the viscosity as a function of shear rate. Conclusions (1) A simple analytical model to calculate the average shear stress and shear rate at the surface of a sphere has been developed. (2) Viscosity measurements of Carbopol-960, CMC, polyethylene oxide and polyacrylamide solutions in the intermediate shear rate range taken with the falling-ball viscometer give good agreement with those obtained from other viscometers. (3) For (highly) viscoelastic fluids (i.e. 3 500 to 10 000 wppm polyacrylamide aqueous solutions) the terminal velocity was found to be a function of the time interval between successive balls. (4) The falling-ball viscometer has limited utility as a device for measuring the characteristic diffusion time and viscosity of viscoelastic fluids. To extend the range of the instrument, geometrically accurate balls with densities slightly higher than that of water are needed. Acknowledgements The authors wish to express appreciation to Mr E.Y. Kwack for his assistance in the experimental program. In addition, the first author thanks Dr S.T.J. Peng at the Jet Propulsion Laboratory, Pasadena, for useful discussion on the diffusion process of viscoelastic fluids. Norhenclature A d rl?
f 4 g n,m r R Rc Re,
undetermined constant in trial stress function, eqn. (5) r8 component of rate of strain tensor in spherical coordinates, dij = vi,j + vj,i total drag coefficient of a slowly falling sphere, f = 24X/Re,, total drag force, eqn. (7) gravitational constant power-law model constants, eqn. (4) radial distance in a cylindrical coordinate system measured from the center of a sphere radius of the sphere radius of the! falling ball viscometer cylinder generalized Reynolds number, pv$“(2R)“/m
73
Surface of a sphere terminal velocity at infinity measured terminal velocity time-dependent terminal velocity for viscoelastic fluids non-Newtonian correction factor in the drag coefficient of a sphere,
f = 24X/Re,,
dimensionless radial distance, y = R/r second invariant of rate of strain tensor, II = dijdij undetermined constant in trial stress function, eqn. (5) average shear rate at the surface of a sphere, eqns. (13) and (14) undetermined constant in trial stream function, eqn. (2) apparent viscosity, eqn. (15) normalized viscosity angular coordinate in a spherical coordinate system measured from the rear stagnation point characteristic time fluid density sphere density re component of trial stress tensor in spherical coordinates average shear stress at the surface of a sphere, eqns. (10) and (11) trial stream function for the upper bound, eqn. (2) References
7 8 9 10
11 12 13 14 15
R.P. Chhabra and P.H.T. Uhlherr, Rheol. Acta., 18 (1979) 593. M. Gottlieb, J. Non-Newtonian Fluid Mech., 6 (1979) 97. P.H.T. Uhlherr, T.N. Le and C. Tiu, Can. J. Chem. Eng., 54 (1979) 497. R. Hill and G. Power, Q. J. Mech. Appl. Math., 9 (1956) 311. F.H. Leslie, Q. J. Mech. Appl. Math., 14 (1961) 36. K. Ada&i, N. Yoshioka and K. Sakai, J. Non-Newtonian Fluid Mech., 3 (1977/1978) 107. G. Astarita, J. Non-Newtonian Fluid Mech., 2 (1977) 343. M.L. Wasserman and J.C. Slattery, AIChE J., 10 (1964) 383. Y,I. Cho and J.P. Hartnett, J. Non-Newtonian Fluid Mech., 12 (1983) 243. Y.I. Cho and J.P. Hartnett, Theoretical analysis and experimental procedure in the falling ball viscometer for power law fluids, presented at the 72nd AIChE Annual Meeting, San Francisco, 1979. Y.I. Cho, Ph.D. Thesis, A study of non-Newtonian flows in the falling ball viscometer, University of Illinois at Chicago, 1980. W.Y. Lee, Viscosity measurements of viscoelastic fluids in falling ball viscometer, unpubl. M.S. Thesis, University of Illinois at Chicago, 1982. H. Faxen, Ark, Mat. Astron, Fys., 17 (1923) 28. R.M. Turian, AIChE J., 13 (1967) 999. Y.I. Cho, J.P. Hartnett and E.Y. Kwack, Chem. Eng. Commun., 6 (1980) 141.
74 16 Y.I. Cho and J.P. Hartnett, Viscoelastic effects in the faIIing baII viscometer, presented at the Golden Jubilee Meeting, The Society of Rheology, Boston, 1979; also in J. Rheol., 24 (1981) 891 (abstract). 17 Y.I. Cho and J.P. Hartnett, L&t. Heat Mass Transfer., 6 (1979) 335. 18 R.E. Powell and H. Eyring, Nature, 154 (1944) 427. 19 E.Y. Kwack and J.P. Hartnett, Int. J. Heat Mass Transfer, 25 (1982) 1445.
61
NON-NEWTONIAN VISCOSITY MEASUREMENTS IN THE INTERMEDIATE SHEAR RATE RANGE WITH THE FALLING-BALL VISCOMETER
Y.I. CHO *, J.P. HARTNETT
and W.Y. LEE
Energy Resources Center, University of Illinois at Chicago, Chicago, Illinois 60680 (U.S.A.) (Received August 2, 1983)
Summary
An attempt to use the falling-ball experiment to measure the non-Newtonian viscosity in the intermediate shear rate range was successfully accomplished by combining the direct experimental observations with a simple analytical model for the average shear stress and shear rate at the surface of a sphere. The viscosity data of aqueous solutions of Carbopol-960, carboxymethyl cellulose, polyethylene oxide and polyacrylamide obtained from the falling-ball viscometer gave good agreement with those from other viscometers, confirming the general applicability of the analytical approach. In the experiments with the highly viscoelastic polyacrylamide solutions the terminal velocity was observed to be dependent on the time interval between the dropping of successive balls. This time-dependent phenomenon was used to determine characteristic diffusion times of the concentrated solutions of polyacrylamide. These values were, in turn, compared with characteristic relaxation times determined by the Powell-Eyring model. The experimental program revealed that the falling-ball viscometer has very limited utility for the measurement of the steady shear viscosity of aqueous polymer solutions.
* Present Address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91103, U.S.A.
62 Introduction
The falling-ball experiment has been widely used to determine the viscosity of Newtonian fluids because of its simple theory and low cost. The use of the falling-ball viscometer for measuring the non-Newtonian viscosity has been of interest to rheologists for some years. However, its application has been limited to very low shear rate ranges where the non-Newtonian behavior can be neglected and the viscosity remains constant at the so-called zero shear viscosity [1,2]. From an engineering point of view, it would be valuable to extend the falling-ball experiment into the intermediate shear rate range where the viscosity of non-Newtonian fluids exhibits considerable variation with shear rate [3]. As noted in several analytical studies which predict the drag coefficients of a slowly moving sphere in a non-Newtonian fluid [4-61, the flow encountered in the falling-ball viscometer is a non-viscometric flow where the shear stress and the shear rate near the wall of the sphere change locally. Therefore, in order to establish the falling-ball experiment as an independent viscometer in the intermediate shear rate range, it is necessary to develop and validate a procedure to average appropriately these locally changing parameters. In brief, the objective of the current study is to develop a simple analytical model to average the locally varying shear stress and shear rate along the wall of a sphere and to experimentally validate the model with some typical non-Newtonian fluids. Average shear stress and shear rate
It is obvious on physical grounds that the average shear stress at the surface of a sphere, 78,ecan be obtained by dividing the viscous drag by the surface area: = 7
ave
(-
JJ
58
sin @>a, dS.
4aR2
wall
0)
Of note is that the contribution of form drag to the calculation of shear stress becomes zero because the direction of the hydrostatic pressure is normal to the surface. Even though there is no exact solution for qe in the case of non-Newtonian fluids, there are several approximate expressions obtained from variational principles in the form of upper and lower bounds [4,7-91. For the upper bound case, Tomita’s stream function modified by Wasserman and Slattery [8] is given by:
63 1c/= -+~R~sin~B[y-~][l
-u”]‘.
(2)
This yields: I-
ave=
(3)
m( v,/R)”
where the power law model was applied: rre = m[+II](“-“‘2d,e.
(4
For the lower bound case, using the trial stress function introduced by Wasserman and Slattery [8], 7r0 = - mArs( v,/R)”
sin 8
(5)
it can be shown that 7ave= $im(v,/R)“.
(6)
The numerical values of u and A in eqns. (3) and (6), respectively, have been reported elsewhere [8,10,11]. Since m in eqns. (3) and (6) can be eliminated, the average shear stress, 7*ve) can be expressed as a function of n only. From the force balance equation, the total drag, F,, can be shown to be: F, = $rR3g( ps - p).
(7)
Also, from the definition of the total drag coefficient, f, which includes both skin drag and form drag it can be shown that F, = 3rrm (2R)2-“vcX,
(~8)
where X is the non-Newtonian correction factor in the total drag coefficient of a slowly moving sphere, i.e., f = 24X/Re,. The numerical values of X have been reported in the literature as a function of the power law index n [8-111. Hence, equating these two:
m=
2% ps- dR”+’ 9vzx
This reduces to the well-known formulation for the determination viscosity of a Newtonian flmd in the falling-ball viscometer: 17=
2 gh 9v,
dR2 *
(9) of the
(94
Using eqn. (9), the average shear stress at the surface of a sphere can be expressed as a function of n only. Approximate expressions for the upper and lower bound average shear stress values are given as a function of n by the following polynomial equations:
64 (Upper bound) 7avc= (0.2827 + 0.8744n + 0.4562n2 - 0.7486~1~) $gR( p, - p),
(LO)
(Lower bound) 7ave= (0.6388 + 0.6418~1 - 0.4344~1~ + 0.1560n3)
$gR( p, - p).
(11)
The average shear rate can be obtained from the following approximation of the power-law model: 7Bve= ~(%,“J
(12)
Using eqns. (10) and (11) together with eqn. (9), it can be shown that (Upper bound) ?W., = (-1.731+
41.28n - 116.0n2 + 123.9n3 - 46.72n4)(
Q/R),
(13)
(Lower bound) Y,,, = ( - 2.482 + 54.35n - 160.1~~2~+ 178.2n3 - 69.04n4)( u,/R).
04)
Therefore, the apparent viscosity q can be expressed as (15)
7) = %W/jlitve’ It is noteworthy that R, ps, p and urnare all measurable parameters. Detemination
of n
To establish the falling-ball experiment as an independent viscometer, the power-law index n must be determined from the falling-ball experiment itself. For most non-Newtonian fluids of interest in engineering applications, n is not a constant. However, if the intermediate shear rate region is divided i&o three or four segments, it can be assumed that n is approximately constant in each division. Since the power-law index, n, does not change over a small range of +, we have d(h T&We) ’ = d(ln Jo,,,) *
(16)
Substituting the above expressions for 78,e and y,,,, it can be shown that over a small segment n =
d(h Rbs - P))
a4 u,/R )) -
(17)
Since both R( p, - p) and u-/R are experimentally measurable quantities, the power-law index n can be determined directly from the falling-ball experiment.
ADAPTOR
/
ELECTRIC HEATER
WATER BATH & C I RCUIATOR u
Fig. 1. Schematic diagram of the falling-ball viscometer.
Experimental study
A schematic diagram of the falling-ball viscometer is shown in Fig. 1. A detailed description of the apparatus may be found elsewhere [ll]. A special adaptor and funnel was designed to drop the individual balls at the center of the cylinder (diameter = 10 cm) filled with the test fluid, a condition which was found to be very important for viscoelastic fluids. Another advantage of this device is that it prevents the attachment of air bubbles to the surface of the balls during the experiment. Spheres with different densities including teflon, nylon, aluminum and stainless steel were chosen to provide experimental data over the desired range of shear rate [11,12]. The diameters of the steel and aluminum spheres ranged from 0.16 to 0.80 cm, while the diameters of the teflon and nylon balls ranged from 0.24 to 0.64 cm. To measure the terminal velocity, a ball was dropped at the center line of the cylinder filled with the test fluid; then the time required for the ball to pass two premarked lines in the lower half of the test section was recorded. The terminal velocity was determined by the distance between the marks and the recorded time. Subsequently, another ball of the same radius and density was dropped in the same manner and the measurement was repeated. The height of the test cylinder (137 cm) was sufficient to ensure the attainment of terminal velocities for all test spheres. To minimize the end effect, the lower 30 cm of the test cylinder were not used in the terminal velocity measurements. For aqueous solutions of the purely viscous non-Newtonian fluid (Carbopol-960) and the weakly or moderately viscoelastic fluids (CMC and polyethylene oxide), the measured terminal velocity of the first ball dropped
66 102
101 0 Gi
,
,,-I
’
,“I
.
..‘-
CARBOPOL 940
90 +0
: :
000 loo I
10-11
4x10-l
100
101 )5,,!sec-l)
lo*
105
Fig. 2. Apparent viscosity vs. average shear rate for aqueous solutions of Carbopol 960 at 23.8”C.
was within one percent of balls of the same radius velocity was corrected for yield the terminal velocity
the terminal velocity measured for all subsequent and density. The average value of the terminal the wall effect using Faxen’s equation [13,14] to at infinity:
08) /[I - 2.104(~~)]~ % = %l.%SS where the values of R/R, in the present study ranged from 0.016 to 0.080. The apparent viscosities of Carbopol-960, CMC and polyethylene oxide were determined using the above-mentioned analytical model in conjunction with the falling-ball measurements. The results presented in Figs. 2-4 are in good agreement with measurements made with the Weissenberg rheogoniometer (WRG, model R-18). The lower bound is best at the lower shear rates
1000 4x 10-2 10-l
loo ~aelsec-ll
lo1
lo*
Fig. 3. Apparent viscosity vs. average shear rate for aqueous solutions of CMC at 24°C.
67
10-11 10-2
10-l
loo
.lO1
YJsec-l)
102
Fig. 4. Apparent viscosity vs. average shear rate for aqueous solutions (WSR-301) at 23°C.
ld polyethylene
oxide
and the upper bound at the higher, when compared with the rheogoniometer data. When the average of these two are used, the viscosity data in the intermediate shear rate range can be obtained from the falling-ball experiment with about 5-20 percent accuracy. It is also of note that the use of the capillary tube viscometer to measure the viscosity in the intermediate shear rate region requires a special device to control the reservoir vacuum pressure, posing some difficulties if the accuracy of the viscosity measurement is to be maintained. Therefore, if one does not have the rotating type rheogoniometer, the falling-ball experiment may be a suitable device to provide engineering estimates of the viscosity in the intermediate shear rate regime. In the course of measuring the apparent viscosity of the aqueous polyethylene oxide solutions with the falling-ball viscometer, a major limitation of the system became apparent. At concentrations of Carbopol-960, CMC and polyethylene oxide below 500 wppm, the terminal velocity of the available balls exceeded the Stokes’ flow limit and it was no longer possible to obtain reliable viscosity measurements. For the (highly) viscoelastic aqueous polyacrylamide solutions, the measured terminal velocity was found to be very sensitive to the time interval between the dropping of successive balls in the cylinder. Therefore, for each polyacrylamide solution, preliminary tests were carried out for each set of balls having the same radius and density to determine the diffusion time of the fluid. The diffusion time (or characteristic time) as used here designates the time interval between successive balls necessary to ensure that the measured terminal velocity of the second and succeeding balls dropped into
68 the viscometer is the same as the terminal velocity of the first ball dropped into the viscometer. For all the polyacrylamide solutions studied a waiting time of approximately thirty minutes between dropping balls was sufficient to ensure that the measured terminal velocity presented the behavior of the undisturbed polymer solution. The resulting average measured terminal velocities were corrected to account for the wall effects using the equation of Cho et al. for (highly) viscoelastic fluids [15]. u, = v,,,
R/R&0.02,
/[LO148 - 0.7375R/R,], %a =&lwa.s
R/R,~0.02.
(19)
Applying these wall corrections together with the above analytical model, the apparent viscosities for polyacrylamide 2000, 3500, 5000 and 10,000 wppm solutions were calculated. e results are shown in Fig. 5, which also presents viscosity data from the Weissenberg rheogoniometer (WRG) and the capillary tube viscometer ( AVS). Again, the lower bound is best at the lower shear rates and the u 7 pe bound at the higher, when compared with data from other @cometers/ In eneral, when the average of the lower and upper bound values are used, th viscosity data for polyacrylamide solutions obtained from the falling-ball e periment gave good agreement with data measured from other viscometer . \ currently available limited the falling-ball The size and density of balls apparatus to concentrations of 2000 wppm and above. Below this concentra-
Fig. 5. Apparent vi 24T.
sity vs. average shear rate for aqueous solutions of polyacrylamide at
69
tion the terminal velocity exceeded the Stokes’ flow limit for all available balls. In order to measure the apparent viscosity of smaller concentrations it is necessary to obtain accurately fabricated spheres with a density slightly higher than that of water. Time-dependent
terminal velocity and characteristic time
The terminal velocity, which depends on the time interval between the dropping of successive balls in the cylinder was studied with aqueous solutions of polyethylene oxide and polyacrylamide. For the polyacrylamide 10000 wppm case, one hundred stainless steel balls of l/8 in, (0.312 cm) I.D. were chosen and a series of tests were conducted in which the time interval between the dropping of the balls was controlled. In the first set of experiments, the time interval between successive balls was held to thirty minutes; in the second set, the time interval was held to fifteen minutes. This process was repeated for various time intervals with the smallest time interval being thirty seconds. The terminal velocity measured for each At, designated as the time-dependent terminal velocity, u,~, is plotted in Fig. 6, demonstrating that the terminal velocity for polyacrylamide 10000 wppm solution in the falling-ball experiment is a function of the time interval between the dropping of the falling-balls. As the time interval was decreased from thirty minutes to thirty seconds, the time-dependent terminal velocity increased by about fifty percent. It is interesting to note that Fig. 6 also shows the existence of an asymptotic value of the terminal velocity for each At. Some preliminary discussions on the time-dependent terminal velocity were given in previous papers [16,17]. The time interval of more than 30 min needed to eliminate the effect of the previous ball is certainly not a measure of the relaxation time in the time tminl
0
10
2030405060708090 time tmin)
Fig. 6. Time-dependent
terminal velocity vs. lag time in the falling-ball experiment.
70 usual sense. Rather, the 30 min are an estimate of the diffusion time required for the disturbed region left by the previous ball to recover its original undisturbed condition. When a ball drops through a viscoelastic fluid medium, it opens or breaks the high molecular weight polymer network matrix locally. As the ball moves down the center line of the cylinder, the solvent (i.e., water) immediately fills the space left by the ball in the center region. Therefore, the fluid in the region of the center lines has a relatively higher concentration of solvent, thus making spheres fall faster in the disturbed condition than in the undisturbed. At the same time, the relatively low concentration of polymer molecules at the tube center results in the diffusion of polymer molecules toward the center line. The existence of an asymptotic value of the terminal velocity at each At suggests that there is an equilibrium between the influx of solvent and the diffusion of polymer molecules for a given value of At. Using the experimental results given in Fig. 6, it is possible to determine the asymptotic value of the apparent viscosity for each time interval, At, and to normalize this asymptotic viscosity with respect to the apparent viscosity determined by the terminal velocity measured in the undisturbed fluid (i.e. the value of u,~ measured at time zero for all runs). The results are shown in Fig. 7 and suggest that the falling ball-viscometer may yield a quantitative estimate of the characteristic diffusion time of the fluid. For example, if it is assumed that the fluid has fully diffused when the apparent viscosity is within 5% of that associated with the original undisturbed state, then the characteristic time of the 10000 wppm polyacrylamide solution is of the order of 19.5 min. The same procedure was used for the 3500 and 5000 wppm polyacrylamide solutions, yielding 3.2 and 10.5 min for the characteristic times as shown in Table 1. It was not possible to measure a characteristic time with the falling-ball viscometer for concentrations of polyacrylamide
SEPARANfAP-2731 10,000~~
0
10
20
30
40
50
60
Atlmm.1
Fig. 7. Normalized vpm.
viscosity vs. time interval for aqueous solution of polyacrylamide
10004
71 TABLE 1 Characteristic (P-E) model
times obtained
Solution
wppm
polyacrylamide (separan AP-273)
Polyethylene (WSR-301)
* Diffusion
from the falling ball viscometer
oxide
2000 3500 5000 10000 5000 10000 20000 30000
time; + Relaxation
(FBV) and Powell-Eyring
x FBV * (min)
P-E Model + (s)
3.2 10.5 19.5 -
8.8 34.5 53.3 133 0.25 3.3 16.4 41.8
time
of 2000 wppm or less. In this range, the characteristic time of the falling-ball experiment is of the order of seconds and cannot be accurately measured. In the case of polyethylene oxide it was not possible to determine a characteristic diffusion time with the falling ball viscometer even at concentrations of 20000 wppm. It is of note that both purely viscous fluids and weakly viscoelastic fluids yielded diffusion times which were too small to be measured (i.e. practically zero) indicating that the diffusion time may be a qualitative measure of viscoelasticity. The characteristic relaxation times of the polyacrylamide and polyethylene oxide solutions were also calculated using the Powell-Eyring model [l&19] along with the viscosity measurements made with the Weissenberg rheogoniometer. These results are given in Table 1. It is of note that the characteristic times calculated from the falling-ball experiment represent a diffusion time, while those from the Powell-Eyring model represent a relaxation time. Physically the diffusion process is much slower than the relaxation process, supporting the data given in Table 1. Since the Powell-Eyring relaxation time for the 30000 wppm polyethylene oxide is greater than that of the 3500 wppm polyacrylamide, it would appear that the falling-ball test should be capable of measuring the characteristic diffusion time for this polyethylene oxide concentration. Unfortunately, the poor visibility of this optically dense solution made it impossible to carry out the necessary observations. These measurements-or lack of them-underscore the limitations of the falling-ball instrument. Further, it should be noted that it is not possible to determine a priori the applicability of the falling-ball viscometer to other highly viscoelastic fluids. For a given polymer, the measurement of the diffusion time following the
72 approach outlined in this section should give some indication. If the diffusion time is in the range of minutes or a few hours the falling-ball viscometer should yield an engineering estimate of the viscosity as a function of shear rate. Conclusions (1) A simple analytical model to calculate the average shear stress and shear rate at the surface of a sphere has been developed. (2) Viscosity measurements of Carbopol-960, CMC, polyethylene oxide and polyacrylamide solutions in the intermediate shear rate range taken with the falling-ball viscometer give good agreement with those obtained from other viscometers. (3) For (highly) viscoelastic fluids (i.e. 3 500 to 10 000 wppm polyacrylamide aqueous solutions) the terminal velocity was found to be a function of the time interval between successive balls. (4) The falling-ball viscometer has limited utility as a device for measuring the characteristic diffusion time and viscosity of viscoelastic fluids. To extend the range of the instrument, geometrically accurate balls with densities slightly higher than that of water are needed. Acknowledgements The authors wish to express appreciation to Mr E.Y. Kwack for his assistance in the experimental program. In addition, the first author thanks Dr S.T.J. Peng at the Jet Propulsion Laboratory, Pasadena, for useful discussion on the diffusion process of viscoelastic fluids. Norhenclature A d rl?
f 4 g n,m r R Rc Re,
undetermined constant in trial stress function, eqn. (5) r8 component of rate of strain tensor in spherical coordinates, dij = vi,j + vj,i total drag coefficient of a slowly falling sphere, f = 24X/Re,, total drag force, eqn. (7) gravitational constant power-law model constants, eqn. (4) radial distance in a cylindrical coordinate system measured from the center of a sphere radius of the sphere radius of the! falling ball viscometer cylinder generalized Reynolds number, pv$“(2R)“/m
73
Surface of a sphere terminal velocity at infinity measured terminal velocity time-dependent terminal velocity for viscoelastic fluids non-Newtonian correction factor in the drag coefficient of a sphere,
f = 24X/Re,,
dimensionless radial distance, y = R/r second invariant of rate of strain tensor, II = dijdij undetermined constant in trial stress function, eqn. (5) average shear rate at the surface of a sphere, eqns. (13) and (14) undetermined constant in trial stream function, eqn. (2) apparent viscosity, eqn. (15) normalized viscosity angular coordinate in a spherical coordinate system measured from the rear stagnation point characteristic time fluid density sphere density re component of trial stress tensor in spherical coordinates average shear stress at the surface of a sphere, eqns. (10) and (11) trial stream function for the upper bound, eqn. (2) References
7 8 9 10
11 12 13 14 15
R.P. Chhabra and P.H.T. Uhlherr, Rheol. Acta., 18 (1979) 593. M. Gottlieb, J. Non-Newtonian Fluid Mech., 6 (1979) 97. P.H.T. Uhlherr, T.N. Le and C. Tiu, Can. J. Chem. Eng., 54 (1979) 497. R. Hill and G. Power, Q. J. Mech. Appl. Math., 9 (1956) 311. F.H. Leslie, Q. J. Mech. Appl. Math., 14 (1961) 36. K. Ada&i, N. Yoshioka and K. Sakai, J. Non-Newtonian Fluid Mech., 3 (1977/1978) 107. G. Astarita, J. Non-Newtonian Fluid Mech., 2 (1977) 343. M.L. Wasserman and J.C. Slattery, AIChE J., 10 (1964) 383. Y,I. Cho and J.P. Hartnett, J. Non-Newtonian Fluid Mech., 12 (1983) 243. Y.I. Cho and J.P. Hartnett, Theoretical analysis and experimental procedure in the falling ball viscometer for power law fluids, presented at the 72nd AIChE Annual Meeting, San Francisco, 1979. Y.I. Cho, Ph.D. Thesis, A study of non-Newtonian flows in the falling ball viscometer, University of Illinois at Chicago, 1980. W.Y. Lee, Viscosity measurements of viscoelastic fluids in falling ball viscometer, unpubl. M.S. Thesis, University of Illinois at Chicago, 1982. H. Faxen, Ark, Mat. Astron, Fys., 17 (1923) 28. R.M. Turian, AIChE J., 13 (1967) 999. Y.I. Cho, J.P. Hartnett and E.Y. Kwack, Chem. Eng. Commun., 6 (1980) 141.
74 16 Y.I. Cho and J.P. Hartnett, Viscoelastic effects in the faIIing baII viscometer, presented at the Golden Jubilee Meeting, The Society of Rheology, Boston, 1979; also in J. Rheol., 24 (1981) 891 (abstract). 17 Y.I. Cho and J.P. Hartnett, L&t. Heat Mass Transfer., 6 (1979) 335. 18 R.E. Powell and H. Eyring, Nature, 154 (1944) 427. 19 E.Y. Kwack and J.P. Hartnett, Int. J. Heat Mass Transfer, 25 (1982) 1445.