Ind. Eng. Chem. Res. 1999, 38, 1729-1735
1729
RESEARCH NOTES Does the Viscosity of Glycerin Fall at High Shear Rates? Prasannarao Dontula, Christopher W. Macosko,* and L. E. Scriven*,† Coating Process Fundamentals Program, Center for Interfacial Engineering and Department of Chemical Engineering & Materials Science, University of Minnesota, Minneapolis, Minnesota 55455
The sole report of glycerin-water viscosities at high shear rates, by Ram, has so far been interpreted as evidence of shear thinning. Two alternative interpretations, both of which invoke experimental artifacts, have now been explored. One of these, viscous dissipation, qualitatively explains the progressive fall in the viscosity of glycerin-water as the shear rate is ramped-up beyond 60 000 s-1 to 90 000 s-1. New measurements with parallel disks separated by a narrow gap indicate that the viscosity of 60 wt % glycerin in water is constant up to 90 000 s-1. In a 1967 communication to Eirich’s Rheology, Ram1 stated that certain glycerin-water solutions are shear thinning at shear rates of 60 000 s-1 and above. No data or experimental details were provided, only seven literature citations.2-8 Neale2 had reported shear thinning in castor oil in a journal-bearing apparatus in which the current drawn by the motor was measured as a function of continually rising journal speed. On the basis of well-conducted experiments, Weltmann3 had shown that poor temperature control and wide-gap fixtures can lead to apparent shear thinning in rheological measurements with oils and inks, though not glycerin. Hagerty4 had observed that the viscosity of concentrated solutions of glycerin in water (under unspecified experimental conditions) fell with time over 5 min and had speculated it was caused by striae visible in these liquids. Dumanskii and Khailenko5 had reported viscosities of glycerin-water solutions that are 10-100 times larger than accepted values, an aberration that was not resolved. Merrill6 and Ram and Tamir7 had mentioned results from Ram’s Sc.D. thesis8 in passing. The seventh citation was of Ram’s8 unpublished Sc.D. thesis at Massachussetts Institute of Technology (M.I.T.), which contains the sole set of experimental data. The thesis can be examined only by application to M.I.T., which we have done; it seems not to have been scrutinized before on this account. Ram measured the torque on the stationary outer cylinder of a narrow-gap (150 µm) concentric cylinder fixture, as the angular velocity of the inner cylinder was ramped-up linearly over anywhere between 10 and 20 s, a standard procedure with the Merrill-Brookfield viscometer9 (also Figure 1a). The maximum shear rate was about 96 000 s-1 at the end of the ramp-up. The ends of the fixture were not sealed; sample liquid leaked out slowly and was continually replenished by a syringe. The temperature of the brass fixtures was controlled to within (1 °C by water circulating at 2 L/min in each annular channel about 1.5 mm beneath the cylindrical surfaces (Figure † E-mail:
[email protected]. Phone: (612) 625-1058. Fax: (612) 626-7246.
1b). The temperature of the fixtures themselves was not measured; presumably it was inferred from the water temperature in the cooling bath. The shear viscosity at each moment of this short test was taken from the instantaneous slope of the “flow curve” of shear stress (from measured torque) versus shear rate (from imposed angular velocity). The reproducibility was reported to be within 2-5% for liquids of viscosity larger than 2.5 mPa s. The inferred shear viscosity of a 62.5 wt % solution of glycerin in water at an unspecified temperature (29 °C according to the now standard measurements of Sheely10) was 9 ( 0.089 mPa s up to 40 000 s-1, i.e., during the first 8.5 s of a 20 s ramp-up; it fell gradually above 55 000 s-1 and then more rapidly to about 8.44 mPa s at 90 000 s-1, i.e., about 6.2% smaller (cf. Figure 51 in Ram’s thesis8). From these data, Ram concluded that glycerin-water solutions above 50 wt % concentration thin at high shear rates. Ram also reported shear thinning in ethylene glycol, aqueous sugar, and solutions of paraffin in decalin (cf. Table XIII in Ram’s thesis8). Today this conclusion is surprising, because no supporting data have been reported, as far as we know, on glycerin-water solutions or other low molecular weight hydrogen-bonding liquids. The possibility that glycerinwater solutions are mildly shear thinning above 60 000 s-1 at room temperature is important, because they are now often employed as Newtonian test liquids in various viscous free-surface coating flows where strain rates often exceed 10 000 s-1, and even 100 000 s-1. Hence, we consider here possibilities that the apparent shearthinning was an experimental artifact, in particular due to instrument eccentricity and shear heating. By this note we also report our own measurements of viscosities of aqueous glycerin up to 100 000 s-1. Effect of Hydrodynamic Instabilities Couette flow with a rotating inner cylinder is susceptible to Taylor instability; the toroidal vortices that form raise the torque. Up to the highest speed possible in Ram’s device, the critical Taylor number above which the secondary flow appears can be exceeded only with liquids whose viscosity is less than 4 mPa s.
10.1021/ie9805685 CCC: $18.00 © 1999 American Chemical Society Published on Web 02/25/1999
1730 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999
Figure 1. Schematic of the Merrill-Brookfield viscometer used by Ram.8 (a) Plan view: the rotor, stator and rotating frame form liquid and air bearings that support the reaction force applied by the knife connected to the torque-measuring member. (b) Radial cross-section A-A of the apparatus: the brass stator and rotor are both cooled by water circulating at 2 L/m in annuli 1.5 mm beneath the surface.
Effect of Eccentricity Figure 1a also shows a plan view schematic of Ram’s device. The outer cylinder (stator) was floated in the tapered frame (clearance unspecified) by compressed air supplied at about 0.17 M Pa from eight equispaced holes around the periphery (size and location unspecified). The stator was restrained by a knife resting against a V-shaped bearing on its surface. The knife actuated a torque transducer via a lever, and the reaction force by the knife deflected the stator by 3° at maximum torque (cf. Appendix D in Ram’s thesis8). From dimensions of the apparatus and measured torque, this reaction force can be estimated to be 1.614 N at 55 000 s-1 and 2.642 N at 90 000 s-1 with 60 wt % glycerin in water. This load was borne by the difference in restoring forces set up in the liquid and air “bearings”. The speeds were such that the gas-bearing number was small and the compressibility of the gas was negligible.11 A reasonable value of the nominal radial clearance in the air bearing is 25.4 µm (0.001 in.); the eccentricity would then be only 0.0747% of the liquid gap at 55 000 s-1 and imperceptibly higher at 90 000 s-1. The effect of eccentricity on the shear stress distribution, drag force, and torque on the walls of the liquidfilled gap is well-known. From Sommerfeld’s lubrication theory12
2x1 - 2 M ) M0 2 + 2 where M0 is the torque that would be measured if cylinders were coaxial, and is the eccentricity pressed as a fraction of the nominal gap between cylinders. The torque is virtually unaffected by centricities below 1%, as in this case.
(1) the exthe ec-
Effect of Shear Heating The sample liquid slowly flows axially through the gap between the cylinders, draining out the bottom around the circumference and being replenished halfway up the outer cylinder wall at one circumferential location. The maximum axial velocity of a 11 mPa s liquid under gravity in the 150 µm gap, about 3 × 10-3 m/s, is negligibly small compared to the maximum imposed azimuthal velocity, 14.4 m/s. Gravity flow thus contributes inconsequentially to shear rate.
However, thermal effects can be significant. Shearing of Newtonian liquid between infinite parallel plates separated by 2h, a good approximation to the actual gap, generates heat. When the speed of the inner cylinder is raised linearly, the rate of heat generation per unit volume, µγ˘ 2, rises quadratically with time if the shear viscosity µ is independent of shear rate γ˘ . If the liquid also has constant thermal properties and the channel walls are maintained at constant temperature (see Figure 2a), the temperature at position x (-h e x e h) within the liquid is given by13
Tl(x,t) )
[∑
4R2µ πFc
∞
(-1)n
n)02n
+1
cos
](
(2n + 1)πx t2 2h
β
-
2t β2
+
)
2(1 - e-βt) β3
(2)
where R is the angular acceleration and β ≡ (2n + 1)2κlπ2/4h2. Here κl is the thermal diffusivity kl/Flcl, about 10-7 m2/s in liquids. All thermal properties of 62.5 wt % glycerin in water are not available; hence, subsequent calculations are carried out for 60 wt % glycerin in water at 20 °C. The time constant β in eq 2 for 60 wt % glycerin in water in a 150 µm gap is large (density Fl of 1156 kg/m3, specific heat cl of 2410 J/(kg K), and thermal conductivity kl of 0.381 J/(s m K) at 20 °C14). The liquid temperature is nearly constant at short times (less than 0.05 °C rise in 5 s) and rises slowly and then rapidly with the square of time. The temperature at the midplane of the 11 mPa s 60 wt % glycerin-water solution would rise by 0.24 °C in the first 11.5 s of a 20 s ramp-up, i.e., at 55 000 s-1, and about 0.65 °C when the shear rate was 90 000 s-1. The estimated temperature rise in shorter ramps is smaller, but the estimated difference at 90 000 s-1 between 10 and 20 s ramps is less than 0.01 °C. However, the plate surfaces did not, in fact, remain at constant temperature, for two reasons. One is the thermal inertia of the brass between them and the coolant passages; the other is the thermal resistance of the same material. If the brass has constant thermal properties and the brass-cooling water interfaces are maintained at constant temperature, the method of Laplace transforms can be used to calculate the temperature Tl in the liquid (0 e x e h) and Tb in the brass (-l e x e 0; see Figure 2b):
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1731
Figure 2. Schematic of the cases studied: (a) liquid layer only with isothermal walls and (b and c) heat transfer through the liquid and brass wall composite. In part b, the outermost surfaces of the brass walls remain at constant temperature, whereas in part c, they are allowed to vary. One wall is ramped-up linearly in speed.
Tl(x,t) )
2R2µσ Fc
∞
1
∑β
by integration across the gap,
×
cos βnl cos qβn(h - x)
(σl + q3h) sin βnl cos qβnh + (σqh + q2l) cos βnl sin qβnh
(
t2
-
κbβn2
Tb(x,t) )
2R2µq2 Fc
dx 1 0 ) - ∫V dv ) ∫02hµ(x,t) τ
n)1 n
∞
1
∑β
2t
+
)
2(1 - e-κbβn t) 2
(κbβn2)2
×
(κbβn2)3
(3)
×
n)1 n
sin qβna sin βn(l + x)
(σl + q3h) sin βnl cos qβnh + (σqh + q2l) cos βnl sin qβnh
(
t2
κbβn2
-
2t (κbβn2)2
)
2(1 - e-κbβn t) 2
+
×
(κbβn2)3
(4)
Here, the βn, n ) 1, 2, 3, ..., are the positive roots of
σ cos βnl cos qβnh - q2 sin βnl sin qβnh ) 0 and q ≡ xκb/κl, σ ≡ qkb/k, and kb and κb are respectively the thermal conductivity and thermal diffusivity of brass. The properties of yellow brass are as follows: density Fb, 8470 kg/m3; specific heat cb, 385 J/(kg K); thermal conductivity kb, 116 J/(s m K).15 With Mathematica,16 the first seven terms of the series in eqs 3 and 4 were used to calculate temperature in the liquid and brass. The temperature at the midplane of a 60 wt % glycerin-water solution was thereby estimated to rise by 0.3 °C after 11.5 s of a 20 s ramp-up, i.e., at 55 000 s-1, and by about 0.75 °C after 18.75 s, i.e., when the shear rate reached 90 000 s-1. The estimated temperature rise in shorter ramps is smaller, but the estimated difference at 90 000 s-1 between 10 and 20 s ramps is less than 0.05 °C. Figure 3a shows how the temperature distribution in the glycerin-water and the confining brass walls evolves during a 10 s ramp-up. The viscosity of glycerin-water solutions falls with temperature, the more so the higher the glycerin concentration. The viscosity of 60 wt % glycerin in water falls probably exponentially from 11 mPa s at 20 °C to about 7 mPa s at 30 °C;10 the temperature coefficient of viscosity is 0.0306 K-1 at 20 °C. The shear stress τ ) -µ ∂v/∂x is uniform across the gap of width 2h. Hence,
V(t) τ
(5)
where V is the velocity of the wall at x ) 0. Taking the temperature, and hence viscosity distributions, to be symmetric about the midplane (eq 3) yields the shear stress. The shear stress τ, and hence the apparent viscosity, would be 0.6% smaller than the true viscosity at 55 000 s-1 and 1.6% smaller at 90 000 s-1. Thus, shear heating explains only a fourth of the decrease reported by Ram (Figure 4). Three terms instead of seven in eq 3 lowered the estimate of the apparent viscosity by less than 0.01%. However, the two brass-cooling water interfaces did not, in fact, remain at constant temperature. The average temperature rise of about 600 cm3 of cooling water used during this short test in Ram’s device was estimated by integrating the flux at the brass-water interface and neglecting the nonuniform distribution of cooling water in the annuli; the estimate was about 15 °C, i.e., 20 times the maximum temperature rise in glycerin-water estimated with eq 3. The temperature of water would, in reality, rise little at first and then sharply, qualitatively mirroring the temperature rise in the sheared liquid; this probable time course differs from the ideal isothermal condition used in the analysis so far (Figure 2b). A more accurate boundary condition (Figure 2c) would be kb dT/dx ) H(Tb - Tw), where Tw is the water temperature and H the heat-transfer coefficient in forced convection. If the brass has constant thermal properties and the cooling water is at constant temperature, heat transfer by forced convection would alter the temperature Tl in the liquid (a e x e b; see Figure 2c for notation) and Tb in the brass (0 e x e a) to (cf. O ¨ zis¸ ik,17 pp 273-298)
Tl(x,t) ) ∞
∑ Xln(x)
n)1
Tb(x,t) ) ∞
∑ Xbn(x)
n)1
( (
g/nR2
g/nR2
t2
-
2t
2
+
) )
2(1 - e-βn t)
βn2
βn4
βn6
t2
2t
2(1 - e-βn t)
βn2
-
βn4
2
+
βn6
(6)
(7)
1732 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999
Figure 3. Calculated temperature distribution with position in the 60 wt % glycerin-water and confining brass walls as the shear rate is ramped-up from 0 to 90 000 s-1 in 10 s. (a) The brass-cooling water interface is at constant temperature (Figure 2b). (b) The brasscooling water interface temperature is allowed to vary with time (Figure 2c).
where the βn, n ) 1, 2, 3, ..., are the positive roots of
sin
βn
βn H xκb cos a) kb βn xκb
a-
xκb
[
x
]
βn βn βn κb H xκb tan (b - a) cos a+ sin a κl κl xκb kb βn xκb
kl kb and
βn H xκb Xbn(x) ) cos x+ sin x xκb kb βn xκb βn
[
Xln(x) ) Cln cos βn
cos Cln ) cos
βn
b
xκl
xκl
x + tan
xκl
[
cos
(b - a) g/n )
N)
βn
βn
βn
b sin
xκl
βn
]
x
xκb
]
xκb
µ N
∫abXln2(x′) dx′
kb κb
k
∫0aXbn2(x′) dx′ + κll∫abXln2(x′) dx′
The cooling water in Ram’s experiments flowed through “flat ducts” (25.4 mm wide, 4.5 mm thick, and about 175 mm long; the thickness was read from Ram’s Figure 578). The flow was just within the transition zone from laminar to turbulent flow (Reynolds number Re based on the hydraulic radius is 2229). For lack of better correlations, the cooling-water flow was considered to be laminar (Re < 2100). The heat-transfer coefficient H can be estimated from Graetz’s solution (Jakob,18 pp 462-4)
(
)
4cpFQ HDe ) 1.85 k πkL
()
HDe µw k µb
0.14
)
[(
1.75
βn H xκb a sin kb βn xκb
a+
water. The quantity within brackets is a modified Graetz number for flow through ducts. The heattransfer coefficient H in eq 8 is based on the arithmetic mean temperature of the cooling water and approaches that based on the logarithmic mean temperature difference for Graetz numbers larger than 2000.18 Equation 8 is similar to the empirical relationship
1/3
(8)
where De ) 4Rh, Rh is the hydraulic radius, Q is the cooling-water flow rate, L is the length of the annular channel, and cp, k, and F are respectively the specific heat, thermal conductivity, and density of the cooling
)
4cpFQ πkL
b
(
+ 0.04
) ]
L Gr × Pr De
0.75 1/3
(9)
b
used to correlate a large number of heat-transfer measurements. McAdams19 reports eq 9 fits the data with a maximum deviation of 60% (pp 235-7). Here Gr is the Grashof number and Pr is the Prandtl number of the flow, and the subscripts b and w denote values in the bulk and those at the wall, respectively. If the Grashof and Prandtl numbers are small and the wall temperature does not significantly differ from the bulk temperature, then free convection effects are negligible and eq 9 reduces to eq 8 with a 6% difference in the numerical constant. In Ram’s8 device at 20 °C, Pr ) 7, Gr ≈ 20, and the Graetz number was 1860. Hence, either eq 8 or eq 9 may be used to estimate the heattransfer coefficient. In subsequent calculations, a heattransfer coefficient H of 1500 J/(s m2 K) estimated with eq 8 was used. With Mathematica,16 the first seven terms of the series in eqs 6 and 7 were used to calculate temperature in the liquid and brass. The temperature at the midplane of a 60 wt % glycerin-water solution would then rise by 1.45 °C in the first 11.5 s of a 20 s ramp-up, i.e., at 55 000 s-1, and about 4.5 °C when the shear rate was 90 000 s-1. The temperature would rise by 1.13 and 3.65 °C, respectively, at the same shear rates during a 10 s ramp-up, while it would rise by 0.85 and 2.8 °C during a 5 s ramp-up. The shear stress τ, and hence the apparent viscosity, during a 5 s ramp-up would be 2% smaller than the true viscosity at 55 000 s-1 and 7.03% smaller at 90 000 s-1 and is enough to account for the deviation in Ram’s experiments.8 The longer the ramp, the greater the apparent shear thinning predicted (Table 1 and Figure 4b). Three terms instead of seven in eq 6 lowered the estimate of the apparent viscosity by less than 0.05%. Doubling the heat-transfer coefficient H raised the estimated apparent viscosity at 90 000 s-1 in Table 1 by 1% during a 5
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1733
Figure 4. Comparison of Ram’s8 data on 62.5 wt % glycerin-water and calculations on 60 wt % glycerin-water. (a) Shear stress versus shear rate: the dotted line corresponds to Newtonian behavior without shear heating. (b) Ratio of the apparent viscosity and viscosity at low shear rate versus shear rate. The circles are representative data points from part a. The dashed lines are theoretical estimates for 60 wt % glycerin-water with an infinite heat-transfer coefficient (Figure 2b) and finite heat-transfer coefficient by forced convection (Figure 2c). Table 1. Comparison of Experiment with Theory: Concentric Cylinders % reduction in apparent viscosity at
ramp heat-transfer duration, coefficient, γ˘ ) γ˘ ) s J/(s m2 K) 55 000 s-1 90 000 s-1 experiment (Ram8) theory
a
10-20 10-20 3 5 10 20
infinite 1500 1500 1500 1500
1.96a 0.6 1.73 2.15 2.94 3.89
6.2a 1.6 5.60 7.03 9.38 11.63
Ram’s experiments on 62.5 wt % glycerin-water.
s ramp-up, by about 2% during a 10 s ramp-up, and by about 4% during a 20 s ramp-up and is still sufficient to explain the decrease reported by Ram. Figure 3 compares the effect of the thermal boundary condition on the temperature distribution in the glycerin-water and confining brass walls during a 10 s ramp-up. Measurements with Parallel Disks We measured shear viscosity in a room at 22.5 °C with an ARES controlled-strain rheometer (Rheometrics Scientific, Piscataway, NJ). The rheometer was equipped with a force-rebalance transducer that measures torques between 0.02 g cm (1.96 µN m) and 2000 g cm (0.196 N m) in two ranges to within 1% accuracy. The liquid was sheared between two titanium alloy disks, each 50 mm in diameter and 1.5 mm thick. Because of the unusually small gaps used (down to 30 µm), the alignment of the actuator (motor) and transducer axes, their concentricity, and gap uniformity were crucial. The total indicated runout of the actuator was measured by a Mitutoyo dial gauge (0.0001 in. or 2.54 µm, least count) to be less than (0.0002 in. The transducer slide axis was parallel to the actuator axis to within 0.0002 in. over 1.5 in. of its travel. The face of one of the disks was perpendicular to the actuator axis to within (0.0004 in. and the other to within (0.0001 in. Actuator and transducer were concentric to within (0.0003 in. The shear viscosity of a silicone oil (nominal viscosity 0.6 Pa s) and 60 wt % glycerin in water was measured at different disk separations between 0.02 and 0.75 mm to quantify the effect of alignment errors. Figure 5b shows the average of the apparent viscosity of these
liquids between 100 and 1000 s-1. The error bars bound the maximum and minimum values; no thinning was observed over this range of shear rates. The apparent viscosity falls as the gap is narrowed and more sharply so at smaller gaps. Two reasons for the drop in viscosity can be attributed to canted fixtures (Figure 5a). First, the measured gap between the disks corresponds to the distance between the points of closest approach on the disks (high points) and is consequently incorrect. Second, the gap is nonuniform; the azimuthal flow is hence a combination of drag flow due to the moving plate and azimuthal pressure gradient-driven flow. The torque reduction owing to the nonuniform gap can be estimated by integrating Sommerfeld’s solution for a flooded journal bearing (eq 1) across the radius. The dimensionless eccentricity, defined by
(r) )
∆h r Rhnom
rises with radius r. Here ∆h is the cantedness at the largest radius R (obtained from perpendicularity measurements), and hnom is the nominal separation between the plates (Figure 5a). Radial flow can be neglected at the first approximation. Figure 5b shows that both of these factors are significant and explain the drop in apparent viscosity at small separations. The theoretical estimates are based on the perpendicularity measurements, which were accurate only up to 2.54 µm. Other factors such as run-out of the actuator axis and departure of the transducer and actuator axes from coaxiality may also affect the flow field and alter the measured torque and thus the inferred viscosity. High shear rates were achieved by ramping the actuator to its maximum speed and down as described by Connelly and Greener.20 The shear rate was raised in two steps: from 0 to 1000 s-1 and from 1000 to 90 000 s-1. Each step was 3 s long. The shear rate was lowered similarly. In order to collect torque data more rapidly (every 10 ms) than normally possible with the ARES rheometer, a software low-pass filter was turned off.21 The motor is rated to attain two-thirds of its programmed speed in less than 3 ms. Figure 6 shows the apparent viscosity versus shear rate during the experiment. The apparent viscosity is constant over the range of shear rates attained and varied by less than (1% with three different sample loadings. Ramps longer than
1734 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999
Figure 5. Apparent viscosity measured with parallel disks. (a) Schematic of the parallel disks, as used. When the disks are canted, the nominal gap is incorrectly measured and the gap is nonuniform. (b) Apparent viscosity of silicone oil (nominal viscosity 0.61 Pa s, circles) and 60 wt % glycerin-water (triangles) at different apparent gaps between the parallel disks. Shear rates varied from 100 to 1000 s-1; error bars bound the maximum and minimum values. The apparent viscosity measured with the largest gap was used to normalize the experimental results with each liquid. The dotted line shows the change in viscosity just due to ∆h, the nominal error in reading the gap. The dashed line shows the estimated total effect of canted disks. The apparent viscosity at a gap of 0.5 mm was used to normalize the theoretical estimates.
Figure 6. Measured shear stress and apparent viscosity of 60 wt % glycerin in water as it is sheared in a 40 µm apparent gap between two 50 mm diameter parallel disks. The shear rate is raised from 1000 to 90 000 s-1. Torque at low rates is smaller than the minimum that can be measured, and hence the spread in viscosity is larger.
5 s showed hysteresis in the measured torque when the shear rate was ramped-up and then ramped-down (also see Connelly and Greener20). After the correction for small gaps interpolated from experimental results (Figure 5b) is applied, the viscosity is 9.66 ( 0.14 mPa s, 1.73% lower than Sheely’s10 reported value at 22.5 °C. Conclusions Ram’s8 measurements of glycerin-water viscosities at high shear rates have been interpreted by others as evidence of shear thinning. Those measurements were of shear torque versus rotation rate in a Couette-type instrument, the Merrill-Brookfield viscometer. Two alternative interpretations have now been explored. Both invoke artifacts that can account for the torque rising progressively less than in proportion to the rotation rate as the mean shear rate in the gap passes 55 000 s-1 and ramps-up to 90 000 s-1. The liquid gap itself and the instrument’s largerradius air-bearing gap (Figure 1a) could not have been appreciably uncentered by the thrust on the latter by the reaction force of the torque-measuring member and hence cannot account for the deviation reported. Heating by viscous dissipation can cause apparent shear thinning as shown by Sukanek and Laurence.22 The liquid in the gap in Ram’s device was certainly
heated by viscous dissipation. Our estimate of the efficacy of the instrument’s provision for heat removal and of the consequent viscosity variation in the gap suggests that the entire reported 6.2% deviation at 90 000 s-1 stems from viscous heating. A 5 s ramp with heat removal by forced convection of cooling water predicts a 7.03% deviation, close to Ram’s8 observations (Table 1). The slower the ramp, the greater the shear thinning predicted. Other liquids in the same gap must have been viscously heated; Ram reported shear thinning in 14 mPa s ethylene glycol at 60 000 s-1. Equation 3 predicts a 3% drop in viscosity of ethylene glycol at the same shear rate at 17.5 °C (c of 2398 J/(kg K), F of 1114.9 kg/m3, µ of 23 mPa s, temperature coefficient of viscosity of 0.041 K-1, and k of 0.241 at 25 °C23,24). More realistic thermal boundary conditions, i.e., forced convection incorporated in eqs 6 and 7, predict higher deviations that also depend on ramp duration. Our measurements with parallel disks separated by narrow gaps indicate that the viscosity of 60 wt % glycerin in water is constant up to 90 000 s-1. Acknowledgment This work was supported by industrial and National Science Foundation funds through the Center for Interfacial Engineering at the University of Minnesota. Literature Cited (1) Ram, A. High-shear Viscometry. In Rheology: Theory and Applications; Eirich, F. R., Ed.; Academic Press: New York, 1967; Vol. 4, p 251. (2) Neale, S. M. The Viscosity of Oils at High Rates of Shear. Chem. Ind. (London) 1937, 37, 140. (3) Weltmann, R. N. Consistency and Temperature of Oils and Printing Inks at High Shearing Stresses. Ind. Eng. Chem. 1948, 40, 272. (4) Hagerty, W. W. Use of an Optical Property of GlycerineWater Solutions to Study Viscous Fluid-flow Problems. J. Appl. Mech. 1950, 17, 54. (5) Dumanskii, I. A.; Khailenko, L. V. Rheological Properties of Glycerine Solutions. Kollidn. Zh. 1960, 22, 277. (6) Merrill, E. W. Non-Newtonianism in Thin liquids: Molecular and Physical Aspects. In Modern Chemical Engineering Series; Acrivos, A., Ed.; Reinhold: New York, 1964; Vol. 1, p 141. (7) Ram, A.; Tamir, A. A Capillary Viscometer for NonNewtonian Liquids. Ind. Eng. Chem. 1964, 56, 47.
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1735 (8) Ram, A. High-Shear Viscometry of Polymer Solutions. Sc.D. Thesis, Massachusetts Institute of Technology, Boston, MA, 1961. (9) Merrill, E. W. A Coaxial Cylinder Viscometer for the Study of Fluids under High Velocity Gradients. J. Colloid Sci. 1954, 9, 7. (10) Sheely, M. L. Glycerol Viscosity Tables. Ind. Eng. Chem. 1932, 24, 1060. (11) Gross, W. A. Fluid Film Lubrication; Wiley-Interscience: New York, 1980. This is an extension and revision of the 1962 publication Gas Film Lubrication by W. A. Gross. (12) Tipei, N. Theory of Lubrication; Stanford University Press: Stanford, CA, 1962. (13) Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids, 2nd ed.; Oxford University Press: Oxford, U.K., 1959; p 131. (14) Perry, R. H.; Green, D. W.; Maloney, J. O. Perry’s Chemical Engineers’ Handbook, 6th ed.; McGraw-Hill: New York, 1984. (15) Baumeister, T.; Avallone, E. A.; Baumeister, T., III. Mark’s Standard Handbook for Mechanical Engineers, 8th ed.; McGrawHill: New York, 1978. (16) Wolfram, S. Mathematica, 2nd ed.; Addison-Wesley: Reading, MA, 1991. (17) O ¨ zis¸ ik, M. N. Boundary Value Problems in Heat Conduction; Dover: New York, 1989. This is a republication of the 1968 International Textbook Company edition.
(18) Jakob, M. Heat Transfer Vol. 1; Wiley: New York, 1949. (19) McAdams, W. H. Heat Transmission, 3rd ed.; McGrawHill: New York, 1954. (20) Connelly, R. W.; Greener, J. High-shear Viscometry with a Rotational Parallel-disk Device. J. Rheol. 1985, 29, 209. (21) Mackay, M. E.; Liang, C.-H.; Halley, P. J. Instrument Effects on Stress Jump Measurements. Rheol. Acta 1992, 31, 481. (22) Sukanek, P. C.; Laurence, R. L. An Experimental Investigation of Viscous Heating in Some Simple Shear Flows. AIChE J. 1974, 20, 474. (23) Marie, C.; Bodenstein, M.; Bruni, G.; Cohen, E.; Wilsmore, N.-T.-M. Annual Tables of Constants and Numerical Data; GauthierVillars: Paris, France, 1925/26; Vol. 7, p 89. (24) Marie, C.; Bodenstein, M.; Bruni, G.; Cohen, E.; Wilsmore, N.-T.-M. Annual Tables of Constants and Numerical Data; GauthierVillars: Paris, France, 1927/28b; Vol. 8, pp 31 and 59.
Received for review September 2, 1998 Revised manuscript received January 12, 1998 Accepted January 13, 1998 IE9805685
