Particle motion in sheared non-Newtonian media

Handbook of Conveying and Handling of Particulate Solids A. Levy and H. Kalman (Editors) 9 2001 Elsevier Science B.V. All rights reserved.

459

Particle motion in sheared n o n - N e w t o n i a n media K. C. Wilson Department of Civil Engineering, Queen's University at Kingston, ON Canada K7L 3N6 Some particles do not settle in a medium with a yield stress. Applying a transverse shear rate causes these particles to settle, and the experiments reported here indicate that a dimensionless measure of settling increases approximately as the square of the dimensionless transverse shear rate. 1. INTRODUCTION Many slurries of interest to industry have as their principal component a homogeneous mixture of fine particles in water (or some other liquid). Such mixtures typically display nonNewtonian flow characteristics. In addition, industrial slurries often have a fraction (perhaps only a small one) of larger particles, and it is important to know if these larger particles will be supported by the non-Newtonian medium, or will settle out. From an engineering viewpoint, the prediction of settling is very important, since the resulting deposition can lead to greatly increased power requirements for slurry pumping, and may induce system instability. In practice, the non-Newtonian materials mentioned above are often viscoplastic. For a material of this sort no strain rate is produced until the applied shear stress x exceeds the yield stress Xy. Hence, if a discrete particle is placed in a quiescent viscoplastic medium, the particle will not settle unless it is heavy enough to produce a shear stress within the medium that exceeds Xy. If, however, the medium is sheared by some other mechanism -- for example, rotation of a viscometer bob, or pressure difference along a pipeline -- particle settling is initiated or enhanced. The following sections of this paper describe a device used to develop this sort of shear, and present initial experimental findings.

2. BACKGROUND For viscoplastic materials the rheograms (i.e. plots of shear stress x versus strain rate dU/dy) are very often approximated by the Bingham model, a straight line of the form x = "CB+ riB dU/dy

(1)

Here the slope of the line, riB, is the tangent or Bingham viscosity; and the stress intercept, "oR, is the Bingham yield stress.

460 As opposed to the case of flow in a circular tube, where the wall shear stress is uniform, the shear stress set up on the surface of a spherical particle is non uniform. Nevertheless, the mean surficial shear stress, denoted by Xm, forms a useful stress measure. This mean shear stress is defined as the submerged weight force divided by the surface area of the sphere, which is rtD 2 where D is the sphere diameter. The submerged weight force is the product of the sphere volume ~D3/6 and (lOs - lOf)g,where g is gravitational acceleration and 9s and lOfare the densities of the solid and fluid phases, respectively. Thus the mean surficial shear stress is given by aTm = (lOs- lOf)gD/6

Analogous to the pipe-flow case, a shear velocity, V*, can be defined

(2) as

~/'l:m/lo f .

This will

be used below. A useful dimensionless parameter, denoted by L, compares the Bingham yield stress to Xm, thus = 6 xB/[((p~- pf)gD]

(3)

The value of L at which settling begins in a quiescent medium, say ~q , will not be unity; and must be obtained experimentally. In their experiments Ansley and Smith [ 1] found that tomato sauce (ketchup) obeyed the Bingham equation very closely. Using various particles in this medium, they found Lq ~ 0.53. For media that diverge somewhat from Bingham behaviour at small shear rates, the value of Lq is somewhat larger. Extensive experiments by Duckworth and his co-workers (for example, Duckworth et al.[2]) showed typical values near 0.6. They also showed that slurries in which large particles did not immediately settle remained stable for long durations, measured in terms of weeks. It is known, however, that settling is enhanced or initiated when the slurry is subjected to an externally imposed strain rate. Early experiments were performed by Highgate and Warlow [3] using spheres in a pseudoplastic fluid that was sheared in the space between coaxial cylinders. As the fluid had no yield stress, the particles settled slowly in the quiescent case. When shear was applied by rotating the outer cylinder, the settling velocity increased; reaching five times the initial value at high strain rates. Subsequently Thomas [5] used a similar apparatus, filling the annular space with chinaclay slurries that approximated Bingham behaviour. The spheres that he used were stationary under static (quiescent) conditions, but when one of the cylinders was rotated they settled, and their settling velocities increased with increasing strain rate. Thomas noted that his results should be applicable to slurries containing particles in a viscoplastic medium. These could be non-settling when stationary, but on being pumped through a pipeline, settlement would occur in the sheared outer annulus of the flow, though it should not occur in the unsheared central core. This behaviour has recently been recorded by Pullum and Graham [4], who made tomographic scans through a test pipeline using magnetic resonance imaging (MRI).

461 3. E X P E R I M E N T A L W O R K

These MRI scans indicate the potential industrial importance of particle settling through viscoplastic media. The general trends had been investigated by Thomas [5] but further experimentation was clearly required. It was decided to construct a cup and bob similar to those of a conventional rotary viscometer, but modified so that the gap between them increases with depth, and thus the strain rate decreases with depth. This apparatus was constructed at Curtin University, Western Australia. A vertical section is shown on Fig. 1. A brief description, together with some preliminary results, was presented by Wilson [6]. Since that time, a second bob has been fabricated. It is also shown on Fig. 1. The particles tested were glass ballotini (Ps = 2500 kg/m 3) of various diameters from 0.70 mm to 1.19 mm. The test media were solutions of various concentrations of long-chain molecules in water. They were transparent, and rheograms showed that the Bingham approximation was appropriate. It was verified that for all cases the particles remained stationary in an unsheared sample of the medium. Shear strain was then applied by rotating the bob at specified angular speeds from 1 to 4 revolutions per minute, and for each run the particle location was observed at selected times. It was found that particle motion ceases at a level near the bottom of the bob, and no further settling occurs, even if the testing continues for 24 hours. This is an important finding, because it indicates that it is the transverse strain rate, rather than the transverse stress itself that promotes settling.

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59 Note- all dimensions in mm

Fig. 1. Vertical section through apparatus (schematic)

462 For the settling portion of each run, particle location was plotted versus time, on a logarithmic basis. The plot for a typical run (0.725 mm particle in 2% floxit medium) is shown on Fig. 2. The particle settles with a rate that decreases with depth, and the data can be approximated, piecewise, by straight lines on the logarithmic plot. The slope of each line segment was determined, as were the geometric mean values of depth and time. These were used to obtain a representative fall-velocity, Vt, for each segment. The representative shear rate, dU/dr, was based on the size of the gap between cup and bob at the geometric mean depth and the velocity difference at this level. As expected, Vt tended to increase with increasing dU/dr. This trend was complicated, however, by the effects of the various media used and the different particle sizes; thus it was decided to employ dimensionless variables. The shear rate was made dimensionless by multiplying dU/dr by D/V*. The settling velocity Vt was used to calculate the equivalent viscosity ~eq of a Newtonian fluid (of the same density as the medium) which would produce Vt if the particle were dropped through it in quiescent conditions. For slow motion with no inertial effects (as would be the case for these experiments) ~teqis inversely proportional to Vt. The ratio lqB/geq thus represents a suitable dimensionless measure of Vt. Figure 3 shows, on a logarithmic basis, a plot of the two dimensionless ratios for the results obtained to date. There is some experimental scatter, but the trend is clear. As shown by the line on the figure, the dimensionless measure of settling velocity increases approximately as the square of the dimensionless shear rate. More data points are required before this correlation can be recommended for design purposes, and additional experiments are now being planned. It is worth noting that the present experiments produce equivalent viscosities ~l,eq that are much larger than riB. ~,

1E+02

1E+02

1E+03

1E+04 Time (s)

Fig. 2: Logarithmic plot of location

versus

time for typical run

1E+05

463

1.00E-03 .................................................................................

-

1 . 0 0 E - 0 4

__

9

"

9

rib =2.7 ( - D ~ d__~_r/ 2 1.00E-05 + ..............~ - - + 1.00E-03

....~-+-H+~- ...... +---+--~-+--~--~+1.00E-02 D dU

1.00E-01 9 small gap-

big gap

V* dr Fig. 3: Logarithmic plot of rlB/~Leqv e r s u s dimensionless transverse shear rate 4. CONCLUSION For non-Newtonian media with a yield stress some particles do not settle in quiescent conditions, and this observation has led to the concept of stable slurries. However, an externally-applied shear in the medium may initiate particle settling, with possible deleterious effects for pipeline transport. It was decided to add to the rather limited existing data in this area by experimenting with a novel cup-and-bob apparatus in which shear rate decreases with depth. The experimental media were transparent, and approximated Bingham-plastic behaviour. The particles that were used did not settle under quiescent conditions. It was found that the particles settled, albeit slowly, wherever the transverse shear rate was non-zero. Settling velocity tended to increase with strain rate. For the present experimental results it was shown that the viscosity ratio rlB/~teq, which is proportional to settling velocity, was very small. These results also indicate that the viscosity ratio is approximately proportional to the square of the dimensionless transverse shear rate. REFERENCES

1. R.W. Ansley and T.N. Smith, Motion of spherical particles in a Bingham plastic, AIChEJ., 13(6), pp. 1193-1196, 1967. 2. R.A. Duckworth, L. Pullum, G.R. Addie and C.F. Lockyear, The pipeline transport of coarse materials in a non-Newtonian carrier fluid. Proc. Hydrotransport 1O, BHRA Fluid Engineering, Cranfield, UK pp 69-88, 1986.

464 3. D.J. Highgate and R.W. Whorlow. The viscous resistance to motion of a sphere falling through a sheared non-Newtonian liquid, Brit. J. AppL Phys. 18, pp 1019-1022, 1967. 4. L. Pullum, and L.J.W. Graham, A new high-concentration pipeline test loop facility, Proc Hydrotransport 14, BHR Group, pp 505-514, 1999. 5. A.D. Thomas, Settling of particles in a horizontally sheared Bingham plastic, First Nat'l. Conf. on Rheology, Melbourne, Australia, pp 89-92, 1979. 6. K.C. Wilson, The rocky road of pipeline rheology, Rheology in the Mineral Industry II, Oahu, Hawaii, USA, pp 5-10, 1999. NOMENCLATURE D g r U Vt V, y ~n )~

particle diameter gravitational acceleration radial distance velocity settling velocity shear velocity = ~/x m / P f normal distance Bingham viscosity (Eq. 1) see Eq. 3

m m/s 2 m m/s m/s m/s m Pa.s

)gq ~teq pf Ps x XB Xrn "l~y

)~ at which settling begins equivalent viscosity Pa.s density of fluid kg/m 3 density of solid kg/m 3 shear stress Pa Bingham shear stress (Eq. 1) Pa see Eq. 2 Pa yield stress Pa

ACKNOWLEDGEMENTS The author's stay in Westem Australia was made possible by a C.Y. O'Connor Fellowhship, facilitated by Prof. R.R. Horsley of Curtin University. Subsequent work in Canada was funded by an NSERC grant. Laboratory experiments at Curtin were carried out by Dr. T. Kealy, and at Queen's by Ms. A.M. Bach-Jacobsen. The author wishes to thank these individuals and organizations for their contributions to this work.