Powder Technology, 76 (1993) 165-174 165

The settling of spheres in clay suspensions

B. J. Briscoe, P. F. Luckham and S. R. Ren Particle Technology, Department of Chemical Engineering~ Imperial College, London (UK)

(Received July 17, 1992; in revised form January 25, 1993)

Abstract

This paper describes the results obtained for a falling ball experiment in opaque aqueous bentonite suspensions. The flow behaviour of the bentonite suspensions has been effectively described by a Bingham flow model which incorporates a yield stress and a plastic viscosity and by a power-law model in terms of an apparent viscosity and a shear rate. A modified particle Reynolds number, based on this power-law model, is formulated for the comparison with the classic drag coefficient correlation derived from the Bingham model and shows the general applicability of the drag coefficient correlation method for non-Newtonian fluids. Both correlations are also compared with the predictions of a finite element method solution based upon a Bingham response. The limitations of the correlation method have been quantitatively addressed to the falling of spheres under dynamic, or incomplete gelatinous, conditions. The wall effect and the influence of the ball surface roughness on the falling sphere characteristics for the same fluid are also described.

Introduction

In the description of the free-settling of particles in Newtonian fluids, drag coefficient correlations formu- lated with particle Reynolds number have been widely and successfully used. The classic works, and the recent developments in this field, have been critically reviewed by Clift et al. [1], Flemmer and Banks [2], and Khan and Richardson [3]. Usually, the drag coefficient, Cd, for a sphere of diameter, d, is defined within the frame- work of the Stokes' law,

Ca 4 (p , -p )gd 8Fg = 3pV= - ,rrd2pV2 (1)

where Ps is the density of the sphere, and p the density of the fluid, V is the terminal velocity of the sphere and g is the gravitational acceleration. Fg is the effective external force acting on the sphere. The drag coefficient includes the non rheological parameters of the fluid and its response to the deformation. The other pa- rameter, the particle Reynolds number, Re, for a New- tonian system, provides the interrelationship between the viscosity of the fluid, ~7, and the settling velocity of the sphere,

pdV Re = ~ (2)

In general, the correlation between the drag coefficient and the Reynolds number is written in the form,

Ca =xRY (3)

where x and y are the correlation parameters. Various correlations have been obtained with different regions of flow; these are shown schematically in Fig. 1. There are also many modifications available which attempt to improve the accuracy of the above correlations [1--4].

These ideas, for the Newtonian systems, have also been adopted to deal with the non-Newtonian fluids where the behaviour is more difficult to interpret since the viscosity of the fluid is at least shear rate-dependent. Many constitutive relationships may be invoked to pre- scribe the interrelationship between the imposed stress, r, and the rate of shear, ~,; Bingham and power-law fluid descriptions are popular first order approximations.

I Laminar Transition Turbulent region I region region

cd ~ 1

I C 18.5

C 2 4 I . e

I x

i i 10 -3 0.2 500 105

~- Re

Fig. 1. Schematic diagram of the correlation between the drag coefficient and the Reynolds number for Newtonian response.

0032-5910/93/$6.00 © 1993- Elsevier Sequoia. All rights reserved

166

In dealing with a simple Bingham fluid characteristic (~'= ~'y + ~Tp~Y), the central problem is to separately in- clude the influences of the yield stress, ~'y, and the plastic viscosity, ~Tp~, on the drag coefficient. Several formulations and modifications either for the drag coef- ficient or/and the particle Reynolds number have been reported [5-7]. A typical example of these relationships has been presented by Ansley and Smith [5] for a Bingham fluid. They postulated a flow pattern of a solid and rigid sphere moving in a Bingham plastic fluid. The normal pressure and the shear stress fields imposed on the plastic material, by the motion of the sphere, induce the material into flow in an envelope surrounding the sphere. Within the envelope, the motion of the sphere and the displaced fluid are steady. As the sphere-envelope system passes through the rigid plastic material, it instantaneously causes localized flow transformations at a distinct boundary. The shape of this 'yield envelope' has been deduced from a consid- eration of the slip-line fields created by stresses in the isotropic plastic material, as the form of truncated toroid. Based on this flow pattern, the relative effect of the yield stress on the total drag was determined as 7~r2dZry/8. The viscous drag (contributed in terms of the plastic viscosity) on the sphere was taken as 3~rdV~qp~. A non-dimensional expression, analogous to the Reynolds number, which was formulated in terms of the ratio of the inertia force to the drag force, was defined as,

1 Nm = (4)

77rT"y ~pl Jr" dpV 24pV 2

where Nru is the so-called dynamic parameter which is then correlated with the drag coefficient, Ca, as Cd =aNR~ b. The first term of the denominator in eqn. (4) is equivalent to the conventional Reynolds number for a Newtonian system (eqn. (2)) where 77 is replaced by the plastic viscosity, ~Tp~- The contribution from the yield stress term, that is the response descriptor for the system as ~,~0, is added as the quantity 7~-~-y/ 24pV z. A good correlation between the drag coefficient and the dynamic parameter was shown for falling ball experiments using a tomato sauce by the same authors.

On the other hand, Beris et al. [7] have presented a numerical solution to the problem for the Bingham fluid. They predicted the shape of the yield surface of the creeping flow around a sphere and the velocity and the pressure fields in the fluid portion of a Bingham material using a finite-element/Newtonian algorithm. They proposed that the shape of the flow was again a toroidal: Fig. 2. The size of the fluid region, as well as the size of the small static gap region at each pole, were found to be governed by the yield stress of the

Fig. 2. Schematic diagram of the mobilised fluid envelope sur- rounding a falling sphere in a Bingham plastic fluid.

Bingham fluid. Their computational results were de- scribed by the correlation of two dimensionless pa- rameters; a so-called Stokes drag coefficient (Cs) and a Bingham number (NRz). These parameters were de- fined as

c~= F~ 37rd~Tel V (5)

and

NR2= ryd (6) 7/pl V

where Fg is the external force acting on the sphere as in eqn. (1). This numerical solution is undoubtedly an efficient way to deal with this complicated problem. It provides a good approximate solution for the flow surrounding a sphere in an ideal Bingham plastic fluid and also may ultimately provide a useful description for more complex fluids. A comparison of experimental results for spheres settling in concentrated bentonite suspensions under stagnant conditions has been pre- sented elsewhere by the current authors [8] and will be further discussed later in the present paper. The notion of a mobile envelope moving within a structured plastic domain conveys the idea of a distinct boundary which is consistent with the Bingham hypothesis, The 'front' is mobilised when the yield stress ~'y is exceeded. The plastic character of the fluid is eventually restored at the tailing boundary. The Ansley and Smith and Beris et al. approaches implicitly suppose an instan- taneous restoration of the gel structure and hence the contribution of the plastic component is not retarded; in this context the term 'retardation' would convey the idea of a time or strain-history dependent plastic yield stress. Similarly, we would also suppose that the strength of the plastic characteristic throughout the medium would also exhibit thixotropy in the general sense. There is evidence that the plastic response of these suspensions is a function of the lasped time after the introduction

167

of a shear deformation. These time dependent facets are discussed further in the conclusion of this paper.

In the study of particle settling in systems which approximate to the power-law fluid description ( r= K~n), the requirement is to formulate an equivalent Reynolds number which matches with the shear-de- pendent viscosity of the fluids. The idea of a distinct sheared envelope is not introduced directly as is the case for the Bingham model. Instead, the strain rate induced is related to the local value of the imposed stress (see later) and presumably a gradient of strain evolves in the medium surrounding the falling ball. In this way the mobilised envelope will be generated but without a distinct boundary. There usually have been two approaches adopted with this model. In the study of cuttings transport in oil-well drilling process, many researchers have calculated the Reynolds number using an effective viscosity of the fluid in certain positions in the wellbore (e.g. the wall of a well) as an equivalent of the Newtonian viscosity [9-11]; that is, the stress and strain rate are specified directly. As an alternative, several investigators have suggested using an effective shear rate induced by the particle to calculate the equivalent Newtonian viscosity of the fluid around the particle [12-17]. The most popular formulation of the particle Reynolds number for the power-law fluids is

pd,,V 2-,, (7) NR3-- K

The correlation between the drag coefficient (eqn. (1)) and this particle Reynolds number has proven to be effective for many power-law fluids including polymer solutions [13, 17].

As an alternative, the power-law viscosity-shear rate model,

K '7= U (8)

where m>0, may be invoked. Then, based on the assumption of the formation of a sheared-envelope of mobilised fluid forming around the falling sphere [4, 5, 18], a general form of the average equivalent shear rate, 5'o, across the sheared fluid regime, imposed by a settling sphere, may be expressed as

V :Ye-- Kvd (9)

where K~, is a coefficient related to the size and shape of the envelope. The equivalent average viscosity around the sphere may then be obtained by applying eqn. (8) assuming a power-law response. From the expression of the classic Reynolds number (eqn. (2)), a modified particle Reynolds number, NR4, by substitution, is given as

p d V pd 1-mV1 += U~4 = - (10)

In the above equation, the only unknown term is K r There are many ways which may be used to determine the value of K~; unfortunately, the results are quite different. Based on the stream functions analysis, equiv- alent values of K~ between 0.5 and 0.752 have been obtained [19]. As is indicated in eqn. (7), other workers have preferred to take K~ simply as unity. Physically, K, is a characteristic factor of the sheared-envelope but there is no reason for the fluid envelope to remain the same in size or shape when the fluid component is undergoing shear under all conditions. Therefore, K, may not be a constant. If we also take arbitrarily K~= 1, here, the modified particle Reynolds number is then written as

NR4 =NR3__ P d l - r ' V l +m K , m = l - n (11)

The accuracy of the above expression is generally satisfactory for shear-thinning fluids which are described by the Cross model in the power-law regime.

There have also been attempts to create general equations using the drag coefficient and the particle Reynolds number to explicitly estimate the velocity of a falling-sphere under various conditions [17, 20].

There is no doubt that the correlations reviewed above are acceptable for many particle settling processes in Bingham and power-law fluids and the effect of the motion of the fluid (i.e. particles settling in dynamic conditions) has been addressed by many researchers [8, 13, 161.

In this paper, we present a modified particle Reynolds number formulated for the power-law regime of the Cross model for viscosity [21]; the characteristics of the Cross model are shown in Fig. 3. The experimental results obtained with bentonite suspensions, which are

Viscosity

I

1"1 o I

[ . ,

I Shear Rate

Fig. 3. Schematic representation of shear-thinning behaviour (the Cross model).

168

either described by the Bingham flow model or the power-law model in terms of the apparent viscosity versus shear rate, are described in order to quantify the drag coefficient correlations for non-Newtonian fluids with different flow models. The limitations of these methods for shear-thinning and thixotropic fluids is also quantitatively shown by falling-ball experiments in bentonite suspensions with deliberately controlled shear histories. The influence of ball surface conditions (e.g. roughness) and the container wall on the drag coefficient is also described.

Experimental method

The falling-ball apparatus is schematically shown in Fig. 4. The ball was released on the top of the fluid manually and its location was continuously monitored by an ultrasonic technique whilst it falls down inside the tube.

The balls used in the experiments were made of different materials, including 'Plasticine' (a mixture of mineral oil and clay), marble and various metals. A large range of densities and sizes of balls were made by coating a layer of 'Plasticine' material onto plastic (polyethylene), marble or metal balls. The density of the balls used ranged from 1 150 kg m -3 to 8 000 kg m -3. The measurements were carried out in a 75 mm (ID) tube at room ambient. In order to ensure a similar shear-history in each measurement, and to minimize the thixotropy characteristics of the clay suspensions,

a fixed stirring procedure (e.g. using a rotational stirrer or by turning a ball up and down several times) was applied to the sample before the ball was released. After stirring, the ball was immediately released within 30 s (i.e. the standing time of the sample after pre- shearing was less than 30 s). The importance of re- producing the dynamic condition, or shear-history, of these fluids will be discussed in a later part of this paper.

Water-based bentonite (ABS 350s, ECC Int., UK) suspensions were used. The clay concentrations ranged from 4--10% (wt.). Their rheological parameters were separately measured using a Bohlin VOR concentric cylinder rheometer (Bohlin Reologi, Sweden).

All the experiments were conducted at 25 °C.

Results and discussions

The experimental data obtained from the Bohlin rheometer, for various bentonite suspensions fitted graphically by the Bingham and power-law flow models, are shown in Fig. 5 and Fig. 6, respectively. The plots of the data for both models suggest that bentonite suspension is a sensible fluid for comparing the falling- sphere correlations derived from these two different rheological models.

Wall effects The wall effect was expressed by a relative drag

increase coefficient, ~w, which is defined as

Cdw-Cdn 'e~= Caw (12)

20

F - x

"= 15

..~1o r.~

• . • i , , •

• 2 0 ' 4'o 8'0 8'o 1 o o 1 1 4 o 6 0 Shear rate (l/s)

Clay content (wt)

t, 10%

• 9%

o 7%

• 4%

Fig. 5. R h e o g r a m plot ted as shear stress against shear rate for Fig. 4. Schematic diagram of the falling-ball apparatus, ben ton i t e suspensions with various clay contents .

10

.01 1 1 0 1 00 1000

Shear rate (l/s)

"7-

<

Clay content (wO

A 10%

• 9%

0 7%

• 4%

Fig. 6. Rheogram plotted as apparent viscosity against shear rate for bentonite suspensions with various clay contents.

where Cdw is the drag coefficient where a ball falls in a restricted way in a tube where the wall effect is significant, and Cd, is the drag coefficient when a ball fails in a large size of tube where the wall effect is negligible. The condition given by grw = 1 signifies that the ball ceases to fall because of wall constraint in a limited size of tube, and ~w = 0 signifies that the wall has no effect on the falling of the ball.

The ratio of the ball to the tube diameter (Re =d/ D) was correlated with the relative drag increase coef- ficient, g t . The experimental results are illustrated in Fig. 7, where the parameter g r against the radius ratio Re is plotted. The relationship between the two terms (that is ~w versus Re) was fitted logarithmMy as

gt W = 15Re 5"7 (13)

This relationship has been used to facilitate the correction of the data where the wall effect provides a restriction on the settling velocity.

Drag coefficient (Ca) correlations The data used to generate the general correlations

were collected under similar pre-shearing conditions for the samples. That is, the ball was released within 30 s after the cessation of the stirring. These muds will be referred to later as being in a 'dynamic condition'. The applied shear rate during the initial stirring pro- cedure was much higher than the computed equivalent shear rate imposed by the ball during the process of falling. The correlation result of Ca and NR4 is shown in Fig. 8. The correlations of Ca with NR1 are shown

169

1.0

'~ 0.8'

L) ,~ 0.6.

0 . 4 ' =

0.2'

0.0

A

i

0.3 0.4 0.5 0.6 v . v i

0.7 0.8 d/D ratio

Fig. 7. Experimental results of the wall effects; for falling ball in bentonite suspensions with various ball and tube dimensions.

~_~. 100 ~ / x

lO

.1 . . . . . . . . i . . . . . . . . i . . . . . . . . J . . . . . .

.1 1 10 100 1000 Modified Reynolds Number

Fig. 8. Drag coefficient correlation (Cd) with the modified Reynolds number (NR4) for falling-ball experiments in bentonite suspensions.

in Fig. 9. The analytical argument presented earlier shows excellent agreement with experimental results. A strong correlation between the drag coefficient and the modified particle Reynolds number (eqn. (10)) is readily evident in Fig. 8. Most of the data for NR4 fall into the range from 1 to 100, in which laminar or transitional flows are usually assumed. The estimated equivalent shear rates from eqn. (9) are in the range of 5-50 s - 1. The results of the two correlations, namely, Cd versus NR4 and Ca versus Nm are quite similar. They are

23.2 C d = NR40.74 (power-law model, K:, = 1) (14)

and

170

100

~ 10"

1

.1 . . . . . . . . i . . . . . . . . i . . . . . . . . i . . . . . .

.1 1 10 100 1000

Dynamic Parameter (Dy)

Fig. 9. Drag coefficient correlation (Cd) with the dynamic pa- rameter (NR1) for the same falling ball experimental data shown in Fig. 7.

21.2 Cd = NRlO.76 (Bingham model, Ansley and Smith) (15)

where the difference is within the errors introduced by the fitted theological parameters adopted in the two flow models; e.g. the error to % is between 0.3-1.0 (Pa) for various clay concentrations, and the error for n usually ranged from 0.02-0.06.

It will be realised that the arbitrary selection ofKv= 1 has produced similar numerical forms for the two correlations. Further, comparing the correlation derived for Newtonian fluids [22],

185 Cd----- R ~ (Newtonian model) (16)

the correlation obtained in the present study for ben- tonite suspensions (eqn. (14)) seems to be a reasonable approach for the mud in a dynamic condition where the Reynolds number calculated for the fluid motion falls into the laminar or transition regions (1 <NR4 < 120).

Stokes coefficient (Cs) correlation The experimental data, associated with the finite-

element prediction from Beris et al., for the correlation between the Stokes coefficient, C~ (eqn. (5)), and the Bingham number, NRz (eqn. (6)), are shown in Fig. 10. A good correlation between Cs and NRZ is shown where NR2 is larger than 100, but for the lower NR2 region, the Stokes coefficient becomes independent of the Bingham number. The correspondence between the finite-element predictions (solid line in Fig. 10) and the experimental data is also good in the larger NR2

E

0 0

9 cO

1000"

100

10

, / O

0 0 0 0

0 E x p e r i m e n t a l

- - F E A s o l u t i o n

. . . . . . . . i . . . . . . . .

0 100 1000 Bingham number

Fig. 10. Stokes coefficient correlation with the Bingham number for the falling ball experimental data shown in Fig. 7.

region, but it is poor when the value of NR2 decreases. We note that the low region of NR2 corresponds to a small value of the yield stress of the sample and a large value of the falling velocity of the ball, where the response of the fluid around the falling ball will deviate. In these circumstances, the deviation from the assumption of the perfect rigid-viscous model adopted in the finite-element analysis will be the most pro- nounced. It may be that, when ~-y is small, ~Tp~ is underestimated in the extrapolation of the Bingham parameters, and so the Cs may be overestimated in our calculations. The best fitting of the data (not including the data at low values of NR2) gives a power- law form of the relation between Cs and NR2 as,

C~ = 4.7NR2 °'76 (17)

A comparison of the various correlations The Bingham and power-law approximations, and

indeed the Newtonian limit, provide similar interre- lationships between the common drag coefficient def- inition (eqn. (1)) and the appropriate Reynolds number; in the formulation of eqn. (3),x = 20 + 3 andy = 0.7 + 0.1. The inference is that, for these systems at least, the viscous component may be described as a Newtonian characteristic contribution to the Reynolds number in the form,

where r/* is a viscous term generally defined. Similarly, the plastic component (yield behaviour)

has the form,

171

24 p(V z] (19) R e p ~ - - ~ \ , / . y ]

In terms of R~v and R~p, we may have a general relationship for both the Bingham model and the power- law model, that is,

[ . \(o.7o±o.1)

where

1 1 1 - - = - - + - - ( 2 1 )

Re* R~ Rep

In the Newtonian limit, l /Rep = 0, and Rev = pdV/rl. For the Bingham case, the Ansley and Smith approach, Rev =dp(V/%~) and Rep = (24/77r)p(V2/ry), is a good de- scription.

The similarity of the predictions from various cor- relations (Cd versus NR1 and Cd versus NR4) has been shown in Figs. 8 and 9. Therefore, we may conclude that all the drag coefficient correlations described above are numerically identical despite their different arith- metical forms.

Particle settling in dynamic conditions The laminar or transitional settling characteristics of

spheres in bentonite suspensions are closely described either by the correlation of the drag coefficient (eqn. (1)) with the modified particle Reynolds number (eqn. (11), power-law model), or the correlation between the drag coefficient and the dynamic parameter (eqn. (4), Bingham model), and the Stokes drag coefficient (eqn. (5)) versus the Bingham number (eqn. (6), Bingham model). The former correlation is more generally used than the latter. Moreover, the simple expression (eqn. (11), power-law form) for NR4, formulated in terms of V, is convenient for the correlation to be used to calculate the settling velocity of particles. The previously reported data and the present experimental results appear to show that the particle settling problem in non-Newtonian fluids may be solved in a similar way to that which is effective for Newtonian fluids by choosing suitable rheological parameters. However, it should be noted that the falling ball experiments are usually conducted in stagnant fluids, while most particle trans- port operations are dynamic processes. That is, the fluid itself, rather than the ball, is in motion. For Newtonian fluids, there may be no substantial difference between the particle settling kinetics under stagnant or in dynamic conditions. However, there has been some controversy for non-Newtonian fluids. Some re- searchers [12, 24] claim, on the basis of their exper- imental results, that the particle settling was basically independent of the fluid velocity. Others [13, 16] argue

that a particle should settle faster in dynamic fluids than in stagnant ones as a consequence of the theological response of the fluid surrounding the particle changing under dynamic conditions. Because the rheological pa- rameters used in the correlation are mostly obtained in a continuous shear measurement, which may cor- respond to a dynamic condition, the accuracy of the falling ball correlation in stagnant conditions therefore needs to be addressed.

For the bentonite suspensions, it has been found that their shear-thinning and thixotropic properties have a significant effect on the results of the falling ball correlations. Figure 11 shows the correlations of Ca with N R 4 f r o m t '~vo sets of data measured for different shear-histories for the same mud. The data used to fit Curve 1 were obtained from the falling ball experiments when the mud sample was left quiescent for 30 and 40 rain after being stirred. The data in Curve 2 cor- respond to experiments in which the balls were dropped within 30 s after the suspension was sheared (the condition dealt with mainly in this paper). There is a major difference between the two data sets mainly reflected in the pre-exponential term. Obviously, the correlation represented by Curve 2 is a closer repre- sentation of a dynamic condition than that of Curve 1. It may be concluded therefore that the particle settles faster in a dynamic condition; as one may expect. Further evidence for the same conclusion has been found in the rolling-ball geometry, where the accumulated shear strain of the pre-stirring may be controlled by rotating the tube at different inclined angles. These results demonstrate that there is a major difference between the rolling velocities of the ball when the mud is pre- stirred compared to the case without pre-stirring [25]. However, a further increase of the pre-stirring rate has no very significant effect on the falling velocities of the

~" 100

lo (2) 'S

8

1

.1

Mud 7% (w/w)

. . . . . . . . i . . . . . . . 10

Modified Reynolds Number

Standing Time after stined

& 30 sec

O 30 rnin

• 40 rain

00

Fig. 11. Correlation of the drag coefficient with the modified particle Reynolds number, experimental results for a 7% bentonite mud with different shear-histories.

172

200"

Plasticine I ~, covered b a l l /

'~ 100 Q

¢,0

le Ball

0 10 2O

Sample Standing Time (rain) After Sheared

Fig. 12. Drag coefficient against sample standing time after applying a fixed shearing; showing the effect of ball surface roughness and tt~e shear history on the falling ball velocity in bentonite suspensions.

balls when it is greater than the shear rate induced by the moving balls themselves.

Effect of ball surface conditions When a ball falls in an ideal Newtonian fluid, the

drag resistance on the ball is caused only by the hydrodynamic effects, and no slip between the ball and the fluid occurs. However, when a ball moves in a non- Newtonian fluid, it has been assumed that slip, or at least partial slip, may occur between the ball surface and the fluid. It is obvious that the slip may cause the drag force on the ball to increase due to the sliding friction dissipated between the solid particles and the surface of the ball. There have been few experimental studies carried out to investigate this effect. In this paper, an experiment is described which was designed to investigate this phenomenon by studying the effect of the surface condition (or roughness) of a falling ball on its velocity profile.

Two groups of marble balls were chosen. The balls in one group has a smaller diameter than those in the

other group. The smaller balls were coated by a thin layer of 'Plasticine' material to make their weights and diameters the same as those of the larger marble group. The marble balls had smoother surfaces than the 'Plas- ticine' coated balls. The falling ball experiments were carried out in an 8% bentonite suspension and also in a 95% glycerol solution to serve as comparison for Newtonian fluids. The surface characteristics of these balls and the experimental results are listed in Table 1.

It can be seen that the effect of surface roughness of the ball on the falling velocity is quite significant in the bentonite suspension but is much less so in the Newtonian-like glycerol solution. This effect seems to be enhanced when the degree of gelation of the mud increases, i.e. the time period the mud is left standing after an applied shear. The plot of the drag coefficient against the standing time of the mud sample after the fixed stirring is described in Fig. 12. This figure shows that as the standing time increases, the gel structure of the mud being gradually restored, the difference between the drag coefficient for the two balls becomes greater. The slip is apparently induced by the devel- opment of the structural character of the flocculated clay particles.

Conclusions

The basic requirement in the development of em- pirical drag coefficient correlations for particle settling in non-Newtonian fluids, as well as Newtonian fluids, is to formulate a dimensionless parameter which is equivalent to the Reynolds number derived for New- tonian fluids. There are many approaches proposed based on different flow models and various consider- ations of the shear-dependent viscosity of non-New- tonian fluids. The dynamic parameter (NR1, eqn. (4)) correlation has been shown to be a favoured approach for Bingham-plastic fluids, and the modified particle Reynolds number (NR4, eqn. (11)) correlation for-

TABLE 1. The effect of ball surface conditions on the falling velocity of balls in various media

Balls Weight Diameter Medium (g) (mm)

Velocities ~ measured at various standing times after shearing

30 s 10 rain 15 min

Marble 165:0.02 235:0,05 mud 8% 54.0 19.5 11.7 Plasticine 16+0.02 235:0.05 mud 8% 40.9 13.3 8.1

Marble 16 5:0.02 23 5:0.05 glycerol 26.3 Plasticine 16 5:0.02 23 5:0.05 glycerol 24.2

aThe mean values of falling ball velocity (cm s -z) were taken on the basis of at least 3 measurements in mud, and 5 measurements in glycerol.

mulated in this study has been identified as an effective model for general shear-thinning fluids. The experi- mental results have shown that the agreement between Cd the two correlations in the laminar or transition flow Cdw regions is excellent for the bentonite suspensions. This coincides with the fact that the theological properties Cdn of the bentonite suspensions can be closely described by both the Bingham model and viscosity-shear rate D power-law model in the large shear rate regions (> 10 d s-1). Generally, the modified particle Reynolds number G correlation may be found to be more convenient in g applications with shear-thinning fluids than those in- K volving the dynamic parameter. However, the practical Kv use of both correlations may be limited by the underlying m influence of the motion of the fluid itself (i.e. the occurrence of the dynamic condition as it is usually Nm termed). Because of the difficulty in simulating the behaviour of falling balls in dynamic fluids, the pre- NR2 sheared procedure adopted in this experiment is a good NR3 simulation of the dynamic conditions which occur in drilling fluids, for example. NR4

The correlation between the Stokes coefficient and the Bingham number, NR2, provides a good agreement n with the finite-element predictions in the large Bingham number regions. The poor quality of the correspondence Rd in the low Bingham number region probably arises Re from the adoption of an inappropriate flow pattern Re* surrounding the ball in the finite-element analysis as Rcp a result of the use of a rigid-viscous rheological response which does not accurately prescribe the theology of Rev the system. This error appears to be exaggerated where the plastic component is relatively small. V

The experiments have shown that there is a substantial difference between the particle settling behaviour in stagnant fluids and in dynamic fluids. The surface Y roughness of a ball has been found to have a major ~/ retarding effect on the failing of the ball in a highly Ye structured suspension. This is probably due to the occurrence of slip between the ball surface and the suspension under certain conditions. In addition, sig- nificant container wall effects are noted and an empirical correction procedure has been introduced to correct for their influence.

173

List of symbols

drag coefficient drag coefficient when a ball falls in a limited size of tube drag coefficient when a ball falls in a large size of tube where the wall effect is negligible diameter of tube diameter of ball shear modulus gravity acceleration consistency index of a power-law fluid shear rate coefficient related to the ball-liquid envelope size and shape power-law index for viscosity versus shear rate relationship dynamic parameter based on a Bingham re- sponse Bingham number based on a Bingham response particle Reynolds number based on a shear stress power-law response modified particle Reynolds number based on a viscosity power-law response power-law index for shear stress versus shear rate relationship ratio of the ball to the tube diameters Reynolds number generalised Reynolds number plastic component of generalised Reynolds number Newtonian viscous component of generalised Reynolds number terminal velocity of a falling ball

Greek letters

~7 r/*

"r/pl p ps

7y ~w

shear strain shear rate average equivalent shear rate imposed by a falling ball apparent viscosity of a fluid viscous term generally defined plastic viscosity defined in Bingham flow model density of fluid material density of solid ball shear stress yield stress defined in Bingham model relative drag increase coefficient related to the wall effect

Acknowledgements

Thanks are due to Mr M. Glaese for his initial work in the falling-sphere experimental system. The financial support from the British Council and the National Education Commission of P.R. China to one of the authors (SRR) is gratefully acknowledged.

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174

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