particle transport in gravity concentration and the bagnold effect.pdf

M/nera/J Eng/neer/ng, Vol. 5, No. 2, pp. 205-221, 1992 0892-6875/92 $5.00 + 0.00 Printed in Great Britain Pergamon Press pie

PARTICLE TRANSPORT IN GRAVITY CONCENTRATORS AND THE BAGNOLD EFFECT

P.N. HOLTHAM

Centre for Minerals Engineering, University of New South Wales, P.O. Box 1, Kensington, NSW 2033, Australia

(Received 4 July !991; accepted 5 August 1991)

ABSTRACT

The literature in a number of disciplines concerned with particle transport by water streams has been reviewed, from which it is clear that a better understanding of such transport exists than is reflected in the gravity concentration literature, which uses as a basis the 1939 analysis o/Gaudin. The review highlights the importance o/the work of Bagnold on particle-particle interactions in sheared flows, which has been widely cited in the gravity concentration literature. There appears however to have been no evaluation of the Bagnold flow regime for any gravity concentration device. The experimental evaluation of the/low regime on two different spirals represents a preliminary attempt at remedying this situation.

Keywords Gravity Concentration; Particle Transport; Bagnold Force, Spiral Concentrators

INTRODUCTION

Gravity concentration devices such as the spiral, pinched sluice and Reichert cone can be regarded as sediment transport systems in which particles are sorted according to their density (and size) as they pass through the device. In nature, rivers perform the same functions of transportation and sorting on a much larger scale, and the mechanics of sediment transport by natural streams has been an important research topic for many years in the disciplines of sedimentary geology, civil engineering and fluid mechanics. This has led to the development of a very extensive literature, some of which is relevant to the understanding of gravity concentration processes, and this is reviewed below.

Three important differences between natural streams and gravity concentration devices should however be noted at the outset. Firstly, rivers and streams transport sediment over loose rough beds of similar material, with sediment being picked up (eroded)and deposited depending on the local hydrodynamic conditions. Gravity concentration devices on the other hand have fixed, generally smooth beds, and treat approximately constant flow rates of solids and water, with deposition of the solids and erosion (or wear of the bed) being undesirable. Secondly, rivers have a very shallow slope and the down slope component of the particle immersed weight can be neglected, this is not the case with units such as the spiral for example. Thirdly, the depth of flow in gravity concentration devices is generally very much less than in rivers, hence it is difficult in such devices to sample, or take measurements over the flow depth.

205

206 P.N. HOLTHAM

PARTICLE TRANSPORT BY FLUIDS

Single Particle Motion

In his classic 1939 textbook Gaudin [1, p281 ] analysed the forces acting on a single particle rolling or sliding in a two dimensional laminar film along a smooth bed, and the study of particle transport by fluids in the gravity concentration literature does not appear to have advanced much beyond this point.

In practice, the motion of particles being transported b y fluids is considerably more complex than described by ~Gaudin's analysis, Experimentally, the study of sediment transport is difficult; it is impossible to follow the motion of a single particle within a moving three dimensional dispersion of other particles except at very low particle concentrations. Francis [2] used a multi-exposure photography technique to follow the motion of a single particle over a fixed rough bed, acknowledging that this approach ignored the effects of particle concentration. Francis divided the particle motion into three modes:

I. rolling or sliding, in which the particle always maintained contact with the bed (analysed by Gaudin [1 ]);

. saltation, in which the particle made jumps up into the fluid following a ballistic trajectory before once again coming into contact with the bed;

. suspension, in which the particle made longer and higher trajectories which differed from those of saltation in that the upper parts appeared wavy due to support from turbulent eddies;

with the mode adopted by the particle being established by the transport stage:

transport stage ffi u* / u* o (1)

where u* (defined as ~/(f/p)), with r the bed shear stress and p the fluid density) is the shear velocity of the observation, and u* 0 is the critical shear velocity for the initiation of motion if the particle forms part of a co-planar fully mobile bed. The value of u* 0 was determined from Shields' criterion (Yalin, [3] pg0). By the simple expedient of increasing the fluid viscosity, Francis demonstrated that saltation persisted in laminar flow, hence this mode of particle motion cannot be attributed to fluid turbulence.

The work of Francis was later extended by Abbott and Francis [4], who provided quantitative data on the relationship between mode and transport stage. It was found more appropriate to use u* 1 instead of u*9 in the definition of transport stage, u* 1 being defined as the critical shear velocity for the initiation of particle movement when the particles were placed on top of a fixed bed - this will also be a more realistic criterion in the case of a gravity concentration device, although still likely to be an overestimate. Obviously, u* 1 < u* 0. The mean forward grain speed was shown to be linearly dependent on u* for a particular stream depth over a range of transport stages from 0.75 to 3.05. (Motion at a transport stage less than 1.0 was due to the observed particle protruding above the surrounding bed). At the lowerend the particle was rolling90 - 100% of the time, at the higher end of the range the particle was in suspension almost 95% of the time. Figure 1 is redrawn from [4] and implies that particles of different density can be 'transport' sorted in a down stream direction in the manner summarised by Gaudin [1, p280] and subsequent gravity concentration texts, and as suggested by Slingerland [5] in his examination of natural gravity concentration.

The transport of a single particle by turbulent flow along the bottom of an open channel was also investigated photographically by Sumer & Oguz [6] (for the smooth bottom case) and Sumer & Deigaard [7] (for the rough bottom case). The particle lift off and subsequent mode 3 type motion, as well as the mode 3 motion observed by Abbott & Francis [4] being

Particle transport in gravity concentrators 207

related to the burst theory for the structure of turbulence in boundary layers (Often & Kline [8]).

,60] i Parucie 140 Density

-] + 1"2°1 ~ f " ~ / " I

4 ° 1"541 . , / / , ¢ / t I > 1001 * 1"761 ~ / f I .~ 80"] x 2.601 A~"~r',e / t " I

6o

20

0 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Shear Velocity u* (an/s)

Transport Sorting of Different Density Particles (after Abbott and Francis [4])

Fig.1

High Concentrations of Particles

The observation of single particle motion in either laminar or turbulent flow is too simplistic. For the case where particle concentrations are high, Leeder [9] modified and added to Francis' original modes:

I. 2. 3.

4.

5.

6.

rolling (as before); uninterrupted saltation (the saltation of Francis); uninterrupted partly suspensive saltation in which the normal ballistic saltation trajectory is modified by the effect of fluid turbulence (the suspension of Francis); interrupted partly suspensive saltation, as 3 above, but with the addition of upward acceleration due to inter-particle collision; interrupted suspension, in which the particle is maintained in suspension by both fluid turbulence and inter-particle collisions; uninterrupted suspension, in which there is true suspension of the particle by fluid turbulence;

These modes are illustrated in Figure 2. In sediment transport nomenclature, the division of particle transport between suspended load and bed load can now be defined in terms of whether the particles are operated on by fluid momentum transfer alone or a combination of solid and fluid momentum transfer.

Suspended load can be defined as those particles held in true suspension against gravity by random eddy currents of turbulence having velocity components normal to the bed greater than the terminal settling velocity of the particles relative to the fluid surrounding them, i.e particles in mode 6 only. The particles may remain out of contact w i t h the bed indefinitely, depending on the nature of the turbulence. Modes 1 to 5 are all modes of bed load transport.

In a gravity concentration device, where a load of heterogeneous particles (in terms of both size and density) is being transported, all six modes can co-exist. The particles will have

208 P.N. HOLTHAM

different values of critical shear velocity u*0, but the shear velocity u* at a particular point will be common to all particles, resulting in different transport stages for different particles. Thus large dense particles may be in the rolling or saltation modes, while small less dense particles are in the true suspension mode.

1. Roiling 2. Uninterrupted Saltation

3. Uninterrupted, Partly Suspensive Saltation

4. Intm'rupted, Partly Suspensive Saltation

5. Interrupted Suspension

collision events

6. Uninterrupted Suspension

Fig.2 Particle Transport Modes (after Leeder [9])

With regard to modes 3 to 6, and the effect of turbulence on particle transport, it has been observed that turbulence is suppressed by high concentrations of particles (Bagnold [10]). This phenomenon is discussed in more detail by Hetsroni [11], whose theoretical analysis suggests that particles with low particle Reym~lds numbers (based on relative velocity and size) ,tend to suppress turbulence, while the presence of particles with Reynolds numbers greater than about 400 tend to enhance turbulence. More experimental data is required to fully quantify the effect.

Limiting Stage of Unsuspended Transport

Of interest in gravity concentration processes is the limiting stage for unsuspended particle transport. Particles maintained entirely in suspension may not be fully subjected to the sorting process and will tend to follow the stream in which turbulence is greatest. Bagnold [12] developed a theoretical criterion for establishing the onset of suspended transport, showing that a particle should become liable to suspension at a transport stage of approximately 0.8vt/u*0, where v t is the particle terminal velocity in water. Figure 3 (redrawn from [ 12])'with u* 0 again calculated from Shields' criterion, shows that for cluartz particles greater than about 700 #m diameter, vt/u* 0 has an approximately constant value of 4.5, and hence such particles could be expected to become suspended at a transport stage of about 3.6, a value similar to that found experimentally by. Francis [2]. It can also be seen that the ratio vt/u* 0 decreases rapidly as the particle size decreases, until for quartz particles finer than 100 # m some suspension should occur a t the threshold of bed movement. Applying Bagnold's suspension criterion to gravity concentration devices suggests that minus 100#m quartz particles will be transported entirely in mode 6, turbulent suspension. However, as also noted previously, turbulence is considerably dampened by the presence of solids, resulting in the onset of turbulence being delayed until a much higher transport stage is reached.


10

i . . . . . . . . . . . Particle Density[ 2.65

o 1.5

.1 .01

Fig.3

. . . . . . . . | . . . . . . . . | . . . . . . . . m . . . . . . . .

.1 1 10 I00 Particle Size (nun)

Bagnold's Suspension Criterion (after Bagnold [12])

The Bagnold Effect

Transport modes 4 and 5 described above occur with high particle concentrations and the rheology of suspensions of large solid particles in fluids at high shear rates is of interest to many other disciplines, as well as mineral processing. Where particle concentrations are high (such as in all practical gravity concentration devices) there is a significant increase in the shear resistance of the particulate f l u id and considerable distortion of the velocity distributions compared with those of the fluid phase alone.

Some of the earliest experiments dealing with this type of flow were carried out by Bagnold [10] who, motivated by an interest in the physics of sediment transport in river beds, measured the shear and normal stresses developed over a range of shear rates when neutrally buoyant wax spheres suspended in water were sheared in a coaxial-cylinder Couette flow apparatus. Depending on the value of a dimensionless shear rate group B, the Bagnold number, analogous to Reynolds' number, Bagnold defined two limiting flow regimes, the macroviscous (B _< 40) and the particle-inertia (B >_ 450) separated by a transition region (40 < B < 450), where B is given by:

B = (aA°-SDa//~)(du/dz) (2)

and du/dz is the mean shear rate, a the particle density, D is the particle diameter, # the fluid viscosity and A, the linear concentration, is the ratio of the particle diameter to the mean free separation distance between particles. As a function of the volume concentration, C, A is given by:

A = l/((C*/C)l/3-1) (3)

where C* is the maximum possible particle concentration. More recent work (Savage and McKeown, [13]) suggests that the values of B defining the boundaries of the different regimes may differ from those given by Bagnold [10].

In the macroviscous regime, stresses are transmitted by interstitial fluid friction and are therefore dependent on fluid viscosity but are independent of particle density, and based on experimental observations in the Couette apparatus, Bagnold [ 10] proposed the empirical relationships,

shear stress rzx = A3/2/~(du/dz) (4)

210 P.N. HOLTHAM

normal or dispersive stress rzz = 1.3rzx (5)

i.e the stresses are linearly dependent on shear rate, and are independent of particle size and density. Bagnold attributed the normal stress to an anisotropy in the spatial particle distribution (Figure 4a).

In the particle-inertia regime, Bagnold argued that the interstitial fluid plays a minor role and the dominant effects arise from the succession of particle-particle collisions as the particles of one layer overtake those of an adjacent slower layer (see Figure 4b). By considering the mean paths of particles undergoing rapid shear, Bagnold recognised that both the momentum transfer per collision and the frequency of particle collision are proportional to the mean shear rate, resulting in normal and shear stresses quadratic in the mean shear rate.

a) possible statistically preferred distribution of particles in the macroviscous regime

0000 0000 0000

b) particle-pertide collision in shea_,x~ layers in the particle-inertia regime

z I '~zz T 0 0 , . , , . , ,..,

[~ter layer O , slower layer

Fig.4 Particle-Particle Interactions in Fluid Transport (after Bagnold [ 10])

Bagnold experimentally verified this quadratic stress dependence, as well as the strong dependence of the stresses on the volume concentration of particles for B > 450 and 1.4 <

< 14, and proposed the following relationships (Bagnold, [10], Hanes and Inman, [14]):

normal stress rzz ffi o(AD)2(du/dz) 2 (6)

shear stress rzx ffi °(AD)2(du/dz) 2 (7)

i.e the stresses are indePendent of fluid viscosity but are dependent on particle size, density and concentration, and hence are likely to be significant in gravity concentration processes. Bagnold himself noted that for a given shear rate du/dz, the normal stress being proportional to D 2 suggests that when a range of particle sizes of constant density is sheared together in a gravity flow the larger particles should tend to move towards the zone of least shear strain, i.e the free surface, and the smaller particles towards the zone of greatest shear strain, i.e the bed [10].

Bagnold did not explicitly consider particles of different density, however the likely significance of the Bagnold force for the segregation of particles has been very widely noted in the gravity concentration literature. Burr [ 15, p97], and Sivamohan and Forrsberg [ 16] use


equation 6 to show that sorting in the particle-inertia regime places coarse light particles above fine dense particles, while in the macroviscous regime equation 5 was used to show that fine light particles are maintained in the faster flowing upper layers of the flow. The limited experimental evidence of Bonsu [17] describing particle distributions vertically through the pulp on a Humphreys spiral is suggestive of Bagnold's particle-inertia normal stress in operation (Table 1). Sallenger [18] used an identical argument to apply Bagnold's results to natural sizing and gravity concentration of quartz and heavy mineral assemblages and found the data suggested that the Bagnold normal stress could be contributing to the concentration process.

TABLE I Recovery of Particles in Upper and Lower Pulp (feed pulp density 320/0 w/w, pulp depth g mm, sample 7 cm from inner rim)

(after Bonsu [17])

Size (l~m) Recover T to Lower Pulp (%) Recover}, to Upper Pulp (%)

-1200 +850 18.3 81.7

-850 +600 29.7 70.3

-600 +425 61.4 38.6

-425 +300 82.9 17.1

-300 +212 83.3 16.7

-212 +150 78.6 21.4

-150 +106 75.0 25.0

-106 +75 100.0 0.0

Bagnold's results have stood for some 30 years under the conditions for which they were formulated: steady, uniform, simple shear flow of neutrally buoyant spherical particles. However in cases involving more complex flows, application of Bagnold's relationships leads to unrealistic constraints. In particular, in Bagnold's formulation, the stresses vanish for vanishing mean velocity gradient because there is no source for particle velocity variations other than mean shear (Hanes and Inman [14]). Only relatively recently have attempts been made to extend Bagnold's work both experimentally and theoretically. Savage and Jeffrey [19] developed a 'kinetic' model for the rapid shear flow of cohesionless spheres by explicitly considering the fluctuating velocity component in calculating the momentum transfer due to binary particle collisions. The normal and shear stresses predicted theoretically are of the right order of magnitude and consistent with Bagnold's experimental results with respect to particle density, diameter and the mean shear rate. The variations with particle concentration are predicted correctly up to volume concentrations of 0.5, but for values greater than 0.5 the experimental stresses increase much more rapidly. The results of the most recent experimental work (in special shear cell apparatus) of Hanes and Inman [14] and Savage and MeKeown [13] confirm that shear and normal stresses are developed in particulate-fluid flow and that at sufficiently high shear rates the stresses are quadratically dependent on the mean shear rate at a given volume concentration, which supports Bagnold's hypothesis that the stresses result from particle collisions.

Experimental evidence of the significance of the Bagnold stresses in a more realistic engineering application has been obtained by Nasr-el-Din et al [20] who measured lateral concentration variations in horizontal slurry pipelines, thereby eliminating the effect of particle immersed weight. Figure 5a-b redrawn from [20] show the lateral concentration distributions of fine and coarse quartz sand. At high particle concentrations, there is clear evidence of particle migration towards the pipe centre line which was attributed by Nasr- el-Din et al to the effect of the normal or dispersive stress component, the effect being more noticeable with the coarser particles, as expected from Bagnold's analysis.

212 P . N . HOLTHAM

a) Fine Sand (Ds0 = 190 ~ra)

A

t3

60"

50 1

30"

20"

10"

0 o 0.0 0.2 0.4 0.6 0.8 1.0

Dimensionless Pipe Radius

Fig.5

b) Coarse Sand (Dso = 900 gin)

~ 3(1

~ 20

0 ~ 0.0 0.2 0.4 0.6 0.8 1.0

Dimensionless Pipe Radius

Lateral Concentration Distributions in a Horizontal Slurry Pipeline (after Nasr-e l -Din et al [20])

Figure 6 (also redrawn f rom [20]) shows the vertical distribution of coarse polystyrene where the normal stress occurs in combination with the force of gravity. At the lowest concentration there is a significant di f ference in concentration between top and bottom of the pipe, and at this concentration (0.09) lateral variation was negligible. At a concentration of 0.21, a steeper concentration gradient occurs near the top of the pipe where the normal stress acts in combination with gravity.

Near the bottom of the pipe, gravity is insufficient to maintain a negative concentration gradient and the normal stress causes a concentration reversal. At the highest concentration, the profile is almost symmetrical with the normal stress becoming dominant compared with other effects.

Of more relevance, support for the significance of the Bagnold normal stress in gravity concentration can be found f rom examination of the operation of the pinched sluice or the Reichert cone. Both of these devices give adequate segregation only at high pulp densities (i.e high volume concentration of particles) in which Bagnold's part icle-inert ia conditions might be expected to apply and hence in which a significant normal stress would be developed.


Coarse Polystyrene (Ds0 ffi 1400 gut)

1.0

0.8

i5 0.6-

o.4. vol . II cone. II

÷ 9 11 0.2. * %11

0 0 . . . . *

0 10 20 30 40 50 60 Vol. Conc. C (%)

Fig.6 Vertical Concentration Distributions in a Horizontal Slurry Pipeline (after Nasr-el-Din et al [20])

Abdinegoro and Partridge [21] carried out a sampling programme on a pinched sluice, and their results are summarised in Figure 7a-b, which should be compared with Figure 6. The profiles suggest that a normal stress is developed only at high concentrations, as theoretically required. From the original data it is possible to estimate the value of the Bagnold number B. The feed particle size was not completely specified but if a Ds0 of 500/~m is assumed, for the conditions under which the sluice was operated the value of B falls within the macro-viscous (low feed concentration) and low transitional (high feed concentration) regimes, as the mean shear rate was quite low (typically only 30 s-l). If the feed was in fact f iner than assumed, the value of B would decrease, Bagnold [10] himself suggests that the dispersive force effect in the particle-inertia regime should disappear when the particle size is less than about 200 #m.

Subasinghe and Kelly [22] also carried out experimental work on a pinched sluice and obtained results which could only be explained in terms of the existence of the Bagnold normal stress.

Modelling of Particle Transport

Much of the very extensive sediment transport literature is devoted to the development of completely general models, relating the bulk flow rate of sediment to the prevailing hydrodynamic conditions, for solids which are generally assumed to be of quartz density. A number of such models have been reviewed by Yalin [3].

The requirements of gravity concentration modelling are different, the prediction of the bulk flow rate through a device is not required, what is required is a description of how the interaction between particles and fluid at a given pulp flow rate and pulp density leads to segregation by density and size, i.e the requirement is a model capable of predicting the particle concentration distributions within the fluid (in the manner of Figure 7) as a function of particle size and density and pulp flow rate and solids concentration.

The classical approach to the modelling of particle transport by suspension in turbulent flow (the suspended load) is through a diffusion process, with expressions of the form (Allen [23]):

CV s ffi _~s(dC/dz ) (8)

which can be integrated, giving:

214 P.N. HOLTI-IAM

C = CrefeXp(-vt(Z-Zref)/es} (9)

where C is the solids concentration, v t the particle terminal fall velocity, es the diffusion coefficient and Cre f the reference concentration at a known height from the bed Zre f. Equations 8 and 9 refer to the suspended sediment as a whole, without distinction as to particle size (or density, again quartz density particles are assumed).

a) Feed concentration 4 % solids by volume

I [ + llmenite

3

1

0 0 10 20 30 40

Mineral Distribution %

b) Feed concentration 25 % solids by volume

nmenite 5 o Quartz

4

0

v

t~

Fig.7

! !

0 10 20 30 4O

Mineral Distribution %

Particle Concentration Distributions in a Pinched sluice (particle size 180/~m, feed flow rate 1.2m3/hr)

(after Abdinegoro and Partridge [21])

Models based on this approach are very successful, yet suffer from the serious limitation that they give no indication of the nature and origin of the upward acting force which by Newton's first law must be equal and opposite to the immersed weight of the particles (Bagnold [12]). Equation 9 also shows that the suspended load is only predictable using a diffusion model when a reference concentration at a reference height has somehow been determined. Referring again to equation 9, the particles are characterised kinematically in terms of their terminal settling velocity (as in 'classical' gravity concentration theory) and


not dynamically. As put rather picturesquely by Bagnold, the diffusion theory treats the particles as little fishes of zero immersed weight perpetually swimming downwards at their terminal velocity, and with some assumptions about the turbulence, succeeds in predicting the decrease in concentration of little fishes with distance from the boundary. The theory fails to predict any limit to the total weight of fishes which a given turbulent flow can carry in suspension. From this viewpoint, Bagnold [ 12] developed an alternative dynamical model of sediment transport, which again only yielded total sediment transport rate, and not particle concentration distribution. However his objections to the diffusion theory remain valid.

Wilson and Pugh [24] combined the diffusion model with Bagnold's normal stress model to generate theoretical concentration distributions with depth for various particle sizes. Good agreement between predictions from the model and the experimental measurements of Nasr-el-Din et al [20] was obtained.

Summary

The transport of particles by fluids in open channels has been very extensively studied in disciplines other than mineral engineering, and it is clear that gravity concentration researchers will have to consider more than the simple rolling/sliding along the bottom model proposed by Gaudin [ I ] which appears to represent the limit of knowledge in mineral processing texts. Models based on a diffusion mechanism and also incorporating Bagnold's normal stress have been developed by other disciplines and show good agreement with measured concentration profiles for quartz density particles. The concentration profile approach, extended to include a range of particle densities, appears to be a profitable avenue of research for gravity concentration processes involving shear flow.

Bagnold's original paper [10] has been very widely cited in the gravity concentration literature as having relevance to the particle stratification mechanism, but the findings do not appear to have been applied other than qualitatively. As far as the author is aware, the only attempt to evaluate the Bagnold number B for a gravity concentration device and hence determine which flow regime applies is that of Burch [25]. In addition there is very little data in the literature giving size, density and concentration distributions in typical gravity concentration devices, thus the extent to which the Bagnold force contributes to the particle sorting process by either size or density is unknown.

The objective of the experimental work described below, which forms part of a larger study of fluid and particle motion on spirals, was to obtain a first estimate of the Bagnold number and hence the flow regime on two metallurgically different spirals, in order to determine the likely significance of the Bagnold force on particle stratification.

DETERMINATION OF THE BAGNOLD FLOW REGIME ON SPIRALS

General Procedure

A fully instrumented experimental rig was used consisting of a Vickers FGL mineral spiral and a Mineral Deposits LD9 coal spiral capable of both open and closed circuit operation. Each spiral was fitted with a splitter box, dividing the discharge into 8 streams of known width, and a mult i -point computer controlled pulp depth gauge. Full details of the spiral rig have been given elsewhere [26].

An open circuit experimental run was carried out on each spiral under automatic control, at a flow rate of 6 m3/hr on the LD9 spiral and 4 m3/hr on the FGL spiral, and a nominal pulp density in each case of 15 % solids by mass of quartz sand. The size distribution of the sand is shown in Figure 8. Difficulties with choking of the sampling device limited the feed pulp density to a value considerably lower than would be used in industrial practice. Duplicate depth measurements were made and triplicate stream samples taken. The stream samples were weighed wet and dry, then dry screened to determine the size distribution of

100

each. As a check on the sampling procedure, the feed pulp density and feed size distribution were back calculated from the stream samples for comparison with the set point feed pulp density and measured feed size measured, in both cases the automatic control system used ensured good agreement was achieved.

r 75

50 ..~

U 25

0 10

• f

216 P.N. HOLTHAM

f

Fig.8

10( 1000 10000 Size (ttm)

Particle Size Distribution of Quartz Sand

Results

Although the feed pulp density on both spirals was low, the stream pulp density in the inner trough regions was very high, especially in the case of the LD9 spiral, where a maximum of 73 % solids by mass was achieved (a volume concentration of 50.5 %). The particle concentrations by volume in each stream are compared in Table 2. These high solids concentrations were reflected in the depth profiles, where in both cases the depth was noticeably greater than when operating the spirals with clear water [26], contradicting the findings of Dallaire et al [27], who noticed no difference in depth between pulp and clear water. The pulp depth profiles on each spiral are plotted in Figures 9a and b.

TABLE 2 Particle Concentrations in each Stream (% v/v)

LD9 spiral

FGL spiral

S t r e a m

1 2 3 4 5 6 7 8

49.0 50.5 46.1 13.0 3.1 1.7 0.7 0.4

25.4 37.0 41.5 42.7 39.6 29.5 7.8 1.1

Figure 9a shows that the LD9 spiral exhibited a pronounced bulging of the pulp surface in streams 1, 2 and 3, which may be a manifestation of the Bagnold normal stress. This effect is not so noticeable on the FGL spiral (Figure 9b); there is a slight bulge extending as far as stream 7, reflecting the fact that higher concentrations are maintained further into the outer streams than on the LD9 spiral, however the maximum concentrations on the FGL spiral are lower. It is interesting to note that both Hanes and Inman [14] and Savage and Jeffrey [19], who investigated the Bagnold normal stress experimentally, found that the stress was only weakly dependent on particle volumetric concentrations up to 50 %, but above this figure the stress increased rapidly. The experimental values in streams 1, 2 and 3 on the LD9 spiral are approaching this value, while the stream concentrations on the FGL spiral all fall well below. This again may suggest that the Bagnold normal stress is causing

a) LD9 Coal spiral

160 ] 120

b) FGL Mineral spiral

0 40 80 120 160 200 240 280 Radial Distance from Column Wall (mm)

160

t

0 0

• • • | i , J JJ . iS " ! ~ " e " " " ' t~ ' i ,~ • I

40 80 120 160 200 240 Radial Distance from Column Wall (ram)


the observed pulp dilation and hence may contribute significantly to particle stratification on the spiral.

• i |

28O

Fig.9 Pulp Depth Profiles (feed pulp density 15% quartz sand by mass)

The mean stream velocities on each spiral were calculated from the mean stream depth (and hence cross-sectional area) and the stream volumetric flow rate in the manner described in Holtham [26], the results are shown in Table 3. In the case of both spirals, the mean stream velocities in the inner trough are greater than those in the clear water case [26]. This appears to be a reflection of the greater pulp flow rate in these streams resulting from the presence of the quartz sand.

The DSO of the particles in each stream is given in Table 4. The difference in the stream Dso between the two spirals is noticeable. The FGL spiral exhibits a regular pattern of DSo increasing outwards (with the exception of stream $) and this has been observed previously on this spiral with a different feed (Holtham and Stitt [28]). The LD9 does not follow this pattern, and examination of the size analysis data shows that the size distributions in each stream are by no means regular.

218 P. N. HOLTHAM

TABLE 3 Mean Stream Velocities (m/s)

LD9 spiral

FGL spiral

S t r e a m

1 2 3 4 5 6 7 & 8

0.3 0.3 0.3 0.2 0.3 0.5 2.5

0.1 0.1 0.2 0.3 0.3 0.4 1.7

TABLE 4 Stream Particle Dso (/~m)

LD9 spiral

FGL spiral

* es t ima ted

S t r e a m

1 2 3 4 5 6 7 8

560 560 375 750 880 95 45* 28*

120 215 300 380 495 680 720 560

The experimental data above now permit an estimate to be made of the value of the Bagnold number B (defined by equation 2) in each stream of both spirals, and hence the Bagnold flow regime.

The Bagnold Flow Regime

If the maximum concentration C" (loose packing) of natural quartz particles in water is taken to be 0.558 (Allen, [23] p34), the particle density a to be 2650 kg/m 3, the water viscosity/~ to be 0.00106 Pa s, D to be the particle Ds0 values (/~m) and C the measured fractional volumetric concentrations in each stream, then after calculation of ~, using equation 3, the only unknown in equation 2 is the shear rate du/dz.

Values of the shear rate in the primary flow direction du/dz can be calculated for values of B corresponding to the boundaries of the transition between the macro-viscous and particle inertia flow regimes (B = 40 and B ffi 450 respectively, using Bagnold's suggested values). The assumption is made that particle concentration, and particle diameter throughout the depth of the flow are uniform. In practice it is apparent from the discussion earlier that the Bagnold stress will affect both the concentration and size gradients.

For comparison, values of mean shear rate du/dz in streams 1 to 6, and 7 and 8 combined were estimated from the experimental stream velocity data (in Table 2) by assuming that the depth profiles can be described by the laminar equation (in the inner trough region):

u = (r0h/#){(y/h)(1-½[y/h])} (4)

and the smooth turbulent equation (in the outer trough region):

u/u* = 2.5 ln[yu*/~,] + 5.5 (5)

where y/h is the fractional depth and ~ the kinematic viscosity. The justification for the use of these equations was discussed in Holtham [26]. However, as noted earlier, when particles are suspended by a flowing fluid, the vertical gradient of particle concentration may affect the flow dynamics sufficiently to cause a deviation of the velocity profile from that described by equations 4 and 5. In the case of turbulent flow (equation 5) any change in the


profile from the logarithmic form was attributed in the past to an alteration of the Karman constant ~ f rom the clear water value of 0.4, hence changing the factor 2.5 (1/~) in equation 5 (Yalin [3], p22). More recently however, it has become possible to explain the effect of particle concentration on the observed logarithmic velocity profiles physically without recourse to a varying ~; [29]. Soulsby and Wainwright [29] show that an estimate of u* in particle laden flows in which the effects of the particles have been neglected will always be an overestimate compared with the true value, and they provide a criterion for establishing when in practice the particle effect can be neglected.

Values of du /dz calculated from the experimental stream concentrat ion and size data for Bagnold numbers of 40 and 450 using equations 2 and 3 are compared in Tables 5 and 6 with experimental estimates of the mean shear rate (over fractional depth 0.1 to 0.5) from the assumed flow profiles. In making the estimates, the effect of the particle concentration on u* was neglected, as a result u* is overestimated, hence the actual values of du /dz are likely to be slightly lower than those shown in the tables.

TABLE 5 i

DS0 (tun)

Vol Conc. C ~1/2

(du/dz)Bffi40 (s -1)

(du/dz)B=450 (s "1)

(du/dz)expt. est. (S "1)

LD9 Spiral: Evaluation of Bagnold Flow Regime

Stream

1 2 3 4 5 6 7

560 560 375 750 88O 95 45 28

0.49 0 . 5 1 0.46 0.13 0 . 0 3 0.02 0.01 0.00

4.69 5.34 3.86 1.27 0.79 0.67 0.55 0.49

11 10 29 22 27 2667 5195 41649

122 107 321 250 300 3 0 0 0 0 58442 470000

230 195 195 53 60 43 200

TABLE 6 FGL Spiral: Evaluation of Bagnold Flow Regime

DS0 (~an)

Vol Conc. C xl/2

(du/dz)B-_.40 (s "1)

(du/dz)B=450 (s -1)

(du/dz)expt. est. (s -1)

Stream

1 2 3 4 5 6 7

120 215 300 380 495 680 720 560

0.25 0.37 0.42 0.43 0.40 0.30 0.08 0.01

1.82 2.60 3.09 3.25 2.86 2.05 1.04 0.61

571 133 57 34 23 17 30 83

6429 1500 643 385 257 190 333 938

80 80 125 170 170 69 125

The data in Table 5 show that the flow regime on the LD9 spiral varies from macro-viscous in the outer trough (streams 6 to 8), through transitional (streams 4 and 5) reaching the particle-inertia regime in streams l and 2, and, in view of the uncertainties associated with the shear ra te determination, possibly in stream 3 also. Thus in the inner trough, a significant Bagnold stress may be expected, which will markedly affect the coarser particles in the stream. The calculation provides confirmation that the bulge in the pulp profile in streams 1 to 3 can (at least in part) be attributed to a Bagnold normal stress. The question as to what proportion of the bulge is due to the Bagnold stress and what proportion to the normal concentration of solids remains to be determined. It is worth noting however that

MINE---5 /2- - -G

220 P.N. HOLTHAM

similar bulges are observed in non-Newtonian fluids such as polymer melts, which are also attributed to the development of a normal stress (Tanner [30, p5]).

The flow regimes on the FGL spiral are completely different (Table 6). Under the experimental conditions used, none of the streams exhibit particle-inertia type flow. Streams 1 and 2 are in the macro-viscous regime, all other streams are transitional.

The explanation for the difference in flow regimes between the two spirals appears to lie in the difference between the stream Ds0 values and the values of solids concentration achieved in each stream. During preliminary development of the depth gauge on the FGL spiral, pronounced pulp bulges were noticed in the middle streams when treating a somewhat coarser feed than used here, and as would be expected those streams had the highest pulp density values.

CONCLUSION

A review of the literature from a number of disciplines has shown that there exists a considerable body of work concerning the transport and sorting of particles by both size and density in shear flows. The work by Bagnold on particle-particle interaction in such flows which is widely cited but only applied qualitatively in the gravity concentration literature has also attracted the attention of a number of researchers elsewhere, and it is clear that the Bagnold effect is real and may contribute significantly to particle stratification in gravity concentration devices at high feed pulp densities.

The limited experimental work reported in this paper has shown that the conditions under which the Bagnold normal force occurs are likely to exist on the spiral, especially at feed pulp densities used in practice. At the low feed pulp density used, solids concentrations by volume at some points on the trough of each spiral were sufficient to take the flow near to or into the particle-inertia regime. The results must however be treated with some caution in view of the uncertainty associated with the choice of the velocity profiles used, and the effect of particle concentration on the resultant shape of the profiles.

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