The mixing and segregation of cohesionless particulate materials Part I. Failure zone formation

Powder

Technology.

21 (1376)

17 - 2s

1.

The Mixing and Segregation of Cohesionless Part I. Failure Zone Formation

Particulate

lMateriaIs

D. J. STEPHENS* and J_ BRIDGWATER Department

of Engineering

Science.

Oxford

UniL-ersity.

Parks

Road,

Oxford

OS1

3PJ (Ct.

Britain)

(Received October 2’7,19’7’7)

SUMMARY The mixture quality of cohesionless particulate materials is controlled. at least in part. by the detailed processes occurring in failure zones. In the f&t part of this work, the construction of an annular shear cell to simulate a failure zone and the determination of the velocity distributions in the zone is describedFor a given material it is found that, within the accuracy of measurement, the central portion of the failure zone has a constant rate of strain. The zone depth can be characterised conveniently by the central velocity gradient_ This depth could not be described by a single value common to the two particle sizes studied in either dimensional or dimensionless form. The use and significance of these results in esamining mising is considered in Part II.

INTRODUCTION The mixing of powders is common in the chemical, pharmaceutical and agricultural industries. yet, despite its importance, little progress has been made in overcoming the many difficulties [ 1, 2]_ Consequently the design of powder mixers relies largely on practical experience and empirical relations rather than an understanding of principles. When mising cohesionless powders of different particle size, the quality of the mixture, however measured, initially increases with mixing time until a maximum is reached, although not corresponding to a perfectly ran-

*Present address: ICI CrrrporateLaboratory, P-0. Boz 1X, The Heath, Runcorn, Cheshire, WA7 4QE_ 0032-591017810021-0017$2.25

dom misture [3] _ If mixing is continued, the quality then decreases to a constant value [4] _ The time for these changes and their magnitude will depend upon the type of miser and the powders used. Conventionally, powder mixing processes have been investigated using small-scale models of misers. This has proved to be unsatisfactory, mainly because the scale-up of powder mising systems is not straightforward. To maintain geometric similarity between a miser and powder when a system is scaled down, even if the practical problem of making smaller particles can be overcome, will frequently result in changes in the physical properties of the materials_ Thus, while a powder may originally have been cohesionless, reducing its particle size in proportion to the reduction in miser size may result in it becoming cohesive_ Xlternatively, the miser size may be reduced but the original particle size maintained_ This is the most commonly used approach, as it enables the actual powder system to be investigated. However, this does not maintain geometric similarity between the miser and powders, so although the mising in the model can be successfully studied, it cannot be assumed to be representative of a larger mixer. As well as finding a system that accurately represents the miser and powders, there are also problems in testing the performance of a miser and the interpretation of results [ 5]_ Misture quality, or degree of misedness, is usually determined as a function of the variance of samples taken from the mixture. The results of any sampling procedure may be affected by the type of sampler, the choice of sampling positions and the sample size. The quality required from a mixing system is determined by the use of the mixture. This should be reflected in the sampling procedure Printed

in

the

Netherlands

and the choice of mixing index, although the presentation of a single mising index cannot provide the maximum information about a mixture [S] _ The confusion that exists in the measurement of mkture quality is indicated by the many arbitrary mixing indices that ha\-e been proposed [ri] _

although the Iatter are not very satisfactory definitions [ 2]_ For esample, it is possible to identify both macroscopic and microscopic processes as diffusive, although they are fundamentally different, and the separate identification of shear mixing only acknowledges the existence of failure zones where micro-

Powder mixing studies can be split into three generaI categories: large scale, small scale and fundamental_ Large-scale mixers, although of prime importance, have received little direct attention. presumably because of size and the difficulty in obtaining and interpreting measurements, whereas small-scale misers, or models of misers, have received more attention_ However, as has already been pointed out, the results of this type of study are of Iimited application and, in terms of the overall objectives, few are of any significance_ The fundamental mechanisms of powder mising can be examined by reducing the compIesity of the powder behaviour_ This cannot easily be done using models of actual misers because of the interaction of the mechanisms controlling the powder behaviour even in the simplest mixers. so specifically designed equipment must be used enabling particular effects to be isolated_ This approach should be considered as part of a long-term strategy, E slfficient information to enable 2 miser to be

scopic processes may occur. Generally, macroscopic processes may be considered as promoting mixing, and their_importance is determined largely by the design and operation of the mixer. Microscopic processes, however, may cause both mising and segregation and are dependent on the properties of the powders as well as the miser. Cohesionless powders are especially susceptible to segregation_ The microscopic mechanism of strain-induced interparticle percolation has been studied in a reciprocating simple shear cell in which a ,rfranularmaterial could be strained with the rate independent of the position in the cell [12,13] _ Smaller particles placed at the top of the cell moved systematically down through the bed, and the time to pass through the cell was found to be a linear function of the bed height_ The cell baseplate was designed to let small particles leave the bed. allowing the system to be fully automated_ Using this equipment, basic data were obtained on the rate at which small partides passed down through the bed, the percolation velocity, and its dependence upon a range of variables_ The rate at which the material was being strained and the normal load applied were found to have some effect, but by far the most important factor in determining the percolation velocity was the relative size of small and large particles_ Changing the particle diameter ratio from OS0 to 0.27 was found to increase the percolation velocity by 2 factor of fifty. Thus reproducible basic data have been found for individual variables such as particle size difference in a controlled system where other effects can be kept constant_ In 2 powder mixer the only control is from the design and the overall operating conditions_ The flow patterns within a mixer and the size and frequency of formation of failure zones [2] are unknown and rely on the interaction between the powders and the mixer_ The behaviour in failure zones is important and this extension of the simple shear cell work provides the motive here_

designed

from first principles

is unlikely

to be

available in the near future. Fundamental investigations can provide a quantitative evaluation of many of the processes within a miser or powder system, which can then be used to esplain qualitatively the behaviour in 2 real miser, and assist in design and operation_ The fundamental mechanisms may be considered in two categories, macroscopic and microscopic_ Macroscopic mechanisms refer to the movement of coherent clumps or blocks of powder, and correspond ta convective mis‘brgas described by Lacey [S] _ Between the moving blocks there exist regions where the powder is failing 19, IO] _ These regions, known as failure zones, are up to twenty times the particle diameter in depth_ Similar regions occur at free surfaces [ll] and adjacent to walls in flowing powders_ Within these zones, where particles are moving relative to their neighbours, microscopic mechanisms are of importance_ Interparticle percolation is an example of a mfcroscopic process, as are diffusive and shear miving as described by Lacey,

19 EQUIPAIENT SELECTION To investigate the microscopic processes in a powder mixture, its macroscopic behaviour must be known or controlled_ The main interest in shear testing of soils has been directed towards initial failure, using various forms of shear apparatus, while the strength of soils after initial failure has been of less interest_ Annular shear cells were developed to study the decrease in the shearing resistance in soils that occurred after continued straining in one direction [ 141 but, because of the difficulties of sample preparation, use has been restricted to remoulded soils and has not been widespread [ 151 _ Whereas with soils the initial failure is of primary interest, for granular materials the behaviour after initial failure is also of -great importance, and this is readily investigated in ;t? annular cell_ Another advantage of the annular cell is that it is capable of indefinite strain in one direction, thus avoiding the problems of the reciprocating action of the simple cell. Lens-shaped failure zones, as found in the simple shear box developed by Jenike [ 161, are also avoided. Consistent sample prepaxxtion is difficult in the Jenike box, and it is not satisfactory at stresses of less than 5 kN/m”, which are common in powders [ 17]_ With an annular cell, preparation of powder samples is straightforward and operation at low stresses possible_ Novosad [lS] built an annular shear cell based on the design of Hvorzlev [ 141 to determine strengths and angles of friction of vular materials_ The inner diameter of the annulus was 60 mm and the outer diameter 120 mm, and it could hold a sample 30 mm deep. The cell walls were divided at the level of the centre of the powder sample, the lower pa;+, of the cell being restrained and the upper part rotated. This positioning of the division in the cell walls encouraged failure within the body of the powder rather than against either of the confining rings. The design of the cell used by Bridpvater and Bag&x [ 19, 203 is similar to that of Novosad, having an annulus 30 mm wide and holding a sample up to 50 mm deep. In this cell, the lower section is rotated with the upper fried, but the main difference is the much greater cell diameter, 203 and 26’7 mm inner and outer annulus diameters. One problem inherent in annular cell design is that material at the inner wall

moves less and is therefore strained less than that ar. the outer wall. The ratio of inner wall velocity, or strain rate, to outer wail velocity was 0.176, compared with 0.50 for the Novosad ccl!. These two cells both determined shear stress by measuring the twisting moment between the upper and lower sections of the cell. Their drawback has been their limited accuracy, and that insufficient is known about the powders and techniques used to be sure of what effects the cell may be having on the results_ Annular shear cells have also been built for powder studies by Carr and Walker [l’i] and Scarlett and Todd [Zl] _ Both used an annular trough into which the powder is put, and the cell lid, when placed on top of the powder, is also inside the trough. This arrangement encouraged the powder to fail adjacent to the cell lid, although to overcome this Carr and Walker fitted vanes on the cell lid to grip the powder. Consequently the powder failure will be influenced to an unknown estent by the construction of the cell lid, and will not necessarily be representative of the properties of the powder alone. The upper ring of Scarlett and Todd’s cell was divided into three concentric sections, each independently mounted, so that the effect of the variations in the flow profile across the annulus could be reduced_ However, in this cell, as for Carr and Walker‘s cell, the shear stress was measured as the twisting moment on the cell lid, which may not be the same as the shear stress in the body of the powder. The development of the annular shear cell for both soil and powder studies has been reviewed by Bishop et al_ [ 151, who also described in defti a cell built to study the failure properties of soils. To develop a system closer to the behaviour in a powder miser, yet retaining sufficient control to enable basic approach to be maintained, an annular shear cell was designed and built for the particular purpose of esamining particle movement rather than stresses_ The strain rate was dependent on vertical position, and, although this makes the obtaining of basic data difficult, it represents more closely a naturally forming failure zone.

Fig. 2. Str‘tpped view of annular shear cell showing the lower part of the annulus. CELL

DESIGX

AND

OPERATION

To study the microscopic mising of powders, a failure zone must be formed within the powder with the minimum of restriction or interference from the cell. From previous designs and the requirements peculiar to this work, key points in the annular cell design 1221 (Fig- I) can be identified: (a)Overalisizeofcell

(b) Width of annuius (c) Depth of annulus (d) Position of cell wall division (e) Alignment of upper and lower wall sectioizs (f) Cell lid and top ring alignment (g) Application oI^nonna.I Ioad (h) Materials of constrxtion (i) Drive system (j) Loading and unloading of powder (k) Monitoring and recording of results The overall size of the cell was determined as a compromise between the size necessary to restrict the effect of the cell on the poxvders, such as orde,ring of the packing, to minimise the variation in strain rate across the an-

nulus, and the availability of construction materials. The cell was constructed around a central shaft varying in diameter from SO mm at the bottom to 40 mm at the top, and bolted to a baseplate_ The annulus (Fig. 2) was 45 mm wide. having inner and outer diameters of 217 and 307 mm, the ratio of the velocity at the inner walI to that at the outer wall being 0.70. To ensure that a complete failure zone could form, the cell was built to allow for a maximum bed depth of 200 mm, although this could be adjusted to any depth. A layer of coarse sintered bronze was mounted on the inner surface of the upper and lower confining rings_ As well as providing a surface rough enough to prevent the powder slipping, the sintered sheet enabled the confming rings to be made porous so that humid air might be passed through the simple ccl: to reduce electrostatic charge on the particles_ The cell walls were made from four lengths of stainless steel pipe, allowing both abrasive powders to be used without damage to the cell and corrosion problems to be avoidedBoth the inner and outer surfaces of each section were machined accurately_ The txyo lower

wall sections were press-fitted into aluminium alloy discs and mounted on a pair of taper roller bearings fitting onto the central shaft_ The inner wall section in the upper part of the cell was mounted on a brass bush fitted onto the central shaft_ To mount the outer wall section in the upper part of the cell, it was fitted to a ring and located in position by the outer shell of the cell_ This was an aluminium alloy casting bolted to the cell baseplate_ PTFE pads were positioned in the side of the casting_ backed by springs and adjusting screws, and set into the top of the casting as bearings for the outer upper wall section_ The clearances between the wall sections were set to less than 0.1 mm by locking the outer upper section of the cell wall into position on the outer casting and, using locking rings set on a screw thread cut on the central shaft, the lower part of the cell was adjusted vertically until minimum clearance between the upper and lower section of the cell was achieved_ The clearance between the inner wall sections was then set by adjustment of the locking rings below the upper section of the cell. The cell lid, consisting of the upper confining ring mounted below a cylindrical section fitted onto an alloy disc, was located on the central shaft by a recirculating ball bearing that allowed both vertical and rotational movement. The weight of this section was used to apply a normal load to the powder and was counterbalanced using a lever arm and weights mounted on the apparatus frame_ This arrangement allowed normal stresses up to 5.18 kN/m’ to be applied to the powder. During experiments the upper wall sections were kept stationary, and the lower section of the cell rotated. To prevent the cell lid rotating, and to measure the twisting moment on it, it was attached to a spring balance mounted on the apparatus frame_ For loading and unloading of the cell, the upper and lower sections were locked together using a vertical bar through the inner wall section mountings, and a pin across the gap dividing the outer wall sections_ With the lid removed, the cell could then be rotated without disturbing the powder_ The cell was driven by a 0.2 h-p. d-c. motor with a speed range of 5 to 3000 r-p-m. The drive was transmitted via a 36:l reduction gearbox, and further reduced by 6:l on the

ring gear mounted on the bottom of the cell, producing a maximum cell speed of 12 r.p.m. The mctor unit operated on a feedback signal from a generator, mounted on the same shaft as the motor, to maintain a set speed. The control unit monitored the motor speed and the torque developed_ If allobvance is made for the frictional losses in the drive, the motor torque gives an immediate measure of the shear strength of the powder in the cell. Torque and speed outputs from the control unit were fed to a U.V. oscillograph fcr contin-uous recording_ The bed height in the cell was also monitored using a position transducer mounted on the top of the cell to measure the vertical movement of the cell lid. To eliminate high frequency oscillations of the torque output on the oscillograph.. a 0.5 s time constant was built into the interface circuitry_ Consequently very fast torque, or shear stress, changes could not be recorded. The cell was filled by locking the upper and lower sections together and pouring the powder into the rotating cell through a distributor mounted on the stationary outer casing of the cell. The distributor design was based on a motionless miser investigated by Cooke [ 23]_ It consisted of a rectangular tube approsimately-42 mm X IS mm and about 220 mm long. In the lower 60 mm there were ten bars across the inside of the tube which broke up the downward flow of powder and distributed it across the width of the annulus. During filling, the distributor was adjusted vertically so that it was always about 10 mm above the powder surface. This technique produced a level surface across the annulus and provided a consistent and simple means of loading the cell. Powder was removed from the cell using a vacuum system run by a constant pressure compressed air supply. _A rectangular nozzle, ‘7 mm across and the width of the annulus, was mounted on the end of a pipe and positioned just above the powder surface_ Compressed air was directed upwards into the pipe just above the nozzle, creating a vacuum and lifting the particles at the surface of the powder. The entrained particles were separated in a small cyclone and collected in sample bottles. The complete unit was mounted on the side of the cell outer casing with a vertical scale attached so that its position could be

22

adjusted a-,d set at the required level_ Using this system, layers of powder of a known depth, usually 2 mm, could be removed from the bed. This left a flat surface on the powder remaining in the cell, and proved to be a consistent method of removing the powder in controlled quantities. The vacuum system was also used to level the powder when necessary during the loading and to level the upper surface before the cell lid was placed on the powderVELOCITY

level of the split in the cell ~a&, uersus horizontal displacement x, the latter being equivalent to the horizontal velocity as a percentage of the cell velocity. This can be regarded as a direct measure of the velocity profile across the faiIure zone (Figs. 3 and 4). The

DISTRIBUTION

Two groups of esperiments have been carried out, using glass beads of mean diameter 1.91 and 4.00 mm, to obtain basic information about behaviour in the annular shear cell, namely (a) the shape and position of the powder failure zone, and (b) the shear stress for a range of normal stresses and strain rates_ To identify the shape and position of the failure zone, the cel1 was fiIIed and the top surface of the powder Ievelled. As bed preparation, the powder was strained for a few revolutions under esperimental conditions_ _A vertical column of tracer particles was then inserted and the cell r!!n for one revoiution. To insert such tracers, a solid bar (10 mm d&n_) was pushed down through the bed at the centre of the ann~h~s, a close fitting thinwalled tube was then slid over the bar into the powder and the bar removed_ The tracer partfties were poured into the t-ube, which was then removed slowly, being tapped gently to ensure that the particles did not stick to the tube_ To determine the position of the tracers after the experiment, the ring holding the top outer wall section was marked off into one hundred divisions_ The powder was then systematicaliy removed in layers, and at each level the circumferential movement of the tracer particles was recorded as a percentage of the total movement of the cell_ The spread of the tracer was allowed for by recording the extremes of the distribution at each level- Unstrained material below the failure zone was dermed to have a position of 070, and above the faihue zone lOO%, corresponding to the movement relative to the bottom of the cell. The results can be presented directly as a plot of vertical position z, measured from the

i

0

20

10

50

DE.plecenent

I IO0

eo

I

t-1.1

Fig. 3. Failure zone identification, velocity profile I-ES)_Vertical position against circumferential tracer

displacement as per cent of one cell revoiution. 1.91 mm_ diam. glass. norma! stress 2.7 kN/m-, cell speed 2 r_p.m.

0

20

LO

60

O~splocemen~

ec _

100

1V.l

4_ Failure zone identification (Fl5). 4.00 mm ghss, normal stress 2.7 kN/m2, cell speed 0.5 r_p.m.

Fig_

23

gradient of the velocity profile (dr/dz) is proportional to the strain rate at that point within the failure zone, so that the strain rate profile across the failure zone can be determined from the velocity profile. if we consider the failure zone as the region within which the material suffers strain, then it corresponds to the range of vertical positions for which the displacement of the tracers is other than 0 or 100%. The exact boundaries of the failure zone are arbitrary, but its approximate position is determined readily_ To enable results to be compared and to be used later, the results of the failure zone identification experiments must be quantified_ Therefore an arbitrary but consistent means of analysing the results is introduced_ Each horizontal tracer distribution is reduced to a single position at the centre of the distribution, and then only those points in the range 10 - 90% of the total horizontal cell movement are considered (number of mean tracer positions, Table 1). Xn empirically linear line is then fitted to these points (Figs. 3, 4, Table 1). From the slope and intercept with the vertical asis of this line, the failure zone depth and position are found by extrapolation, and an average strain rate calculated_ For esample, from Fig. 3, dx/dz = 24.1 mm/100% Hence, failure zone depth, h, = 24.1 mm. The total cell movement is 823 mm in 30 s, and the strain per revolution is 34.1 rev-l. Hence, mean strain rate, 7, = 1.14 s-l _ Although this does not give a full representation of the behaviour in the failure zone, it gives a set of average values which are sufficient to characterise the failure zone (Table l)_ In those esperiments where there are similar quantities of material above and below the split in the cell wall, the failure zone is approsimately in the centre of the bed. When the position and depth of the material bed are altered (Eli, E12, E13), movement occurs adjacent to the top or bottom boundary rings, and an incomplete velocity profile is obtained_ However, that part which does exist closely matches the corresponding part of tbe complete velocity profile obtained from the deeper beds_ It is clear, therefore, that the position of the failure zone is largely determined by the split in the cell walls. The shear stress is highest at the cell wall split because of the frictional force between the material and the cell walls either side of

the split. If the strength of the powder is approsimately equal throughout the bed, then failure will occur at the cell wall split. Once the powder has failed, we can expect the failure zone to expand towards some characteristic failure zone depth. The shear stress necessary to maintain strain in an already failing material is less than that necessary to initiate failure_ Therefore it is probable that the centre of the failure zone will stay in the region of the cell wall split. The velocity profile can be split into three regions. There is a central region of high strain rate where the tracers are widely spread: secondly, there are transition regions above and below this. Thirdly, between the transition regions and the bed confining rings there are blocks of unstrained material, so that in this case the depth of the failure zone is controlled by the powder and not the limitations of the cell. The spread in Figs. 3 and 4 show that the tracer may be distributed over as much as 50% of the circumference at a single level. This large apparent horizontal spread is probably the result of vertical movements of the particles from the layers above and below-. X movement of a few millimetres vertically is sufficient to create the observed horizontal spread_ A similar displacement horizontally would not be significant in comparison with the total horizontal movement_ The esperimental method will also increase the distribution, both because of the difficulty of deciding if a tracer is at the material surface or not, and because the bed is analysed every 2 mm vertically, which is equivalent to O-5 or 1 particle diameter of the powder_ At this scale, a compromise between a continuous and a discrete interpretation may be necessary. It is clear that the idealised concept of layers of particles moving over each other during straining of the powder is unsatisfactory, especially in the central region of the failure zone where the particle movement is greatest_ If it is assumed that the vertical movement of particles is directly related to the strain rate, then the large spread of tracer in the centre of the failure zone and the much smaller spreads towards the edges are as would be expected. The experimental data are not sufficiently precise to define clearly the boundaries of the failure zone, if such boundaries exist. It can

21

25

be interpreted as a smooth reduction of strain rate towards the unstrained regions of the bed but should probably be considered as an intermittent reduction related to the particle size_ A particle moving only a small fraction of the total horizontal cell movement would be expected to complete that movement in a number of discrete stages rather than moving continuously_ The transition region is slightly deeper at the top of the failure zone than at the bottom (Figs. 3,4). Otherwise the shape of the velocity profile is similar above and below the centre of the failure zone, and consequently the strain rate profile will be approximately symmetrical about the centre. The experimental results (E9, ElO, E16) suggest that increasing the normal load on the cell reduces the depth of the failure zone. There is a significant difference between the normal load at the top and bottom of the failure zone due to the weight of powder, typically about 0.5 kN/m’_ This could account for the upper transition region being deeper than the lower region. The difference may also be related to the equipment construction. Espansion of the bed while it is being strained is allowed for by the upper ring of the cell being able to float vertically, so that on espansion the whole bed above the failure zone moves upwards, whereas the powder below the failure zone remains stationary_ This movement of the powder may have a loosening effect on its structure so that the failure zone expands upwards more readily than downwards. With the apparatus and powders used for these experiments, it is not possible to determine whether either or both of these effects are significant. The majority of failure zone identification esperiments were performed after the powder had been prepared by straining for several revolutions of the cell, but some were done without any previous straining to form comparable pairs of experiments (El4 + ElO, E4 f E3). The previously unstrained powder produced a narrower failure zone and a strain rate approximately 25% higher than for the prepared powder. These experiments give an average over the first revolution of the cell, equivatent to a strain of about 30, so cannot indicate how fast the failure zone builds up to its full depth after initial failure.

To test for consistent failure zone formation across the annulus, the failure zone identification esperiment was performed using tracers loaded in columns close to the walls (E5 - ES)_ From these experiments and others, and when running the cell with the lid removed, some difference in behaviour was apparent in a layer about two particle diameters wide at the cell walls. During the loading it was not possible to ensure that all the tracers were initially adjacent to the walls, so any subsequently found more than two particle diameters away were neglected as being unrepresentative of the behaviour at the walls. In both the prepared and unprepared powders, the faiiure zone is shallower and the strain rate greater at the walls than in the centre of the failure zone. The failure zone depth is similar at both inner and outer walls, so the strain rate is highest at the outer wall, 2.65 s-l compared with 1.47 s-i (Table 1). Xt a constant cell speed, the strain rate is higher for the smaller material, and the failure zone depth correspondingly less_ For the three pairs of esperiments El + E9, E3 + ElO, E4 + E14, the strain rate is approsimately 60% greater for the 1.91 mm particles and the failure zone depth is typically 24 mm compared with 36 mm for the 4.00 mm particles. If this is espressed in dimensionless form, as a number of particle diameters, then the depths are 12.6 and 9.0 for the 1.91 and 4.00 mm particles respectively, so that the depth of the failure zone cannot be defined simply as a single value in terms of a number of particle diameters_ Increasing the normal load on the powder reduces the failure zone depth, hence increasing the strain rate (E9, ElO, E16). Xt lower normal loads the contribution due to the weight of powder in the cell becomes more significant, the normal load at the top of the failure zone becoming a much smaller fraction of the normal load at the bottom- This is consistent with the greater difference in the transition region velocity profiles at low normal load, when the upper transition region is much deeper than the lower, which seems to be insensitive to changes. Lowering the cell speed from 2 to 0.5 r.p.m. (E15) caused the average strain rate to drop by the same ratio, 0.76 to 0.19 s-l, and the shape of the velocity prof& was -unchanged.

TABLE

2

Shear stress measurements _

Particle diameter

Srrain rate T (s-l)

Cell speed (r-p-m.)

d (mm)

for glass particks Lid stress us. shear stress. slope

illi

-1_oo

O-1

-t_oo -1-00 a.00 -1-00 -1.00 a.00

0.5 20 -I_0 6.0 s-0 10-O

-f_OO 0.1 3.0 f-0

1.91 l-91 1.91

S-0 10.0

I-91

SHEAR LAR

STRESS

Slcpe

i/0

-4ngle of internal friction

(ded

(HI/m-) -1.21 -0.95

0.33 0.32

0.45

-l-IT

0.Z

1252 3 3s ___ 3-0-l 3.80

0.51 O-56 0.5s 0.5s

-1.33 -1.15 -0.96 -0.9-8

0.2-I 0.2-l 0.25 0.25

-1.10

0.3-i

15

O-3-3 0.16 0.25

-1.05 -I_!% -

029 029 -

16 16

0.19 0.2’7 0.20

-1.4s - 1.35 -158

0.25 0.3--l 0.2-t

1-l 13 13

-1.3-1

0.26

14

0.33

Mean

SHEAR

Shear stress L’S. normal stress Interc42~t. i = 0

O_i6

0.25

0.19

ualue~r

0.05i 1.1-l ‘> “S ___ 3.-&z -2.56 5.iO

6-O

cell

1s 1s 15 13 13 l-1 14

0.038

Nean

1.91 1.91 1.91

in the annuIar

values:

JIEASUREMENT

IN THE

XNNU-

CELL

The strength of a powder, measured as a shear stress, can be determined for a range of normal loads (normal stresses) and cell speeds (strain rates) in the annular shear cell_ Xlthough this particular cell has limitations in this respect, it is necessary, before looking at mixing, to esamine this matter. The cell was ioaded and run under test conditions for a few revolutions to eliminate any effects associated with the loading and initial failure. The variation in shear stress with normal load was determined at each strain rate, starting with the cell lid removed and then progressively increasing the normal load_ Xt each setting the cell was alloxed to run until a steady cxrdition was reached. The velocity profile is askumed to maintain the same form throughout t’_re range of cell speeds, so that a doubling of cell speed gives a doubling of the average strain rate. The bed expansion is measured as a clrange from the height of the unstrained bed. _A.straight hnz can be fitted to the shear stress uersus normal stress data at constant cell speed, aud this will give an angle of internal friction for the powder, tan-l (5/e)-

The results for the angle of internal friction (TabIe 2) are generally lower than those reported elsewhere for similarly sized glass particles [ X3]_ The powder in the upper part of the failure zone and the proportion of the normal stress transmitted to the cell walls mean that the values of normal stress used may be significantly different from the actual normal stress in the centre of the failure zone. This would be reflected in the slope of shear stress uersus normal stress pIot (i/a)_ A corollary is that top lid shear stresses 7, (Table 2) are less than 7. Although the absolute values obtained may be suspect, as the normal stress is increased at a constant cell speed the depth of the failure zone is reduced, increasing the strain rate. Therefore the slope of a plot of shear stress uersus normal stress at constant strain rate will be greater than that plotted at constant cell speed. As the cell speed is increased, there is a small decrease in the shear stress for a constant normal stress and in the measured angle of internal friction. The highest cell speed used was a factor of 100 greater than the lowest, but this caused a reduction in the angle of internal friction of only about 30% Bridgwater 1201 found that the shear stress increased

27

slightly with increasing strain rate, but that the form of the results was very dependent upon the normal stress and the materials used. The maximum strain rate used in these esperiments corresponds to the minimum used by Bridgwater, so the results are not readily compared_ For the 4.00 mm glass, straining causes an initial vertical expansion of the bed by a height equivalent to about one particle diameter_ The bed height then changes slightly with changes in strain rate and normal stress. At each of the strain rates used, the bed contracts by about 02 particle diameters as the normal stress is increased from 1.35 to 5.18 kN/m’_ This change is small in relation to the overall bed depth or the failure zone depth, but is about 20% of the expansion that occurs at initial failure. For the same increase in normal stress, the failure zone depth also decreases by about 20% (Table 1). An increase in cell speed by a factor of 100, from O-1 to 10 r-p-m., at constant normal stress causes an espansion of about 0.15 particle diameters_ For the 1.91 mm glass the change in bed height vvith normal stress is similar to the A.00 mm glass, but no consistent change in the bed height with strain rate is apparent. CONCLUSION

The design of the cell and the operating techniques have been kept simple, although & a result some of the esperiments were tedious, requiring separation of tracer particles by hand. This enables results to be generated without the long delays that would have resulted from the development of more sophisticated techniques, of doubtful use before a phenomenon is understood_ It permits easy changes between types of esperiment so that a broadly based investigation can be carried out_ In characterising the performance of the cell, the formation of a consistent failure zone was demonstrated_ The velocity distribution in the failure zone was estimated for a range of conditions and may be characterised by an estrapolated failure zone depth. For the two sizes of particle used, the shapes of the velocity profiles were similar, but the failure zone depth could not be described by a single value common to both particie sizes, as either a di-

mensional (h) or dimensionless depth (h/d). Decreasing the particle size reduced the failure zone depth but the dimensionless depth increased_ Increasing the normal load decreased the failure zone depth and therefore increased the strain rate at a given cell speed. The particular application of the basic information presented here to the mising and segregation of free-flowing solids is considered in Part II of this work [24] _ ACKKOWLEDGEMCNTS

The work was made possible by a research grant from the Science Research Council and by the provision of an S.R.C. Research Studentship for D.J.S. The cell construction was carried out by Mr. R. _A_Ducker. LIST OF SYMBOLS

particle diameter, mm failure zone depth, mm (horizontal displacement)/(cell displacement) vertical position from the split in cell walls (positive upwards), mm mean strain rate, s-r normal stress applied to top surface of material, kN/m” shear stress at split in cell walls, l-X/m’ shear stress at lid, kN/m” REFERENCES R. L. Brown, The fundamental principles of segregation, J. Inst. Fuel. 13 (1939) 15 - 19. J. Bridgwater. Fundamental powder mising mechanisms, Powder Technol., 15 (19’76) 216 - 235. J. BI_ Coulson and N. K. blaitra. The mixing of solid particles, Ind. Chem., 36 (1950) 55 - SO_ N. Harnby, A comparison of the performance of industrial solids misers using segregating materials. Powder Techcol.. 1 (1967) 94 - 102. L. T: Fan, S. J. Chen and k. A. Watson, Solids mixing. Ind. Eng_ Chem.. 62 (71(1970) 53 - 69. P. V. Danckwer& The definitidn~and measurement of some characteristics of mixtures, Appl. Sci. Res., A3 (1953) 279 - 296. L. T. Fan and R. H_ Wang, On mixing indices, Powder Technol., 11 (1975) 27 - 32. P. M. C. Lacey, Developments in the theory of particle mixing. J_ Appl. Chem., 4 (1954) 257 268. R H. Roscoe, The influence of strain in soil mechanics, Geotechnique, 20 (1970) 129 - lTO_

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P. L. Bransby. P. &I. Blair-Fish and R. G_ James, An investigation of the flow of granular materials, Powder TechnoL. 5 (1971172) 1 - 17_ R_ Hogg, D. A. Augenstein and C. L. Hwang, Segregation in flowing powders, AIChE 68th Annu. Meet., Los Angeles, 1975, paper SSb_ A. M_ Scott and J_ Eridgwater. Interparticle percolation: a fundamental solids mixing mechanism. Ind_ Eng_ Chem_ Fundam.. 11(19’75)22 - 2’7. J. BridEwater. M. H. Cooke and A. &I. Scott. Interparticle percolation: equipment cfevelopment and mean percolation velocities, Trans. Inst. Chem. Eng_. in press. XI. J. Hvorztev, Torsion shear tests and their place in the determination of the shearing resistance of soils. Proc. Am_ Sot. Test. Mater., 39 (1939) 999 - 1023_ A_ W_ Bishop. G_ E_ Green, V_ Ii_ Garga. A. Xndressen and J_ D. Brown, -4 new ring shear nppnratus and its application to the measurement of residual strength. Geotechnique, 21 (1971) 273 328. X. \V_Jenike. Gravity flow of bulk solids, Utah Univ. Eng. Esp. Stn. Buli. 108 (1961). J_ F. Carr and D. JI. Walker, An annular shear cell for granular materials. Powder Technot., 1 (196i/ 6s) 369 - 353.

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J_ Novosad. Studies on granular materials. 2. Apparatus for measuring the dynamic angle of internal and external friction of granular materials, Collect. Czech. Chem. Commun., 29 (1964) 2697 - 2701. J. Bridgwater and D. F_ Bags&r. Annular shear cell design and operation: considerations arising from some detailed studies, paper presented at 3rd CHISA Congr., Marienbad, Czechoslovakia, 1969. J_ Bridgwater, Stress-velocity relationships for particulate solids, ASME paper 72-hIH-21 (lSi2). B. Scarlett and A. C. Todd, A split ring annular shear cell for the determination of the shear strength of a powder, J_ Phys. E, Ser. 2, 1 (1968) 655 D. J_ Stephens, hIking and segregation of powders in failure zones, D. Phil. Thesis, Univ. of Osford. 1976. &I. H. Cooke and J_ Bridgwater. New motionless mixers for solids, Inst_ Chem. Eng_ Symp. Ser., -25 (197’7) G-5-l to G-5-8. D. J. Stephens and J_ Bridgwater. The mixing and segregation of cohesionless particulate materials. Part II. Microscopic mechanisms for particles differing in size, Powder Technol., 21 (1978) 29 - -14.