Interparticle forces in fluidisation: a review

Powder Technology 113 Ž2000. 261–268 www.elsevier.comrlocaterpowtec

Interparticle forces in fluidisation: a review J.P.K. Seville ) , C.D. Willett 1, P.C. Knight School of Chemical Engineering, The UniÕersity of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 24 December 1998; received in revised form 12 October 1999; accepted 28 April 2000

Abstract Because of the balance of forces in fluidised beds, particle interactions can have a strong effect on their microscopic and macroscopic behaviour, leading to agglomeration and defluidisation. Three types of particle interactions are reviewed: van der Waals forces, liquid bridge forces and sintering. Sintering is qualitatively different in its effects because it is a time-dependent process. The observed effects of these three types of interactions on fluidisation behaviour are described and explained in terms of simple models. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Fluidised beds; Van der Waals forces; Liquid bridges; Sintering

1. Introduction — the nature of interparticle forces By definition, a state of fluidisation exists when the force of gravity on a set of particles is balanced by the drag arising from the flow of the fluidising gas. It follows, therefore, that small interparticle forces, which may not be noticeable in other circumstances, may have observable consequences at the point of fluidisation and beyond. Interparticle forces can occur due to a variety of causes; those of interest in this paper are van der Waals interactions, liquid bridges and sintering. 1.1. Van der waals forces AVan der Waals forcesB is a collective term taken to include the dipolerdipole, dipolernon-polar and nonpolarrnon-polar ŽAdispersionB . forces arising between molecules w11x. Though other intermolecular forces can occur, such as hydrogen bonding, these are related to the specific chemical nature of the materials; van der Waals forces always exist. Some confusion has arisen between the expressions for intermolecular and interparticle van der Waals forces. Although intermolecular forces decay with

) Corresponding author. Tel.: q44-121-414-53-22; fax: q44-121-41453-77. E-mail address: [email protected] ŽJ.P.K. Seville.. 1 Present address: Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, UK.

molecular separation, a, as ay7 , when the pair potentials are integrated between macroscopic bodies, such as spherical particles, the resulting force is much less sensitive to separation, decaying as ay2 in the case of sphere–sphere interaction:

Fvw s

AR 12 a 2

Ž 1.

where R is the sphere radius, A is the Hamaker Žmaterials-related. constant and a is the surface separation, which takes a minimum value of the order of the intermolecular spacing. Suitable values for the variables give the lines plotted in Fig. 1. It will be apparent that intermolecular forces depend more on the particle surface properties than on the bulk, so that it may be more plausible to assume Žor measure. a surface roughness and use this to determine the curvature. The van der Waals force then depends on this local curvature and is independent of R. This result is also plotted in Fig. 1, and suggests, for the set of variables chosen here, that spherical particles of diameter of order 100 mm should exhibit interparticle van der Waals forces to equal their single particle weight. If the gross particle radius is taken as the controlling factor, as in Eq. Ž1., the corresponding diameter is 1 mm, which is less plausible. Particles of 100-mm sizes are commonly found adhering to surfaces and resisting the force of gravity; 1-mm particles are not!

0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 0 0 . 0 0 3 0 9 - 0

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predicted boundaries between groups which are shown in Fig. 2. Although much more complex hydrodynamic explanations have also been proposed, Molerus’s explanation fits the facts, particularly since a number of workers w19x have shown how enhancing the interparticle forces can move the observed behaviour from B to A to C. Molerus attempted to obtain a value for the ratio of particle weight to interparticle force at the transition from group B to group A, from literature values of Hamaker constants; this resulted in an estimate of 0.16. However, as noted above, van der Waals forces are in practice determined by surface properties and a more realistic value, taking into account experimental measurements for interparticle forces w19x might fall in the range 2.0–3.0

1.2. Liquid bridges

Fig. 1. Comparison of the magnitude of interparticle forces Ždashed lines indicate asperity-to-plane contact.. Theoretical interparticle forces for single-point contact between equal spheres Žin air., with particle weight plotted for comparison w22x. van der Waals Ži. As6.5=10 -20 J Žquarts. Žii. values presented for interparticle separations of

Capillary

Electrostatic

Weight

˚ and 4.0 A˚ 1.65 A Žiii. dashed lines assume asperity-to-plane contact with asperity radius 0.1 mm Ži. g s 72.8=10 -2 N m -1 Žwater. Žii. values are maximum Ž b 0. Žiii. dashed lines indicate asperity contact as above Ži. maximum force Žopposite sign. Žii. ´ r s1; ´ 0 s8.9=10 -12 C 2 N -1 m -2 Žiii. charge density s10 mC m -2 Dp s 3=10 3 kg m -3

™

The famous group classification diagram due to Geldart w9x has been of great value in allowing easy prediction of fluidisation properties in terms of particle size and density, but the reasons for the different types of behaviour have remained controversial, particularly the transition from group B type Žbubbling at minimum fluidisation. to group A type ŽAuniformB bed expansion with separation of the points of minimum fluidisation and minimum bubbling.. Molerus w16x suggested that this transition corresponds to a particular value of the ratio between interparticle force and particle weight: in group A, these forces are of comparable magnitude; in group B, interparticle forces are insignificant by comparison with weight. This approach results in the

Liquid bridges are more interesting than van der Waals forces from a practical point-of-view, since their magnitude can be adjusted by altering the amount of free liquid and its properties, particularly surface tension and viscosity. They are of practical importance in agglomeration processes, driers, and in some types of reactors and bioreactors. They are also more complex than van der Waals forces in that they exhibit both dynamic and static forces and are dissipative of energy. ŽVan der Waals forces are conservative, although they can lead to surface deformation, which may not be.. The static liquid bridge force arises from the sum of the surface tension force and the force arising from the pressure deficit in the liquid bridge w22x ŽFig. 3.: Fls s 2p r 2 g q p r 22 D P

Ž 2.

where D P is the reduction in pressure within the bridge with respect to the surrounding pressure. The magnitude of this force is difficult to compute exactly, even for spheres, because the bridge forms a

Fig. 2. The famous group classification of Geldart w9x for the four broad types of fluidisation behaviour.

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263

Fig. 3. Liquid bridge between two equal spheres. as Half particle separation.

gas–liquid interface of constant curvature in order to satisfy the Laplace equation: D Psg

1

1 y

r1

r2

Ž 3.

Fig. 4. The effect of dimensionless half particle separation distance, ar R, on the response of the attractive force as a function of bridge half-filling angle, b , calculated from the toroidal approximation.

The liquid bridge also dissipates energy by viscous flow, away from the contact area on approach and vice

This results in a bridge shape ŽFig. 3. in which r 1 is a variable for a given bridge volume, so that r 2 must also be a variable. However, the torodial approximation due to Fisher w8x, in which r 1 is taken as constant, enables a simple and reasonably accurate result to be obtained. At contact, the maximum static liquid bridge force is Fls ,max s 2p Rg

Ž 4.

which is plotted in Fig. 1 and again compared with the force which would arise if the contact were dominated by surface asperities of dimensions independent of gross particle diameter. For water, the static liquid bridge force is rather larger than the maximum van der Waals force. A curiosity about Fisher’s result for surface contact is that it predicts a decreasing liquid bridge force for increasing liquid loading, the opposite of what one would intuitively expect. This apparently paradoxical behaviour has been confirmed by careful laboratory experiments w5,15x down to very low bridge volumes. Peitsch w18x attempted to resolve the problem by suggesting that all real contacts are rough and that an effective surface separation needs to be considered. If this is done, separations of order 10y3 of the sphere radius are enough to reduce the predicted liquid bridge force considerably for bridge half-angles below 58, and the force now shows a maximum at a bridge volume which increases with the separation as shown in Fig. 4. It is generally assumed that the static Žor low relative velocity. liquid bridge force is conservative, but Willett w24x has recently shown, both experimentally and theoretically, that this is not the case. If the contact angle is non-zero and the surface is AroughB, both of which are often true, the contact line may be ApinnedB and the forcerseparation curves on approach and departure follow different paths ŽFig. 5., leading to hysteresis and energy dissipation.

Fig. 5. Demonstrating the effect of an advancing and receding contact angle, w , on the interparticle force due to a liquid bridge between two equal spheres, Ža. schematically and Žb. experimentally and theoretically Ždashed lines., for a polyethylene glycol liquid bridge during four 100-mm oscillations. V ) s Liquid bridge volumerparticle volume w24x.

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versa. The viscous force always opposes relative movement, unlike the surface tension force. During separation, the reduction in pressure around the point of closest approach may easily lead to cavitation in the liquid w3x. The force is given, to a first approximation, by Reynolds’ lubrication equation w1,4x:

question of whether a fluidised bed can be adequately described in terms of independent particles in free-flight between collisions, this approach predicts defluidisation when

Fld s 6pm R 2 Õra

where St ) is the critical value of the Stokes number, e is the coefficient of restitution for the particles, a0 is taken as 2r3 of the liquid layer thickness and a1 is the characteristic roughness dimension. In general a1 is unknown, but can in principle be obtained by plotting St ) against ln a0 . A further difficulty with use of this equation is that it requires a measurement of the collision velocity, which is not generally known. Ennis et al. w7x assumed, quite reasonably, that

Ž 5.

which implies a singularity at contact. In practice, the surfaces are rough, so that there exists a non-zero minimum separation, a 0 , andror they deform. The ratio of viscous force to static force has been calculated, as a function of separation rate, for the following variable values: b s 58 and 308, a s 1 and 10 mm, 2 R s 922 mm, g s 0.072 N my1 Žwater at 258C., and m s 0.001 kg m sy1 Žwater at 258C.. The results are plotted in Fig. 6, showing that, in this case, viscous forces can become significant by comparison with capillary forces for separation rates above about 1 cm sy1 if the particle separation is small, rising to about 1 m sy1 for larger separations. Barnocky and Davis w3x have considered in detail the conditions under which particles impacting on a thin liquid layer will be captured, using Reynolds’ lubrication equation to estimate the forces during the impaction process. They showed that whether capture occurs depends on the value of an impaction Stokes number, mÕr6pm R 2 , where m is the particle mass. This approach was also used by Ennis et al. w7x in developing a predictive approach to deciding whether surface-wet particles will coalesce in agglomeration processes, including fluidised beds. For spheres that show significant roughness, it was predicted that defluidisation would occur when the particles within the bed no longer had enough kinetic energy to escape capture on inpact with their neighbours. Leaving aside the

St ) s Ž 1 q ey1 . ln Ž a 0ra1 .

Õ s a Ž U y Umf .

Ž 6.

Ž 7.

where a is a constant of proportionality that depends principally on the particle size. Ennis et al. w7x tested this approach by plotting a modified Stokes number, mŽU y Umf .r6pm R 2 , against the logarithm of the layer thickness for a series of experiments performed by Gluckman et al. w10x, where liquid loading but not viscosity was varied. Good agreement was reported, although this has not been confirmed by other workers in the field. The third, and possibly most significant, energy dissipation mechanism is the stretching and eventual rupture of the bridge; in a fluidised bed, bridges can be imagined to be continually rupturing and reforming. The energy thus dissipated depends on the rupture distance, which takes the very simple form w14x amax s Ž 0.5 q 0.25w . V 1r3

Ž 8.

where V is the liquid bridge volume.

1.3. Sintering The forces arising from sintering are quite different in kind from those discussed above. Sintering is a time-dependent process in which material migrates, due to diffusion, viscous flow or some other mechanism or combination of mechanisms, to the region of contact to form a AneckB. The size of the neck increases with time according to an equation of the form: x

ž / R

Fig. 6. Ratio of viscous to capillary forces as a function of separation rate Žfor water 258C; spheres of diameter s922 mm..

2

s kt

Ž 9.

where x is the neck radius at time t . In the Frenkel equation for viscous sintering w13x, for example, k s 3gr2 R m.

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In a fluidised bed, a sinter neck of sufficient size may become permanent, resisting the breakage forces due to the movement of solids and gas, and such growth may eventually lead to defluidisation. In general, the rate of material migration is a sensitive function of temperature. In viscous sintering, as above, migration is driven by surface energy minimisation Žwhich is relatively independent of temperature. and opposed by viscosity Žwhich is an Arrhenius function of temperature..

Fig. 7 shows the dramatic effect on the average particle velocity Žfor 1-mm dry ballotini. of addition of a very small amount of liquid, given here as the equivalent thickness of coating, assuming all the liquid to be well-dispersed over the available surface. ŽParticles were pre-coated in a mechanical mixer and experiments were repeated with different batches to ensure uniformity of coating.. As predicted by Baeyens and Geldart w2x, the particle velocity Õp is a linear function of fluidising velocity:

m s m 0 exp Ž ErRT .

Õp s a Ž U y Umb .

Ž 10 .

The result is that sintering occurs much faster at higher temperatures. This provides the explanation for high temperature sintering of ash in combustion and gasification processes and in reduction of iron w12x. It may be noted that the effect of temperature on sintering is therefore quite different from its effect on dynamic liquid bridge forces, where higher temperatures and reduced viscosity leads to lower forces ŽEq. Ž5.., lower energy disruption in collision and therefore lower probability of agglomeration and growth Žprovided the fractional liquid loading remains constant..

Ž 11 .

where Umb is the minimum velocity for bubbling. However, Baeyens and Geldart’s own measurements show a value of a which is about an order of magnitude less than that indicated here; the difference is due to the fact that they were only able to measure particle velocities at the wall, whereas Õp here is an average value for the bed as a whole. The minimum bubbling velocity, Umb , can be obtained either by direct observation or by extrapolating the lines in Fig. 7 to the axis; the two methods give similar results. It is unclear whether there is an effect of viscosity here; Simons et al. w23x report no effect of viscosity on the

2. Effects of liquid loading on fluidisation It is clear that the addition of thin layers of liquid to a fluidised bed makes the particles more cohesive, and, from Fig. 1, the maximum liquid bridge force can equal the particle weight for particles of order 100 mm and below. One consequence of this is to increase the settled bed voidage at minimum fluidisation w19,20x, so increasing the minimum fluidisation velocity. The cohesive effect stabilises a structure of higher voidage than would otherwise be possible. These early studies also showed that addition of a small quantity of liquid was enough to cause a separation between the points of minimum fluidisation and minimum bubbling, with a stable region of expansion in between. More recently, Willett w24x has examined the effect of involatile oil coating on fluidisation of spherical lead–glass ballotini for the cases of constant and linearly increasing liquid loading, the latter simulating the conditions of an agglomeration process, for example. In both cases, the particle motion was monitored by following a single radioactive tracer particle using the technique of Positron Emission Particle Tracking ŽPEPT; Parker et al. w17x.. This allows a particle which is representative of the bulk Žin this case identical. to be followed at velocities of up to a few metres per second, locating the position of order 50 times per second. Successive locations can be used to obtain velocities, using a variety of smoothing algorithms to allow for location errors. In this case, further processing was performed to obtain a 2-min moving average.

Fig. 7. Average bed activity for constant fluidising gas velocity and liquid content for two different liquid viscosities: Ža. 50 cS Silicone fluid and Žb. 200 cS Silicone fluid Ž1-mm glass ballotini. w24x.

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3. Sintering Seville et al. w21x have modelled particle sintering in fluidised beds using two assumptions: 1. Sintering occurs in the quiescent zones in which interparticle motion is limited. The time spent in the quiescent zones is a function of the excess fluidising velocity. 2. The force applied to sintered agglomerates by the bubble movement is independent of the excess fluidising velocity. Thus, for a given bed, the critical size of sinter neck will be independent of fluidising velocity. One measure of the characteristic time in a quiescent zone is the average time t bb , between the passage of bubbles, which is given by:

t bb s

2

Db

3 Ž U y Umf .

Ž 12 .

where D b is the diameter of the bubbles. Another measure of the characteristic time is the time t bo required to turn the contents of the bed over once. From the Atwo-phaseB theory of fluidisation, this time is given by:

t bo s

Fig. 8. The time-average bed mobility as a function of liquid addition and gas velocity of a bed of 1-mm glass ballotini for Ža. 50 cS Silicone fluid and Žb. 200 cS Silicone fluid. The indicated points represent the data from Fig. 7 for the case when the liquid content is constant w24x.

minimum velocity required for bubbling, over a similar viscosity range. Fig. 8 shows the moving average velocity of the tracer particle as liquid is continuously added to the bubbling fluidised bed, with fixed fluidising air velocity. Also shown are some sample data points from Fig. 7, which coincide very well with the measured data, indicating that the liquid can be considered to be well dispersed over the particle surfaces. The bed activity decreases with added liquid Ži.e., less vigorous bubbling.; from Eq. Ž11. this can result from a change in a or Umb . Since Fig. 7 appears to show little variation in a , this reduction in bed activity, or defluidisation, must be attributable to an increase in Umb . Since Umb is determined by the maximum stable voidage of a settled bed, it appears that the structure of the defluidised bed is controlled by static forces. This is consistent with the suggestion of Donsi and Massimilla w6x, that interparticle forces are responsible for stabilising cavities or voids in the bed, and that expansion can occur beyond the point of minimum fluidisation by nucleation and growth of such cavities.

Hmf

k Ž U y Umf .

Ž 13 .

where Hmf is the bed height at minimum fluidisation, k is the ratio of the volume of solids moved by bubbles to the volume of bubbles. The critical time, ts for sintering, when the extent of sintering Ž xrR . is sufficiently large for the agglomerate not to be broken by the passage of bubbles, depends on the mode of sintering. For viscous sintering, for example, the Frenkel equation Žsee Section 1.3. gives:

ts s

x

ž / R

2

2 Rm 3g

Ž 14 .

If now, the time for sintering is equated with one of the characteristic times for bed movement, an expression is

Fig. 9. The effect of temperature on the minimum fluidising velocity of low density polyethylene granules of particle size 2 mm Žmelting temperature 100–1258C. for one bed height w21x.

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267

Fig. 10. The data of Fig. 9 and other similar data replotted to determine the value of ErR in Eq. Ž16. for: Ža. no account of bed height and Žb. including the effect of bed height w21x.

obtained relating the critical excess fluidising velocity to the surface viscosity: x

ž / R

2

2 Rm 3g

K1

s

Ž Umfs y Umf .

Ž 15 .

where Umfs is the minimum fluidising velocity under sintering conditions and K 1 is equal either to Hmfrk or to 2 D br3. Substituting for the dependence of viscosity on temperature using Eq. Ž10. and rearranging gives:

Ž Umfs y Umf . s

K1 K2

m 0 exp Ž ErRT .

Ž 16 .

where K 2 characterises the critical size of the sinter neck. Eq. Ž16. provides an expression for the temperature dependence of the minimum fluidising velocity under sintering conditions. The surface viscosity can be measured independently and hence the equation provides predictive capability. ŽA similar analysis can be applied for the case of sintering by diffusional processes w12x.. Fig. 9 shows how the minimum fluidising velocity varies with temperature for low-density polyethylene granules of 2-mm diameter Žmelting range 100–1258C.. Under non-sintering conditions, Umf increases slightly with temperature in the range 19–758C. The magnitude of Umf at higher temperatures, had sintering not occurred, was obtained by the linear extrapolation shown in the figure and was taken to be 0.75 m sy1 . At about 1008C, sintering became significant and the minimum fluidising velocity under sintering conditions, Umfs , had to be increased. Analysis of the data by means of Arrhenius type plots ŽEq. Ž14.. is shown in Fig. 10a and b. Data are shown for four bed heights in the range 110–265 mm using Eqs. Ž12. and Ž13. as alternatives. The respective values of ErR were found from the gradients to be 32 = 10 3 and 39 = 10 3 Ky1 . In order to test the theory, an independently determined value of ErR is required for the viscous flow of the

material. Seville et al. w21x report dilatometry data for this, giving good agreement with the activation energy from the fluidisation experiments, and confirming the value of this approach. 4. Conclusions The three types of interparticle interaction, namely the van der Waals, liquid bridge and sintering forces have been discussed theoretically. In each case an example of some interesting macro scale phenomena has been provided. Relatively simple theories of the single particle–particle interaction have been successful in describing these observed effects at the assembly scale. For the van der Waals force, Molerus showed that sensible assumptions regarding interparticle force were sufficient to predict the BrA transition in the Geldart chart. The dramatic effect of added liquid on the velocity of particles in a bubbling fluidised bed was shown experimentally and is again adequately explained by an increase in the capillary liquid bridge forces of the dense phase material Žthus allowing a greater voidage and increased gas throughput.. For the case of particle sintering, the simple model of viscous sintering for the polyethylene particles was used to describe the effects on the fluidised particles and showed striking agreement with experimental results. Nomenclature a Particle half-separation distance A Hamaker constant Bubble diameter Db e Coefficient of restitution E Activation energy F Interparticle force Bed height at minimum fluidisation Hmf k Constant in Eq. Ž9.

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K 1, K 2 r1 r2 R T U Õ Õp V x a d DP g w k

m t bb bo ls ld max mf mb s vw St

Constants in Eq. Ž16. Liquid bridge meridional radius of curvature Liquid bridge neck radius Spherical particle radius; gas constant Temperature Fluidising air velocity Particle–particle relative velocity Particle velocity Liquid bridge volume Sinter neck radius Ratio of particle velocity to excess gas velocity Average liquid coating thickness Reduced hydrostatic bridge pressure Liquid–vapour surface energy Three-phase contact angle Ratio of the volume of solids moved by bubbles to the volume of the bubbles Fluid dynamic viscosity Time Between passing bubbles Bed turnover Liquid, static Liquid, dynamic Maximum value At the point of minimum fluidisation At the point of minimum bubbling Sintering van der Waals Stokes number

Acknowledgements The authors are indebted to a variety of sponsors, including the Engineering and Physical Sciences Research Council, the Biology and Biological Sciences Research Council, Unilever Research, BP and NEDO for support of parts of this work; to a number of excellent research students: Matthias Stein, Roland Schiftner, Heike

Silomon-Pflug, Michael Kemmerich, Aidan McCormack, Mark Leaper, Amran Salleh; and to their collaborators Professors Masayuki Horio and Hidehiro Kamiya. References w1x M.J. Adams, B. Edmundson, in: B.J. Briscoe, M.J. Adams ŽEds.., Tribology in Particulate Technology, Adam Hilger, Bristol, 1987. w2x J. Baeyens, D. Geldart, Fluidization et ses Applications, ENSIC, Toulouse, 1973, p. 182. w3x G. Barnocky, R.M. Davis, Phys. Fluids 31 Ž6. Ž1988. 1324. w4x A. Cameron, Basic Lubrication Theory, Ellis Harwood, Chichester, 1981. w5x N.L. Cross, R.G. Picknett, Proc. Int. Conf. The Mechanism of Corrosion by Fuel Impurities, Marchwood Engineering Laboratories, Marchwood, U.K., 1963, p. 383. w6x G. Donsi, L. Massimilla, AIChE J. 19 Ž6. Ž1973. 1104. w7x B.J. Ennis, G.I. Tardos, R. Pfeffer, Powder Technol. 65 Ž1991. 257. w8x R.A. Fisher, J. Agric. Sci. 16 Ž1926. 492. w9x D. Geldart, Powder Technol. 7 Ž1973. 285. w10x M.J. Gluckman, J. Yerushalmi, A.M. Squires, Fluidization Technology, Hemisphere, New York, 1975. w11x J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1991. w12x P.C. Knight, J.P.K. Seville, H. Kamiya, M. Horio, Chem. Eng. Sci. Ž2000. in press. w13x W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to Ceramics, Wiley, New York, 1976, p. 492. w14x G. Lian, C. Thornton, M.J. Adams, J. Colloid Interface Sci. 161 Ž1993. 138. w15x G. Mason, W.C. Clarke, Chem. Eng. Sci. 20 Ž1965. 859. w16x O. Molerus, Powder Technol. 33 Ž1982. 81. w17x D.J. Parker, C.J. Broadbent, P. Fowles, M.R. Hawkesworth, P. McNeil, Nucl. Instrum. Methods Phys. Res., Sect. A 326 Ž1993. 592. w18x W.B. Pietsch, Nature 217 Ž1968. 736. w19x J.P.K. Seville, in: B.J. Briscoe, M.J. Adams ŽEds.., Tribology in Particulate Technology, Adam Hilger, Bristol, 1987. w20x J.P.K. Seville, R. Clift, Powder Technol. 37 Ž1984. 117. w21x J.P.K. Seville, H. Silomon-Pflug, P.C. Knight, Powder Technol. 97 Ž1998. 160. w22x J.P.K. Seville, U. Tuzun, ¨ ¨ R. Clift, Processing of Particulate Solids, Blackie Academic, London, 1997. w23x S.J.R. Simons, J.P.K. Seville, M.J. Adams, Proc. 6th Int. Symp. on Agglomeration, Nagoya, Japan, 1994, p. 117. w24x C.D. Willett, PhD Thesis, University of Birmingham, 1999.