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Powder Technology 182 (2008) 127–129www.elsevier.com/locate/powtec

Editorial

Editorial for Special issue on Granular Temperature

In all but the most trivial of particulate systems, the velocityof a particle, u, at any instant in time invariably differs from theaverage velocity of the particles around it, ⟨u⟩. This difference,the fluctuating (or peculiar) velocity, c=u−⟨u⟩, is quantified bythe granular temperature, which for a monodisperse threedimensional granular material is

h ¼ 13hcd ci

The granular temperature was first formally defined byOgawa in the late 1970s as part of one of the earliest kinetictheory based models for inelastic granular systems [1], and it isnow recognised as an important parameter in these theories. It isalso playing an increasing role in other contexts including heattransfer [2], attrition [3], and granulation [4] in particulatesystems.

Studies of published literature on granular temperatures anddiscussions with delegates at a number of recent conferences onparticulate technology suggested that the level of awareness ofgranular temperature and its importance is very mixed. Further,while its use in theoretical contributions is fairly consistent,experimental evaluation using the different techniques availablecovers such a wide range of conditions and time-scales that it isfar from clear that the same quantity is being measured in eachcase. It is against this background that we sought the productionof this special issue of Powder Technology.

The special issue starts with an article by Professor IsaacGoldhirsch of Tel Aviv University, who we asked specially tointroduce the concept of granular temperature and address someof the misconceptions that have appeared in the literature fromtime to time. The paper has been deliberately kept at a level whichwe hope is accessible to those whomay not be so familiar with themore theoretical aspects of granular temperature and, hence,hopefully provide a bridge to the more mathematical papers thatappear in the literature.

The next eight papers report studies aimed at determininggranular temperatures experimentally for a range of differentsystems. In the first of these, Breault et al. use laser Dopplervelocimetry (LDV) to show that, at least for the region adjacent tothe wall of a circulating fluidised bed (CFB) riser, particle velo-cities are not normally distributed, that the granular temperature isnon-isotropic, and that the granular temperature of particle

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clusters is an order of magnitude less than that of dispersedparticles.

In the next contribution, Cody et al. present granular tem-peratures obtained from their acoustic shot noise method for alarge range of different particles in a gas fluidized bed (FB). Theyalso cover measurements by other workers using pressure spectraand diffusing wave spectroscopy. They find that all the data isreduced very well as a function of Us/Umf by a scaling they have

proposed previously [5]:ffiffiffih

p=Us

� �D=D0ð Þ, where D is the par-

ticle diameter,Us the superficial velocity, andD0 a scaling length.They have found that this scaling length naturally arises in twoseparate kinetic theories for dense granular systems, and that it is

given by D0 ¼ k A2f =q2pg

� �1=3, where the pre-factor, k, is a

constant that appears to be independent of the particlematerial butwhich transitions relatively quickly as the boundary between theGeldart classifications is crossed. If this finding is in fact shown tohold more widely, then the scaling would greatly aid the morerigorous design of granular systems.

In the fourth paper of this special issue, Holland et al. useNuclear Magnetic Resonance (NMR) to measure the granulartemperature and velocity distribution in a gas-fluidized bed.They demonstrate that the granular temperature can be highlyanisotropic and in general spatially inhomogeneous, and depen-dent on both the geometry and dimensions of the bed.

Wildman and Huntley review their considerable body of ex-perimental and theoretical work on the vibro-fluidized bed in thenext paper of this special issue. Their work using positron emis-sion particle tracking (PEPT) shows that the granular temperaturein vibro-FBs may also be anisotropic, and that it is not the samefor the different particles in a bi-disperse vibro-FB. Comparisonbetween model results and their experimental work has alsorevealed the potential importance of boundaries in determininggranular temperature.

In the sixth contribution, Zivkovic et al. use diffusing wavespectroscopy (DWS) to study a dense three-dimensional vibro-fluidized bed. They found that the granular temperature increaseswith the square of the peak velocity of the vibrating base of thebed. Their DWS data also suggests, however, that granulartemperature may not be sufficient to describe particle motion indense vibro-FBs in which the amplitude of the vibrations issmaller than the particle diameter.

128 Editorial

In the first of three papers that report the use of video imageanalysis based methods, Hsiau et al. determine particlevelocities and granular temperatures in two-dimensional dryand “damp” vibro-fluidized beds. Their work confirms theanisotropy of granular temperature in these units, and showsthat the granular temperature is reduced as the viscosity of theliquid in the bed increases.

In the eighth contribution here, Hounslow et al. apply video-image analysis to determine the velocity and velocity fluctua-tion fields at the free surface of a high shear granulator fittedwith a three-bladed impeller rotating about a vertical axis. Theyfound that the fluctuations were anisotropic, with the largestcomponent being in the direction tangential to the impellerblades. They also found the fluctuations to be generally welldescribed by a Gaussian distribution.

In the final experimental contribution to this special issue,Armanini et al. also use velocity and granular temperature dataobtained from video image analysis to assess the use of a kinetictheory featuring granular temperature to describe liquid-solidflows down inclines. These workers show that introduction ofthe granular temperature does improve the ability to model suchflows in comparison to Bagnold's theory, but also shows thatthere is further room for improvement.

As is demonstrated in the next four papers, granulartemperature is a primary output of the distinct element model(DEM). Rosato et al. continued their work to generate a shakingcondition in a vibro-fluidized bed where granular temperatureand solid fraction profiles agreed with kinetic theory. Theyfound that under these circumstances, the calculation ofdiffusion coefficient could be simplified and simulation resultscoincided with the kinetic theory prediction on diffusion.

In the eleventh paper of this special issue, McNamara andFalcon show that granular temperatures predicted by DEM aredependent on the particle–particle interaction law used — thissuggests that care should be exercised in selecting this law,which is often dictated by the inadequacy of data characterisingthe interaction (in which case a minimalist approach is taken) orcomputational considerations (in which case “soft” interactionsare usually used).

Experimental measurement of granular temperature is oftendifficult or even impossible — this must certainly be the casein many industrial contexts. In such cases, DEM may well beable to provide granular temperature data as demonstrated byLiu et al. and Godlieb et al., who use advanced DEMmodels todetermine the granular temperature and other quantities for,respectively, a large rotating drum and a high pressure fluid-ised bed reactor used in the production of low densitypolyethylene.

Because of their relatively low computational expense,continuum models such as the classical two-fluid model play anessential role in the analysis and design of particulate systems.Kinetic theories of (inelastic) granular systems provide arelatively simple means of evaluating the transport coefficientsfor these continuum models given details of the particle–particle interactions. However, their reliance on the “(molecu-lar) chaos” assumption means their application to anythingbeyond dilute, slightly dissipative, granular flows must at least

be open to debate. Goldhirsch et al. show that kinetic theoriescan capture behaviours not typical of gases, such as the absenceof equipartition in bi-disperse systems. They also show thatgranular temperature may have a role to play in the descriptionof elasticity of granular solids. However, their paper suggest thedebate is still very open for dense systems.

In the next contribution, Brey and Prados use a simple modelto investigate some static and dynamical aspects of the conceptof compactivity introduced by Edwards to characterize thesteady state of dense granular systems, which is related to aconfigurational granular temperature — this relationship isdiscussed with a view to eventually putting the Edwards theoryon a more microscopic basis similar to that currently enjoyed bykinetic theory.

The remaining papers in this special issue involve applica-tion of kinetic theory-based Eulerian models to granularsystems. In the first of these papers, Huilin et al. introduce anew model for the riser of circulating fluidized beds that treatsisolated particles and particles in clusters as separate phases, inaddition to that of the fluidizing gas. Comparison of the resultsfrom their model with experiment is encouraging, but they wereunable to compare their granular temperature predictions due toa lack of appropriate published experimental data— the data ofBreault et al. in this special issue may be very helpful in thisregard.

In the next contribution to the special issue, Benavides andvan Wachem detail a new model based on kinetic theory thatincludes turbulence. Comparison of results from the model forturbulent pipe flow with experimental data show a generallygood correspondence, particularly for the granular temperature.

In the final two contributions to this special issue, Davis andKoenders and Eskin apply kinetic theory based models thatinclude granular temperature to two very different applications—filtration, and slurry flow in fractures. These papers reinforce onceagain that granular temperature has a currency well beyond thetraditional fields of application, and that it can lead to improvedmodelling capabilities in these novel contexts.

It is clear from many of the experimental papers presentedhere that data must be obtained at a rate somewhat faster thanthe average time between collisions (i.e. the ballistic regimeshould be well resolved) when seeking to determine granulartemperature — failure to do this may explain the rather highgranular temperatures presented by some workers elsewhere.The experimental results also make it clear that one should notassume in general that the granular temperature is isotropic,normally distributed or the same for all components in amulticomponent system — this would suggest that cautionshould be exercised in taking too far the analogy between thegranular and thermodynamic temperatures, but both the initialpaper by Goldhirsch and the more theoretical contributions inthe second half of this issue indicate that provided this is donecarefully, such experimentally observed behaviour can indeedbe recovered from kinetic theory based models.

Given the growing body of granular temperature data, thetime is right for real quantitative comparison of experimentaldata from the various methods to be undertaken so as to buildconfidence in the data being produced — this has in part been

129Editorial

started in the contribution from Cody et al. with some en-couraging results. However, accurate comparison probablyrequires the same experimental set-up to be used, a realchallenge because of the varying limitations of each method —for example, the small vessel size and particle requirements (i.e.strongly proton-bearing) for NMR would in most cases runcounter to those required in DWS where the particle-to-vesselsize ratio and particle type must be selected to ensure sufficientlight scattering to obtain the diffusive limit without excessivelight absorption. This does not mean, however, that suchcomparisons should not be attempted, only that great care isrequired. Finally, the growing body of experimental granulartemperature data means those who develop models are now ableto not only validate their predictions but, because of the centralrole of granular temperature in the closure of these models,understand better the reasons for poor predictions or, indeed, ifgood predictions are just fortuitous.

References

[1] S. Ogawa, Multitemperature theory of granular materials, in: Cowin S.C.,Satake M. (Eds.), Proceedings of the US-Japan seminar on continuummechanical and statistical approaches in the mechanics of granular mate-rials; 1978; Sendai, Japan: Gakajutsu Bunken Fukyu-Kai: Tokyo, Japan,1978, pp. 208–217.

[2] M.L. Hunt, Discrete element simulations for granular material flows:effective thermal conductivity and self-diffusivity, Int. J. Heat Mass Transfer40 (1997) 3059–3068.

[3] R.W. Lyczkowski, J.X. Bouillard, State-of-the-art review of erosionmodeling in fluid/solids systems, Prog. Energy Combust. Sci. 28 (2002)543–602.

[4] H.S. Tan, M.J.V. Goldschmidt, R. Boerefijn, M.J. Hounslow, A.D. Salman,J.A.M. Kuipers, Building population balance model for fluidized bed meltgranulation: lessons from kinetic theory of granular flow, Powder Technol.142 (2004) 103–109.

[5] G.D. Cody, D.J. Goldfarb, G.V. Storch, A.N. Norris, Powder Technol. 87(1996) 211–232.

Mark J. Biggs*Don. H. Glass

Edinburgh, November 2007⁎Corresponding author.

E-mail address: m.biggs@ed.ac.uk (M.J. Biggs).