hcsd rheology transp japan

Korea-Australia Rheology Journal March 2001 Vol. 13, No. 1 47

Korea-Australia Rheology JournalVol. 13, No. 1, March 2001 pp. 47-54

Rheology and pipeline transportation of dense fly ash-water slurry

Hiromoto Usui*, Lei Li and Hiroshi Suzuki Department of Chemical Science and Technology, Kobe University

Rokkodai-cho 1-1, Nada, Kobe 657-8501, Japan

(Received March 30, 2001)

Abstract

Prediction of the maximum packing volume fraction with non-spherical particles has been one of theimportant problems in powder technology. The sphericity of fly ash particles depending on the particlediameter was measured by means of a CCD image processing instrument. An algorithm to predict themaximum packing volume fraction with non-spherical particles is proposed. The maximum packing vol-ume fraction is used to predict the slurry viscosity under well dispersed conditions. For this purpose,Simha’s cell model is applied for concentrated slurry with wide particle size distribution. Also, Usui’smodel developed for aggregative slurries is applied to predict the non-Newtonian viscosity of dense flyash - water slurry. It is certified that the maximum packing volume fraction for non-spherical particlescan be successfully used to predict slurry viscosity. The pressure drop in a pipe flow is predicted by usingthe non-Newtonian viscosity of dense fly ash-water slurry obtained by the present model. The predictedrelationship between pressure drop and flow rate results in a good agreement with the experimented dataobtained for a test rig with 50 mm inner diameter tube. Base on the design procedure proposed in thisstudy, a feasibility study of fly ash hydraulic transportation system from a coal-fired power station to acontrolled deposit site is carried out to give a future prospect of inexpensive fly ash transportation tech-nology.

1. Introduction

The deposit of a large amount of fly ash and bottom ashdischarged from coal-fired power stations is a serious prob-lem. The amount of fly ash is larger than bottom ash. Con-siderable amount of recycle use is available, mainly byadding fly ash to cement. However, the addition of fly ash tocement is limited because the production rate of cement isleveled off, and also the concentration of fly ash in cementis limited. Thus, it is expected that large amount of fly ashmust be transported from power station to waste materialdeposit site in the near future. The deposit site is not nec-essarily located near the power station. Long distance trans-portation of fly ash with reduced cost must be considered.As a possible technique of fly ash transportation, hydraulictransportation of fly ash-water slurry is investigated in thisstudy. The design of a pipeline transportation system mustbe accomplished by means of the design procedure based onprecise knowledge of rheological behavior of dense fly ashslurries.

Suspensions generally show the complex flow behav-

ior. These complexities may be caused by interactionsbetween solid particles forming the internal structure,particle size distribution, and non-spherical characteris-tics of suspended particles. Recently, Chryss-Bhatta-charya (1999) discussed the prediction method of themaximum packing volume fraction for non-sphericalparticles. But his discussion was limited to the case ofuniform sphericity. Generally speaking, sphericity is thefunction of particle size, and more sophisticated treat-ment of sphericity must be carried out in future studieson suspension rheology. The sphericity of fly ash par-ticles was measured for different particle size ranges bymeans of microscopic CCD camera and automatic par-ticle image processor. Based on these experimentalresults, a model to predict the maximum packing volumefraction is proposed. On the other hand, one of thepresent authors (Usui 1995; 1999; 2000) has been dis-cussing the viscosity prediction method of concentratedslurries with wide particle size distribution and withagglomerative nature. The first aim of this paper is topresent the viscosity prediction method for non-sphericalsuspensions which is based on Usui’s suspension rhe-ology model. The second aim is to propose a possibleway of long distance transportation of large amount ofthe ash by pipeline.

*Corresponding author: [email protected]© 2001 by The Korean Society of Rheology

Hiromoto Usui, Lei Li and Hiroshi Suzuki

48 Korea-Australia Rheology Journal

2. Experiments

The dense fly ash-water slurry used in this study was pre-pared as follows.Sample fly ash was discharged from Mat-suura Power Station in Japan where Blair Athol coal wasburned. Two kinds of dense fly ash-water slurries; Slurry-A and B were prepared. Slurry-A; 70 wt% fly ash-water slurry without additive.Slurry-B; 70 wt% fly ash-water slurry with 0.3 wt%/slurrypolystyrene sulfuric acid (PSS) as a dispersing additive.The solid volume fraction corresponding to this concen-tration was φ=0.51. The particle size distribution of pul-verized coal was measured by means of the LaserDiffraction Particle Size Analyzer, Horiba-SeisakushoModel 920. Particle size distribution of fly ash was in goodagreement with the Rosin-Rammler distribution given byD(%)=100 exp(-0.061dp

1.24), where D(%) is cumulativeoversize distribution. Although the particle size distributionis continuous, the particle size distribution must be dividedinto several parts for the convenience of the followingmodel predictions. Usui et al. (2000b) discussed the effectof the subdivided particle size distribution on the accuracyof viscosity prediction. The conclusions obtained from thediscussion were as follows. The most important point wasthe number of minimum sized particles should be muchlarger than the number of other size particles. The secondpoint was that the number of subdivisions was rather insen-sitive, and the number of ten to twenty subdivided partswas enough to give an accurate estimation of slurry vis-cosity. Thus, in this study, the particle size was divided intotwelve parts. The detailed method is given as follows. Thelaser diffraction particle size analyzer gave 52 segments ofparticle size and the probability of each particle size. Suc-cessive four segments data were averaged to give the num-ber mean particle diameter by . Since thesummation of the first four segments did not give enoughnumber of minimum sized particles, first and second seg-ments were combined to give the minimum sized particles.The probability density and the number of particles for

each size band are listed in Table 1. The sphericity wasmeasured by means of a flow particle image analyzer,FPIA-2000, Sysmex Co. The well dispersed particle imageis taken by a CCD camera, and the sphericity,Ψ, is definedby following equation.

(1)

The image fly ash particles is shown in Fig. 1. The sphe-ricity data were automatically accumulated and the aver-aged value of sphericity for each particle size band isshown in Table 1. The sphericity of fly ash is a weakfunction of particle size. The apparent viscosity of flyash-water slurries was measured by a remoter (IwamotoSeisakusho Co., Ltd. Model IR-200) with coaxial rotat-ing cylinders. The temperature of the test samples wasmaintained as 298K.

di nkdk nkk∑⁄

k∑=

Ψ diameter based on area of particle eimag( )diameter based on perphery of particle eimag( )

----------------------------------------------------------------------------------------------------------------------------------=

Table 1. Particle size distribution and sphericity of tested fly ash

Particlesize (µm)

Probabilitydensity (-)

Number based particle size distribution

(1/m3-slurry)

Sphericity(-)

1.02 0.162 1.50Ø 1017 0.93

3.15 0.059 1.84Ø 1015 0.97

4.13 0.067 9.25Ø 1014 0.97

5.43 0.079 4.79Ø 1014 0.97

7.14 0.098 2.62Ø 1014 0.919

9.35 0.124 1.47Ø 1014 0.919

12.24 0.138 7.34Ø 1013 0.919

16.0 0.122 2.90Ø 1013 0.919

20.9 0.088 9.37Ø 1012 0.919

27.2 0.045 2.16Ø 1012 0.919

35.4 0.014 3.03Ø1011 0.919

45.0 0.002 1.91Ø 1010 0.919

Fig. 1.Photograph of fly ash particles.



3. Maximum packing volume fraction for non-spherical particle system

It is necessary to predict the maximum packing volumefraction of a non-spherical particle system if we want topredict the slurry viscosity of general suspension systems.Usui’s suspension rheology model (Usui, 1999; 2000)needs the value of maximum packing volume fraction esti-mated for the test sample dispersed in the slurry. Asalready mentioned above, the fly ash particles have particlesize distribution, and the sphericity depends on particlesize. Several theories to predict the maximum packing vol-ume fraction have been proposed in the past, e.g. Lee(1970), Yu (1991) and Suzuki-Ohshima (1983). Each the-ory predict almost the same value of the maximum packingvolume fraction, but they are a little different from eachother. Since we want to obtain a rough estimation of themaximum packing volume fraction, we would rather selecta simple procedure to estimate the maximum packing vol-ume fraction. Thus, Lee’s theory has been adopted in thisstudy.

As the first step of the calculation, the continuous par-ticle size distribution is subdivided into two parts, i.e.,d0~dk and dk+1~dmax as shown in Fig. 2. The particlediameter determined by laser diffraction method is inter-preted as the volume based diameter. On the other hand,the circularity of the two dimensional trajectory image ofa particle is conventionally determined by an image pro-cessing instrument (e.g. a flow particle image analyzer,FPIA-2000, Sysmex Co). The well dispersed particleimage is taken by a CCD camera, and the circularity,Ψ,is defined by Eq.(1). It is assumed that the circularity,thus obtained, can be used as sphericity for the case ofthree dimensional calculation. The sphericity data were

automatically accumulated and the averaged value ofsphericity for each particle size band. The diameternewly defined as di,nons=di/Ψi is the diameter whichinclude the effect of non-sphericity. Using the non-spher-ical diameter and their probability density distribution,we calculate the maximum packing volume fractionwhen the particles, dk+1,nons~dmax,nons, are packed into theunit volume. This packing condition may be imagined asshown in Fig. 2(a). Since the particle size distribution isvery wide, small particles may be packed into the space,which is vacated by the effect of non-sphericity of largerparticles. The void space shown in Fig. 2(a) is packed bythe smaller particles, d0,nons~dk,nons. The small area isenlarged in Fig. 2(b) where the non-spherical smallerparticles are packed under the maximum packing con-dition. The maximum packing volume fraction for theoccupation of the void in Fig. 2(a) gives the amount ofsmaller particles, d0~dk. This amount of smaller particlesmust coincide with those calculated by the particle sizedistribution. If this condition is not satisfied, the borderof the two parts, i.e. the number k, is changed, and theabove procedure is repeated until good convergence isobtained.

The calculation procedure is summarized as follows. 1. Give the discrete particle size distribution and the

sphericity data.2. Assume the critical number, k, to separate the particles

into two parts as shown in Fig. 2(c), and obtain theratio of solid volume fractions; φL/φS, where φL and φS

correspond to the solid volume fractions of thesmaller particles, d0~dk , and the larger particles, dk+1~dmax, respectively.

3. Estimate the maximum packing volume fraction,φp,max,L for the larger particles, dk+1,nons~dmax,nons, usingLee’s algorithm.

4. Calculate the solid volume fraction, φp,L,solid which cor-respond to the value of φp,max,L by the following equa-tion.

(2)

Then, calculate the net solid volume fraction, a whichcorresponds to the smaller particles, d0~dk, by,

α = φp,L,solid(φS/φL) (3)

5. Estimate the maximum packing volume fraction, φp,max,S,to fill up the volume; 1-φp,L,solid, by the non-sphericalsmaller particles; d0,nons~dk,nons.

6. Calculate the net solid volume fraction, β=φp,S,solid, bythe following equation,

φp L solid, ,

π6--- di

3 Nik 1+

max

∑

π6--- dnons i,

3 Nik 1+

max

∑--------------------------------φp max L, ,=

Fig. 2.Maximum packing by non-spherical particles.



(4)

7. Compare the values of a and β. If the two values arenot the same, new critical number, k, is reassumed, and theprocedures 2~7 mentioned above are repeated to give agood convergence.

8. Once the critical number k is determined, the max-imum packing volume fraction is calculated by φp,max,nons=φpLS,solid+ φp,S,solid.

4. Prediction of slurry viscosity for well dis-persed condition

Usui (2000) has proved that Simha’s model was valid forthe viscosity prediction of suspensions, which had bothmono-modal and tetra-modal particle size distribution ofspherical silica particles. Thus the same procedure isapplied for the case of non-spherical fly ash particle sus-pension with continuous particle size distribution. The cal-culation procedure of maximum packing volume fractionwas explained in the previous section of this paper. Thedependence of a(=φL/φS) and β(=φp,L,solid/φp,S,solid) on k isindicated in Fig. 3 for the case of fly ash particles. It is evi-dent that a and β are equal at k=3. The value of k, whichsatisfies the condition mentioned above, may not neces-sarily be an integer, but we use an integer because of theconvenience of calculation. The viscosity predicted bySimha’s model using k=3 is shown in Fig. 4 by the brokenline. The calculation procedure of Simha’s cell model topredict the completely dispersed suspension for given solidvolume fraction, particle size distribution and maximumpacking volume fraction was reported by Usui et al.(2001). The viscosity level shown by the broken line isinterpreted as the lower limit of 70 wt% fly ash-waterslurry with completely dispersed condition. The experi-

mental data shown in Fig. 4 are much higher than thislower limiting level because of the agglomerative nature offly ash particles. We will discuss the effect of agglom-eration in the next section.

5. Prediction of non-newtonian slurry viscosityand discussion

The fly ash particles agglomerate with each other whenthe shear rate decrease, and the apparent viscosity increase.The particle size distribution in this study is shown in Table1. The particle size is subdivided into twelve discreteranges. Usui et al. (2000) proposed the viscosity predictionmethod for slurries with discrete particle size distribution.The basic idea was the combination of Simha’s cell modeland Usui’s non-Newtonian slurry viscosity model. But,Simha’s cell model needs the maximum packing volumefraction. We estimate it by the packing model for non-spherical particles proposed in this study. The model pro-posed by Usui concentrates the effect of the agglomerativenature of suspended particles to the cluster formation byvery small primary particles. The number of primary par-ticles contained in a cluster, n, is chosen as a thixotropyparameter. The shear break-up process was determined bytaking the force balance on the two spherical clusters insimple shear flow. Usui proposed a rate equation in whichthe breakup of the cluster and both Brownian coagulationand shear coagulation processes were taken into account.The rate equation is given by

(5)

Since the solid concentration is known, the unknownparameters in Eq. (5) are n, F0 and ε. The dependence of

β φp S solid, , 1 φp L solid, ,–( )φp S max, ,

π6--- di

3 Ni0

k

∑

π6--- dnons i,

3 Ni0

k

∑------------------------------= =

dndt------ 4αbkbTN0

3η0

------------------------ 4αsφnγ·π

------------------- 3πd03n

4F0Nb

--------------- n1 ε–---------- 1–

ηγ·2–+=

Fig. 3.Solid volume fraction ratio and maximum packing vol-ume fraction ratio depending on subdividing number, k.

Fig. 4.Apparent viscosity of fly ash-water slurries. Solid lineindicates slurry viscosity predicted by Eq.(5) withSimha’s cell model. Dotted line indicates viscosity levelpredicted by Simha’s cell model for well dispersed non-spherical particle suspension. Dotted line is viscositylevel predicted by Simha’s cell model for the well dis-persed spherical particle suspension.



ε on the number of primary particles contained in a clusteris assumed to be given by the following equation (Usui,1999).

ε = εmax{1-n-0.4} (6)

where εmax is the void fraction at n=î. Using the rela-tionship given by Eq. (10), the unknown parameters arereduced to two. When a set of apparent viscosity and shearrate data is given experimentally, the number of primaryparticles contained in a cluster is calculated as follows.

1. Assume the value of n, and estimate the size and num-ber of cluster by Eq. (10).

2. Calculate the maximum packing volume fraction byφp,max=0.63Ψ3.

3. Use Simhaís cell model (Simha, 1952) to predict theslurry viscosity at a given shear rate.

4. If the predicted viscosity is not the same as the exper-imentally determined one, the value of n is reassumeduntil a good convergence is obtained.

The number, n, of the primary particles contained in acluster is shown in Fig. 5. The data obtained for Slurry-Bindicate that the cluster is almost deflocculated at highershear rate range. The distribution of n obtained for Slurry-A does not reach to unity at higher shear rate range. More-over, it does not show the tendency to be leveled off. Thismeans that the experimental data obtained for Slurry-Adoes not show the Newtonian flow characteristics even athigher shear rate range of this experiments. Eq. (5) is usedto predict the inter-particle bonding energy of the primaryparticle. Also the same equation was used to predict thenon-Newtonian Slurry viscosity. Fig. 6 shows the depen-dence of F0 on the shear rate. The averaged values of inter-particle bonding energy were estimated for 1.02µm fly ashparticle as F0=2.38Ø10-15 [J] (without additive) and F0=1.06Ø10-17 [J] (with 0.3 wt% PSS), respectively. Thenslurry viscosity is predicted for the known particle size dis-

tribution. The predicted viscosity is shown in Fig. 4 by thesolid line. This diagram indicates that the non-Newtonianslurry viscosity of a non-spherical particle system can bepredicted by the slurry viscosity model proposed in thisstudy. The increase in viscosity from the broken line levelobserved for experimental data in Fig. 4 is attributed to theeffect of agglomeration of fly ash particles.

6. Design of hydraulic pipeline transportation offly ash slurry

A model to predict the non-Newtonian viscosity of densefly ash-water slurries was proposed in the previous sectionof this paper. Once the value of inter-particle bondingenergy, F0 is determined for slurry at certain solid volumefraction, the relationship between shear rate and shearstress can be predicted even for different solid concen-trations with the same fly ash. Using the shear rate-shearstress relationship, the flow rate in a slurry pipeline at afixed pressure drop can be easily calculated by followingrelationships.

(7)

(8)

Kawasaki Heavy Industries Co. Ltd. is now doing the flyash slurry transportation experiments using the test rig with50 mm inner diameter stainless tube. The concentrationrange of test slurries is between 70 wt% and 75 wt%. Theirresults are compared with the predicted pressure drop ofthis study in Fig. 7. Since the reciprocal slurry pump isused in their test rig, the flow is pulsative. Thus, the exactcomparison is difficult. However, as shown in Fig. 7, thepresent model to predict hydraulic pipeline transportationof fly ash slurries gives a reasonable agreement with theexperimental results. We can conclude that the rheological

dPdx------ 2τwall

R-------------=

Q 8RdP dx⁄--------------- τ2γ· τ( ) τdτcentre

τwall∫=

Fig. 5.Number of primary particles contained in cluster as func-tion of shear rate.

Fig. 6. Inter particle bonding energy determined for each vis-cosity measured for different shear rate.



model proposed in this study for non-spherical particle sus-pensions with particle size distribution is successfully usedfor the hydraulic pipeline transportation of dense slurries.

The power consumption of slurry transportation pipelinewas estimated for the following design conditions.

1. Discharge rate of fly ash from a power station =640,000 ton/year, or 88.9 ton/h,

2. Particle size distribution, slurries, density, slurry tem-perature are the save as the present study,

3. Slurry concentration = 70 wt%,4. Horizontal pipeline length = 10 km, pipe inner

diameter = 125 mm.The predicted results are;Total pressure drop = 11.5 MPa,Power consumption = 500 kW, with pumping efficiency= 0.5.

Flow rate = 0.0220 m3/sAverage velocity of pipe flow = 1.8 m/s.Based on the pipe line transportation design mentioned

above, the cost estimation of hydraulic transportation andits comparison with the other transportation method, e.g.belt conveying or track conveying, should be done in thefuture study.

7. Sedimentation stability of dense fly ash-waterslurries

Bunn-Chambers (1993) and Ward et al. (1999) investi-gated the hydraulic transportation of dense fly ash-waterslurry. The fly ash transportation pipeline from Bayswaterpower station in NSW, Australia to the deposit site (9.5 kmfrom the power station) has been successfully operated(Ward et al., 1999). The flow rate is 300 ton/hr. Fly ash-water slurry (70 wt%) without additive is transported inthis system. Since the fly ash particles sediment very easily,it cause a very high pumping power when the pipeline sys-tem is restarted. This is a serious problem because a largerpump with over estimated power consumption must be

installed. There are several techniques to avoid the sedi-mentation of fly ash particles, when the pipeline operationis stopped. Purge of slurries by water from pipeline systemis not realistic since the volume of pipeline is quite large ifthe pipeline length is long, e.g. 123 m3 for the designexample mentioned in the previous section of this study.The use of pig causes the same problem. The use of sta-bilizing additive to prevent the sedimentation of fly ashparticles seems to be appropriate if the cost of additives isreduced. However, the addition of stabilizing additivecauses an increase in slurry viscosity. The combined use ofdispersing additive to reduce the slurry viscosity may bethe solution for this problem.

Several natural or synthetic polysaccharides wereselected as the stabilizer for fly ash-water slurries. They arerhamsan gum S-194 (trade name: Alcaligrnes-ATCC31961, Dainippon Pharmaceutical Co. Ltd.), rhamsan gumS-130 (trade name: Alcaligrnes-ATCC 31555 DainipponPharmaceutical Co. Ltd), xanthan gum (trade name Van-zan, R. T. Vanderribdt Co.) and carboxyl methyl cellulose(trade name CMC, Daicel Chemical Co. Ltd.). The con-centration of those stabilizer was adjusted so that the vis-cosity of fly ash slurry with 0.3 wt% naphthalene sulfuricacid formalin condensate (NSF) becomes the same level offly ash slurry-A without additive. The experimental resultsare shown in Fig. 8. Using these fly ash-water slurries withdifferent additives, but at the same viscosity level, the grav-itational sedimentation tests were carried out. Test slurrieswere contained in a glass column (38 mm diameter and200 mm height). After one day or one week since the sed-imentation test started, the slurry was sampled from thesampling hole attached on the sidewall of glass column.The solid concentration of each sample was measured, andthe results are shown in Fig. 9. It is evident that the addi-tives can act as the stabilizer. Rhamsan gum, S-194, is thebest to prevent the sedimentation of fly ash particles. Theslurry with 0.02 wt% S-194 showed soft pack slurry even

Fig. 8.Apparent viscosity of 70 wt% fly ash-water slurries withvarious kind of additives.

Fig. 7.Comparison of pressure drop estimation with the exper-imental results.



after 7 days. Rhamsan gum, S-130, is the second and CMCfollows. Xanthan gum was less effective to prevent the sed-imentation. We recommend the use of Rhamsan gum S-194 or S-130 as a stabilizer. CMC may be also recom-mended.

The results mentioned above indicate that the slurry canbe suddenly changed to non-sediment if we can put theadditives into pipe flow by means of suitable method. Forthis purpose we propose the additive injection systemshown in Fig. 10. This injection system must be operatedperiodically just before the pipeline system is to be shutdown. Since the average velocity in the pipe is 1.8 m/s asshown in the previous feasibility study of this paper, all thefly ash slurry can be replaced by no-sediment slurry within31 minutes if we use three additive injection station asshown in Fig. 10. The cost of the additive for one shutdownis roughly estimated as 1,500 US$. The increase of total

cost by using the stabilizing additives does not cause a seri-ous problem. Moreover, the decrease of steady pumpingcost by adopting the optimum power consumption ofpumping system may cause the reduction of total cost.Thus we conclude that the hydraulic fly ash slurry trans-portation system with periodical addition of stabilizer isworth to be considered in the practical slurry transportationsystem in the future.

8. Conclusions

A model to predict the maximum packing volume frac-tion for a non-spherical particle suspension is proposed.Simha’s cell model is applied for the suspension with par-ticle size distribution. It is certified that the maximumpacking volume fraction for non-spherical suspension issuccessfully used to predict the slurry viscosity under com-pletely dispersed conditions. Also, a thixotropy model pro-posed for dense slurries by Usui is combined with themaximum packing volume fraction for non-spherical par-ticle system to give more accurate prediction of non-New-tonian rheological characteristics of slurries. The modelresults in the estimation of inter-particle bonding forcebetween primary particles in a cluster, and the power con-sumption and flow rate relationship in hydraulic slurrypipeline transportations system is predicted. A possibleway to reduce the total cost of slurry pipeline system bymeans of a periodical addition of stabilizer is proposed.

Acknowledgement

The authors acknowledge the financial support providedby Grant-in-Aid for Scientific Research (No.11650782)from the Ministry of Education, Culture, Science and Tech-

Fig. 10.Hydraulic transportation system for dense fly ash-water slurry with the use of stabilizing additive.

Fig. 9.Sedimentation stability test results obtained for 70 wt%fly ash-water slurries with various kind of additives.



nology, Japan. Also the authors appreciate the valuable dis-cussion provided by Mr. Yutaka Takahashi on the practicalslurry transportation system. The experimental assistanceby Mr. Yohei Tsukamasa is appreciated.

Nomenclature

a = radius of spherical particle [m]b = cell radius [m]d0 = diameter primary particle which create

the cluster [m]F0 = inter-particle bonding energy [J]kB = Boltzmann constant [J/K]n = number of the primary particles

contained in a cluster [-]Nb = number of chains to be broken-up

during the cluster break-up process [-] N0 = total number of minimum sized (primary)

particles less than critical [-]n = number of the primary particles

contained in cluster [-]t = time [s]T = temperature [K] αb = Brownian coagulation constant (=0.58) [-]αS = shear coagulation constant (=0.6) [-]

= shear rate [s-1]ε = void fraction [-]η = slurry viscosity [Pa.s] η0 = solvent viscosity [Pa.s] Ψ = sphericity [-] ρ = density [kg/m3]τ = shear stress [Pa]φ = volume fraction of solid particles [-] φp,max = maximum packing volume fraction [-]

<Subscirpts>nons. = value obtained for non-spherical particles

p,max = maximum packing L = value obtained for dk+1~dmax

S = value obtained for d0~dk

solid = solid volume fraction

References

Bunn, T. F and A. J. Chambers, 1993, Experiences with densephase hydraulic coveying of vales point fly ash, Int. J. Storing,Handling and Processing Powder. 5, 35-44.

Chryss, A. G. and S. N. Bhattacharya, 1999, Maximum packingconcentration of coal water mixtures, Coal Preparation-A Mul-tinational J. 21, 83-92.

Lee, D. I., 1970, Packing of spheres and its effect on the viscosityof suspension, J. Paint Technol. 42, 579-584.

Simha, R., 1952, A Treatment of the viscosity of concentratedsuspensions, J. Appl. Phys. 23, 1020-1024.

Suzuki, M. and T. Oshima, 1983, Estimation of the co-ordinatenumbers in a multi-component mixtures of spheres, PowderTechnology. 35, 159-166.

Usui, H., 1995, A Thixotropy model for coal-water mixtures, J.Non-Newtonian Fluid Mech. 60, 259-275.

Usui, H., 1999, A rheological model for an agglomerative slurryof mono-modal silica particles, Kagaku Kogaku Ronbunshu.25, 459-465.

Usui, H., K. Kishimoto and H. Suzuki, 2000a, Viscosity pre-diction of agglomerative slurries with particle size distribution,Kagaku Kogaku Ronbunshu. 26, 423-430.

Usui, H., T. Kobayashi and H. Suzuki, 2000b, Prediction of flowcharacteristics of whipped cream under pressurized condition,J. Chem. Eng. Japan. 33, 785-792.

Ward, A., T. Bunn and A. J. Chambers, 1999, The Bayswater flyash transportation system, Coal Preparation J. 21, 125-147.

Yu, A. B. and N. Standish, 1991, Estimation of the porosity ofparticle mixtures by a linear-mixture packing model, Ind. Eng.Chem. Res. 30, 1372-1385.

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