Numerical analysis of fine powder flow using smoothed particle method and experimental verification

Chemical Engineering Science 57 (2002) 227–237www.elsevier.com/locate/ces

Numerical analysis of !ne powder $ow using smoothed particle methodand experimental veri!cation

Tomohiro Suginoa, Shinichi Yuub; ∗

aIshikawajima-Harima Heavy Industries Co. Ltd (IHI), 1 Shin-Nakahara-Cho, Isogo-Ku, Yokohama, JapanbDepartment of Mechanical Engineering, Kyushu Institute of Technology, Sensuicho 1-1, Tobata, Kitakyushu 804-8550, Japan

Received 21 May 2001; received in revised form 20 September 2001; accepted 28 September 2001

Abstract

Numerical simulation of velocity and stress !elds for $owing powder composed of an in!nite number of particles presents a mostdi/cult problem in powder technology.

The distinct element method (DEM) is useful for determining each particle trajectory which involves multi-body interactions. However,total particle cannot be computed using DEM. The particle number which can be calculated for a three-dimensional spherical systemwould be in hundreds of thousands. A description of $ow characteristics for a small amount of powder would not be practical. Simulationwithin a tank would thus be virtually impossible. The authors have conducted numerical simulation of $owing powder using the smoothedparticle (SP) method through application of continuum dynamics. The authors’ group is the !rst to contrive to apply this method topowder $ows. In the SP method, partial di9erential equations that govern $ow !elds are transformed to ordinary di9erential equationsof the Lagrangian-type for particle motion. Numerical analysis of ordinary di9erential equations is much simpler compared to partialdi9erential equations. Lagrangian analysis is suitable for determining the characteristics of discrete particles.

The equation of powder pressure exerted on tank bed due to di9erences in density and constitutive equations that consider yield stresshave been used as basic equations and the latter were obtained by the authors’ group using DEM. These equations provide clari!cationof the rheological characteristics of powder $ow.

Glass beads (particle diameter: 100 �m) were used in the present study as test powder stored in a tank and discharged by gravitationalforce.

Calculated velocity distribution, free surface in the tank and the rate of discharge were compared with experimental data and a goodagreement was noted. Based on the results of this study, the SP method in conjunction with the model equation for yield stress appearsquite useful for simulating the $ow of !ne powder. ? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Smoothed particle method; Numerical simulation; Yield stress; Constitutive equations; Fine powder; Gravity $ow

1. Introduction

For powder storage equipment design, properties and fea-tures of a given powder should be clari!ed. The boundarybetween $owing and stagnating zones should be clearly de-termined for this purpose. Powder stored in a tank is dis-charged by gravitational force. When the powder is heatedby the heating medium, the discharge of su/ciently heatedpowder alone is desirable but it is di/cult to predict whatportion of the powder in the tank will actually be discharged.

∗ Corresponding author. Tel.: +81-93-884-3174; fax: +81-93-871-8591.

E-mail addresses: tomohiro [email protected] (T. Sugino),[email protected] (S. Yuu).

The smoothed particle (SP) method based on continuumdynamics is a Lagrangian model requiring no spatial meshand thus is appropriate for !ne powder $ow simulation in-volving many particles.

In the SP method, physical properties, such as velocityand density, may be determined from continuum equationsby interpolating points which can be thought of as imaginaryparticles that fall within the kernel function range.

By the SP method, Lagrangian motion of interpolatingpoints or imaginary particles in the continuum may be deter-mined, thus making possible the avoidance of grid distortionoften encountered with other methods.

The SP method is applicable to astrophysics andhydrodynamics for solving numerous problems in in-compressible $ow. Monaghan (1994, 1996) used this

0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S 0009 -2509(01)00383 -9

228 T. Sugino, S. Yuu /Chemical Engineering Science 57 (2002) 227–237

method in regard to free surface incompressible $owfor a bursting dam, bore formation and others. The re-sults indicated that the SP method is useful for sim-ulating free surface $ow. Monaghan and Kocharyan(1995) formulated a two-phase $ow of dusty gas us-ing the SP method. The gas and dust motion initiatedby the dust moving into a static region of gas hasbeen simulated. Takeda, Shoken, and Sekiya (1994) ap-plied the SP method to low Reynolds number viscous$ow simulation for comparison with the !nite di9erencemethod.

Recently, Gutfraind and Savage (1998) numericallysimulated the wind-driven motion of fractured ice in awedge-shaped channel using the SP method. The rheologyused in the SP method is based on the Mohr–Coulomb yieldcriterion without cohesive force. They considered that theice $oes were incompressible material. When one uses !nitedi9erence methods on !xed Eulerian grids, moving bound-aries cannot be clearly de!ned by the arti!cial di9usion andsetting the boundary conditions on the moving boundarycan be a di/cult task. Gutfraind and Savage have shownthat one can avoid these problems by using the SP method.

Oger and Savage (1999) calculated the motion ofbroken-ice !elds $oating on the water surface and movingunder the e9ect of wind forces. In their work, cohesive inter-action between the ice $oes is introduced to simulate theirthermal sintering (freezing and=or breakage of the $oes). Agood agreement was found between the numerical resultsby the SP method with the Mohr–Coulomb rheological andanalytical solutions.

Shimosaka, Kano, and Hidaka (1997, 1998) calculatedthe granular $ow using Smoothed Particle Element Method.They calculated only the bulk density using the smoothingprocess and no other physical property. In the Lagrangiancalculation of imaginary particle in the continuum, the inter-action forces, any other physical property and its derivativeshould be calculated by the smoothing process in order toconsider the interaction e9ect in the e9ective range in thecontinuum.

The author’s group (Yuu, Waki, & Umekage, 1998) hassimulated powder $ows in a hopper using the SP method.The constitutive equations derived using distinct elementmethod (DEM) simulation by Yuu, Hayashi, Waki, andUmekage (1997) have been used. In the study of Yuu et al.(1998) we have decided experientially the stagnant zone inthe hopper since one cannot calculate that zone using theconstitutive equations alone.

In the present SP simulation, we have succeeded in calcu-lating the stagnant zone applying the Bingham plastic $uidcalculation method to the constitutive equations derived byYuu et al. (1997). We have also considered the pressure dif-ference between the static and dynamic states in the presentstudy.

The author’s group is the !rst to contrive to apply theSP method to powder $ows and to extend the calculationmethod to be able to describe well the experimental powder

$ows. Two-dimensional simulation results obtained with theSP method are veri!ed based on comparison with experi-mental results.

2. The SP equations

The main feature of the SP method is the replacementof the continuum to imaginary particles which behave ac-cording to the Lagrangian equations of motion. The physi-cal properties of the continuum are de!ned by the integralinterpolating over the particles which are distributed usingthe kernel function. Spatial derivatives of these propertiesmay be replaced by those of the kernel function.

Basic equations for the SP method are the momentumequation, equation of state and constitutive and continu-ity equations. All these are used in the present study for atwo-dimensional !eld.

In the SP method, the physical property � is determinedfrom the kernel function W . Smoothed 〈�(r)〉 for �(r) isexpressed as

〈�(r)〉i =∫D�(rj)W (ri − rj; h) drj: (1)

Approximation of the integral interpolant is given by

〈�(r)〉i =N∑j=1

mj�j

jW (ri − rj; h); (2)

where W (ri − rj; h) is an interpolating kernel with∫W (ri − rj; h) = 1: (3)

The Gaussian function was used as an interpolating ker-nel in this study. In order to satisfy the principle of actionand reaction between particles, the kernel function shouldbe symmetric, that is, W (ri − rj) = W (rj − ri). Then thetwo-dimensional kernel function takes the form

W (rij) =12�

{e−(ri−rj)2=h2

i

h2i

+e−(ri−rj)2=h2

j

h2j

}: (4)

h is given as

hi = �(m0

bi

); hj = �

(m0

bj

); (5)

where su/xes i and j show the reference and the surroundingparticles. � is a parameter of the range of the kernel function.�=1:4 provides a good velocity !eld for powder $ow usingglass beads. The e9ects of other particles with centers whichare within the radius 3hi are taken into account to obtain thesmoothed value.

Substitution of velocity vj = �j into Eq. (2) gives

〈v〉i =∑j

mj

jvjW (rij; h): (6)

T. Sugino, S. Yuu /Chemical Engineering Science 57 (2002) 227–237 229

The gradient of �(r) may be obtained as

〈∇�(r)〉i =N∑j=1

mj�j

j∇W (rij; h): (7)

2.1. Static stress

The Rankine static stress model was used for the initialstress distribution of the powder in the tank. The expressionsof the stress tensor components, when the major principalstress orientation is supposed to be in the vertical directiony are

�x0 =(

1 − sin�1 + sin�

) bgH; (8)

�y0 = b gH (9)

and

�xy0 = 0: (10)

2.2. Pressure–density relation

To determine the pressure generated by density di9er-ences due to motion in powder, the following equation isapplied:

P = K bgH( b 0

− 1): (11)

The constant K for the powder is not clear but should bechosen so as to be in agreement with experimental results.

2.3. Continuity equation

The density is calculated using the continuity equationtransformed as

d dt

= −∇( v) + v∇ : (12)

2.4. Constitutive equations

Yuu et al. (1998) have calculated the forces acting on eachpowder particle in the powder bed using DEM and obtainedthe shear and the normal stresses by locally averaging theseforces on the plane in the powder bed which assumed con-tinuum. This is essentially the same method of the moleculardynamics which gives the stress !eld in the $uid. The ex-perimental relationships were also obtained under the sameconditions. In order to obtain the relation between stressesand rates of the strain in the experiment, shear stresses andstrain rates were measured in a shear $ow. Normal stressesand strain rates were measured in a hopper when the powderdischarged. The comparison of calculated and experimen-tal stress–strain rate relationships shows a good agreement

Fig. 1. Rheological plastic $ow.

and both dynamic shear and dynamic normal stresses areexpressed as linearly dependent on strain rates over a fairlywide strain rate region. The following equations obtainedby Yuu et al. show the stress–strain rate relationships in theparticulate matter:

�x = �x0 +√�2x0 + �2

y0 A1Dxx; (13)

�y = �y0 +√

�2x0 + �2

y0 A1Dyy (14)

and

�xy = �xy0 +√

�2x0 + �2

y0 A2Dxy (15)

where Dxx, Dyy and Dxy are the deformation rates.The coe/cient A2 is much larger than the coe/cient A1.

This means that shear deformation occurs more easily than

normal deformation in the particulate matter.√�2x0 + �2

x0 A1

and√�2x0 + �2

x0 A2 which have the dimension of viscosityare essentially viscosities when the particulate matter is as-sumed continuum. They were adjusted to give the best agree-ment between the calculated and the experimental results.

To calculate the stagnant zone on the powder bed, theconstitutive equations of a Bingham-type plastic model areused with those above.

In a Bingham model (Fig. 1), shear stress tensor � and de-formation rate tensor D are related by the following relation:

� = �c + �0D; (16)

where �0 and �c are plastic viscosity and yield stress.


Eqs. (13)–(15) can be expressed as

�x =�x0 + �y0

2+(�x0 − �x0 + �y0

2

)

+√�2x0 + �2

y0 A1Dxx; (17)

�y =�x0 + �y0

2+(�y0 − �x0 + �y0

2

)

+√�2x0 + �2

y0 A1Dyy (18)

and

�xy =(�x0 + �y0

2

)+√

�2x0 + �2

y0 A2Dxy; (19)

where the static shear stress is assumed to be equal to ((�x0+�y0)=2) and to the yield shear stress.

Yield stresses �c which are equivalent to deviatoricstresses are obtained by the following equations:

�xc =(�x0 − �x0 + �y0

2

); (20)

�yc =(�y0 − �x0 + �y0

2

)(21)

and

�xyc =(�x0 + �y0

2

): (22)

Identifying viscosity coe/cients, �0 of Bingham modelare obtained by the following equations:

�0x =√�2x0 + �2

y0 A1; (23)

�0y =√�2x0 + �2

y0 A1; (24)

and

�0xy =√�2x0 + �2

y0 A2: (25)

When we calculate a velocity !eld of a continuum usingBingham plastic model, the viscosity becomes in!nite atthe deformation rate D = 0. Then we cannot calculate thestagnant zone in the powder bed. To be able to calculate astagnant zone, the deformation rates are usually divided intotwo regions as shown in Fig. 1. Dc is the boundary valueof the deformation rate D. The viscosity in the region ofthe deformation rate which is smaller than Dc is very highbut not in!nite as shown in Fig. 1. Then we can calculatethe whole !eld including the stagnant zone where D ; 0approximately.

Using Eqs. (16)–(25) and Fig. 1, we obtain the followingequations:

�′x = �x − �x0 + �y0

2= �xc + �0xDxx = �xDxx;

�x = � if D6Dc;

�x =�xcDxx

+ �0x if D¿Dc; (26)

Fig. 2. Computational domain.

Table 1Computational conditions

Internal friction angle � (rad) 0.209Constants in Eqs. (13)–(15) A1, A2 (−) 0:1 × 10−3; 0:5 × 10−4

Representative value of Dc (s−1) 10.0deformation rateMass of initial imaginary m0 (kg) 0:01 × 10−3

powderConstant in Eq. (11) K (−) 50.0Initial bulk density of powder 0 (kg m−3) 1150.0Constant in Eq. (5) � (−) 1.4

Fig. 3. Experimental apparatus.

�′y = �y − �x0 + �y0

2= �yc + �0yDyy = �yDyy;

�y = � if D6Dc;

�y =�ycDyy

+ �0y if D¿Dc (27)


Fig. 4. Comparison of particle con!gurations determined by calculation and experiment.


and

�′xy = �xy − �xy0 = �xyc + �0xyDxy = �xyDxy;

�xy = � if D6Dc;

�xy =�xycDxy

+ �0xy if D¿Dc: (28)

The value of Dc in this work was taken as Dc = 10 s−1. Avery small value of Dc can lead to numerical instabilities.The value was chosen so that a suitable solution might beobtained.

2.5. Momentum equations

These equations for powder beds of x- and y-componentsmay be expressed asdudt

= − 1 b

@P@x

− 1 b

(@�x@x

+@�xy@y

)(29)

anddvdt

= − 1 b

@P@y

− 1 b

(@�xy@x

+@�y@y

)− g: (30)

Substituting Eqs. (17)–(19) into Eqs. (29) and (30) givesdudt

=− 1 b

@P@x

− 1 b

@@x

(�x0 + �y0

2

)

− 1 b

@@x

{(�x0 − �x0 + �y0

2

)

+√�2x0 + �2

y0 A1Dxx

}

− 1 b

@@y

{(�x0 + �y0

2

)+√

�2x0 + �2

y0 A2Dxy

}(31)

anddvdt

=− 1 b

@P@y

− 1 b

@@y

(�x0 + �y0

2

)

− 1 b

@@x

{(�x0 + �y0

2

)+√

�2x0 + �2

y0 A2Dxy

}

− 1 b

@@y

{(�y0 − �x0 + �y0

2

)

+√�2x0 + �2

y0 A1Dyy

}− g (32)

where P is given by Eq. (11).

3. Calculation conditions

Fig. 2 shows the computational domain whose size isthe same as that of the experimental apparatus. Horizontalwidth of the tank is 189 mm and slit width, 40 mm. Theimaginary particles are initially !lled to a uniform height

Fig. 5. Velocity vectors at various times in upper part of tank.

of 100 mm in the upper part of the tank and the cen-ters of imaginary particles are separated from each otherby 3 mm. The imaginary particle number calculated was2079.

In the smoothing process, we use the partition shown inFig. 2 to increase the calculation e/ciency. The size of apartition is 10:5 mm × 10:5 mm. A partition contained 16particle centers at the initial state since the distance betweenthe particle centers is 3 mm. As shown in Fig. 2, the refer-ence particle interacted with particles in nine partitions in-cluding eight neighboring partitions.


Fig. 6. Velocity distribution of the y-component at various heights of powder bed (t = 0:2 s).

To satisfy the boundary condition, the imaginary com-putational domain whose width is one partition was placedoutside the !xed boundary, as shown in Fig. 2. The particlecon!guration in the outer imaginary partition is symmetricwith that in the inner partition. The outer particle velocitycomponent perpendicular to the wall boundary is set to beopposite to that of the inner particle. Then the velocity per-pendicular to the wall boundary becomes zero on the bound-ary. Other parameters for outer particles were made to beequal to those in inner particles.

Table 1 shows the conditions under which the presentsimulation was carried out.

4. Experiment

The experimental apparatus which consists of a tank189 mm long, 400 mm high and 189 mm wide for powderobservation is shown in Fig. 3. Glass beads, used as thetest powder, were discharged from the lower slit 40 mmin width. Mean particle diameter was 100 �m and particle

density, 2500 kg=m3. Glass beads were placed horizontallyat a depth of 100 mm in the tank along with black-coloredbeads similarly placed at a height of 50 mm from the bot-tom to facilitate the visualization (videos) of powder $ow.The glass beads were allowed to fall by opening the slitgate. The total weight of glass bead was 5400 g.

A video camera (shutter speed 1=1000 s, 30 frames=s)was used to observe powder motion. The velocity of theblack-colored bead was measured by the video camera us-ing a !xed background grid of 1 cm squares.

The powder discharged was collected in a saucer and itsweight was measured by a load cell under the tank, at afrequency of 10 Hz. The rate of discharge was computedfrom the initial powder weight and the weight of the powderdischarged.

5. Results and discussion

Fig. 4(a) shows results obtained with the SP methodusing Lagrangian equations. Con!gurations, which would


Fig. 7. Velocity distribution of the x-component at various heights of powder bed (t = 0:2 s).

indicate the location of the powder in the tank, aremeant to provide some indication of powder movement.The powder starts to run from the slit (40 mm) at thebottom of the tank at time zero. At 0:2 s, the pow-der at the top begins to sink and $ows sparsely fromthe outlet, while that near the corner remains essen-tially stationary. The height of the powder bed at thecenter is about 35 mm from the bottom. At 0:4 s, thepowder near the bottom retains its initial location. Theyield stress is determined from the height of the pow-der bed (Eqs. (8)–(10), (20)–(22)), and thus becomeszero near the surface. Then, the powder near the sur-face moves easily. At 0:6 s, the powder bed height hasdecreased further and at 1:0 s, the discharge of pow-der has diminished. The stagnant zone of powder in thecorner of the tank was found to be the same as indi-cated by experiment. Fig. 4(b) shows experimental re-sults for powder in the tank. The calculated $ow patternsand shape of stagnant zone and those measured are ba-sically the same. The simulation using the SP methodclearly shows the real powder $ow pattern, which would

not be able to be simulated using Eulerian continuummethod.

Fig. 5 shows powder velocity vectors. At 0:2 s, the pow-der near the free surface reaches the slit outlet. Velocityvectors over the slit become greater and more random indirection. The powder near the corners is virtually stag-nant. At 0.4 and 0:6 s, the powder near the center of thetank $ows toward the outlet, accumulates on the slit andthen is discharged. At 1:0 s, the stagnant zone becomeslarger.

Fig. 6 shows velocity distribution for the y-componentas determined by calculations at various heights (10, 30,50 mm) at 0:2 s. At y = 50 mm, a non-zero velocity is ob-served at 25 mm from the walls. The width of the veloc-ity distributions becomes smaller on reaching the nozzleoutlet.

Fig. 7 shows velocity distribution for the x-component.The x-component of velocities shows the same tendency asthe result of the y-component. These velocity pro!les showthe position of the motionless powder and allow to identifythe stagnant zone in the tank.


Fig. 8. Comparison of calculated and experimentally determined velocity(t = 0:1 s).

In Fig. 8, calculated and experimental velocities ofpowder are compared at 0:1 s. Calculated maximum ve-locities in the x direction are larger than the measuredvalues. As shown in Fig. 4, the gradient of the cal-culated $ow pattern is somewhat larger than that ofthe experimental data. This causes the di9erence be-tween the calculated and the measured x-componentvelocities.

Calculated velocity in the y direction was essentiallythe same as the experimental result for the center powder$ow. Experimental y-direction velocity was almost zerowithin 50 mm from the side wall. However, calculated val-ues in this region are slightly larger than zero. The largecalculated velocities at the side walls are caused by the rep-resentative value of deformation rate Dc which should besmaller.

Fig. 9 shows density distribution obtained by calculation.Powder density was initially 1150 kg=m3. On approachingthe outlet, the di9erence between the center and the sur-face densities could be seen. The dark areas in the !guresindicate the powder particles to be densely contacting with

each other. Lower density regions above the slit and near thepowder surface showed high-velocity vectors, as indicatedin Fig. 5.

Fig. 10 shows rates of discharge. The !nal exper-imental value was about 78%, indicating 22% of thepowder to be still in the tank. Calculated dischargerates are in fairly good agreement with the experimen-tal data. The experimental discharge rate steadily in-creased. At 0.25–0:5 s, calculated results always ex-ceeded those obtained experimentally. The large calcu-lated velocity of powder near the side wall, as shownin Fig. 8, corresponds to this. The calculation curveshowed satisfactory correspondence to experimentaldata.

6. Conclusions

The numerical simulation of powder particle $ow in a tankis important to design a tank containing an in!nite numberof particles.

The simulation in the present study which was con-ducted using the SP method well describes the real pow-der $ows in a tank. The comparison of the calculatedand the experimental $ow patterns shows a close agree-ment. To verify the simulated results, calculated glassbead velocities were compared with experimental data.Calculated velocities in the x and y directions were vir-tually the same as determined experimentally. Exper-imental results for discharge rate were essentially thesame as those calculated. Then, the results show thatthe simulation applied the SP method to the powder$ow, making it possible to express the $ow mechanismof powders correctly when the applicable constitution,static stress and pressure density equations are given.Therefore, when these equations for submicron parti-cle powders are derived using DEM, it is possible tosimulate correctly the complex $ow of very small parti-cle powder using the SP method. This study shows thatthe SP method is an e9ective means for powder $owsimulation.

Notation

A1,A2 constants in Eqs. (13)–(15), dimensionlessD deformation rate, s−1

Dc representative value of deformation rate, s−1

g gravitational acceleration, mm=s2

h standard deviation of kernel function, mmH depth of powder, mmK constant in Eq. (11), dimensionlessm0 mass of initial imaginary powder, kgP pressure, Pat time, s


u; v x and y components of powder velocity, mm=s

W kernel function, mm−2

x; y coordinates, mm

Greek letters

�0 plastic viscosity, Pa s

�′ maximum viscosity de!ned by Eqs. (26)–(28), Pa s

� internal friction angle, rad

Fig. 9. Density distribution.

� constant in Eq. (5), dimensionless

0 initial bulk density of powder, kg=m3

b bulk density of powder, kg=m3

�x, �y x and y components of normal stress, Pa

�x0, �y0 x and y components of static normal stress,Pa

�c yield stress, Pa

�xy shear stress, Pa

�xy0 static shear stress, Pa


Fig. 10. Rate of powder discharge from tank outlet.

References

Gutfraind, R., & Savage, S. B. (1998). Flow of fractured icethrough wedge-shaped channels: Smoothed particle hydrodynamics anddiscrete-element simulations. Mechanics of Materials, 29, 1–17.

Monaghan, J. J. (1994). Simulating free surface $ows with SPH. Journalof Computational Physics, 110, 399–406.

Monaghan, J. J. (1996). Gravity currents and solitary waves. Physica D,98, 523–533.

Monaghan, J. J., & Kocharyan, A. (1995). SPH simulation of multi-phase$ow. Computer Physics Communications, 87, 225–235.

Oger, L., & Savage, S. B. (1999). Smoothed particle hydrodynamicsfor cohesive grains. Computer Methods in Applied Mechanics andEngineering, 180, 169–183.

Shimosaka, A., Kano, J., & Hidaka, J. (1997). New simulation method of$ow behavior of granular materials using smoothed particle element.Proceedings of the Joint ASME, ASCE and Summer Meeting, 626,Northwestern University.

Shimosaka, A., Kano, J., & Hidaka, J. (1998). Computer simulation ofgranular $ow using Smoothed Particle Element Method. Proceedingsof the world congress on particle technology, Vol. 3 (pp. 1–11). PaperNo. 340.

Takeda, H., Shoken, M., & Sekiya, M. (1994). Numerical simulationof viscous $ow by smoothed particle hydrodynamics. Progress inTheoretical Physics, 92, 939–960.

Yuu, S., Hayashi, A., Waki, M., & Umekage, T. (1997). The stress–strainrate relationship for $owing coarse particle powder beds obtained bythe 3-dimensional distinct element method and experiments. Journalof the Society for Powder Technology, Japan, 34, 212–220.

Yuu, S., Waki, M., & Umekage, T. (1998). Numerical simulation of thevelocity and stress !elds for a $owing powder using the smoothedparticle method and experimental veri!cation. Journal of the Societyfor Powder Technology, Japan, 11, 113–128.

Documents

Numerical analysis of fine powder flow using smoothed particle method and experimental verification