research article numerical investigation of the effect of...

Research ArticleNumerical Investigation of the Effect of Bottom Shape onthe Flow Field and Particle Suspension in a DTB Crystallizer

Hao Pan Jun Li Yang Jin Bo Yang and Xing Li

Department of Chemical Engineering Sichuan University Chengdu 610065 China

Correspondence should be addressed to Jun Li lijunscueducn

Received 16 December 2015 Accepted 8 February 2016

Academic Editor Raghunath V Chaudhari

Copyright copy 2016 Hao Pan et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The influence of the bottom shape on the flow field distribution and particle suspension in a DTB crystallizer was investigated byComputational Fluid Dynamics (CFD) coupled with Two-Fluid Model (Eulerian model) Volume fractions of three sections weremonitored on time and effect on particle suspension could be obtained by analyzing the variation tendency of volume fractionThe results showed that the protruding part of a119882 type bottom could make the eddies smaller leading to the increase of velocityin the vortex Modulating the detailed structure of the119882 type bottom to make the bottom surface conform to the streamlines canreduce the loss of the kinetic energy of the flow fluid and obtain a larger flow velocity which made it possible for the particles inthe bottom to reach a better suspension state Suitable shape parameters were also obtained the concave and protruding surfacediameter are 032 and 0373 times of the cylindrical shell diameter respectively It is helpful to provide a theoretical guidance foroptimization of DTB crystallizer

1 Introduction

DTB crystallizer is a kind of crystallizer of high efficiencywhich is widely used in chemical food and pharmaceuticalindustries [1] With draft tube and baffles hydrodynamics inthe crystallizer become different with common stirred tankAfter years of experiments and operations DTB crystallizerhas been proved to be of good performance Crystals of largersize (the maximum can reach 600ndash1200 120583m) can be pro-duced It has a higher production strength while less crystalswill adhere to the inner wall of the crystallizer making itbecome one of the main forms of continuous crystallizersThe existence of a draft tube in a DTB crystallizer makesit benefit in requiring a much lower indenter to realize theinner circulation than thatwithout a draft tube and the powerconsumption can be reduced by 20when the system reachesa certain suspension state [2 3]

Obviously hydrodynamics in a DTB crystallizer have astrong impact on crystallization process In some extremelycomplex problems hydrodynamics are hard to get byexperiments With the advances in computer technologyand numerical techniques Computational Fluid Dynamics(CFD) simulations which are able to obtain fundamental

physical quantities with less time and cost have been increas-ingly used to solve complex fluid mechanical problems [4 5]

For DTB crystallizer most researches about CFD sim-ulations focus on the influences of structures (the impellerthe installation height of the draft tube etc) on the flowfield distribution the state of particle suspension and powerconsumption [6ndash9] Oldshue [6] investigated the suspen-sion process using a stirred tank with a draft tube insideand the results showed that two critical velocities shouldbe concerned for the solid-liquid suspension operation toensure that the particles would not deposit in the bottomSha and Palosaari [5] studied the influences of the struc-tures production discharge location and mixing intensityon the continuous crystallization process Jaworski et al[7] compared the results between LDA measurements andCFD predictions to study the effects of size location andpumping direction of pitched blade turbine impellers onflow patterns Zhong et al [8] calculated the critical justoff-bottom suspension impeller speed of solid-liquid systemin a DTB crystallizer Besides the impeller draft tube andinstallation location the bottom shape of aDTB crystallizer isalso an important factor that can affect the hydrodynamics ofthe fluid inside which can influence the particle suspension

Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2016 Article ID 6862152 11 pageshttpdxdoiorg10115520166862152

2 International Journal of Chemical Engineering

Z

X

Z

YX

H=165

mm

h=100

mm

T = 150mm

Figure 1 Geometry and computational grid of the investigated DTB crystallizers

and crystallization processes [6 9] In the former studies thebottom shape of the DTB crystallizer was mostly flat baseor dished bottom few researches focused on the advantagesof 119882 type bottom The curvature of the surface of a 119882 typebottom is more conformed to the streamlines of fluid motiongenerated by the combined effect of rotating impeller bafflesand draft tube which has advantages in particle suspensionand reducing the deposition probabilities of particles at thebottom Thus production of larger size and more uniformparticle size distribution can be available

Suitable 119882 type bottoms which were better for crystal-lization process were investigated Five bottoms of differentshapes were presented by modulating the diameters of theconcave and protruding parts Flow field distribution andparticle suspension of all five DTB crystallizers with different119882 type bottoms were simulated by commercial software Flu-ent Three representative sections were monitored on timeand graphs of volume fractions varied with flow time couldbe obtained By comparing the simulation results suitableshape parameters of119882 type bottom could be available whichmight help to provide better hydrodynamic conditions for thecrystallization process and particle suspension

2 Geometry of the Crystallizers andParameter Variations

Figure 1 shows the geometry of the investigated DTB crys-tallizers the computational model is a proportional modelto the self-made crystallizer used in the experiments Allthe structure parameters are the same except the bottomshapeThe total height of the crystallizer is 165mm the underpart is made up of a 100mm cylinder with 150mm innerdiameter and a bottom of 30mm high and the top part is anexpanding section the maximum diameter is 190mm Thedraft tube is 100mm high with an inner diameter of 100mmand the installation height is 30mm above the bottom Fourinstalled stationary baffles begin 15mm above the bottomof the crystallizer and extend vertically over 110mm Thediameter of the impeller is 90mm The computational gridapplied in theCFDcalculations is also showed in Figure 1TheCFD results may of course depend strongly on the employed

Table 1 The bottom shapes and structure parameters of the DTBcrystallizers

Number Bottom shapes 1198631

1198632

I mdash mdash

IID1

D20320119879 0373119879

III 0373119879 0373119879

IV 0220119879 0507119879

V 0500119879 0060119879

IndashV are DTB crystallizers with different bottom shapes and IndashV in Figures3 5 7 8 and 9 are corresponding to those in Table 1

mesh In order to check grid-independency two differentmeshes employing roughly 32000 and 60000 grid cells havebeen compared for aDTB crystallizer And the results showedthat there is little difference in both flow field distribution andsolid particle volume fraction distributionTherefore the onewith less grid cells is chosen for quicker simulations

The specific bottom shapes and detailed parameters arelisted inTable 1 and it shows the vertical planes of the bottomsat 119884 = 0 I is a flat bottom IIndashV are 119882 type bottoms V is atoroidal surface bottom selected from a design manual [10]and IIndashIV are bottoms developed fromVby some appropriatemodulations 119863

1is the diameter of the concave part of the

bottom and 1198632is the diameter of the protruding part of the

bottom

3 Computational Model

The solutions of the complete equations are computed bythe commercial Software ANSYS-Fluent The well knownstandard 119896-120576 turbulence model is adopted in the singlephase simulation and this model is based on the turbulentkinetic energy and diffusivity Assuming that the flow field

International Journal of Chemical Engineering 3

is completely in the turbulent state turbulent transportequations corresponding to 119896 and 120576 are presented as follows

120597 (120588119896)

120597119905+120597 (120588119896119906

119894)

120597119909119894

=120597

120597119909119895

[(120583 +120583119905

120590119896

)120597119896

120597119909119895

] + 119866119896+ 119866119887

minus 120588120576 minus 119884119872+ 119878119896

120597 (120588120576)

120597119905+120597 (120588120576119906

119894)

120597119909119894

=120597

120597119909119895

[(120583 +120583119905

120590120576

)120597120576

120597119909119895

]

+ 1198621120576

120576

119896(1198661198961198623120576119866119887) minus 11986221205761205881205762

119896

+ 119878120576

(1)

According to Launder and Spaldingrsquos recommended val-ues and experimental values the constants in the model areequal to the numerical values below [11]120590

119896and 120590120576are Prandtl

number of 119896 and 120576 respectively

1198621120576= 144

1198622120576= 192

119862120583= 009

120590119896= 10

120590120576= 13

(2)

An MRF [12] approach is applied to calculate the flowfield distribution of single liquid fluid in the crystallizer Forthe liquid-solid two phases flow besides the MRF modelEulerian model [13] which can simulate every single phasein the multiphase fluid flow is also chosen as the total solidphase volume fraction is more than 10 The continuityequation is formulated as

120597

120597119905(120572119902120588119902) + nabla sdot (120572

119902120588119902

997888rarrV119902) = 0 (3)

Themomentumexchange between twophases is based onthemomentum exchange coefficient For liquid-solid systemthe momentum exchange coefficient 119870

119904119897is based on the

drag coefficient and Syamlal-OrsquoBrien model [14] is used tocalculate 119870

119904119897when the solid phase shear stress is defined by

Syamlal and OrsquoBrien

119870119904119897=3120572119904120572119897120588119897

4V2119903119904119889119904

119862119863(Re119904

V119903119904

)10038161003816100381610038161003816

997888rarrV119904minus997888rarrV119897

10038161003816100381610038161003816

Re119904=

120588119897119889119904

10038161003816100381610038161003816


10038161003816100381610038161003816

120583119897

119862119863= (063 +

48

radicRe119904V119903119904

)

2

(4)

Combining with the sedimentation velocity of the par-ticles the axial velocity distribution and the impeller input

power it is possible to compare the advantages and disadvan-tages of each crystallizer Ignoring the influence between par-ticles the terminal velocity of the particles can be formulatedsimply

119906119905= radic

4

3

119889119901(120588119901minus 120588) 119892

119862119863120588

(5)

To characterize 119862119863

and fluid dynamics in a globalmanner themodified version of the Reynolds number is used[15]

Re =119873119863imp

2120588

120583 (6)

The calculated Reynolds number for default operatingcondition is approximately 108000 Therefore a turbulencemodel is obviously required to describe hydrodynamics andit can be figured out that the drag coefficient 119862

119863asymp 044

Substituting it into (5) the terminal velocity appears

119906119905= 174radic

119889119901(120588119901minus 120588) 119892

120588 (7)

Shear rate is another important factor for crystallizationprocess In the rotation system similar toDTB crystallizer theshear rate is a function of 120576 (turbulent dissipation rate) and V(kinematic viscosity) [16 17]

= (120576

V)

12

(8)

The torque of the impeller can be calculated by theempirical equations [18]

119879119902=

intΩ(120583119898

10038161003816100381610038161003816100381610038161003816

2

) 119889Ω

2120587119873

120583119898= 120583(1 minus

120601

120601max)

(9)

Finally the input power of the impeller can be calculated[19]

119875 = 2120587 times 119879119902times 119873 (10)

In all simulations the momentum continuity and turbu-lent transport equations are numerically solved by the ldquoSIM-PLErdquo algorithm to predict the time-dependent variations offlow and all simulations are unsteady All the simulations areunder the condition of 800 rpmAnd the boundary and initialconditions are showed in Appendix The results are used topredict particle volume fraction distribution and the state ofsuspension inDTB crystallizers with different bottom shapes

4 Results and Discussions

41 Hydrodynamics In the single liquid phase simulationthe flow time could be seen in Figure 2 The calculation


minus02

00

02

04

06

08

10

12

14

16

18

20Ve

loci

ty m

agni

tude

(ms

)

minus05 00 05 10minus01

yR

5 s10 s15 s

(a)

VxVyVz

0 2010 40 5030

Flow time (s)

minus12

minus10

minus08

minus06

minus04

minus02

0002040608101214

Velo

city

(ms

)(b)

Figure 2 Velocity varied with time at different locations

minus005

000

005

010

015

020

025

030

035

Axi

al v

eloci

ty (m

s)

02 04 06 08 1000

ZH

IIIIII

IVV

(a)

minus01

00

01

02

03

04

05

06

07

Axi

al v

eloci

ty (m

s)

02 04 06 08 1000

ZH

IIIIII

IVV

(b)

Figure 3 Axial velocity distributions In (a) the radius is 60mm and in (b) the radius is 65mm

time spent to reach this pattern was about 40 minutes butit took much longer to calculate the solid-liquid two-phasesimulations (about 5 hours)

Figure 2(a) showed the velocity magnitude distributionof a chosen line at different times and we could see that thevelocity distribution at 15 s was almost the same as that at 10 sFigure 2(b) showed how the velocities (119881

119909 119881119910 119881119911) changed

with flow time of one specific point It could be obviouslyseen that after about 12 s 119881

119909 119881119910 and 119881

119911mainly remained

unchanged Therefore we could reach a conclusion that theflow fluid reached a steady state after about 12 s

The axial velocity distributions at different positions inthe crystallizer were showed in Figure 3 The way the flowfield influences particle motions and the state of particle sus-pension could be reflected indirectly Figures represented thevelocities of straight lines from bottom to top between drafttube and outside wall of the crystallizer Since a periodicityand symmetry condition along the azimuthal direction could


Y

Z

11

07

05

03

01

09

minus11

minus05

minus15

minus01

minus07

minus13

minus03

minus09

Axial velocity (ms)

Figure 4 Axial velocity distribution below the draft tube in the flatbottom crystallizer

be invoked for the flow field it was reasonable to determinethe position of the straight lines by the value of radius (119903) InFigure 3 and all the following figures I represents flat bottomand IIndashV represent119882 type bottoms

Figure 3(a) showed the axial velocity of the line at 119877 =

60mm the positive values meant the fluid flows upward andvice versa The axial velocity of I was lower than that of IIndashVwhen 119885119867 lt 02 that is the distance from the bottom wasapproximately 30mm which represented the region belowthe draft tube In this area the axial velocity of I was lowerthan 01ms while the axial velocities in IIndashV were obviouslylarger than I and probably reached 02ms When 119885119867 gt

02 the variation tendency of axial velocities in IndashV basicallyvaried similarly Nevertheless at 119877 = 65mm the variationtendency of axial velocities in IndashV basically remained thesame and the values could exceed 02ms when 119885119867 waslarger than 01 Based on (7) the terminal velocity of theparticles (119906

119905) could be calculated to be 0157msThe absolute

velocity (119906119901= 119906 minus 119906

119905) of the particles was directly related to

the hydrodynamics of the fluid Only 119906 ge 0157ms couldmake it possible for the particles to move with the fluid andsuspend in the crystallizer Comparing 119906

119905with Figure 3 it was

obvious to see that at 119877 = 60mm under the draft tube flowvelocities in IIndashVwere larger than 119906

119905 particles in IIndashV should

be more likely to reach the state of complete suspension thanthose in I

Figure 4 showed the axial velocity distribution of thelower part in crystallizer I As the legend showed the positivevaluemeans the fluidmoves upward otherwise the fluid flowsdownward It was obvious to know that just below the drafttube the axial velocities in all five cases were downward andthe axial velocities of the near crystallizer wall annulus regionwere all greater than 0157ms Hence what needed to bediscussed was the near draft tube annulus region Becauseof the interaction between the two phases the axial velocitywould decreasewhen the solid particles were added under thesame conditions [20] In the near draft tube annulus region

Table 2 Impact of stirring speed on input power of the impeller

Stirring speedrpm TorqueNsdotm PowerWsdotmminus3

800 01947 16311000 04987 5219

axial velocity was lower than 0157ms it could be seenthat the axial velocity would get smaller so that the flowingfluid was unable to carry the particles Therefore completesuspension of particles was not achievable in this area On thecontrary particlesweremore likely to deposit and accumulatein the bottom which could affect the crystallization processand the size of ultimate production As was discussed aboveabout Figure 3 the axial velocities of IIndashV in the same regionwere greater than 0157ms which was benefit for the particlesuspension and crystal growthThe state of suspension couldbe improved by increasing stirring speed accompanying theincrease of power consumption Therefore the performanceof the flat bottom DTB crystallizer was not as good as the119882type bottom crystallizers

A comparison of the input power of the impeller withdifferent stirring speed which was calculated by (10) wasshowed in Table 2 The input power of the impeller increasedas the stirring speed increases but the rates of increasevaried a lot The input power appeared as an increase of32-fold while the stirring speed merely increased by 125-fold With the increase of the stirring speed the metastableregion shrank which made it easier to generate more finegrains And the increase of flow-shear stress also enhancedthe possibility of collisions between crystals which resultedin the increase of the second nucleation rate [21] Hence dueto the extra consumption of energy and bad influence on thecrystallization process flat bottom was inappropriate to use

Figure 5 showed the streamlines in the vertical planes(119884 = 0) of different crystallizers under the same conditionsThe more intensive the streamlines were the larger theflow velocity was The distributions of the streamlines wereroughly the same in five vertical planes Eddies created byrecirculation of the fluid existed below the impeller in theoutlet and inlet of the draft tube in all five crystallizersbut eddies were not the same size and the intensities of thestreamlines differ delicately Eddies below the impeller hadthe greatest influence on the particles suspension Particlesmoved with the fluid and there were two main motion pathstill the particles arrived at the outlet of the draft tube Firstparticles moved into the annulus area and then continuedto move upward Second particles moved into the eddiesbelow the draft tube spinning around or accumulating inthe bottom It was obvious to see that eddies in I and V arebigger than others making it more probable for the particlesto move according to the second way Streamlines in theseeddies were less intensive which meant the velocities werelower and more particles would deposit relatively On thecontrary eddies in II and III were smaller thusmore particleswouldmove in the first wayMeanwhile streamlines in II weremore intensive which meant the velocity was higher and theaqueous carrying capacity of particles could be greaterThere-fore 119882 type bottoms should be better than flat bottom for


I II III

IV V

Figure 5 Comparison of instantaneous streamlines neglecting the velocity component perpendicular to the cutting plane (vertical plane119884 = 0)

the crystallization process and fine distinctions in the shapeparameters of the bottom could affect the hydrodynamics andmake a difference in the particle suspension

42 Particle Suspension In order to elaborate the detaileddifferences between the different 119882 type bottoms furthersimulations were investigated Particle size distribution of theultimate production had a certain requirement in industryIn this work the average size of the particles was 500120583mTherefore monosized particles of 500 120583m were chosen toinvestigate the state of suspension in each crystallizer withdifferent bottom shapes The rotating speed of impellersremained 800 rpm and other physical property parametersof liquid phase remained the sameThe total volume fractionof solid particles was 10 vol In order to observe the volumefraction of the particles in the bottom of the crystallizersintuitively volume fractions of three sections weremonitoredon timeThe three sections were showed in Figure 6 SectionsA and B were transversal surfaces of the crystallizer atdifferent heights while section C was part of the longitudinalsurface of the crystallizer at 119884 = 0

By monitoring the volume fractions in section A andsection B we could figure out the state of suspension in thebottom of the crystallizers The value of the volume fractionin section A reflected the amount of the particles in thebottom of the crystallizers to some extent Because belowsection A the volume fraction should be larger theoretically

Simultaneously the comparison betweenA and B showed thehomogeneity of the particle distributions in whole crystal-lizers The smaller the difference of values between A and Bwas the more homogeneous the particles were distributedThe results were showed in Figure 7

In order to investigate the state of suspension of theparticles in the bottom the monitoring of section C in allfive DTB crystallizers were showed in Figure 8 as a furtherillustration Because of the periodicity and symmetry of thefluid flow in the DTB crystallizer the equilibrium value of thevolume fraction in section C could account for the particlesuspension more accurately More particles deposited in thebottom when the value of volume fraction got larger and theflow-ability of the fluid gotworsemore particles accumulatedin the bottom which resulted in the nonuniform distributionof the particles and the ultimate size of productions wasuneven

As the results showed in Figures 7 and 8 it was obvious tofigure out that in all the five DTB crystallizers with differentbottom shapes the volume fractions of the solid particlesarrived at an equilibrium state after stirring for 100 s whichmeant that under the condition of 800 rpm the influence ofthe bottom shape on themixing efficiency was tiny enough toneglect Although the time to attain an equilibrium remainedthe same the numerical values of volume fraction in the stateof equilibrium in section A section B and section C differeda lot


B

A

h = 55mm

h = 25mm C

h = 25mm

Figure 6 The chosen sections to monitor the volume fractions

As expected it was observed that volume fractions ofsection B were larger than those of section A in II and IIIwhen the mixing of the particles reached an equilibriumstate while in I and IV the volume fractions of section Bwere smaller than those of section A which meant moreparticles accumulated in the bottom part of the crystallizersIn V the two values were equaled In II and III the volumefractions of section B were larger than those of section Awhich meant less particles deposited in the bottom but moreparticles suspended in the middle part of the crystallizers Itwas better to see that the volume fraction value in section Aremained smaller than that in section B which meant morecrystals suspended in the middle part of the crystallizer Andin case V the volume fraction of section A equaled the valuein section B the particle distribution was more uniform butthe volume fraction value was a little larger than that of II andIIITherefore we could see that the suspension state in case Vwas not the best one As discussed above in II and III eddiesunder the rotating impeller were smaller than the othersthus less particles moved according to the second way andthe volume fraction of section A got smaller In I IV and Veddies got much bigger and the velocities inside the vortexesgot lower the ability for the flow fluid to carry the particlesmoving with it became weaker resulting in the accumulationof particles in the bottom

The consequences of the monitoring of the volumefractions of particles in section A and section B in liquid-solid two-phase simulations were conformed to the resultsin the investigations of hydrodynamics in single phase sim-ulation It had been surprising to see the great differencesof hydrodynamics and deposition probabilities caused by thedifferent bottom shapes To investigate this issue further itwas essential to continue to monitor the volume faction ofthe particles in section C to verify the suspension propertybetween II and III

Figure 8 shows the volume fractions variedwith flow timein section C in all the crystallizers with different bottoms It

was more intuitive to see the amount of particles accumulatein the bottom part of the whole crystallizer The volumefractions in I and V were larger than the others and itwas corresponding to the results of single phase simulationThe volume fractions of III and IV were smaller than thoseof I and V Although the numerical values of III and IVwere basically the same the volume fraction of section A inIII was smaller than IV which means the total amount ofparticles in the bottom of III was larger than IV Thereforethe bottom shape of III was not good enough to improvethe hydrodynamics and promote the suspension propertyIt was obvious to see that the volume fraction of II wasmuch smaller than the others which meant the influence ofthe bottom shape on the hydrodynamics was positive and itwas benefit to the suspension of the particles More particlesin II moved in the first way mentioned above which wasgood to the particle suspension and has advantages in thecrystallization process All the conclusions are conformedto the results in Section 41 The reason why bottom shapehad a great influence was that119882 type bottoms with differentstructures and parameters affected the fluid flow distributionThe protruding part of the bottom broke the eddies makingthem become smaller so that less particles would move intothe eddies in the bottom part of the crystallizers and just spanin the bottom which prevented the solid particles gatheringand aggregating Therefore it was important to design anappropriate119882 type bottom

As mentioned above we have discussed the state ofparticle suspension in all five DTB crystallizers with the totalsolid particles which amount to 10 vol The results showthat II is better than the others in the suspension propertyunder this condition In order to make further verificationthat the hydrodynamics of II is superior to the others inparticle suspension volume fractions of section C vary withflow time under the conditions of the solid particles whichamount to 15 vol and 20 vol depicted in Figure 9


A h = 25mmB h = 80mm

006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

50 100 150 2000

Flow time (s)


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

I II


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

100 2000 50 150

Flow time (s)


006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

100 2000 50 150

Flow time (s)

III IV


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

V

Figure 7 Volume fractions vary with flow time in sections A and B


006

008

010

012

014

016

018

020

022

Volu

me f

ract

ion

40 80 120 160 2000

Flow time (s)

IIIIII

IVV

Figure 8 Volume fractions vary with flow time in section C

012

015

018

021

024

027

030

033

036

Volu

me f

ract

ion

200 400 600 800 10000

Flow time (s)

II 15 volII 20 vol

III 15 volIII 20 vol

Figure 9 Volume fractions vary with flow time in section C withdifferent initial total particle volume fractions

The above statements are confirmed that the suspensionproperty of III is second to II Therefore the state of particlesuspension is only compared between II and III In Figure 9 itis obvious to see that when the initial particle volume fractionincreases to 15 or 20 the total flow time for the volumefractions in section C arriving at an equilibrium state getslonger than that of 10 vol of particles Whether the initialparticles amount to 15 vol or 20 vol the volume fraction ofII is always smaller than that of III whichmeans less particlesin II deposit in the bottom region Therefore more particlesmove upward with the fluid the hydrodynamics in II makethe particles reach a better suspension state which preventsthe accumulation and agglomeration of the particles andmore uniformly distributed production is available Never-theless with the initial particle volume faction increasing the

Table 3 Unevenness of the particle distribution

Volumefraction

IIunevenness

IIIunevenness

10 5904 673315 4418 477620 2976 3424

Table 4 Input power of the impeller with different amount ofparticles

Volumefraction

II IIITorqueNsdotm PowerWsdotmminus3 TorqueNsdotm PowerWsdotmminus3

10 01852 1551 01857 155515 01988 1665 01997 167220 02122 1777 02139 1791

difference between II and III gets narrow and it is knownthat the advantages of hydrodynamics generated by bottomshape like II could only be reflected in a low particle contentAs the amount of particles gets too large the advantages ofthe hydrodynamics of flow fluid made by the special bottomshape are not enough to offset the impact created by theinteraction of the two phases which means it is necessary toremove the particles from the crystallizer in time to make thetotal particle amount maintain a good suspension property

The analysis above majorly focuses on the distribution ofthe solid particles in the bottom part of the DTB crystallizerThe emphasis of the study is to investigate the way theshape of bottom affects the hydrodynamics and furtherinfluences the particle suspension Therefore it helps toprevent the excessive accumulation of the particles whichis helpful to avoid the bad influence on the crystallizationprocess and equipment operation In order to investigate thesuspension uniformity in the whole crystallizer the conceptof unevenness is introduced and the unevenness is definedas follows

119872 = [1

119899

119899

sum

119894=1

(119862119894minus 119862

119862

)

2

]

05

(11)

The unevenness of the particle distribution of II and IIIunder different conditions is showed in Table 3

From Table 3 it is apparent to see that the evenness of theparticle distribution of II is always smaller than that of IIIwhich means the particles are better distributed in II

Table 4 shows the input power of the impeller with differ-ent amount of particles and as expected power consumptionof II is always less than that of III which further prove that IIis better than the others not only in the good performance inparticle suspension but also in the lower cost of energy

5 Conclusion

Numerical simulations are conducted to investigate the effectof bottom shape on the hydrodynamics and particle sus-pension It is found that the bottom shape has significant


Table 5 Boundary and initial conditions of the simulations

Crystallizer parts Boundary conditions

Baffle Wall motion stationary wallShear condition no slip

Impeller Wall motion moving wallShaft Wall motion moving wallDraft tube Wall motion stationary wall

Walls of crystallizer Wall motion stationary wallShear condition no slip

Rotating region 800 rpmInitial particle volumefraction 10 vol of the whole crystallizer

influence on the flow field distribution Under the effect ofthe rotating impeller agitation vortexes can be created by thefluid flows in a restricted space as a DTB crystallizer andthe vortexes then become an important factor affecting theparticle suspension The 119882 type bottom can counteract theimpact brought by the eddies to some extent The presenceof the protruding part can destroy the main eddies underthe draft tube making them become smaller than beforeWith the vortexes being smaller less particles will move intothe vortexes when they arrived at the outlet of the drafttube more particles will move into the annulus region andcontinue moving upward Meanwhile the smooth surface ofthe concave part of the bottom is much more conformedto the streamlines which decrease the energy consumptionof the fluid flow This is how the bottom shape affect thehydrodynamics and consequently affect particle suspensionand crystallization process Comparing the simulation resultsof crystallizers with different bottoms it can be known thatdifferent 119882 type bottoms have different effects on hydrody-namics it is important to find suitable shape parameters ofthe bottom for suspension advantaged hydrodynamics andless energy consumption In general the CFD simulationis capable of providing theoretical guidance for design andoptimization of DTB crystallizer

Appendix

See Table 5

Nomenclature

119889119901 Particle diameter mm

119867 Crystallizer height mmℎ Draft tube length mm119879 Cylindrical shell diameter mm119863 Draft tube diameter mm1198631 Concave surface diameter of the

bottom mm1198632 Protruding surface diameter of the

bottom mm119863imp Impeller diameter mm119873 Impeller rotating speed rpm

119896 Turbulent kinetic energy m2sdotsminus2119870119904119897 Momentum exchange coefficient sminus1

119862119863 Drag coefficient

119875 Power Wsdotmminus3119879119902 Impeller torque Nsdotm

119906 Fluid flow velocity msdotsminus1119906119901 Particle velocity msdotsminus1

119906119905 Terminal velocity msdotsminus1

119899 Number of sampling locations119872 Unevenness119862119894 Particle volume fraction

119862 Weighted average particle volumefraction

119905 Time sV119903119904 Particlersquos terminal velocity msdotsminus1

1198621120576 1198622120576 119862120583 Standard coefficients for 119896-120576 turbulencemodel

Greek Letters

120588 Density kgsdotmminus3120583 Viscosity Pasdots120576 Turbulent dissipation rate m2sdotsminus3Φ Particle concentrationΦmax Maximum particle concentration120592 Kinematic viscosity m2sdotsminus1 Shear rate sminus1120572119904 Solid phase volume fraction

120572119904 Liquid phase volume fraction

120590119896 120590120576 Standard coefficients for 119896-120576 turbulence model

Dimensionless Number

Re Reynolds number Re = 1205881198731198892120583

Subscripts

119904 Solid phase119897 Liquid phase119901 Particles

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y C Huang J Tang and R Q Xie ldquoApplications of com-putational fluid dynamics in chemical engineeringrdquo ModernChemical Industry vol 27 no 5 pp 65ndash68 2007

[2] S J Shiue and C W Wong ldquoStudies on homogenizationefficiency of various agitators in liquid blendingrdquoTheCanadianJournal of Chemical Engineering vol 62 no 5 pp 602ndash6091984

[3] F J Wang Computer Fluid Dynamics AnalysismdashPrinciples andApplications of CFD Tsinghua University Press Beijing China2004


[4] C R Liu A D Huhe and W J Ma ldquoNumerical and exper-imental investigation of flow over a semicircular weirrdquo ActaMechanica Sinica vol 18 no 6 pp 594ndash602 2002

[5] Z Sha and S Palosaari ldquoModeling and simulation of crystal sizedistribution in imperfectly mixed suspension crystallizationrdquoJournal of Chemical Engineering of Japan vol 35 no 11 pp 1188ndash1195 2002

[6] J Y Oldshue Fluid Mixing Technology edited by Y C HuangM L Ling Trans Chemical Industrial Press Beijing China1991

[7] Z Jaworski K N Dyster and A W Nienow ldquoThe effect ofsize location and pumping direction of pitched blade turbineimpellers on flow patterns LDA measurements and CFDpredictionsrdquoChemical Engineering Research and Design vol 79no 8 pp 887ndash894 2001

[8] L Zhong X B Huang and Z G Jia ldquoCFD modeling ofsolids just suspended impeller speed in stirred tanksrdquo Journalof Beijing University of Chemical Engineering vol 30 no 6 pp18ndash22 2003

[9] D H Xie ldquoDTB type crystallizerrdquo Chemical Engineering ampMachinery vol 21 no 1 pp 55ndash57 1994

[10] Z P Chen X W Zhang and X H Lin Stirring and MixingEquipmentDesignManual Chemical Industry Press 1st edition2004

[11] B E Launder and D B Spalding Lectures in MathematicalModels of Turbulence Academic Press London UK 1972

[12] J Y Luo R I Issa and A D Gosman ldquoPrediction of impeller-induced flows in mixing vessels using multiple frames ofreferencerdquo Icheme SymposiumSeries vol 136 pp 549ndash556 1994

[13] D A Drew and R T Lahey In Particulate Two-Phase FlowButterworth Heinemann Boston Mass USA 1993

[14] M Syamlal and T J OrsquoBrien ldquoComputer simulation of bubblesin a fluidized bedrdquoAIChE Symposium Series vol 85 no 270 pp22ndash31 1989

[15] B Ashraf Ali G Janiga E Temmel A Seidel-Morgenstern andDThevenin ldquoNumerical analysis of hydrodynamics and crystalmotion in a batch crystallizerrdquo Journal of Crystal Growth vol372 pp 219ndash229 2013

[16] S A Altobelli R C Givler and E Fukushima ldquoVelocityand concentration measurements of suspensions by nuclearmagnetic resonance imagingrdquo Journal of Rheology vol 35 no5 pp 721ndash734 1991

[17] F E Kruis and K A Kusters ldquoThe collision rate of particles inturbulent flowrdquoChemical Engineering Communications vol 158pp 201ndash230 1997

[18] L Fradette P A Tanguy F Bertrand F Thibault J-B Ritzand E Giraud ldquoCFD phenomenological model of solid-liquidmixing in stirred vesselsrdquoComputers and Chemical Engineeringvol 31 no 4 pp 334ndash345 2007

[19] F A Holland and F S Chapman Liquid Mixing and Processingin Stirred Tank Reinhold Publishing Corporation New YorkNY USA 1966

[20] Y Y Bao X B Huang L T Shi and Y CWang ldquoThe influenceof solid particles on fluid velocity in a stirred tankrdquo ChemicalEngineering (China) vol 30 no 5 pp 29ndash33 2002

[21] A Van Hook Crystallization Theory and Practice Chapman ampHall London UK 1961

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Z

X

Z

YX

H=165

mm

h=100

mm

T = 150mm

Figure 1 Geometry and computational grid of the investigated DTB crystallizers

and crystallization processes [6 9] In the former studies thebottom shape of the DTB crystallizer was mostly flat baseor dished bottom few researches focused on the advantagesof 119882 type bottom The curvature of the surface of a 119882 typebottom is more conformed to the streamlines of fluid motiongenerated by the combined effect of rotating impeller bafflesand draft tube which has advantages in particle suspensionand reducing the deposition probabilities of particles at thebottom Thus production of larger size and more uniformparticle size distribution can be available

Suitable 119882 type bottoms which were better for crystal-lization process were investigated Five bottoms of differentshapes were presented by modulating the diameters of theconcave and protruding parts Flow field distribution andparticle suspension of all five DTB crystallizers with different119882 type bottoms were simulated by commercial software Flu-ent Three representative sections were monitored on timeand graphs of volume fractions varied with flow time couldbe obtained By comparing the simulation results suitableshape parameters of119882 type bottom could be available whichmight help to provide better hydrodynamic conditions for thecrystallization process and particle suspension

2 Geometry of the Crystallizers andParameter Variations

Figure 1 shows the geometry of the investigated DTB crys-tallizers the computational model is a proportional modelto the self-made crystallizer used in the experiments Allthe structure parameters are the same except the bottomshapeThe total height of the crystallizer is 165mm the underpart is made up of a 100mm cylinder with 150mm innerdiameter and a bottom of 30mm high and the top part is anexpanding section the maximum diameter is 190mm Thedraft tube is 100mm high with an inner diameter of 100mmand the installation height is 30mm above the bottom Fourinstalled stationary baffles begin 15mm above the bottomof the crystallizer and extend vertically over 110mm Thediameter of the impeller is 90mm The computational gridapplied in theCFDcalculations is also showed in Figure 1TheCFD results may of course depend strongly on the employed

Table 1 The bottom shapes and structure parameters of the DTBcrystallizers

Number Bottom shapes 1198631

1198632

I mdash mdash

IID1

D20320119879 0373119879

III 0373119879 0373119879

IV 0220119879 0507119879

V 0500119879 0060119879

IndashV are DTB crystallizers with different bottom shapes and IndashV in Figures3 5 7 8 and 9 are corresponding to those in Table 1

mesh In order to check grid-independency two differentmeshes employing roughly 32000 and 60000 grid cells havebeen compared for aDTB crystallizer And the results showedthat there is little difference in both flow field distribution andsolid particle volume fraction distributionTherefore the onewith less grid cells is chosen for quicker simulations

The specific bottom shapes and detailed parameters arelisted inTable 1 and it shows the vertical planes of the bottomsat 119884 = 0 I is a flat bottom IIndashV are 119882 type bottoms V is atoroidal surface bottom selected from a design manual [10]and IIndashIV are bottoms developed fromVby some appropriatemodulations 119863

1is the diameter of the concave part of the

bottom and 1198632is the diameter of the protruding part of the

bottom

3 Computational Model

The solutions of the complete equations are computed bythe commercial Software ANSYS-Fluent The well knownstandard 119896-120576 turbulence model is adopted in the singlephase simulation and this model is based on the turbulentkinetic energy and diffusivity Assuming that the flow field



120597 (120588119896)

120597119905+120597 (120588119896119906

119894)

120597119909119894

=120597

120597119909119895

[(120583 +120583119905

120590119896

)120597119896

120597119909119895

] + 119866119896+ 119866119887

minus 120588120576 minus 119884119872+ 119878119896

120597 (120588120576)

120597119905+120597 (120588120576119906

119894)

120597119909119894

=120597

120597119909119895

[(120583 +120583119905

120590120576

)120597120576

120597119909119895

]

+ 1198621120576

120576

119896(1198661198961198623120576119866119887) minus 11986221205761205881205762

119896

+ 119878120576

(1)


119896and 120590120576are Prandtl


1198621120576= 144

1198622120576= 192

119862120583= 009

120590119896= 10

120590120576= 13

(2)


120597

120597119905(120572119902120588119902) + nabla sdot (120572

119902120588119902

997888rarrV119902) = 0 (3)






119870119904119897=3120572119904120572119897120588119897

4V2119903119904119889119904

119862119863(Re119904

V119903119904

)10038161003816100381610038161003816


10038161003816100381610038161003816

Re119904=

120588119897119889119904

10038161003816100381610038161003816


10038161003816100381610038161003816

120583119897

119862119863= (063 +

48

radicRe119904V119903119904

)

2

(4)



119906119905= radic

4

3

119889119901(120588119901minus 120588) 119892

119862119863120588

(5)



Re =119873119863imp

2120588

120583 (6)


119863asymp 044


119906119905= 174radic

119889119901(120588119901minus 120588) 119892

120588 (7)


= (120576

V)

12

(8)


119879119902=

intΩ(120583119898

10038161003816100381610038161003816100381610038161003816

2

) 119889Ω

2120587119873

120583119898= 120583(1 minus

120601

120601max)

(9)


119875 = 2120587 times 119879119902times 119873 (10)





minus02

00

02

04

06

08

10

12

14

16

18

20Ve

loci

ty m

agni

tude

(ms

)


yR

5 s10 s15 s

(a)

VxVyVz

0 2010 40 5030

Flow time (s)

minus12

minus10

minus08

minus06

minus04

minus02

0002040608101214

Velo

city

(ms

)(b)


minus005

000

005

010

015

020

025

030

035

Axi

al v

eloci

ty (m

s)

02 04 06 08 1000

ZH

IIIIII

IVV

(a)

minus01

00

01

02

03

04

05

06

07

Axi

al v

eloci

ty (m

s)

02 04 06 08 1000

ZH

IIIIII

IVV

(b)




119909 119881119910 119881119911) changed


119909 119881119910 and 119881





Y

Z

11

07

05

03

01

09

minus11

minus05

minus15

minus01

minus07

minus13

minus03

minus09

Axial velocity (ms)







velocity (119906119901= 119906 minus 119906










800 01947 16311000 04987 5219





I II III

IV V









B

A

h = 55mm

h = 25mm C

h = 25mm









006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

50 100 150 2000

Flow time (s)


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

I II


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

100 2000 50 150

Flow time (s)


006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

100 2000 50 150

Flow time (s)

III IV


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

V



006

008

010

012

014

016

018

020

022

Volu

me f

ract

ion

40 80 120 160 2000

Flow time (s)

IIIIII

IVV


012

015

018

021

024

027

030

033

036

Volu

me f

ract

ion

200 400 600 800 10000

Flow time (s)

II 15 volII 20 vol





Volumefraction

IIunevenness

IIIunevenness

10 5904 673315 4418 477620 2976 3424


Volumefraction


10 01852 1551 01857 155515 01988 1665 01997 167220 02122 1777 02139 1791



119872 = [1

119899

119899

sum

119894=1

(119862119894minus 119862

119862

)

2

]

05

(11)




5 Conclusion










Appendix

See Table 5

Nomenclature














Greek Letters






Subscripts




References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120597 (120588119896)

120597119905+120597 (120588119896119906

119894)

120597119909119894

=120597

120597119909119895

[(120583 +120583119905

120590119896

)120597119896

120597119909119895

] + 119866119896+ 119866119887

minus 120588120576 minus 119884119872+ 119878119896

120597 (120588120576)

120597119905+120597 (120588120576119906

119894)

120597119909119894

=120597

120597119909119895

[(120583 +120583119905

120590120576

)120597120576

120597119909119895

]

+ 1198621120576

120576

119896(1198661198961198623120576119866119887) minus 11986221205761205881205762

119896

+ 119878120576

(1)


119896and 120590120576are Prandtl


1198621120576= 144

1198622120576= 192

119862120583= 009

120590119896= 10

120590120576= 13

(2)


120597

120597119905(120572119902120588119902) + nabla sdot (120572

119902120588119902

997888rarrV119902) = 0 (3)






119870119904119897=3120572119904120572119897120588119897

4V2119903119904119889119904

119862119863(Re119904

V119903119904

)10038161003816100381610038161003816


10038161003816100381610038161003816

Re119904=

120588119897119889119904

10038161003816100381610038161003816


10038161003816100381610038161003816

120583119897

119862119863= (063 +

48

radicRe119904V119903119904

)

2

(4)



119906119905= radic

4

3

119889119901(120588119901minus 120588) 119892

119862119863120588

(5)



Re =119873119863imp

2120588

120583 (6)


119863asymp 044


119906119905= 174radic

119889119901(120588119901minus 120588) 119892

120588 (7)


= (120576

V)

12

(8)


119879119902=

intΩ(120583119898

10038161003816100381610038161003816100381610038161003816

2

) 119889Ω

2120587119873

120583119898= 120583(1 minus

120601

120601max)

(9)


119875 = 2120587 times 119879119902times 119873 (10)





minus02

00

02

04

06

08

10

12

14

16

18

20Ve

loci

ty m

agni

tude

(ms

)


yR

5 s10 s15 s

(a)

VxVyVz

0 2010 40 5030

Flow time (s)

minus12

minus10

minus08

minus06

minus04

minus02

0002040608101214

Velo

city

(ms

)(b)


minus005

000

005

010

015

020

025

030

035

Axi

al v

eloci

ty (m

s)

02 04 06 08 1000

ZH

IIIIII

IVV

(a)

minus01

00

01

02

03

04

05

06

07

Axi

al v

eloci

ty (m

s)

02 04 06 08 1000

ZH

IIIIII

IVV

(b)




119909 119881119910 119881119911) changed


119909 119881119910 and 119881





Y

Z

11

07

05

03

01

09

minus11

minus05

minus15

minus01

minus07

minus13

minus03

minus09

Axial velocity (ms)







velocity (119906119901= 119906 minus 119906










800 01947 16311000 04987 5219





I II III

IV V









B

A

h = 55mm

h = 25mm C

h = 25mm









006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

50 100 150 2000

Flow time (s)


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

I II


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

100 2000 50 150

Flow time (s)


006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

100 2000 50 150

Flow time (s)

III IV


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

V



006

008

010

012

014

016

018

020

022

Volu

me f

ract

ion

40 80 120 160 2000

Flow time (s)

IIIIII

IVV


012

015

018

021

024

027

030

033

036

Volu

me f

ract

ion

200 400 600 800 10000

Flow time (s)

II 15 volII 20 vol





Volumefraction

IIunevenness

IIIunevenness

10 5904 673315 4418 477620 2976 3424


Volumefraction


10 01852 1551 01857 155515 01988 1665 01997 167220 02122 1777 02139 1791



119872 = [1

119899

119899

sum

119894=1

(119862119894minus 119862

119862

)

2

]

05

(11)




5 Conclusion










Appendix

See Table 5

Nomenclature














Greek Letters






Subscripts




References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus02

00

02

04

06

08

10

12

14

16

18

20Ve

loci

ty m

agni

tude

(ms

)


yR

5 s10 s15 s

(a)

VxVyVz

0 2010 40 5030

Flow time (s)

minus12

minus10

minus08

minus06

minus04

minus02

0002040608101214

Velo

city

(ms

)(b)


minus005

000

005

010

015

020

025

030

035

Axi

al v

eloci

ty (m

s)

02 04 06 08 1000

ZH

IIIIII

IVV

(a)

minus01

00

01

02

03

04

05

06

07

Axi

al v

eloci

ty (m

s)

02 04 06 08 1000

ZH

IIIIII

IVV

(b)




119909 119881119910 119881119911) changed


119909 119881119910 and 119881





Y

Z

11

07

05

03

01

09

minus11

minus05

minus15

minus01

minus07

minus13

minus03

minus09

Axial velocity (ms)







velocity (119906119901= 119906 minus 119906










800 01947 16311000 04987 5219





I II III

IV V









B

A

h = 55mm

h = 25mm C

h = 25mm









006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

50 100 150 2000

Flow time (s)


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

I II


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

100 2000 50 150

Flow time (s)


006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

100 2000 50 150

Flow time (s)

III IV


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

V



006

008

010

012

014

016

018

020

022

Volu

me f

ract

ion

40 80 120 160 2000

Flow time (s)

IIIIII

IVV


012

015

018

021

024

027

030

033

036

Volu

me f

ract

ion

200 400 600 800 10000

Flow time (s)

II 15 volII 20 vol





Volumefraction

IIunevenness

IIIunevenness

10 5904 673315 4418 477620 2976 3424


Volumefraction


10 01852 1551 01857 155515 01988 1665 01997 167220 02122 1777 02139 1791



119872 = [1

119899

119899

sum

119894=1

(119862119894minus 119862

119862

)

2

]

05

(11)




5 Conclusion










Appendix

See Table 5

Nomenclature














Greek Letters






Subscripts




References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Y

Z

11

07

05

03

01

09

minus11

minus05

minus15

minus01

minus07

minus13

minus03

minus09

Axial velocity (ms)







velocity (119906119901= 119906 minus 119906










800 01947 16311000 04987 5219





I II III

IV V









B

A

h = 55mm

h = 25mm C

h = 25mm









006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

50 100 150 2000

Flow time (s)


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

I II


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

100 2000 50 150

Flow time (s)


006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

100 2000 50 150

Flow time (s)

III IV


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

V



006

008

010

012

014

016

018

020

022

Volu

me f

ract

ion

40 80 120 160 2000

Flow time (s)

IIIIII

IVV


012

015

018

021

024

027

030

033

036

Volu

me f

ract

ion

200 400 600 800 10000

Flow time (s)

II 15 volII 20 vol





Volumefraction

IIunevenness

IIIunevenness

10 5904 673315 4418 477620 2976 3424


Volumefraction


10 01852 1551 01857 155515 01988 1665 01997 167220 02122 1777 02139 1791



119872 = [1

119899

119899

sum

119894=1

(119862119894minus 119862

119862

)

2

]

05

(11)




5 Conclusion










Appendix

See Table 5

Nomenclature














Greek Letters






Subscripts




References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










I II III

IV V









B

A

h = 55mm

h = 25mm C

h = 25mm









006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

50 100 150 2000

Flow time (s)


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

I II


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

100 2000 50 150

Flow time (s)


006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

100 2000 50 150

Flow time (s)

III IV


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

V



006

008

010

012

014

016

018

020

022

Volu

me f

ract

ion

40 80 120 160 2000

Flow time (s)

IIIIII

IVV


012

015

018

021

024

027

030

033

036

Volu

me f

ract

ion

200 400 600 800 10000

Flow time (s)

II 15 volII 20 vol





Volumefraction

IIunevenness

IIIunevenness

10 5904 673315 4418 477620 2976 3424


Volumefraction


10 01852 1551 01857 155515 01988 1665 01997 167220 02122 1777 02139 1791



119872 = [1

119899

119899

sum

119894=1

(119862119894minus 119862

119862

)

2

]

05

(11)




5 Conclusion










Appendix

See Table 5

Nomenclature














Greek Letters






Subscripts




References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










B

A

h = 55mm

h = 25mm C

h = 25mm









006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

50 100 150 2000

Flow time (s)


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

I II


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

100 2000 50 150

Flow time (s)


006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

100 2000 50 150

Flow time (s)

III IV


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

V



006

008

010

012

014

016

018

020

022

Volu

me f

ract

ion

40 80 120 160 2000

Flow time (s)

IIIIII

IVV


012

015

018

021

024

027

030

033

036

Volu

me f

ract

ion

200 400 600 800 10000

Flow time (s)

II 15 volII 20 vol





Volumefraction

IIunevenness

IIIunevenness

10 5904 673315 4418 477620 2976 3424


Volumefraction


10 01852 1551 01857 155515 01988 1665 01997 167220 02122 1777 02139 1791



119872 = [1

119899

119899

sum

119894=1

(119862119894minus 119862

119862

)

2

]

05

(11)




5 Conclusion










Appendix

See Table 5

Nomenclature














Greek Letters






Subscripts




References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

50 100 150 2000

Flow time (s)


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

I II


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

100 2000 50 150

Flow time (s)


006

007

008

009

010

011

012

013

014Vo

lum

e fra

ctio

n

100 2000 50 150

Flow time (s)

III IV


006

007

008

009

010

011

012

013

014

Volu

me f

ract

ion

50 100 150 2000

Flow time (s)

V



006

008

010

012

014

016

018

020

022

Volu

me f

ract

ion

40 80 120 160 2000

Flow time (s)

IIIIII

IVV


012

015

018

021

024

027

030

033

036

Volu

me f

ract

ion

200 400 600 800 10000

Flow time (s)

II 15 volII 20 vol





Volumefraction

IIunevenness

IIIunevenness

10 5904 673315 4418 477620 2976 3424


Volumefraction


10 01852 1551 01857 155515 01988 1665 01997 167220 02122 1777 02139 1791



119872 = [1

119899

119899

sum

119894=1

(119862119894minus 119862

119862

)

2

]

05

(11)




5 Conclusion










Appendix

See Table 5

Nomenclature














Greek Letters






Subscripts




References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










006

008

010

012

014

016

018

020

022

Volu

me f

ract

ion

40 80 120 160 2000

Flow time (s)

IIIIII

IVV


012

015

018

021

024

027

030

033

036

Volu

me f

ract

ion

200 400 600 800 10000

Flow time (s)

II 15 volII 20 vol





Volumefraction

IIunevenness

IIIunevenness

10 5904 673315 4418 477620 2976 3424


Volumefraction


10 01852 1551 01857 155515 01988 1665 01997 167220 02122 1777 02139 1791



119872 = [1

119899

119899

sum

119894=1

(119862119894minus 119862

119862

)

2

]

05

(11)




5 Conclusion










Appendix

See Table 5

Nomenclature














Greek Letters






Subscripts




References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















Appendix

See Table 5

Nomenclature














Greek Letters






Subscripts




References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article numerical investigation of the effect of...

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