CFDAnalysisofPressureLossesandDepositionVelocitiesin...

Research ArticleCFD Analysis of Pressure Losses and Deposition Velocities inHorizontal Annuli

Rasel A Sultan 1 Mohammad Azizur Rahman2 Sayeed Rushd3 Sohrab Zendehboudi4

and Vassilios C Kelessidis5

1BID Group Technologies LTD Deltech Manufacturing Division Prince George BC Canada2Texas AampM University at Qatar Doha Qatar3King Faisal University Al Ahsa Saudi Arabia4Memorial University of Newfoundland St Johnrsquos NL Canada5Petroleum Institute Abu Dhabi UAE

Correspondence should be addressed to Rasel A Sultan ras380munca

Received 31 October 2018 Accepted 5 January 2019 Published 3 February 2019

Academic Editor Evangelos Tsotsas

Copyright copy 2019 Rasel A Sultan et al (is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Estimation of pressure losses and deposition velocities is vital in the hydraulic design of annular drill holes in the petroleumindustry (e present study investigates the effects of fluid velocity fluid type particle size particle concentration drill stringrotational speed and eccentricity on pressure losses and settling conditions using computational fluid dynamics (CFD)Eccentricity of the drill pipe is varied in the range of 0ndash75 and it rotates about its own axis at 0ndash150 rpm(e diameter ratio ofthe simulated drill hole is 056 Experimental data confirmed the validity of current CFD model developed using ANSYS162 platform

1 Introduction

In the petroleum industry predicting frictional pressurelosses and settling conditions for the transportation ofdrilling fluids in the annuli are important for drilling op-erations Inaccurate predictions can lead to a number ofcostly drilling problems A few examples of such problemsare loss of circulation kicks blockage wear abrasion andimproper rig power selection (e existing empirical modelsbecome less accurate as those involve many simplified as-sumptions CFD simulations help to minimize such as-sumptions by using the physics-based NavierndashStokesequations to model the hydrodynamics of the flow systemCurrent work is focused on developing a comprehensiveCFD model which is capable of considering the effects of allimportant drilling parameters such as fluid velocity fluidtype particle size particle concentration drill pipe rotationspeed and drill pipe eccentricity

2 Literature Review

(e estimation of pressure loss in an annulus is more dif-ficult compared with pipe flow due to the complexities inhydraulics resulting from the complex geometry [1 2] Froman empirical perspective the issue is usually addressed byreplacing pipe diameter in the pipe flow models with anldquoeffective diameterrdquo of annulus A number of definitions ofldquoeffective diameterrdquo have been proposed till date Howeverit is difficult to select a definition for a field application asthose were developed andor applied empirically A com-parison of multiple definitions in predicting pressure lossesis presented by Anifowoshe and Osisanya [3] Other issuesthat make the estimation of pressure losses in drilling holesdifficult are the eccentricity and the rotational speed of innerdrill pipe Many studies have been done on the flow ofnon-Newtonian fluids in annuli to introduce empiricalanalytical models which allow to take these effects into

HindawiInternational Journal of Chemical EngineeringVolume 2019 Article ID 7068989 17 pageshttpsdoiorg10115520197068989

account [1 3ndash10] (e results of the previous studies showthat the annular pressure losses for non-Newtonian (powerlaw) fluids flowing in a drill hole depend on drill pipe ro-tation speed fluid properties flow regimes (laminartransitionalturbulent) diameter ratio eccentricity andequivalent hydrodynamic roughness

Using commercially available CFD packages likeANSYS FLUENT to predict pressure losses for the annulartransportation of drilling fluids is comparatively a newapproach Sorgun and Ozbayoglu [9] demonstrated thebetter performance of CFD model compared with theexisting empirical models in predicting frictional pressurelosses Sorgun [8] investigated the effect of pipe eccentricityon pressure loss tangential velocity axial velocity andeffective viscosity by using CFD Erge et al [6] presented aCFD modeling approach which is applicable to estimatefrictional pressure losses in an eccentric annulus with innerpipe rotation while circulating yield power law fluidsHowever most of these CFD studies were limited to thelaminar flow of a single phase in hydro dynamically smoothannuli

Annular flow of drilling fluids containing cuttingsie slurry has not been studied in sufficient detail Examplesof works in the field of annular slurry flow are available inreferences [11ndash14] (e focus of these studies was to un-derstand the hydrodynamics of the slurry flow in annulusfrom real-time experiments and to produce empiricalmodels based on data analysis Recently different re-searchers [15ndash17] have used CFD in studying the trans-portation of slurry in annuli Ofei [15] examined the effect ofthe rheological parameters of the carrying fluids on thevelocity of solid Sorgun and Ulker [16] compared thepredictions of pressure losses obtained using artificial neuralnetwork (ANN) and CFD Both methods produced com-parable results Sun et al [17] studied the effects of in-clination rotational speed and flow rate on the distributionof solid concentration and the frictional pressure lossSimilar to the single-phase annular flow works most of theseCFD studies were limited to the laminar slurry flowconditions

3 Methods

31 CFD Simulation In the current work the CFD simu-lation model of the annular slurry flow is developed usingANSYS Fluent 162 platform Following previous works[18ndash20] a multi-fluid granular model is used to describe theflow behavior of a fluid-solid mixture(e granular version ofEulerianmodel is selected as themultiphasemodel (AppendixC)(is is because high solid volume fraction is expected to beused for this study and the granular version captures thehydrodynamics of high concentration slurries consisting ofvarying grain sizes It allowsmodeling ofmultiple separate butinteracting phases (e phases can be liquids gases or solidsin nearly any combination(e Eulerian treatment is used foreach phase in contrast to the EulerianndashLagrangian treatmentthat is used for the discrete phase model

(e description of multiphase flow as interpenetratingcontinua incorporates the concept of phasic volume fractions

which represent the space occupied by each phase Each phasesatisfies the laws of conservation of mass and momentumindividually (e conservation equations are modified byaveraging the local instantaneous balance for each of thephases [21] or by using the mixture theory approach [22] Adetailed description is available in Appendix A

For Eulerian multiphase calculations the phase-coupledSIMPLE (PC-SIMPLE) algorithm is used for the pressure-velocity coupling PC-SIMPLE is an extension of theSIMPLE algorithm to multiphase flows [23 24] (e ve-locities coupled by phases are solved in a segregated fashion(e block algebraic multigrid scheme used by the density-based solver is used to solve a vector equation formed by thevelocity components of all phases simultaneously [25](ena pressure correction equation is built based on the totalvolume continuity Pressure and velocities are then cor-rected so as to satisfy the continuity constraints

To ensure stability and convergence of iterative processa second-order upwind discretization was used for mo-mentum equation and first upwind discretization wasemployed for volume fraction turbulent kinetic energy andits dissipation Upwinding refers to the face value derivedfrom quantities in the cell upstream or ldquoupwindrdquo relative tothe direction of the normal velocity When first-order ac-curacy is desired quantities at cell faces are determined byassuming that the cell-center values of any field variablerepresent a cell-average value and hold throughout the entirecell the face quantities are identical to the cell quantities(us the face value is set equal to the cell-center value of theupstream cell when first-order upwinding is selected Incontrast when second-order accuracy or second-orderupwinding is desired quantities at cell faces are computedusing a multidimensional linear reconstruction approach[26] In this approach higher order accuracy is achieved atcell faces through a Taylor series expansion of the cell-centered solution about the cell centroid (e total simu-lation process is shown in Figure 1

32 Turbulence Model Selection Turbulent quantities forfluid flow are computed using Reynolds stress model [27ndash29] (Appendix D) Abandoning the isotropic eddy-viscosityhypothesis the RSM closes the Reynolds-averagedNavierndashStokes equations by solving transport equationsfor the Reynolds stresses together with an equation for thedissipation rate [30] Here five additional transport equa-tions are required in 2D flows and seven additional transportequations must be solved in 3D (please see Appendix B forfurther details) (e turbulence model was selected for thecurrent work analyzing the relative performance of differentmodels A typical example of the analysis is presented inFigure 2 In this figure difference refers to the percentiledifference between the experimental measurements ofpressure loss and the corresponding CFD predictions Inmost cases the difference was less than 10 when RSM wasused A list of the important results is presented in Table 1

33 Length Independence Study (e length of the flowdomain is considered long enough to achieve the fully

2 International Journal of Chemical Engineering

developed flow Minimum entrance length considered forthe flow development is 50Dh where hydraulic diameterDh ODndashID [36 37] An example of the length independenttest is shown in Figure 3 (e simulation results were notdependent on length after 3m from the inlet

34 Mesh Analysis (e computational grids for anannular section are generated using ANSYS Fluent andthe meshing is finalized on the basis of proper mesh

independency check Multiple layers of inflation near wallare added from both inner and outer walls to compute thecharacteristics of different parameters near wall moreprecisely Shear stress between wall surface and fluids ismuch higher and this inflation helps to create densermeshing near wall An example of computational griddistribution and mesh independence test is shown inFigures 4 and 5 Mesh independent results could beproduced for more than 800000 number of nodes All theresults presented in the current work were obtained usingaround 900000 nodes

(e values of dimensionless wall distance (y+) werechecked during near wall cells generation in consider-ation of the convergence requirement of y+ for cellsadjacent to the wall (e value of y+ depends on the wallshear stress fluid density hydraulic diameter and mo-lecular viscosity

y+

y

μρτw

radic (1)

where y is the distance from the wall to the cell center μ themolecular viscosity ρ the fluid density and τw the wallshear stress Eventually y+ depends on the mesh resolutionand the flow Reynolds number Default standard wall

Sketching geometry

Developing mesh (with proper mesh independence checking and adding inflation near wall) and defining boundaries (inlet outlet outer wall and inner

wall)

Activating gravity effect (assuming gravitational acceleration 981ms2

downward)

Selecting model for multiphase coupling (the Eulerian nongranular model for fluid-fluid coupling and the Eulerian granular model for fluid-solid

coupling) and model for turbulence closure (Reynolds stress model)

Selecting material for simulation (water and sand are selected with proper density and viscosity value required)

Defining boundary conditions (inlet fluid velocity dispersed-phase input volumetric concentration both wall roughness and inner wall rotation)

Selecting solution method (phase-coupled SIMPLE or PC SIMPLE method selected for multiphase flow stability and convergence) and defining convergence rate (10ndash5 is selected explanation is given in Figure 5)

Initialize the solution and run the calculation

Tria

l and

erro

r with

mes

h in

depe

nden

ce an

alys

is

Analyze the result

Figure 1 Simulation process diagram

0

10

20

RSM SST k-ω Standard k-ω

d

iffer

ence

Figure 2 Performance of turbulence models in predicting pressureloss (data source Kaushal et al 2005 [31])

International Journal of Chemical Engineering 3

functions are generally applicable if the first cell centeradjacent to the wall has a y+ value larger than 30 [38] In viewof the minimum requirement (y+gt 30) the value of y+ wasmaintained above 45 in our study

341 Convergence Rate Analysis An optimum convergencerate of 10minus5 was selected for the termination of iterationFigure 6 shows an example of the analysis used to find outthe optimum convergence rate within 10minus6ndash10minus4 (e sim-ulation results varied when the convergence value was

Table 1 Analysis of turbulence model performance

Reference Experimental conditions Difference

[100(experimentminussimulation)experiment]RSM SST k-ω Standard k-ω Standard k-ε

Kaushal et al(2005) [31]

Pipe orientation horizontal

8 15 17 1748

Pipe dia 00549mPipe material of construction stainless steel (density 8030 kgm3

smooth wall)Fluid single-phase water (density 9982 Kgm3 viscosity

0001003 kgm-s)Fluid velocity 2ms

Camccedili (2003) [32]

Annulus orientation horizontal and concentric

9 17 20 23Annulus dimensions 00432m ID 0123m OD

Material of construction aluminum (smooth wall)Fluid single-phase waterFluid velocity 02ms

Kelessidis et al(2011) [33]



Material of construction Plexiglas (smooth wall)Fluid single-phase waterFluid velocity 112ms

Skudarnov et al(2004) [34]


minus2 minus28 minus3 minus32

Pipe diameter 0023mPipe material of construction stainless steel (wall roughness 32 microm)

Fluid two-phase solid-liquid slurryLiquid water (density 9982 Kgm3 viscosity 0001003 kgm-s)

Solid glass spheres (double species with densities of 2490 kgm3 and4200 kgm3 50 by 50 volume mixture particle diameter (dm)

140 microm volumetric concentration (Cv) 15)Fluid velocity 1724ms

Ozbelge andBeyaz (2001) [35]

Annulus orientation vertical and concentric

minus6 14 15 minus151

Annulus dimensions 0125m OD 0025m IDMaterial of construction stainless steel (density 8030 kgm3 smooth

wall)Fluid two-phase solid-liquid slurry

Liquid water (density 9982 Kgm3 viscosity 0001003 kgm-s)Solid feldspar (mean particle diameter 023mm mean density

2500 kgm3 volumetric concentration Cv 180)Fluid velocity 0135ms



minus4 8 13 16





0

200

400

600

0 1 2 3 4 5 6

Pres

sure

gra

dien

t(P

am

)

Annulus length (m)

(A)

(B)

Figure 3 Length independence analysis (A) 75 eccentric 150RPM inner pipe rotation speed and 400 kgmin flow rate (B)concentric stationary inner pipe and 200 kgmin flow rate


greater than 10minus5 However the results did not change at allfor the values less than 10minus5

35 CFD Model Validation As the preliminary step theCFDmodeling approach is validated with respect to the dataavailable in open literature Few examples of the validationare presented as follows

36 Single-Phase Flow through Annuli Sets of experimentaldata from Kelessidis et al [33] and Camccedili [32] are comparedin Figure 7 with the proposed CFD model (e geometry istaken from Kelessidis et al (2011) where the inner diameter(ID) is 004m outer diameter (OD) is 007m and length is5m (horizontal concentric annuli) Fluid is taken as waterand wall material is Plexiglas (hydro dynamically smoothwall εa 0) In Camccedili (2003) the inner diameter is 00432m

(a)

(b)

(c)

(d)

Figure 4 Mesh distribution (a) Concentric inner pipe (number of nodes 901595) (b) 25 eccentric inner pipe (number of nodes 897540)(c) 50 eccentric inner pipe (number of nodes 909730) (d) 75 eccentric inner pipe (number of nodes 919432)


0

250

500

3 4 5 6 7 8 9 10 11

Pres

sure

gra

dien

t (P

am

)

Number of nodes times100000

Concentric and stationary inner pipe

Flow rate = 200kgminFlow rate = 300kgmin


Figure 5 Mesh independence analysis

0

50

100

150

200

250

300

350

100E ndash 06 100E ndash 05 100E ndash 04

Pres

sure

gra

dien

t (Pa

m)

Convergence rate



Figure 6 An example of optimum convergence rate analysis

1

10

100

1000

10000

001 01 1 10

Pres

sure

gra

dien

t (Pa

m)

Velocity (ms)

CFD results [32]CFD results [33]Data from Camccedili 2003 (00432m inner diameter and 0123m outer diameter)Data from Kelessidis et al 2011 (004m inner diameter and 007m outer diameter)

Figure 7 Comparison of simulated pressure gradient with experimental data fromKelessidis et al (2011) and Camccedili (2003) at different inletvelocities


outer diameter is 0123m and length is 5m Wall material isaluminum (smooth wall)

In Figure 7 the graph represents log-log scale (epressure gradient increasing rate in log-log scale is almostlinear for both cases (e average percentage errors ofsimulation results from Kelessidis et al (2011) and Camccedili(2003) are 988 and 846 respectively which indicates avery good agreement (it was estimated that the error of theexperimental data is approximately plusmn10)

37 Two-Phase (Solid-Liquid) Flow through AnnuliPressure gradient (Pam) profile of water-sand slurry flowthrough vertical concentric annuli is compared with Ozbelgeand Beyaz [35] experimental data in Figure 8 For the CFDsimulation the liquid phase is considered as water (density9982 kgm3 and viscosity 0001003 kgm-s) and the solidphase is taken as feldspar (mean particle diameter 023mmand mean density 2500 kgm3) Length 5m outer diameter0125m inner diameter 0025m inlet velocity range of00738ndash0197ms and overall slurry volumetric concentra-tion range of 10ndash18 with 023mm grain size (dp) areconsidered as boundary conditions A smooth pipe ofstainless steel (density 8030 kgm3) is used for the simulation(e pipe is assumed to be vertical ie gravity effect is in-cluded and gravity acceleration is directed opposite to theoutlet No slip condition for liquid and solid phases is used atthe walls Figure 9 shows the comparison of simulated andexperimental two-phase frictional pressure drop throughvertical annuli at different mixture velocity and at differentvolume concentration of slurry for 023mm mean sandparticle diameter (dp) Simulated results are in good agree-ment with experimental values with 262 average error

4 Results and Discussion

After achieving good validation of proposed model withexperimental data a parametric analysis is done to observethe effect of the changing flow rate drill pipe rotation ec-centricity and particle size (detailed data tables are pre-sented in Appendix (A and B)) (e parameters used for theanalysis are presented in Table 2

41 Effect of Flow Rate (e effect of fluid flow on maximumbed concentration is analyzed in Figures 10ndash13 Water withsand particle mixture (slurry) is used as operating fluid Fourdifferent conditions are taken into consideration with fixsand inlet concentration (20) and sand particle size(01mm) (e conditions are as follows

(i) Concentric annuli with stationary inner pipe(ii) Concentric annuli with 150 rpm rotating inner pipe(iii) 50 eccentric annuli with stationary inner pipe(iv) 50 eccentric annuli with 150 rpm inner pipe

From each case of the analysis it is observed that bedconcentration near the bottom wall decreases when the flowrate increases Because of gravitational force and horizontalorientation sand particles have the tendency to gather near

bottom wall and create a flow blockage Decreasing flowenhances the process From the analysis it is found thatbelow 9lowast 104 Reynolds number bed concentration nearbottom wall is more than 25 in all cases and when it goes

050

100150200250300350400

0 005 01 015 02 025

Pres

sure

gra

dien

t (Pa

m)

Mixture velocity (ms)

CFD results (volumeconcentration 18 vv)CFD results (volumeconcentration 10 vv)

Experiment (volumeconcentration 18 vv)Experiment (volumeconcentration 10 vv)

Figure 8 Comparison of simulated two-phase frictional pressuregradient at different mixture velocities and volume concentrationof slurry with Ozbelge and Beyaz 2001 (0125m outer diameter0025m inner diameter dm 023mm) [dm sand particle di-ameter and Cv sand volumetric concentrations]

0

100

200

300

400

500

600

0 50 100 150 200

Pres

sure

loss

(Pa

m)

Rotational speed (rpm)

Re = 50000

ecc = 0ecc = 25

ecc = 50ecc = 75

Figure 9 Effect of inner pipe eccentricity on pressure loss atdifferent rotational speeds

Table 2 Variable and fixed parameters

Parameters Valueranges UnitOuter diameter 01143 mInner diameter 00635 mLength 5 mWall roughness 0 mmSand concentration 5ndash20 Sand particle diameter 01ndash0005 mmInner pipe rotation 0ndash150 rpmInner pipe eccentricity 0ndash75 Water equivalent Reynolds number 4lowast 104ndash3lowast 105 Unit less


down 4lowast 104 the percentage is above 50 In Figure 14sample contour distribution of sand particles is shown atdifferent flow rates Operating conditions are taken fromFigure 10 Contour distribution of annuli cross section at3m distance from inlet displays the effect of fluid flow andgravitational force on concentration distribution clearly

42 Effect of Drill Pipe Rotation and Eccentricity Inner pipeof annuli can affect pressure loss by changing its eccentricityand rotation (e effect of inner pipe rotation and eccen-tricity is presented in Figures 9 and 15 Single-phase water isused as fluid in this analysis From Figure 15 it can bevisualized that pressure loss increases with the increase ofrotational speed and this trend is same at different flowrates Stationary 50 rpm 100 rpm and 150 rpm rotation ofinner pipe are taken into consideration Increasing trend is

higher at high flow rates Because of rotational speedparticlendashparticle and particlendashwall collision and reboundincrease which results in higher pressure loss

Figure 9 shows the effect of inner pipe eccentricity onpressure loss Concentric 25 eccentricity 50 eccentricityand 75 eccentricity are analyzed in this figure At a fixedflow rate (Re 50000) with the increase of inner pipeeccentricity pressure loss decreases (is trend is applicableduring inner pipe stationary condition With inner piperotation the trend is opposite Because of extra collisionadded by pipe rotation this change occurs

43 Effect of Particle Size In slurry flow sand particle sizehas an effective role on particle blockage near bottom wallof horizontal annular pipe Figure 16 analyzes the effect of

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03

YR

Sand concentration

50 eccentric and 150 rpm rotating inner pipe (20 inlet sand concentration and 01 mm sand

particle diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000

Figure 13 Effect of fluid flow on particle concentration profile(50 eccentric annuli with 150 rpm inner pipe)

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03 04

YR

Sand concentration

50 eccentric and stationary inner pipe (20 inlet sand concentration and 01 mm sand particle

diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000

Figure 12 Effect of fluid flow on particle concentration profile(50 eccentric annuli with stationary inner pipe)

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration

Concentric and 150rpm rotating inner pipe (20 inlet sand concentration and 01mm sand particle

diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000

Figure 11 Effect of fluid flow on particle concentration profile(concentric annuli with 150 rpm rotating inner pipe)

ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration

Concentric and stationary inner pipe (20 inlet sand concentration and 01mm sand particle

diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000

Figure 10 Effect of fluid flow on particle concentration profile(concentric annuli with stationary inner pipe)


Sand volume fractionContour 1

5703e ndash 001

5346e ndash 001

4990e ndash 001

4634e ndash 001

4277e ndash 001

3921e ndash 001

3564e ndash 001

3208e ndash 001

2851e ndash 001

2495e ndash 001

2139e ndash 001

1782e ndash 001

1426e ndash 001

1069e ndash 001

7129e ndash 002

3564e ndash 002

1044e ndash 0180 0035

00175 0053

0070 (m)Z X

Y

(a)


5102e ndash 001

4784e ndash 001

4465e ndash 001

4146e ndash 001

3827e ndash 001

3509e ndash 001

3190e ndash 001

2871e ndash 001

2553e ndash 001

2234e ndash 001

1915e ndash 001

1596e ndash 001

1278e ndash 001

9590e ndash 002

6403e ndash 002

3216e ndash 002

2839e ndash 0040 0035

00175 0053

0070 (m)Z X

Y

(b)

Figure 14 Continued


sand particle size at concentric and stationary inner pipewith 5 inlet sand concentration With the increase of sandparticle size particle deposition near bottom wall increasesbecause individual particle weight increases which

eventually results in this blockage From previous analysiswe have found maximum bed concentration decreases withthe increase in the flow rate (Figures 8 10ndash13) but forparticle size 0005mm maximum bed concentration near


4166e ndash 001

3908e ndash 001

3651e ndash 001

3394e ndash 001

3136e ndash 001

2879e ndash 001

2621e ndash 001

2364e ndash 001

2106e ndash 001

1849e ndash 001

1592e ndash 001

1334e ndash 001

1077e ndash 001

8195e ndash 002

5621e ndash 002

3047e ndash 002

4729e ndash 0030 0035

00175 0053

0070 (m)Z X

Y

(c)


2540e ndash 001

2397e ndash 001

2254e ndash 001

2112e ndash 001

1969e ndash 001

1826e ndash 001

1683e ndash 001

1540e ndash 001

1397e ndash 001

1254e ndash 001

1111e ndash 001

9679e ndash 002

8250e ndash 002

6820e ndash 002

5391e ndash 002

3961e ndash 002

2532e ndash 0020 0035

00175 0053

0070 (m)Z X

Y

(d)

Figure 14 Contour distribution of sand concentration profile (a) Re 4lowast 104 (b) Re 7lowast 104 (c) Re 105 and (d) Re 2lowast 105


bottom wall is almost constant at different flow rates (atmeans for smaller particle size (lt001mm) the effect of theflow rate on particle deposition in negligible

To be noted these mean particle sizes are selected fromparticle size distribution (PSD) charts considered duringCFD analyzes One of the PSD charts is shown in Figure 17with mean particle size 01mm where 01mm particle size isselected from eight different sizes based on cumulativeweight () of each particles To simplify the parametricanalysis process only mean particle sizes are shown

5 Conclusions

In an effort to develop a widely accepted CFD model ofmultiphase flow through drilling annuli the current workrelates the overall plan and the initial progress of the projectIn summary this study can be recounted as follows

(i) A CFD modeling methodology to predict frictionalpressure loss and settling conditions is validated(e validation is presented with examples whichdemonstrate its applicability in complex drillingconditions

(ii) (e effects of following important drilling param-eters on pressure loss and settling conditions aretested fluid flow rate rotational speed and ec-centricity of drill pipe and solid particle size

(iii) Fluid flow rate has the maximum impact on particledeposition In all analyzed conditions with thedecrease of fluid flow particle deposition near thebottom wall increases Specific deposition velocitydepends on specific requirement of maximum de-position near wall which can be calculated using ourapproach

(iv) With the increase of rotational speed and eccentricityof inner pipe the energy loss (pressure loss) of fluidflow increases However at stationary conditionpressure loss decreases with the increase of eccentricity

(v) Preliminary results of the current research programare presented with Newtonian fluid flow (eproject is expected to produce a comprehensiveCFD model capable of considering all importantdrilling parameters with Newtonian and non-Newtonian fluid flow Analysis with non-Newtonian fluid flow is ongoing

Appendix

A Parametric Study Chart with Single-Phase Fluid

(e results of the parametric study chart with the single-phase fluid are given in Table 3

B Parametric Study Chart with Two-Phase Fluid

(e results of the parametric study chart with the two-phasefluid are given in Table 4

0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000

Figure 15 Effect of inner pipe rotation on pressure loss

0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number

Concentric and stationary inner pipe(5 inlet sand concentration)

01mm sand particle size005mm sand particle size0005mm sand particle size

Figure 16 Effect of sand particle size on bed concentration near wall

0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)

Figure 17 Particle size distribution (PSD) of concentric andstationary inner pipe with 5 inlet sand concentration and 01mmaverage particle size


C Description of Multiphase Model

(e volume of phase q Vq is defined by

Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)

(e effective density of phase q is1113954ρq aqρq (A3)

where ρq is the physical density of phase q(e equations for fluid-fluid and granular multiphase

flows are presented here for the general case of an n -phaseflow (e volume fraction of each phase is calculated from acontinuity equation as below

Table 3

Controlled variables CFD resultsReynolds number(Re dHVρμ)

Rotationalspeed (rpm)

Eccentricity()

Pressuregradient (Pam)

20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()

Particle inputconcentration ()

Particlesize (mm)


Maximum bed concentrationnear bottom wall ()

40000 Stationary Concentric 5 005 766966 581270000 Stationary Concentric 5 005 848724 2209100000 Stationary Concentric 5 005 112029 1238200000 Stationary Concentric 5 005 312642 74940000 Stationary Concentric 20 005 767134 581470000 Stationary Concentric 20 005 848557 2204100000 Stationary Concentric 20 005 112029 1236200000 Stationary Concentric 20 005 311821 74740000 Stationary Concentric 5 01 767674 5870000 Stationary Concentric 5 01 8376 421100000 Stationary Concentric 5 01 111226 2591200000 Stationary Concentric 5 01 312031 101540000 Stationary Concentric 20 01 761902 527370000 Stationary Concentric 20 01 837166 4215100000 Stationary Concentric 20 01 111141 2602200000 Stationary Concentric 20 01 312955 101640000 Stationary Concentric 5 0005 759968 52770000 Stationary Concentric 5 0005 848862 514100000 Stationary Concentric 5 0005 112132 51200000 Stationary Concentric 5 0005 314308 50640000 Stationary Concentric 20 0005 759924 52770000 Stationary Concentric 20 0005 848871 514100000 Stationary Concentric 20 0005 11216 51200000 Stationary Concentric 20 0005 311951 50640000 150 Concentric 5 005 812232 577970000 150 Concentric 5 005 888334 2184100000 150 Concentric 5 005 116298 1233200000 150 Concentric 5 005 315197 74840000 150 Concentric 20 005 812265 577470000 150 Concentric 20 005 888189 2165100000 150 Concentric 20 005 116252 1232200000 150 Concentric 20 005 315146 74740000 150 Concentric 5 01 808946 536670000 150 Concentric 5 01 886414 4142100000 150 Concentric 5 01 115553 2581200000 150 Concentric 5 01 316058 101740000 150 Concentric 20 01 81085 514370000 150 Concentric 20 01 886051 4147100000 150 Concentric 20 01 115606 2585200000 150 Concentric 20 01 31498 101640000 150 Concentric 5 0005 799978 52770000 150 Concentric 5 0005 882585 514100000 150 Concentric 5 0005 116234 51200000 150 Concentric 5 0005 316604 50640000 150 Concentric 20 0005 799909 52770000 150 Concentric 20 0005 882953 514100000 150 Concentric 20 0005 116238 51200000 150 Concentric 20 0005 314669 50685000 Stationary 50 5 005 104212 3371100000 Stationary 50 5 005 119823 2603200000 Stationary 50 5 005 325559 996300000 Stationary 50 5 005 684814 77685000 Stationary 50 20 005 10556 3346100000 Stationary 50 20 005 120628 249200000 Stationary 50 20 005 327831 99300000 Stationary 50 20 005 669365 77885000 Stationary 50 5 01 101103 2639100000 Stationary 50 5 01 126345 2356200000 Stationary 50 5 01 317089 991300000 Stationary 50 5 01 647987 683


1ρrq

z

ztaqρq1113872 1113873 + nabla aqρq ϑq

⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)

where ρrq is the phase reference density or the volumeaveraged density of the qth phase in the solutiondomain _mpq characterizes the mass transfer from the pth toqth phase and _mqp characterizes the mass transfer from theqth to pth phase

(e conservation of momentum for a fluid phase q isz

ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875

minusaqnablap + nabla τ

q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶

is the acceleration due to gravity τ

q is the qth

phase stress-strain tensor Fq

⟶is an external body force

F⟶

liftq is a lift force and F⟶

vmq is a virtual mass force(e conservation of momentum for the solid phase is

z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875

minusasnablapminusnablaps + nabla τ

s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)

where ps is the sth solids pressure Kls Ksl is themomentum exchange coefficient between fluid or solidphase l and solid phase s and N is the total number ofphases

Table 4 Continued



Eccentricity()


Particlesize (mm)



85000 Stationary 50 20 01 101664 303100000 Stationary 50 20 01 117916 236200000 Stationary 50 20 01 318033 1062300000 Stationary 50 20 01 663622 7285000 Stationary 50 5 0005 103605 535100000 Stationary 50 5 0005 119255 53200000 Stationary 50 5 0005 318409 515300000 Stationary 50 5 0005 654409 5185000 Stationary 50 20 0005 103573 536100000 Stationary 50 20 0005 119577 53200000 Stationary 50 20 0005 318663 515300000 Stationary 50 20 0005 651947 5185000 150 50 5 005 109577 325100000 150 50 5 005 125955 2406200000 150 50 5 005 334223 1021300000 150 50 5 005 67944 77285000 150 50 20 005 109693 2937100000 150 50 20 005 127749 2262200000 150 50 20 005 335292 985300000 150 50 20 005 684901 75585000 150 50 5 01 107524 2528100000 150 50 5 01 122604 2457200000 150 50 5 01 319613 98300000 150 50 5 01 653321 69485000 150 50 20 01 107157 2525100000 150 50 20 01 124469 2286200000 150 50 20 01 32168 1019300000 150 50 20 01 665604 73285000 150 50 5 0005 109288 535100000 150 50 5 0005 125726 529200000 150 50 5 0005 324663 515300000 150 50 5 0005 656777 5185000 150 50 20 0005 109197 535100000 150 50 20 0005 126121 529200000 150 50 20 0005 324302 515300000 150 50 20 0005 656163 51


D Description of Turbulence Model

(e exact transport equation of Reynolds stress (Rij or uiprimeujprime)

is as follows

DRij

Dt Pij + Dij minus εij +emptyij + θij (A7)

where DRijDt is the summation of the changing rate of Rij

and transport of Rij by convection Pij is the production rateof Rij Dij is the diffusion transport of Rij εij is the rate ofdissipation emptyij is the pressure strain correlation term θij isthe rotation term

(e diffusion term used in simulation is as follows

DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)

where σk 082 μt Cμ(k2ϵ) andCμ 009Production rate Pij of Rij or ui

primeujprime can be expressed as

Pij minus uiprimeumprime

zUj

zxm

+ ujprimeumprime zUi

zxm

1113888 1113889 (A9)

(e pressure-strain term emptyij has emptyij1 or slow pressure-strain term also known as the return-to-isotropy term emptyij2or rapid pressure-strain term and emptyijw as wall-reflectionterm It can be presented as

emptyij emptyij1 + emptyij2 + emptyijw (A10)

whereemptyij1 minusC1

ϵk

uiprimeu primej minus

23δijk1113876 1113877 (A11)

emptyij2 minusC2 Pij minus23δijP1113876 1113877 (A12)

where C1 18 and C2 06(e wall-reflection term emptyijw is responsible for the

normal stresses distribution near the wall (is term ismodeled as

emptyijw equiv C1primeϵk

ukprimeumprime nknmδij minus

32

uiprimeukprimenjnk minus

32

ujprimeukprimenink1113874 1113875

Clk32

ϵd

+ C2prime emptykm2nknmδij minus32emptyik2njnk minus

32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)

where C1prime 05 C2prime 03 nk is the horizontal component ofthe component normal to the wall d is the shortest distanceto the wall andCl C34

μ k where Cμ 009 and k is thevon Karman constant (04187)

(e dissipation rate ϵij or the destruction rate of Rij ismodeled as

ϵij 23δijε (A14)

where

ϵ 2vsijprime sijprime (A15)

and sijprime fluctuating deformation rate

Rotation term is expressed as

θij minus2ωk Rjmeikm + Rimejkm1113872 1113873 (A16)

where ωk rotation vector eikm alternating symbol +1 minus1or 0 depending on i j and k

Nomenclature

CFD Computational fluid dynamicsRSM Reynolds stress modelSST Shear stress transportRANS Reynolds averaged NavierndashStokesSIMPLE Semi-implicit method for pressure linked

equationsdm Sand particle diameterCv Sand volumetric concentrationID Pipe inner diameterOD Pipe outer diameterDH Hydraulic diameterv Fluid velocityvw Water velocityvs Sand velocityε Wall roughnessRe Reynolds numberf Friction factor

Data Availability

(e ldquoANSYS Simulationrdquo data used to support the findingsof this study are available from the corresponding authorupon request

Disclosure

A conference paper [39] by the same authors was presentedwith the primary datasets which are updated andmodified inthis article

Conflicts of Interest

(e authors also declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(e authors are thankful to the Faculty of Engineering andApplied Science of Memorial University of NewfoundlandQatar National Research Fund (the grant NPRP10-0101-170091) Texas AampM University at Qatar and ANSYS Incfor helping and funding this research work

References

[1] A Pilehvari and R Serth ldquoGeneralized hydraulic calculationmethod for axial flow of non-Newtonian fluids in eccentricannulirdquo SPE Drilling amp Completion vol 24 no 4pp 553ndash563 2013

[2] T Reed and A Pilehvari ldquoA new model for laminar transi-tional and turbulent flow of drilling mudsrdquo in Proceedings of


SPE Production Operations Symposium Society of PetroleumEngineers Oklahoma OK USA 1993

[3] O L Anifowoshe and S O Osisanya ldquo(e effect of equivalentdiameter definitions on frictional pressure loss estimation inan annulus with pipe rotationrdquo in Proceedings of SPEDeepwater Drilling and Completions Conference Society ofPetroleum Engineers Galveston TX USA 2012

[4] R M Ahmed M S Enfis H M El Kheir M Laget andA Saasen ldquo(e effect of drillstring rotation on equivalentcirculation density modeling and analysis of field measure-mentsrdquo in Proceedings of SPE Annual Technical Conferenceand Exhibition Society of Petroleum Engineers FlorenceItaly 2010

[5] I T Dosunmu and S N Shah ldquoEvaluation of friction factorcorrelations and equivalent diameter definitions for pipe andannular flow of non-Newtonian fluidsrdquo Journal of PetroleumScience and Engineering vol 109 pp 80ndash86 2013

[6] O Erge E M Ozbayoglu S Z Miska et al ldquoEquivalentcirculating density modeling of Yield Power Law fluids val-idated with CFD approachrdquo Journal of Petroleum Science andEngineering vol 140 pp 16ndash27 2016

[7] S A Hansen and N Sterri ldquoDrill pipe rotation effects onfrictional pressure losses in slim annulirdquo in Proceedings of SPEAnnual Technical Conference and Exhibition Society of Pe-troleum Engineers Dallas TX USA October 1995

[8] M Sorgun ldquoComputational fluid dynamics modeling of pipeeccentricity effect on flow characteristics of Newtonian andnon-Newtonian fluidsrdquo Energy Sources Part A RecoveryUtilization and Environmental Effects vol 33 no 12pp 1196ndash1208 2011

[9] M Sorgun and M Ozbayoglu ldquoPredicting frictionalpressure loss during horizontal drilling for non-Newtonian fluidsrdquo Energy Sources Part A-Recovery Uti-lization and Environmental Effects vol 33 no 11pp 1104ndash1106 2011

[10] R Subramanian and J Azar ldquoExperimental study on frictionpressure drop for nonnewtonian drilling fluids in pipe andannular flowrdquo in Proceedings of International Oil and GasConference and Exhibition in China Society of PetroleumEngineers Beijing China 2000

[11] G Camccedili and T A Ozbelge ldquoDetermination of dilute slurrydensities in a vertical annulus using isokinetic samplingrdquoChemical Engineering Communications vol 193 no 11pp 1482ndash1501 2007

[12] S K Dewangan and S L Sinha ldquoExploring the hole cleaningparameters of horizontal wellbore using two-phase EulerianCFD approachrdquo Journal of Computational Multiphase Flowsvol 8 no 1 pp 15ndash39 2016

[13] T A Ozbelge and A N Eraslan ldquoA computational hydro-dynamic and heat transfer study in turbulent up-flows ofdilute slurries through a concentric annulusrdquo Turkish Journalof Engineering and Environmental Sciences vol 30 no 1pp 1ndash13 2006

[14] T A Ozbelge and G C Unal ldquoA new correlation for two-phase pressure drops in fully developed dilute slurry up-flowsthrough an annulusrdquo Chemical Engineering Communicationsvol 196 no 4 pp 491ndash498 2008

[15] T N Ofei ldquoEffect of yield power law fluid rheologicalproperties on cuttings transport in eccentric horizontalnarrow annulusrdquo Journal of Fluids pp 1ndash10 2016

[16] M Sorgun and E Ulker ldquoModeling and experimental study ofsolid-liquid two-phase pressure drop in horizontal wellboreswith pipe rotationrdquo Journal of Energy Resources Technology-Transactions of the ASME vol 138 no 2 2016

[17] X Sun K Wang T Yan S Shao and J Jiao ldquoEffect ofdrillpipe rotation on cuttings transport using computationalfluid dynamics (CFD) in complex structure wellsrdquo Journal ofPetroleum Exploration and Production Technology vol 4no 3 pp 255ndash261 2014

[18] B J Alder and T E Wainwright ldquoStudies in moleculardynamics II Behavior of a small number of elastic spheresrdquoJournal of Chemical Physics vol 33 no 5 pp 1439ndash14511960

[19] S Chapman and T G Cowling Ie Mathematical Ieory ofNon-Uniform Gases An Account of the Kinetic Ieory ofViscosity Iermal Conduction and Diffusion in GasesCambridge University Press Cambridge UK 1970

[20] M Syamlal W Rogers and T J OrsquoBrien MFIX documen-tation Ieory guide National Energy Technology LaboratoryDepartment of Energy Technical Note DOEMETC-951013and NTISDE95000031 1993

[21] T B Anderson and R Jackson ldquoFluid mechanical descriptionof fluidized beds Equations of motionrdquo Industrial amp Engi-neering Chemistry Fundamentals vol 6 no 4 pp 527ndash5391967

[22] R M Bowen ldquo(eory of mixturesrdquo Continuum Physicsvol 3 part I 1976

[23] S Patankar Numerical Heat Transfer and Fluid Flow CRCPress Boca Raton FL USA 1980

[24] S Vasquez and V Ivanov ldquoA phase coupled method forsolving multiphase problems on unstructured meshesrdquo inProceedings of ASME 2000 Fluids Engineering DivisionSummer Meeting (ASME FEDSMrsquo00) paper FEDSM2000-11281 pp 1ndash6 Boston MA USA 2000

[25] J M Weiss J P Maruszewski and W A Smith ldquoImplicitsolution of preconditioned Navier-Stokes equations usingalgebraic multigridrdquo AIAA Journal vol 37 no 1 pp 29ndash361999

[26] T Barth and D Jespersen ldquo(e design and application ofupwind schemes on unstructured meshesrdquo in Proceedings of27th Aerospace Sciences Meeting American Institute ofAeronautics and Astronautics Reno NV USA 1989

[27] M M Gibson and B E Launder ldquoGround effects on pressurefluctuations in the atmospheric boundary layerrdquo Journal ofFluid Mechanics vol 86 no 3 pp 491ndash511 2006

[28] B E Launder G J Reece and W Rodi ldquoProgress in thedevelopment of a Reynolds-stress turbulence closurerdquo Journalof Fluid Mechanics vol 68 no 03 pp 537ndash566 2006

[29] B E Launder ldquoSecond-moment closure and its use in mod-elling turbulent industrial flowsrdquo International Journal forNumerical Methods in Fluids vol 9 no 8 pp 963ndash985 1989

[30] A J Chorin ldquoNumerical solution of the Navier-Stokesequationsrdquo Mathematics of Computation vol 22 no 104745 pages 1968

[31] D R Kaushal K Sato T Toyota K Funatsu and Y TomitaldquoEffect of particle size distribution on pressure drop andconcentration profile in pipeline flow of highly concentratedslurryrdquo International Journal of Multiphase Flow vol 31no 7 pp 809ndash823 2005

[32] G CamccedilıApplication of isokinetic sampling technique for localsolid densities in upward liquid-solid flows through an annulusMETU PhD dissertation 2003

[33] V C Kelessidis P Dalamarinis and R Maglione ldquoExperi-mental study and predictions of pressure losses of fluidsmodeled as Herschel-Bulkley in concentric and eccentricannuli in laminar transitional and turbulent flowsrdquo Journal ofPetroleum Science and Engineering vol 77 no 3 pp 305ndash3122011


[34] P V Skudarnov C X Lin and M A Ebadian ldquoDouble-species slurry flow in a horizontal pipelinerdquo Journal of FluidsEngineering vol 126 no 1 pp 125ndash132 2004

[35] T A Ozbelge and A Beyaz ldquoDilute solid-liquid upward flowsthrough a vertical annulus in a closed loop systemrdquo In-ternational Journal of Multiphase Flow vol 27 no 4pp 737ndash752 2001

[36] N P Brown and N I Heywood Slurry Handling Design ofSolid-Liquid Systems Springer Science amp Business MediaBerlin Germany 1991

[37] E J Wasp J P Kenny and R L Gandhi ldquoSolid--liquid flowslurry pipeline transportation [Pumps valves mechanicalequipment economics]rdquo Bulk Material Handling vol 1no 4 1977

[38] ANSYS 120121 Documentation Users Guide ManualAnsys Inc Canonsburg PA USA 2009

[39] S Rushd R A Sultan A Rahman and V Kelessidis ldquoIn-vestigation of pressure losses in eccentric inclined annulirdquo inProceedings of ASME 2017 36th International Conference onOcean Offshore and Arctic Engineering American Society ofMechanical Engineers Trondheim Norway 2017


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Submit your manuscripts atwwwhindawicom

account [1 3ndash10] (e results of the previous studies showthat the annular pressure losses for non-Newtonian (powerlaw) fluids flowing in a drill hole depend on drill pipe ro-tation speed fluid properties flow regimes (laminartransitionalturbulent) diameter ratio eccentricity andequivalent hydrodynamic roughness

Using commercially available CFD packages likeANSYS FLUENT to predict pressure losses for the annulartransportation of drilling fluids is comparatively a newapproach Sorgun and Ozbayoglu [9] demonstrated thebetter performance of CFD model compared with theexisting empirical models in predicting frictional pressurelosses Sorgun [8] investigated the effect of pipe eccentricityon pressure loss tangential velocity axial velocity andeffective viscosity by using CFD Erge et al [6] presented aCFD modeling approach which is applicable to estimatefrictional pressure losses in an eccentric annulus with innerpipe rotation while circulating yield power law fluidsHowever most of these CFD studies were limited to thelaminar flow of a single phase in hydro dynamically smoothannuli

Annular flow of drilling fluids containing cuttingsie slurry has not been studied in sufficient detail Examplesof works in the field of annular slurry flow are available inreferences [11ndash14] (e focus of these studies was to un-derstand the hydrodynamics of the slurry flow in annulusfrom real-time experiments and to produce empiricalmodels based on data analysis Recently different re-searchers [15ndash17] have used CFD in studying the trans-portation of slurry in annuli Ofei [15] examined the effect ofthe rheological parameters of the carrying fluids on thevelocity of solid Sorgun and Ulker [16] compared thepredictions of pressure losses obtained using artificial neuralnetwork (ANN) and CFD Both methods produced com-parable results Sun et al [17] studied the effects of in-clination rotational speed and flow rate on the distributionof solid concentration and the frictional pressure lossSimilar to the single-phase annular flow works most of theseCFD studies were limited to the laminar slurry flowconditions

3 Methods

31 CFD Simulation In the current work the CFD simu-lation model of the annular slurry flow is developed usingANSYS Fluent 162 platform Following previous works[18ndash20] a multi-fluid granular model is used to describe theflow behavior of a fluid-solid mixture(e granular version ofEulerianmodel is selected as themultiphasemodel (AppendixC)(is is because high solid volume fraction is expected to beused for this study and the granular version captures thehydrodynamics of high concentration slurries consisting ofvarying grain sizes It allowsmodeling ofmultiple separate butinteracting phases (e phases can be liquids gases or solidsin nearly any combination(e Eulerian treatment is used foreach phase in contrast to the EulerianndashLagrangian treatmentthat is used for the discrete phase model

(e description of multiphase flow as interpenetratingcontinua incorporates the concept of phasic volume fractions

which represent the space occupied by each phase Each phasesatisfies the laws of conservation of mass and momentumindividually (e conservation equations are modified byaveraging the local instantaneous balance for each of thephases [21] or by using the mixture theory approach [22] Adetailed description is available in Appendix A

For Eulerian multiphase calculations the phase-coupledSIMPLE (PC-SIMPLE) algorithm is used for the pressure-velocity coupling PC-SIMPLE is an extension of theSIMPLE algorithm to multiphase flows [23 24] (e ve-locities coupled by phases are solved in a segregated fashion(e block algebraic multigrid scheme used by the density-based solver is used to solve a vector equation formed by thevelocity components of all phases simultaneously [25](ena pressure correction equation is built based on the totalvolume continuity Pressure and velocities are then cor-rected so as to satisfy the continuity constraints

To ensure stability and convergence of iterative processa second-order upwind discretization was used for mo-mentum equation and first upwind discretization wasemployed for volume fraction turbulent kinetic energy andits dissipation Upwinding refers to the face value derivedfrom quantities in the cell upstream or ldquoupwindrdquo relative tothe direction of the normal velocity When first-order ac-curacy is desired quantities at cell faces are determined byassuming that the cell-center values of any field variablerepresent a cell-average value and hold throughout the entirecell the face quantities are identical to the cell quantities(us the face value is set equal to the cell-center value of theupstream cell when first-order upwinding is selected Incontrast when second-order accuracy or second-orderupwinding is desired quantities at cell faces are computedusing a multidimensional linear reconstruction approach[26] In this approach higher order accuracy is achieved atcell faces through a Taylor series expansion of the cell-centered solution about the cell centroid (e total simu-lation process is shown in Figure 1

32 Turbulence Model Selection Turbulent quantities forfluid flow are computed using Reynolds stress model [27ndash29] (Appendix D) Abandoning the isotropic eddy-viscosityhypothesis the RSM closes the Reynolds-averagedNavierndashStokes equations by solving transport equationsfor the Reynolds stresses together with an equation for thedissipation rate [30] Here five additional transport equa-tions are required in 2D flows and seven additional transportequations must be solved in 3D (please see Appendix B forfurther details) (e turbulence model was selected for thecurrent work analyzing the relative performance of differentmodels A typical example of the analysis is presented inFigure 2 In this figure difference refers to the percentiledifference between the experimental measurements ofpressure loss and the corresponding CFD predictions Inmost cases the difference was less than 10 when RSM wasused A list of the important results is presented in Table 1

33 Length Independence Study (e length of the flowdomain is considered long enough to achieve the fully






y+

y

μρτw

radic (1)


Sketching geometry


wall)


downward)







Tria

l and

erro

r with

mes

h in

depe

nden

ce an

alys

is

Analyze the result


0

10

20


d

iffer

ence










8 15 17 1748




























minus4 8 13 16





0

200

400

600

0 1 2 3 4 5 6

Pres

sure

gra

dien

t(P

am

)

Annulus length (m)

(A)

(B)






(a)

(b)

(c)

(d)



0

250

500

3 4 5 6 7 8 9 10 11

Pres

sure

gra

dien

t (P

am

)






0

50

100

150

200

250

300

350


Pres

sure

gra

dien

t (Pa

m)

Convergence rate




1

10

100

1000

10000

001 01 1 10

Pres

sure

gra

dien

t (Pa

m)

Velocity (ms)













050

100150200250300350400

0 005 01 015 02 025

Pres

sure

gra

dien

t (Pa

m)





0

100

200

300

400

500

600

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 50000

ecc = 0ecc = 25

ecc = 50ecc = 75










ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03

YR

Sand concentration


particle diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03 04

YR

Sand concentration


diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000




5703e ndash 001

5346e ndash 001

4990e ndash 001

4634e ndash 001

4277e ndash 001

3921e ndash 001

3564e ndash 001

3208e ndash 001

2851e ndash 001

2495e ndash 001

2139e ndash 001

1782e ndash 001

1426e ndash 001

1069e ndash 001

7129e ndash 002

3564e ndash 002

1044e ndash 0180 0035

00175 0053

0070 (m)Z X

Y

(a)


5102e ndash 001

4784e ndash 001

4465e ndash 001

4146e ndash 001

3827e ndash 001

3509e ndash 001

3190e ndash 001

2871e ndash 001

2553e ndash 001

2234e ndash 001

1915e ndash 001

1596e ndash 001

1278e ndash 001

9590e ndash 002

6403e ndash 002

3216e ndash 002

2839e ndash 0040 0035

00175 0053

0070 (m)Z X

Y

(b)

Figure 14 Continued





4166e ndash 001

3908e ndash 001

3651e ndash 001

3394e ndash 001

3136e ndash 001

2879e ndash 001

2621e ndash 001

2364e ndash 001

2106e ndash 001

1849e ndash 001

1592e ndash 001

1334e ndash 001

1077e ndash 001

8195e ndash 002

5621e ndash 002

3047e ndash 002

4729e ndash 0030 0035

00175 0053

0070 (m)Z X

Y

(c)


2540e ndash 001

2397e ndash 001

2254e ndash 001

2112e ndash 001

1969e ndash 001

1826e ndash 001

1683e ndash 001

1540e ndash 001

1397e ndash 001

1254e ndash 001

1111e ndash 001

9679e ndash 002

8250e ndash 002

6820e ndash 002

5391e ndash 002

3961e ndash 002

2532e ndash 0020 0035

00175 0053

0070 (m)Z X

Y

(d)





5 Conclusions







Appendix





0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000


0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number




0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)





Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






y+

y

μρτw

radic (1)


Sketching geometry


wall)


downward)







Tria

l and

erro

r with

mes

h in

depe

nden

ce an

alys

is

Analyze the result


0

10

20


d

iffer

ence










8 15 17 1748




























minus4 8 13 16





0

200

400

600

0 1 2 3 4 5 6

Pres

sure

gra

dien

t(P

am

)

Annulus length (m)

(A)

(B)






(a)

(b)

(c)

(d)



0

250

500

3 4 5 6 7 8 9 10 11

Pres

sure

gra

dien

t (P

am

)






0

50

100

150

200

250

300

350


Pres

sure

gra

dien

t (Pa

m)

Convergence rate




1

10

100

1000

10000

001 01 1 10

Pres

sure

gra

dien

t (Pa

m)

Velocity (ms)













050

100150200250300350400

0 005 01 015 02 025

Pres

sure

gra

dien

t (Pa

m)





0

100

200

300

400

500

600

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 50000

ecc = 0ecc = 25

ecc = 50ecc = 75










ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03

YR

Sand concentration


particle diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03 04

YR

Sand concentration


diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000




5703e ndash 001

5346e ndash 001

4990e ndash 001

4634e ndash 001

4277e ndash 001

3921e ndash 001

3564e ndash 001

3208e ndash 001

2851e ndash 001

2495e ndash 001

2139e ndash 001

1782e ndash 001

1426e ndash 001

1069e ndash 001

7129e ndash 002

3564e ndash 002

1044e ndash 0180 0035

00175 0053

0070 (m)Z X

Y

(a)


5102e ndash 001

4784e ndash 001

4465e ndash 001

4146e ndash 001

3827e ndash 001

3509e ndash 001

3190e ndash 001

2871e ndash 001

2553e ndash 001

2234e ndash 001

1915e ndash 001

1596e ndash 001

1278e ndash 001

9590e ndash 002

6403e ndash 002

3216e ndash 002

2839e ndash 0040 0035

00175 0053

0070 (m)Z X

Y

(b)

Figure 14 Continued





4166e ndash 001

3908e ndash 001

3651e ndash 001

3394e ndash 001

3136e ndash 001

2879e ndash 001

2621e ndash 001

2364e ndash 001

2106e ndash 001

1849e ndash 001

1592e ndash 001

1334e ndash 001

1077e ndash 001

8195e ndash 002

5621e ndash 002

3047e ndash 002

4729e ndash 0030 0035

00175 0053

0070 (m)Z X

Y

(c)


2540e ndash 001

2397e ndash 001

2254e ndash 001

2112e ndash 001

1969e ndash 001

1826e ndash 001

1683e ndash 001

1540e ndash 001

1397e ndash 001

1254e ndash 001

1111e ndash 001

9679e ndash 002

8250e ndash 002

6820e ndash 002

5391e ndash 002

3961e ndash 002

2532e ndash 0020 0035

00175 0053

0070 (m)Z X

Y

(d)





5 Conclusions







Appendix





0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000


0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number




0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)





Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









8 15 17 1748




























minus4 8 13 16





0

200

400

600

0 1 2 3 4 5 6

Pres

sure

gra

dien

t(P

am

)

Annulus length (m)

(A)

(B)






(a)

(b)

(c)

(d)



0

250

500

3 4 5 6 7 8 9 10 11

Pres

sure

gra

dien

t (P

am

)






0

50

100

150

200

250

300

350


Pres

sure

gra

dien

t (Pa

m)

Convergence rate




1

10

100

1000

10000

001 01 1 10

Pres

sure

gra

dien

t (Pa

m)

Velocity (ms)













050

100150200250300350400

0 005 01 015 02 025

Pres

sure

gra

dien

t (Pa

m)





0

100

200

300

400

500

600

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 50000

ecc = 0ecc = 25

ecc = 50ecc = 75










ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03

YR

Sand concentration


particle diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03 04

YR

Sand concentration


diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000




5703e ndash 001

5346e ndash 001

4990e ndash 001

4634e ndash 001

4277e ndash 001

3921e ndash 001

3564e ndash 001

3208e ndash 001

2851e ndash 001

2495e ndash 001

2139e ndash 001

1782e ndash 001

1426e ndash 001

1069e ndash 001

7129e ndash 002

3564e ndash 002

1044e ndash 0180 0035

00175 0053

0070 (m)Z X

Y

(a)


5102e ndash 001

4784e ndash 001

4465e ndash 001

4146e ndash 001

3827e ndash 001

3509e ndash 001

3190e ndash 001

2871e ndash 001

2553e ndash 001

2234e ndash 001

1915e ndash 001

1596e ndash 001

1278e ndash 001

9590e ndash 002

6403e ndash 002

3216e ndash 002

2839e ndash 0040 0035

00175 0053

0070 (m)Z X

Y

(b)

Figure 14 Continued





4166e ndash 001

3908e ndash 001

3651e ndash 001

3394e ndash 001

3136e ndash 001

2879e ndash 001

2621e ndash 001

2364e ndash 001

2106e ndash 001

1849e ndash 001

1592e ndash 001

1334e ndash 001

1077e ndash 001

8195e ndash 002

5621e ndash 002

3047e ndash 002

4729e ndash 0030 0035

00175 0053

0070 (m)Z X

Y

(c)


2540e ndash 001

2397e ndash 001

2254e ndash 001

2112e ndash 001

1969e ndash 001

1826e ndash 001

1683e ndash 001

1540e ndash 001

1397e ndash 001

1254e ndash 001

1111e ndash 001

9679e ndash 002

8250e ndash 002

6820e ndash 002

5391e ndash 002

3961e ndash 002

2532e ndash 0020 0035

00175 0053

0070 (m)Z X

Y

(d)





5 Conclusions







Appendix





0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000


0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number




0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)





Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(a)

(b)

(c)

(d)



0

250

500

3 4 5 6 7 8 9 10 11

Pres

sure

gra

dien

t (P

am

)






0

50

100

150

200

250

300

350


Pres

sure

gra

dien

t (Pa

m)

Convergence rate




1

10

100

1000

10000

001 01 1 10

Pres

sure

gra

dien

t (Pa

m)

Velocity (ms)













050

100150200250300350400

0 005 01 015 02 025

Pres

sure

gra

dien

t (Pa

m)





0

100

200

300

400

500

600

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 50000

ecc = 0ecc = 25

ecc = 50ecc = 75










ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03

YR

Sand concentration


particle diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03 04

YR

Sand concentration


diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000




5703e ndash 001

5346e ndash 001

4990e ndash 001

4634e ndash 001

4277e ndash 001

3921e ndash 001

3564e ndash 001

3208e ndash 001

2851e ndash 001

2495e ndash 001

2139e ndash 001

1782e ndash 001

1426e ndash 001

1069e ndash 001

7129e ndash 002

3564e ndash 002

1044e ndash 0180 0035

00175 0053

0070 (m)Z X

Y

(a)


5102e ndash 001

4784e ndash 001

4465e ndash 001

4146e ndash 001

3827e ndash 001

3509e ndash 001

3190e ndash 001

2871e ndash 001

2553e ndash 001

2234e ndash 001

1915e ndash 001

1596e ndash 001

1278e ndash 001

9590e ndash 002

6403e ndash 002

3216e ndash 002

2839e ndash 0040 0035

00175 0053

0070 (m)Z X

Y

(b)

Figure 14 Continued





4166e ndash 001

3908e ndash 001

3651e ndash 001

3394e ndash 001

3136e ndash 001

2879e ndash 001

2621e ndash 001

2364e ndash 001

2106e ndash 001

1849e ndash 001

1592e ndash 001

1334e ndash 001

1077e ndash 001

8195e ndash 002

5621e ndash 002

3047e ndash 002

4729e ndash 0030 0035

00175 0053

0070 (m)Z X

Y

(c)


2540e ndash 001

2397e ndash 001

2254e ndash 001

2112e ndash 001

1969e ndash 001

1826e ndash 001

1683e ndash 001

1540e ndash 001

1397e ndash 001

1254e ndash 001

1111e ndash 001

9679e ndash 002

8250e ndash 002

6820e ndash 002

5391e ndash 002

3961e ndash 002

2532e ndash 0020 0035

00175 0053

0070 (m)Z X

Y

(d)





5 Conclusions







Appendix





0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000


0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number




0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)





Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


0

250

500

3 4 5 6 7 8 9 10 11

Pres

sure

gra

dien

t (P

am

)






0

50

100

150

200

250

300

350


Pres

sure

gra

dien

t (Pa

m)

Convergence rate




1

10

100

1000

10000

001 01 1 10

Pres

sure

gra

dien

t (Pa

m)

Velocity (ms)













050

100150200250300350400

0 005 01 015 02 025

Pres

sure

gra

dien

t (Pa

m)





0

100

200

300

400

500

600

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 50000

ecc = 0ecc = 25

ecc = 50ecc = 75










ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03

YR

Sand concentration


particle diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03 04

YR

Sand concentration


diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000




5703e ndash 001

5346e ndash 001

4990e ndash 001

4634e ndash 001

4277e ndash 001

3921e ndash 001

3564e ndash 001

3208e ndash 001

2851e ndash 001

2495e ndash 001

2139e ndash 001

1782e ndash 001

1426e ndash 001

1069e ndash 001

7129e ndash 002

3564e ndash 002

1044e ndash 0180 0035

00175 0053

0070 (m)Z X

Y

(a)


5102e ndash 001

4784e ndash 001

4465e ndash 001

4146e ndash 001

3827e ndash 001

3509e ndash 001

3190e ndash 001

2871e ndash 001

2553e ndash 001

2234e ndash 001

1915e ndash 001

1596e ndash 001

1278e ndash 001

9590e ndash 002

6403e ndash 002

3216e ndash 002

2839e ndash 0040 0035

00175 0053

0070 (m)Z X

Y

(b)

Figure 14 Continued





4166e ndash 001

3908e ndash 001

3651e ndash 001

3394e ndash 001

3136e ndash 001

2879e ndash 001

2621e ndash 001

2364e ndash 001

2106e ndash 001

1849e ndash 001

1592e ndash 001

1334e ndash 001

1077e ndash 001

8195e ndash 002

5621e ndash 002

3047e ndash 002

4729e ndash 0030 0035

00175 0053

0070 (m)Z X

Y

(c)


2540e ndash 001

2397e ndash 001

2254e ndash 001

2112e ndash 001

1969e ndash 001

1826e ndash 001

1683e ndash 001

1540e ndash 001

1397e ndash 001

1254e ndash 001

1111e ndash 001

9679e ndash 002

8250e ndash 002

6820e ndash 002

5391e ndash 002

3961e ndash 002

2532e ndash 0020 0035

00175 0053

0070 (m)Z X

Y

(d)





5 Conclusions







Appendix





0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000


0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number




0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)





Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











050

100150200250300350400

0 005 01 015 02 025

Pres

sure

gra

dien

t (Pa

m)





0

100

200

300

400

500

600

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 50000

ecc = 0ecc = 25

ecc = 50ecc = 75










ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03

YR

Sand concentration


particle diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03 04

YR

Sand concentration


diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000




5703e ndash 001

5346e ndash 001

4990e ndash 001

4634e ndash 001

4277e ndash 001

3921e ndash 001

3564e ndash 001

3208e ndash 001

2851e ndash 001

2495e ndash 001

2139e ndash 001

1782e ndash 001

1426e ndash 001

1069e ndash 001

7129e ndash 002

3564e ndash 002

1044e ndash 0180 0035

00175 0053

0070 (m)Z X

Y

(a)


5102e ndash 001

4784e ndash 001

4465e ndash 001

4146e ndash 001

3827e ndash 001

3509e ndash 001

3190e ndash 001

2871e ndash 001

2553e ndash 001

2234e ndash 001

1915e ndash 001

1596e ndash 001

1278e ndash 001

9590e ndash 002

6403e ndash 002

3216e ndash 002

2839e ndash 0040 0035

00175 0053

0070 (m)Z X

Y

(b)

Figure 14 Continued





4166e ndash 001

3908e ndash 001

3651e ndash 001

3394e ndash 001

3136e ndash 001

2879e ndash 001

2621e ndash 001

2364e ndash 001

2106e ndash 001

1849e ndash 001

1592e ndash 001

1334e ndash 001

1077e ndash 001

8195e ndash 002

5621e ndash 002

3047e ndash 002

4729e ndash 0030 0035

00175 0053

0070 (m)Z X

Y

(c)


2540e ndash 001

2397e ndash 001

2254e ndash 001

2112e ndash 001

1969e ndash 001

1826e ndash 001

1683e ndash 001

1540e ndash 001

1397e ndash 001

1254e ndash 001

1111e ndash 001

9679e ndash 002

8250e ndash 002

6820e ndash 002

5391e ndash 002

3961e ndash 002

2532e ndash 0020 0035

00175 0053

0070 (m)Z X

Y

(d)





5 Conclusions







Appendix





0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000


0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number




0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)





Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03

YR

Sand concentration


particle diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 01 02 03 04

YR

Sand concentration


diameter)

Re = 85000Re = 100000

Re = 200000Re = 300000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000


ndash12

ndash1

ndash08

ndash06

ndash04

ndash02

0

0 02 04 06

YR

Sand concentration


diameter)

Re = 40000Re = 70000

Re = 100000Re = 200000




5703e ndash 001

5346e ndash 001

4990e ndash 001

4634e ndash 001

4277e ndash 001

3921e ndash 001

3564e ndash 001

3208e ndash 001

2851e ndash 001

2495e ndash 001

2139e ndash 001

1782e ndash 001

1426e ndash 001

1069e ndash 001

7129e ndash 002

3564e ndash 002

1044e ndash 0180 0035

00175 0053

0070 (m)Z X

Y

(a)


5102e ndash 001

4784e ndash 001

4465e ndash 001

4146e ndash 001

3827e ndash 001

3509e ndash 001

3190e ndash 001

2871e ndash 001

2553e ndash 001

2234e ndash 001

1915e ndash 001

1596e ndash 001

1278e ndash 001

9590e ndash 002

6403e ndash 002

3216e ndash 002

2839e ndash 0040 0035

00175 0053

0070 (m)Z X

Y

(b)

Figure 14 Continued





4166e ndash 001

3908e ndash 001

3651e ndash 001

3394e ndash 001

3136e ndash 001

2879e ndash 001

2621e ndash 001

2364e ndash 001

2106e ndash 001

1849e ndash 001

1592e ndash 001

1334e ndash 001

1077e ndash 001

8195e ndash 002

5621e ndash 002

3047e ndash 002

4729e ndash 0030 0035

00175 0053

0070 (m)Z X

Y

(c)


2540e ndash 001

2397e ndash 001

2254e ndash 001

2112e ndash 001

1969e ndash 001

1826e ndash 001

1683e ndash 001

1540e ndash 001

1397e ndash 001

1254e ndash 001

1111e ndash 001

9679e ndash 002

8250e ndash 002

6820e ndash 002

5391e ndash 002

3961e ndash 002

2532e ndash 0020 0035

00175 0053

0070 (m)Z X

Y

(d)





5 Conclusions







Appendix





0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000


0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number




0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)





Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



5703e ndash 001

5346e ndash 001

4990e ndash 001

4634e ndash 001

4277e ndash 001

3921e ndash 001

3564e ndash 001

3208e ndash 001

2851e ndash 001

2495e ndash 001

2139e ndash 001

1782e ndash 001

1426e ndash 001

1069e ndash 001

7129e ndash 002

3564e ndash 002

1044e ndash 0180 0035

00175 0053

0070 (m)Z X

Y

(a)


5102e ndash 001

4784e ndash 001

4465e ndash 001

4146e ndash 001

3827e ndash 001

3509e ndash 001

3190e ndash 001

2871e ndash 001

2553e ndash 001

2234e ndash 001

1915e ndash 001

1596e ndash 001

1278e ndash 001

9590e ndash 002

6403e ndash 002

3216e ndash 002

2839e ndash 0040 0035

00175 0053

0070 (m)Z X

Y

(b)

Figure 14 Continued





4166e ndash 001

3908e ndash 001

3651e ndash 001

3394e ndash 001

3136e ndash 001

2879e ndash 001

2621e ndash 001

2364e ndash 001

2106e ndash 001

1849e ndash 001

1592e ndash 001

1334e ndash 001

1077e ndash 001

8195e ndash 002

5621e ndash 002

3047e ndash 002

4729e ndash 0030 0035

00175 0053

0070 (m)Z X

Y

(c)


2540e ndash 001

2397e ndash 001

2254e ndash 001

2112e ndash 001

1969e ndash 001

1826e ndash 001

1683e ndash 001

1540e ndash 001

1397e ndash 001

1254e ndash 001

1111e ndash 001

9679e ndash 002

8250e ndash 002

6820e ndash 002

5391e ndash 002

3961e ndash 002

2532e ndash 0020 0035

00175 0053

0070 (m)Z X

Y

(d)





5 Conclusions







Appendix





0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000


0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number




0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)





Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





4166e ndash 001

3908e ndash 001

3651e ndash 001

3394e ndash 001

3136e ndash 001

2879e ndash 001

2621e ndash 001

2364e ndash 001

2106e ndash 001

1849e ndash 001

1592e ndash 001

1334e ndash 001

1077e ndash 001

8195e ndash 002

5621e ndash 002

3047e ndash 002

4729e ndash 0030 0035

00175 0053

0070 (m)Z X

Y

(c)


2540e ndash 001

2397e ndash 001

2254e ndash 001

2112e ndash 001

1969e ndash 001

1826e ndash 001

1683e ndash 001

1540e ndash 001

1397e ndash 001

1254e ndash 001

1111e ndash 001

9679e ndash 002

8250e ndash 002

6820e ndash 002

5391e ndash 002

3961e ndash 002

2532e ndash 0020 0035

00175 0053

0070 (m)Z X

Y

(d)





5 Conclusions







Appendix





0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000


0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number




0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)





Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




5 Conclusions







Appendix





0

200

400

600

800

1000

1200

0 50 100 150 200

Pres

sure

loss

(Pa

m)


Re = 20000Re = 40000Re = 50000

Re = 60000Re = 80000Re = 100000


0

10

20

30

40

50

60

70

0 100000 200000 300000

Max

imum

bed

conc

entr

atio

n ne

ar b

otto

mw

all (

)

Reynolds number




0102030405060708090

100

0 002 004 006 008 01 012 014 016

Cum

ulat

ive w

eigh

t (

)

Particle size (mm)





Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Vq 1113946aq dV (A1)

where

1113944

n

q1aq 1 (A2)




Table 3



Eccentricity()


20000 0 0 8240000 0 0 15050000 0 0 23060000 0 0 32080000 0 0 557575100000 0 0 92016120000 50 0 10340000 50 0 15850000 50 0 23460000 50 0 32480000 50 0 562112100000 50 0 923520000 100 0 16540000 100 0 20950000 100 0 26960000 100 0 34580000 100 0 595795100000 100 0 93345520000 150 0 27540000 150 0 29350000 150 0 34360000 150 0 40780000 150 0 645529100000 150 0 966120000 0 25 7740000 0 25 14050000 0 25 21460000 0 25 29980000 0 25 555578100000 0 25 90247420000 50 25 9740000 50 25 15050000 50 25 22160000 50 25 30480000 50 25 559412100000 50 25 90642220000 100 25 16940000 100 25 20650000 100 25 25960000 100 25 32780000 100 25 542354100000 100 25 91680320000 150 25 29140000 150 25 29850000 150 25 34060000 150 25 39580000 150 25 582623100000 150 25 9519320000 0 50 8040000 0 50 15050000 0 50 23860000 0 50 34280000 0 50 639231100000 0 50 95578220000 50 50 11240000 50 50 17850000 50 50 26360000 50 50 364

Table 3 Continued



Eccentricity()


80000 50 50 64475100000 50 50 96553420000 100 50 20340000 100 50 25150000 100 50 32360000 100 50 41680000 100 50 689054100000 100 50 10032920000 150 50 35540000 150 50 37350000 150 50 42860000 150 50 50580000 150 50 750208100000 150 50 10560720000 0 75 7140000 0 75 13550000 0 75 21560000 0 75 31080000 0 75 574076100000 0 75 90772220000 50 75 12340000 50 75 19750000 50 75 28860000 50 75 39580000 50 75 680247100000 50 75 10136420000 100 75 23140000 100 75 29650000 100 75 38860000 100 75 50180000 100 75 810132100000 100 75 11665420000 150 75 40140000 150 75 44950000 150 75 52860000 150 75 63180000 150 75 948062100000 150 75 131303


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Table 4



Eccentricity()


Particlesize (mm)





1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


1ρrq

z


⟶1113874 1113875 1113944

n

p1_mpq minus _mqp1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(A4)



ztaqρq ϑq

⟶1113874 1113875 + nabla aqρq ϑq

⟶ϑq

⟶1113874 1113875


q + aqρqg + 1113944n

p11113882Kpq ϑp

⟶minus ϑq

⟶1113874 1113875

+ _mpq ϑpq

⟶minus _mqp ϑqp

⟶1113883 + Fq

⟶+ F⟶

liftq+ F⟶

vmq1113874 1113875

(A5)

where g⟶


q is the qth



F⟶



z

ztasρs ϑs

⟶1113874 1113875 + nabla asρs ϑs

⟶ϑs

⟶1113874 1113875


s + 1113944N

l11113882Kls ϑl

⟶minus ϑs

⟶1113874 1113875

+ _mls ϑls

⟶minus _msl ϑsl

⟶1113883 + Fs

⟶+ F⟶

lifts+ F⟶

vms1113874 1113875

(A6)


Table 4 Continued



Eccentricity()


Particlesize (mm)







is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




is as follows

DRij





DTij z

zxk

μt

σk

zRij

zxk

1113888 1113889 (A8)




zUj

zxm


zxm

1113888 1113889 (A9)




ϵk


23δijk1113876 1113877 (A11)






32


32


Clk32

ϵd


32emptyjk2nink1113874 1113875

Clk32

ϵd

(A13)




ϵij 23δijε (A14)

where






Nomenclature



Data Availability


Disclosure




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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