水平管内cross流体携砂数值模拟（西石油）
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Numerical investigations of liquidsolid slurry flowsin a fully developed turbulent flow regionJ. Ling, P.V. Skudarnov, C.X. Lin, M.A. Ebadian*Hemispheric Center for Environmental Technology, Center for Engineering and Applied Sciences, Florida International University,10555 West Flagler Street, EAS-2100, Miami, FL 33174, USAReceived 6 December 2001; accepted 14 February 2003AbstractIn this paper, a simplified 3D algebraic slip mixture (ASM) model is introduced to obtain the numerical solution in sandwaterslurry flow. In order for the study to obtain the precise numerical solution in fully developed turbulent flow, the RNG Ke turbulentmodel was used with the ASM model. An unstructured (block-structured) non-uniform grid was chosen to discretize the entirecomputational domain, and a control volume finite difference method was used to solve the governing equations. The mean pressuregradients from the numerical solutions were compared with the authors? experimental data and that in the open literature. Thesolutions were found to be in good agreement when the slurry mean velocity is higher than the corresponding critical depositionvelocity. Moreover, the numerical investigations have displayed some important slurry flow characteristics, such as volume fractiondistributions, slurry density, slip velocity magnitude, slurry mean velocity distributions, and slurry mean skin friction coefficientdistributions in a fully developed section, that have never been displayed in the experiments.? 2003 Elsevier Science Inc. All rights reserved.Keywords: Granular flow; Multiphase flow; Numerical analysis; Slurries; Turbulence1. IntroductionSlurry pipeline transportation is a popular mode oftransportation in various industries. It has several ad-vantages, such as its friendliness to the environment andits relatively low operation and maintenance costs. Ingeneral, slurry transportation is divided into three majorflow patterns: (1) pseudo-homogeneous flow (or homo-geneous flow) and heterogeneous flow; (2) heteroge-neous and sliding bed flow (or moving bed flow), and (3)saltation and stationary bed flow (Doron and Barnea,1996). Pseudo-homogeneous flow is a slurry flow patternin which the slurry flows at a very high velocity and allsolid particles are distributed nearly uniformly acrossthe pipe cross-section. With a decrease in slurry flowrate, the heterogeneous flow pattern occurs when thereis a concentration gradient in the direction perpendicu-lar to the pipe axis, with more particles transported atthe lower part of the pipe cross-section, as is the case inmost practical applications. As the slurry flow rate isreduced further, the solid particles accumulate at thebottom of the pipe and form a moving bed layer, whilethe upper part of the pipe cross-section is still occupiedby a heterogeneous mixture. When the slurry flow rate istoo low to suspend all solid particles, a stationary bedlayer at the bottom of the pipe cross-section is observed.This is the saltation and stationary bed flow (Vocaldoand Charles, 1972; Parzonka et al., 1981). The slurryvelocity associated with the formation of a stationarybed layer is called the critical deposition velocity. A bedlayer in the slurry pipeline is unstable and dangerousduring the operation of pipeline transportation. Itprobably enhances pipe wear and causes plugging orblockage of the pipeline. As a result, it should be avoi-ded in design and operation of the pipeline transporta-tion system.Slurry flow is very complex. In a survey of the openliterature on slurry transportation investigations, it wasfound that most investigations were made in labora-tories to determine pressure gradients and critical de-position velocities in slurry flows. Doron et al. (1987)and Doron and Barnea (1993) proposed two-layer and*Corresponding author. Tel.: +1-305-348-3585; fax: +1-305-348-4176.E-mail address: ebadianhcet.fi (M.A. Ebadian).0142-727X/03/$ - see front matter ? 2003 Elsevier Science Inc. All rights reserved.doi:10.1016/S0142-727X(03)00018-3International Journal of Heat and Fluid Flow 24 (2003) 389398/locate/ijhffthree-layer models of the slurry flow, and Wilson andPugh (1988) put forward a dispersive-force modeling inheterogeneous slurry flow, but these models were derivedbased on single-species slurry flow and cannot determinethe slurry density and volume fraction distributions andthe slip velocity between the liquid and solid particles.Nassehi and Khan (1992) provided a numerical methodfor the determination of slip characteristics between thelayers of a two-layer slurry flow, but no comparisons ofexperimental results and numerical solutions were re-ported. In this paper, a simplified two-phase flow alge-braic slip mixture (ASM) model (Manninen et al., 1996)is introduced to obtain the numerical solution in thesandwater slurry flow. A control volume finite differ-ence method (CVFDM) is used to solve the governingequations, and an unstructured (block-structured) non-uniform grid is chosen to discretize the entire com-putational domain. The numerical investigations havedisplayed some important liquidsolid slurry flow char-acteristics, such as volume fraction distributions, slurrydensity, slip velocity magnitude, slurry mean velocitydistributions, and slurry mean skin friction coefficientdistributions in a fully developed section, that have neverbeen displayed in the experiments.2. Calculation models2.1. Governing equationsThe ASM model can model two-phase flow (fluid orparticulate) by solving the momentum equation andcontinuity equation for the mixture, the volume fractionequation for the secondary phase, and an algebraic ex-pression for the relative (slip) velocity. ASM assumesthat a local equilibrium between the phases is reachedover short spatial length scales. This assumption re-quires the dispersed phase (particles) accelerate rapidlyto the terminal velocity (Manninen et al., 1996). Thus,typical dimension of the system should be much longerthan the characteristic length scale of acceleration. Thecharacteristic length scale of acceleration in water ofa particle with a diameter of 1?10?4m and densityof 3300 kg/m3is on the order of 5?10?5m (Manninenet al., 1996). This characteristic length is several ordersof magnitude smaller than dimensions of the system weare modeling (see Section 3.1 for details); thus, ASMmodel assumptions are fully justified.The continuity equation for the mixture in the ASMmodel isootqm ooxiqmum;i 01The momentum equation for the mixture can be ex-pressed asootqmum;jooxiqmum;ium;j ?opoxjooxilmoum;ioxj?oum;joxi? qmgi FjooxiXnk1akqkuDk;iuDk;j2where n is the number of phases, F is the body force, akis the volume fraction of solids, qmis the mixture den-Nomenclature a asecondary phase particle?s acceleration, m/s2C1e, C2e, Clconstantsdpsolid particle diameter, mEempirical constantFbody force, N/m2fmmean friction factorgacceleration of gravity, m/s2Iturbulent intensity levelKturbulent kinetic energy, m2/s2kvvon Karman?s constantkpturbulent kinetic energy at point p, m2/s2Lpipeline length, mPrtturbulent Prandtl number for energySmodulus of the mean rate-of-strain tensorVslurry mean velocity, m/summass-averaged velocity, m/suDkdrift velocity, m/supmean velocity of the fluid at point p, m/s v vqpslip velocity, m/sypdistance from point p to the wall, mDp=DL mean pressure gradient of the slurry flow,Pa/mGreeksakvolume fraction of solidsbcoefficient of thermal expansionqmmixture density, kg/m3qssolid particle density, kg/m3qwwater density, kg/m3edissipation rate of turbulent kinetic energy,m2/s3lmviscosity of the mixture, kg/msspqparticulate relaxation time, sSubscriptsi, j, kgeneral spatial indicesmmixtures, z, wsilica sand, zircon sand, and water390J. Ling et al. / Int. J. Heat and Fluid Flow 24 (2003) 389398sity, and lmis the viscosity of the mixture, which areexpressed asqmXnk1akqkand lmXnk1aklk3 u umand u uDkare mass-averaged velocity and drift veloci-ties, which are expressed as u umPnk1akqk u ukqmand u uDk u uk? u um4The slip velocity is defined as the velocity of thesecondary phase (p) relative to the primary phase (q)velocity: v vqp u up? u uq5The drift velocity and slip velocity are connected by thefollowing expression: u uDp v vqp?Xni1aiqiqm v vqi6The basic assumption in the ASM model is that, toprescribe an algebraic relation for the relative velocity, alocal equilibrium between the phases should be reachedover short spatial length scales. The form of the slipvelocity is v vqpqm? qpd2p18lqfdrag g g ?o u umot!7wherefdrag1 0:15 Re0:687Re610000:0183 ReRe 1000?The volume fraction equation for the secondaryphase isootapqp ooxiapqpum;i ?ooxiapqpuDp;i8This ASM model can be applied in the laminar andturbulent two-phase flows. In practice, since slurrytransportation is in the fully developed turbulent flow,the RNG Ke turbulent model is used with the ASMmodel in this study. The turbulent kinetic energy inRNG Ke turbulent model isootqmk ooxiqmum;ik ooxiaklmokoxi? ltS2? qme9Dissipation rate of the turbulent kinetic energy isootqme ooxiqmum;ieooxiaelmoeoxi? C1eekltS2? C2eqme2k? R10where the coefficients akand aeare the inverse effectPrandtl numbers for k and e, respectively. In the high-Reynolds-number limit, ak aeffi1:393. C1eand C2eareequal to 1.42 and 1.68. b and Prtare the coefficient ofthermal expansion and the turbulent Prandtl number forenergy. S is the modulus of the mean rate-of-straintensor, Sij, which is defined asS ffiffiffiffiffiffiffiffiffiffiffiffi2SijSijpandSij12ouioxj?oujoxi?11R in Eq. (10) was expressed asR Clqmg31 ? g=g01 fg3?e2k12where g S ? k=e, g0? 4:38, f 0:012, and Cl 0:085.2.2. Boundary conditionsNon-slip boundary condition is imposed on the walls,and heat transfer is not considered in the entire com-putational domain. For this paper, in the near-wallzone, the standard wall function proposed by Launderand Spalding (1974) was chosen due to its wide appli-cation in industrial flows. When the mesh is such thaty?611:225 at the wall-adjacent cells, the viscous forceis dominant in the sublayer. The laminar stressstrainrelationship can be applied:u? y?13y?qmC1=4lk1=2pyplm14The logarithmic law for the mean velocity is known tobe valid for y? 11:225 (Fluent Inc., 1996). It can beexpressed asu?1klnEy?15where k is von Karman?s constant, Clis turbulent modelconstant, and kpand ypare the turbulent kinetic energyat point p and distance from point p to the wall,respectively.To simplify the simulations, the mean velocity inletboundary condition and pressure outlet boundary con-dition are imposed on the inlet and outlet of slurrypipeline:ux;inlet constant; uy;inlet uz;inlet 0; andpoutlet constant163. Numerical computation3.1. Physical problems and grid systemThe geometry and physical problem studied in thispaper are as follows:Horizontally straight pipeline length, L1.4 m; innerdiameter of the pipe, d 0.0221 m; range of the volumeJ. Ling et al. / Int. J. Heat and Fluid Flow 24 (2003) 389398391fraction of solid, ak, is 1020%; range of mean velocityof the slurry flow is 13 m/s; the densities of silica sandand zircon sand are 2380 and 4223 kg/m3; mean particlediameter, dp, is 1.1?10?4m.The length of the computational domain, x=d P50,was based on the suggestions from Wasp et al. (1979)and Brown and Heywood (1991) to ensure that fullydeveloped flow results could be obtained in the pipelinecomputational domain. A multi-block unstructurednon-uniform grid system with hexahedral elements wasused to discretize the computational domain, as shownin Fig. 1. This unstructured grid system has five blocksto form the entire computational pipeline. The distri-bution of the grid on the circumference of computa-tional pipeline is uniform, and each hexahedral elementin the grid system contains 27 nodes. The grid inde-pendent study was made to select the optimum griddistribution in this investigation, as shown in Table 1.From Table 1, V was mean velocity of the slurry flow,and Dp=DL and fmwere mean pressure gradient of theslurry flow and mean friction factor in fully developedturbulent flow, respectively. fmcould be obtained fromfm12pZ2p0fhdh17wherefhsw12qmV2mThe grid distribution, 460?400, shown in Table 1could ensure a satisfactory solution for the slurry flow.3.2. Numerical methodIn the numerical investigation of slurry transporta-tion, all governing equations, wall boundary conditions,and inlet and outlet boundary conditions were solved ina Cartesian coordinate system by the CFD solver,FLUENT 5, which used a CVFDM. Heat transfer wasneglected, and the slurry flows were steady state. At thesame time, the slurry flows were in pseudo-homoge-neous flow (or homogeneous flow), and heterogeneousflow and sliding bed flow (or moving bed flow). Thesecond-order upwind scheme was selected as the dis-cretization scheme in the governing equations. TheSIMPLE algorithm from Dormaal and Raithby (1984)was used to resolve the coupling between the velocityand the pressure. To avoid the divergence, the under-relaxation technique was applied in all dependent vari-ables. In the investigation, the under-relation factor forthe pressure, p, was 0.2 and 0.3, that for the velocitycomponents was 0.50.7, and those for the turbulencekinetic energy and turbulence dissipation rate were 0.60.8. The segregated solver was adopted to solve thegoverning equations sequentially. In the segregated so-lution method, each discrete governing equation waslinearized implicitly with respect to that equation?s de-pendent variable. A point implicit (Gauss-Seidel) linearequation solver was used in conjunction with an alge-braic multi-grid method to solve the resultant scalarsystem of equations for the dependent variable in eachcell. The numerical computation was considered con-verged when the residual summed over all the compu-tational nodes at nth iteration, Rn/, satisfied the followingcriterion:Rn/Rm/610?418where Rm/is the maximum residual value of / variableafter m iterations.All the numerical computations in this paper werebased on the grid distribution, 460?400, and carriedout in an SGI Origin 2000 at the Hemispheric Center forEnvironmental Technology (HCET) at Florida Inter-national University (FIU) in Miami.Table 1Grid independent test (V 2 m/s, 576x=d 661, ak0.189, dp0.00011 m, silica sandwater slurry flow)Cross-sectional?axial224?300340?300460?300500?300460?400460?500Total cells67,200108,000138,000150,000184,000230,000Dp=DL (Pa/m)222022052184217720942154fm2.532.4632.3252.3252.3352.335Fig. 1. Unstructured grid of the horizontal pipeline.392J. Ling et al. / Int. J. Heat and Fluid Flow 24 (2003) 3893984. Results and discussion4.1. Comparisons of numerical and experimental dataThe slurry mean pressure gradient is an importantparameter in slurry transportation and pipeline design.To verify and check the numerical results from the ASMmodel, a lab-scale flow loop was constructed at HCET,and extensive experiments for the silica sandwater andzircon sandwater slurry flows were made (Skudarnovet al., 2001). Experimental data from Skudarnov et al.(2001) and other experimental data from Newitt et al.(1955) are compared with the numerical results, asshown in Fig. 2.Fig. 2(a) shows the comparison of the numerical re-sults with the experimental data from Skudarnov et al.and Newitt et al. in silica sandwater slurry flow withthe same pipeline geometry, mean velocity, volumefraction, particle size, and particle density. Fig. 2(b) il-lustrates the comparison of the numerical results withthe experimental data from Skudarnov et al. in zirconsandwater slurry flow. Based on the experimental in-vestigations made by Skudarnov et al., the critical de-position velocities shown in Fig. 2(a) and (b) are 0.97and 1.58 m/s, respectively.It is clear that in Fig. 2(a) all slurry mean velocitiesfor numerical results and experimental data are higherthan the critical deposition velocity, and the numericalresults are in good agreement with the experimental dataalthough the numerical results from the ASM modelresult in a little under-prediction. From Fig. 2(b), thenumerical results are still in good agreement with theexperimental data, and a little under-prediction fromthe numerical results exists when the mean velocities ofthe zircon sandwater slurry flow in the numerical andexperimental investigations are higher than the criticaldeposition velocity. A big discrepancy between the nu-merical results and experimental data takes place whenthe mean velocities of slurry flow in the numerical andexperimental investigations are lower than correspond-ing critical deposition velocities, and the discrepancywould be further increased with a decrease in the slurrymean velocity.In practice, the available flow area in the pipelinewould be reduced, friction loss would be increased, andthe pressure gradient in the slurry flow would be in-creased if the slurry velocity is lower than the corre-sponding critical deposition velocity and a stationarybed of the solids is formed in the experiments. However,the ASM model cannot change its available flow areawhen the slurry flow velocity is lower than the corre-sponding critical deposition velocity. As a result, a bigdiscrepancy would occur for a slurry velocity that islower than the critical deposition velocity. It is evidentthat the ASM model can provide a better prediction ofpressure gradient if the slurry mean velocity is higherthan the critical deposition velocity. Yet the ASM modelwould result in a bigger deviation if the slurry meanvelocity is lower than the critical deposition velocity.4.2. Slurry density, volume fraction, and skin frictionfactorsThe slurry density and volume fraction are importantparameters in slurry flow analysis. In experiments, it isdifficult to measure the slurry density and volume frac-tions at any point of the pipe cross-sectional area, but itis easy to obtain them in numerical investigation.The density distributions of the silica sandwaterslurry flow with different mean velocities are shown inFig. 3. Fig. 3(a) and (b) show the density distributioncontours in the cross-sectional area of the pipeline inV 2 and 3 m/s, and Fig. 3(c) illustrates the densityFig. 2. Comparisons of the numerical solutions from ASM model withthe experimental data from Skudarnov et al. (2001) and Newitt et al.(1955) in fully developed turbulent flow (qw998.2 kg/m3, d 0.0221m, dp0.0000970.00011 m). (a) Silica sandwater slurry, qs2381kg/m, ak20%, (b)zircon sandwater slurry, qz4223 kg/m,ak10%.J. Ling et al. / Int. J. Heat and Fluid Flow 24 (2003) 389398393distributions along the vertical centerline of the pipeline.It is clear from Fig. 3(a) and (b) that the density dis-tribution contours are symmetrical in the horizontaldirection. Because of the effect of the gravitational force,the density distribution contours are asymmetrical in thevertical direction. Since the density of solids is usuallyhigher than that of primary fluid in slurry flows, on thetop of the pipeline, the slurry mean density is relativelysmall and close to the water density, 998.2 kg/m3, and inthe lower part of the pipeline, the slurry mean density isincreased gradually based on the action of gravity. Inthe central part of the pipeline, the slurry mean densityis kept in a small variant range, which is about 12371284 kg/m3, as shown in Fig. 3. However, it should bepointed out that the central area of the slurry flow withthe small variance of the slurry mean density would beenlarged with an increase in the slurry mean velocity.Forthesamesilicasandwaterslurryflowwithqs2381kg/m3,qw998.2kg/m3,ak 20%,d 0.0221 m, and dp0.00011 m, the central area with theslurry mean density in V 3 m/s is bigger than that inV 2 m/s, comparing Fig. 3(a) with (b).Fig. 4 shows the volume fraction distribution con-tours of silica sand and water in the slurry flow withV 2 and 3 m/s, ak20%, qs2381 kg/m3, qw998kg/m3, d 0.0221 m, and dp0.00011 m. Fig. 4 showsthat the volume fractions of silica sand and water aresymmetrical in the horizontal direction, but the volumefractions of silica sand and water have a big gradientin the vertical direction because of the asymmetry ofdensity distribution in the slurry flows.On top of the pipeline, the volume fraction of silicasand is close to zero, and the volume fraction of waterapproaches unity, as shown in Fig. 4. It means that mostslurry flow on top of the pipeline is water. In the centralpart of the pipeline, the volume fractions of silica sandand water are almost kept in a small variant range. Inthe lower part of the pipeline, the volume fractions ofsilica sand are increased gradually, and the volumefractions of water are decreased accordingly. The sum ofFig. 3. Mean density distribution of the silica sandwater slurry flow (qs2381 kg/m3, qw998.2 kg/m3, ak20%, d 0.0221 m, dp0.00011 m).(a) Density contours at V 2 m/s, (b) density contours at V 3 m/s and (c) density distribution along the vertical centerline.394J. Ling et al. / Int. J. Heat and Fluid Flow 24 (2003) 389398volume fractions of silica sand and water is equal tounity at any point of pipe cross-sectional area for thesame slurry flow.The slurry mean skin friction factors from the pipe-line inlet to fully developed turbulent flow region areshown in Fig. 5. All slurry mean skin friction factorswould drop sharply from the pipeline inlet, then, beraised gradually. At x=d P50, the slurry mean skinfriction factors are constant. Fig. 5 illustrates that theslurry mean skin friction factor would be constant in thefully developed turbulent flow. At the same time, Fig. 5shows that the slurry mean skin friction factors will beincreased as the slurry mean velocity and solid particledensity increase. It means that a higher slurry meanvelocity and solid particle density would result in abigger flow loss and slurry mean pressure gradient inslurry pipeline transportation.4.3. Slurry mean velocity profiles and slip velocitymagnitudeThe velocity profile of the slurry flow in the hori-zontal pipeline is mainly affected by the slurry density,volume fraction, and mean velocity. As a result, thevelocity profile of the slurry flow in the cross-sectionalarea of the pipeline is a little bit different from that ofsingle-phase flow. In general, the velocity profile insingle-phase flow is symmetric around the pipe center-line, and liquid density is kept as a constant in the cross-sectional area of the pipeline. However, in the slurryflow, the velocity profile is asymmetrical around the pipecenterline.Fig. 6 shows the mean velocity contours of silicasandwater and zircon sandwater slurry flows withV 3 m/s, ak20%, silica sand density qs2381 kg/m3,zircon sand density qz4223 kg/m3, water densityqw998 kg/m3, d 0.0221 m, and dp0.00011 m.From Fig. 6, it is clear that the slurry mean velocitiesnear the wall drop down sharply due to the strong vis-cous shear stress in the turbulent boundary layer andnon-slip boundary condition on the wall. Like the slurrydensities and volume fractions, the velocity distributionsin the slurry flows are symmetrical in the horizontaldirection, and the velocity profiles are asymmetric in thevertical direction due to the density gradients in theslurry flow.It is clear from Fig. 6(a) and (b) that the maximumslurry velocity center appears at the upper part of thepipeline, not at the centerline. This phenomenon ofslurry flow has been proved with experiments (Roco andFig. 4. Volume fraction contours of silica sand and water (qs2381 kg/m3, qw998.2 kg/m3, ak20%, d 0.0221 m, dp0.00011 m). (a) Silicasand, V 2 m/s, (b) silica sand, V 3 m/s, (c) water, V 2 m/s and (d) water, V 3 m/s.J. Ling et al. / Int. J. Heat and Fluid Flow 24 (2003) 389398395Shook, 1985). On the other hand, comparing Fig. 6(a)with (b), it can be found that the location of the maxi-mum velocity center in zircon sandwater slurry ishigher than that in the silica sandwater when the slurryvelocities, volume fractions, pipeline geometries, andparticle diameters are the same. The figures indicate thatthe location of maximum velocity center with a higherslurry density is higher than that with a lower slurrydensity.Fig. 6(c) shows the velocity profiles of the silica sandwater and zircon sandwater slurry flows with samevelocities and geometries along the vertical centerline ofthe pipeline. Two slurry velocity profiles are asymmetricin the vertical direction, and the velocity profiles in thelower part of the pipe centerline would be lower thanthose in the upper part. This occurs because the densityof solid particles is usually higher than that of liquid; theslurry density in the lower part of pipe centerline shouldbe higher than those in the upper part based on the effectof gravity. As a result, compared with the upper part ofthe pipe centerline, water will spend more energy todrive silica sand in the lower part, which results in alower slurry velocity in this area.On the other hand, with an increase in the slurrydensity, the velocity profile in the upper part of pipecenterline would be increased further and that in thelower part would be reduced accordingly, as shown inFig. 6(c).The slip velocity in an ASM model is defined as thedifference between the liquid velocity and solid particlevelocity and is a function of the density of the slurryflow. Fig. 7 shows the slip velocity magnitudes of thesilica sandwater slurry flow along the horizontal andvertical centerlines based on a non-slip boundary con-dition on the wall. As in Fig. 7(a), the slip velocitymagnitude distribution is symmetric along the horizon-tal centerline since the slurry density along the hori-zontal centerline is symmetric. However, it is evidentfrom Fig. 7(b) that the slip velocity magnitude distri-bution is asymmetric along the vertical centerline. Theslip velocity magnitude in the upper part of the pipelineis relatively larger than that in the other parts since theslurry density in the upper part is relatively smaller. InFig. 5. Slurry mean skin friction factor distributions from the entranceto fully developed turbulent flow region (d 0.0221 m, dp0.00011m). (a) Silica sandwater slurry, qs2381 kg/m3, ak20%, (b) zirconsandwater slurry, qz4223 kg/m3, ak10%.Fig. 6. Velocity contours and profiles of the silica sandwater andzircon sandwater slurry flows (ak20%, qs2381 kg/m3, qz4223kg/m3, qw988.2 kg/m3, V 3 m/s, dp0.00011 m, d 0.0221 m). (a)Velocity isograms of silica sandwater, (b) velocity isograms of zirconsandwater and (c) velocity profiles along the vertical centerline.396J. Ling et al. / Int. J. Heat and Fluid Flow 24 (2003) 389398the central part of the pipeline, the slip velocity magni-tude is kept in a small variant range. In the lower part,the slip velocity magnitude is dropped down sharply dueto the larger slurry density and volume fraction of solidsnear the wall and non-slip boundary condition on thewall. On the other hand, the slip velocity magnitudewould be increased, and the central part along the ver-tical centerline would be decreased, with an increase inthe slurry density, as shown in Fig. 7(b). In any case,based on Fig. 7, it is clear that the effect of the slip ve-locity on the slurry flow is negligible in the heteroge-neous and sliding bed flow (or moving bed flow) becausethe slip velocity magnitude is small.5. ConclusionsThe numerical investigation in slurry flows is valuableand interesting research because it can provide infor-mation that cannot be obtained in experiments. Basedon our numerical investigations, the following conclu-sions can be made:1. The ASM model can provide a good prediction forthe liquidsolid slurry flow if the slurry mean velocityis higher than the corresponding critical depositionvelocity. The numerical predictions for the pressuregradients are in good agreement with the experimen-tal data.2. The slurry density and volume fraction distributionsare symmetric in the horizontal direction, but theywould have a big gradient in the vertical direction.In heterogeneous and sliding bed flow (or movingbed flow), most of slurry flow on the top of the pipe-line is water, while slurry mean density and volumefraction of solids would be increased gradually inthe lower part of the pipeline. Slurry mean skin fric-tion factor would be increased as the slurry meanvelocity and solid particle density increase and wouldbe constant in fully developed turbulent flow.3. Velocity profiles in the slurry flow are symmetric inthe horizontal direction, but they are asymmetric inthe vertical direction. The velocity profiles in theupper part of the pipeline are higher than those inthe lower part. The slip velocity magnitude is usuallysmall, and the effect of the slip velocity on the slurryflow is negligible in the heterogeneous
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