Hydrodynamic Simulation of Horizontal Slurry Pipeline Flow Using

Ind. Eng. Chem. Res. 2009, 48, 8159–8171

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Hydrodynamic Simulation of Horizontal Slurry Pipeline Flow Using ANSYS-CFX Kalekudithi Ekambara, R. Sean Sanders, K. Nandakumar,†,* and Jacob H. Masliyah Department of Chemical and Materials Engineering, UniVersity of Alberta, Edmonton, AB, Canada, T6G 2G6

The behavior of horizontal solid-liquid (slurry) pipeline flows was predicted using a transient three-dimensional (3D) hydrodynamic model based on the kinetic theory of granular flows. Computational fluid dynamics (CFD) simulation results, obtained using a commercial CFD software package, ANSYS-CFX, were compared with a number of experimental data sets available in the literature. The simulations were carried out to investigate the effect of in situ solids volume concentration (8 to 45%), particle size (90 to 500 µm), mixture velocity (1.5 to 5.5 m/s), and pipe diameter (50 to 500 mm) on local, time-averaged solids concentration profiles, particle and liquid velocity profiles, and frictional pressure loss. Excellent agreement between the model predictions and the experimental data was obtained. The experimental and simulated results indicate that the particles are asymmetrically distributed in the vertical plane with the degree of asymmetry increasing with increasing particle size. Once the particles are sufficiently large, concentration profiles are dependent only on the in situ solids volume fraction. The present CFD model requires no experimentally determined slurry pipeline flow data for parameter tuning, and thus can be considered to be superior to commonly used, correlation-based empirical models. 1. Introduction Solid-liquid (slurry) transport has been used for decades in the long-distance transport of materials like coal, mineral ore concentrates, and tailings. Over the past 20 years, the oil sands industry of northern Alberta has become one of the world’s most intensive users of slurry transport. Dense, coarse particle slurries of oil sand ore are transported by pipeline from mining sites to extraction facilities. Pipeline transport is also used to carry waste tailings to the final disposal site. In most cases, slurry pipelines are more energy efficient and have lower operating and maintenance costs than any other bulk material handling methods. Additionally, operations involving slurry flow play a significant role in many other industries, including pharmaceutical manufacturing, nanofabrication, and oil refining. Most engineering models of slurry flow have focused on the ability to predict frictional pressure loss and minimum operating velocity (or “deposition velocity”) for coarse-particle, “settling” slurries. Many models of this type exist and have varying degrees of success in predicting the aforementioned parameters. As noted in the following section, many of these models are phenomenological, meaning that some empirically derived parameters or relationships are required. Additionally, these models tend to provide macroscopic parameters only, for example, frictional pressure drop, deposition velocity, and delivered solids volume fraction for a narrowly sized slurry. Many industrial slurries, however, contain a range of different particle sizes. The location and velocity of these particles at different positions in the flow will drastically affect the pipeline operation. Knowledge of the variation of these parameters with pipe position is crucial if the understanding of mesoscopic processes (e.g., pipeline wear, particle attrition, or agglomeration) is to be advanced. Additionally, analysis of more complex three- or four-phase flows will require models that provide local values of particle concentration and velocity. Finally, accurate predictions of concentration and velocity distributions in more complicated geometries (pumps, hydrocyclones, mixing tanks) * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: +972 2 607 5418. Fax: 780-492-2881. † GASCO Chair Professor, The Petroleum Institute, P.O. Box. 2533 Abu Dhabi, U.A.E.

will require the development and validation of mechanistic computational models. With the advent of increased computational capabilities, computational fluid dynamics (CFD) is emerging as a very promising new tool in modeling hydrodynamics. While it is now a standard tool for single-phase flows, it is at the development stage for multiphase systems. Work is required to make CFD suitable for slurry pipeline modeling and scale-up. In view of the current status on this subject, the application to horizontal slurry pipeline flow of a comprehensive three-dimensional computational fluid dynamics (CFD) model based on the kinetic theory of granular flow has been undertaken. The kinetic theory component of this model is critical because it accounts for the effects of the interactions between particles and between particles and the suspending liquid phase. Simulations have been carried out to investigate the effect of solids volume fraction, particle size, mixture velocity, and pipe diameter on spatial variations of particle concentration and liquid velocity, as well as frictional pressure losses. The model predictions were compared with existing experimental data over a wide range of

Figure 1. Grid structure for the horizontal slurry pipeline simulations.

10.1021/ie801505z CCC: $40.75  2009 American Chemical Society Published on Web 05/29/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009

Table 1. Experimental Data Sets Modeled with Hydrodynamic Simulations measurement technique

source Roco and Shook27

Schaan et al.45

Gillies and Shook12

Gillies et al.13

pipe diameter particle size solids volume particle specific mixture (mm) (µm) concentration (%) gravity (-) velocity (m/s) 50.7

165

51.5 263 495

480 520 1300

6-35

2.65

1.5-4.5

50

85

15-45

2.65

150

90 100

105

420

26-47

264 495

420

103

90

pressure drop

particle concentration

velocity

γ-ray absorption

magnetic flux flow meter

0.8-5.0

pressure transducers γ-ray absorption


2.65

1.8-5.8



10-45

2.65

2.0-8.0


electrical resistivity probe, magnetic flux flow meter

5-50

2.47

1.0-5.0

pressure transducers γ-ray absorption, sampling probe

270 Kaushal and Tomita24

54.9

125 440

pipeline operating conditions: that is, average solids concentrations of 8 to 45% (by volume), uniform particle sizes of 90 to 500 µm, mixture velocities of 1.5 to 5.5 m/s, and pipe diameters of 50 to 500 mm. Before the computational method and simulation results are discussed, a brief review of previous work in this area is presented below.

Figure 2. Particle concentration profile sensitivity analysis (A) effect of forces: (O) exptl, (- · - · - · ) CFD-k-ε model with kinetic theory and drag force; (---) CFD- k-ε model with kinetic theory, drag force, and turbulent dispersion force; (s) CFD-k-ε model with kinetic theory, drag, lift, turbulent dispersion, and wall lubrication force. (B) radial distribution function and kinetic solids viscosity models of Gidaspow35 and Lun and Savage.37

2. Previous Work Durand1 published a pioneering work on the empirical prediction of hydraulic gradients for coarse particle slurry flows. Wasp et al.2 improved the calculation method and applied it to commercial slurry pipeline design. Shook and Daniel3 used the pseudohomogeneous approach to model slurry flow. The unique aspect of this technique is that it allows description of the flow using a single set of conservation equations (as for single-phase flow). The dispersed solids phase is assumed to augment the carrier fluid’s density and viscosity by amounts related to the in situ solids volume fraction. Clearly, this technique is of limited value as it, by definition, assumes the slurry has no deposition velocity. It provides reasonable predictions of friction losses only for relatively fine particles, low solids volume fractions, and for a narrow range of operating velocities. Shook and Daniel4 improved on the pseudohomogeneous approach by considering the slurry as a pseudo single-phase fluid with variable density. However, because of the boundary conditions adopted in their approach, it is difficult to apply their model to actual flow situations. Oroskar and Turian5 used a “constructive energy” approach to calculate the deposition velocity. In their model, they assumed that the kinetic energy of turbulent fluctuations is transferred to discrete particles, which suspends them in the flow. Despite the fact that this model was oversimplified and not intended for dense slurries, predicted deposition velocities compared favorably with the experimental data over a wide range of solids volume fractions. Wilson6 developed a one-dimensional two-layer model wherein coarse-particle slurry flow is considered to comprise two separate layers. Each layer has a uniform concentration and velocity. Because Wilson assumed the particles were very coarse, they were contained in the lower layer (with the upper layer solids concentration being zero). Momentum transfer occurs between the layers through interfacial shear forces. The two-layer model has been extended by a number of researchers.7-13 Doron et al.7 developed a two-layer model for the prediction of flow patterns and pressure drops in slurry pipelines. This model is very similar to that proposed by Wilson,6 except that the lower layer may also be assumed to be stationary. However, the model did not predict the existence of a stationary bed at


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Figure 3. Comparison of predicted and experimentally determined13 concentration profiles for dp ) 270 µm and V ) 5.4 m/s: (A) R j s ) 0.10, (B) R j s ) 0.20, (C) R j s ) 0.30, (D) R j s ) 0.35, (E) R j s ) 0.40, and (F) R j s ) 0.45.

low flow rates, which also reduced the reliability of the pressure drop predictions. Wilson and Pugh8 developed a dispersiveforce model of heterogeneous slurry flow, which extended the applicability of the original Wilson layer model because it accounted for particles suspended by fluid turbulence as well as those providing contact-load (Coulombic) friction. The model was used to predict particle concentration and velocity profiles that were in good agreement with experimental measurements. Nassehi and Khan9 developed a numerical method for the determination of slip characteristics between the layers of a twolayer slurry flow, but no comparisons between experimental results and their numerical solutions were reported. Undoubtedly, the most commonly used version of the twolayer model is the SRC model developed by Gillies and co-workers.10-13 The SRC two-layer model provides predictions of pressure gradient and deposition velocity as a function of

particle diameter, pipe diameter, solids volume fraction, and mixture velocity. This model is “semimechanistic” in that the effect of pipe diameter on pipeline friction loss is specified mechanistically (i.e., does not depend on any empirically determined coefficients). The semiempirical coefficients it contains are based on thousands of controlled experiments done at the Saskatchewan Research Council Pipe Flow Technology Centre. Since the optimum pipeline velocity is usually close to the deposition velocity (Vc), most of the data that were incorporated in the model were obtained at mixture velocities that are just greater than the deposition velocity (Vc e V e 1.3Vc). Doron and Barnea14 extended the two-layer modeling approach to a three-layer model of slurry flow in horizontal pipelines. Their model considered the existence of a dispersive layer, which is sandwiched between the suspended layer and a

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Figure 4. Comparison of predicted and experimentally determined13 concentration profiles for dp ) 90 µm and V ) 3.0 m/s: (A) R j s ) 0.19, (B) R j s ) 0.24, (C) R j s ) 0.29, and (D) R j s) 0.33.

bed. The dispersive layer was considered to have a higher concentration gradient than the suspended layer. A no-slip condition between the solid particles and the fluid was assumed, which is reasonable when the flow is in horizontal or nearhorizontal configurations. The model predictions showed satisfactory agreement with experimental data. Doron and Barnea21 also used a three-layer model to draw flow pattern maps, which can be used to indicate the flow pattern (essentially, the degree of flow heterogeneity). They determined the transition lines between the so-called “flow patterns” and compared these results with experimental data. Ramadan et al.16 also developed a threelayer solid model and applied to simulate slurry transport in inclined channels. The model predictions were compared with experiments, which clearly demonstrated the limitations of this model. A separate (but related) approach to slurry flow modeling also exists, where the one-dimensional Schmidt-Rouse equation11 (or equivalent; see Hunt17) is used to relate the rate of particle sedimentation to the rate of turbulent exchange, as represented by a solids eddy diffusivity, εs: Rsu∞ - εs

dRs )0 dy

(1)

where Rs is the local, time-averaged solids volume fraction, u∞ is the terminal particle settling velocity and y is a vertical position in the pipe. Karabelas18 developed an empirical model to predict the particle concentration profiles based on this formulation. Kaushal and co-workers19-24 developed a diffusion model based on the work of Karabelas,18 where they proposed a modification for the solids diffusivity for coarse particle slurry

flow. They constructed an empirical correlation determining the ratio of the solids diffusivity to the liquid eddy diffusivity. Their function shows that the solids diffusivity increases with increasing solids concentration. However, it does not take into account a significant dependence of the solids diffusivity on both particle size and pipe Reynolds number.25 They also compared their pressure drop data with the modified Wasp model,26 considering the effect of efflux concentration on dimensionless solids diffusivity,20 and found good agreement at higher flow velocities; however, deviations were significant at flow velocities near the deposition velocity. Roco and Shook27-29 developed a similar model for dense slurry pipeline flows. They considered the slurry to be a Newtonian fluid characterized using the mixture density and viscosity. They accounted for turbulent properties of the flow by introducing in the Navier-Stokes equation, an empirical term characterizing the turbulent viscosity. The particle concentration profile was determined using a semiempirical diffusion equation similar to that described above. Model predictions were compared with experimental data for solids volume fractions less than 35%. This model was not reliable at higher solids volume concentrations. Over the years, Roco and co-workers30,31 modified the turbulence model and used higher-order correlations to obtain a better estimate of eddy viscosity. Roco and Mahadevan31 used a one-equation kinetic energy model for turbulent viscosity. The model provides good predictions; however, it contains many empirical parameters. Gillies and Shook11 showed the most significant limitation of this type of model. It is not applicable when the particle size is large enough that the particles cannot be supported by fluid


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As described above, many papers have been published in the past 50 years on the subject of horizontal slurry pipeline flow. From these publications, the following observations can be made: (i) Most of the investigations were conducted using small pipeline loops (D e 55 mm) to determine pressure gradients and deposition velocities. (ii) Many of the earlier studies considered only moderate solids volume concentrations (say up to 26%). (iii) The purpose of many of the models that have been developed is to predict frictional pressure drop and/or minimum operating velocity (i.e., deposition velocity). (iv) Many of the models are 1D or 2D semiempirical models, meaning that they are limited in their ability to describe such characteristics as fluid turbulence, interfacial forces, or the radial variation of particle velocity or concentration. (v) While the SRC two-layer model provides accurate predictions of frictional pressure drop and deposition velocity over a wide range of pipe diameter, particle size, particle concentration, and mixture velocity, it does not provide information about fluid turbulence, local particle velocities, or local particle concentrations. It is also limited in application to straight runs of pipeline having a circular cross-sectional area; in other words, it is not suitable for more complex geometries that are of great interest in many mineral processing industries. In view of these limitations, an attempt has been made to develop a comprehensive computational model to describe the hydrodynamics of horizontal slurry flow based on the kinetic theory of granular flow and using a commercially available CFD package (ANSYS-CFX 10.0). Figure 5. Comparison of predicted and experimentally determined45 concentration profiles for dp ) 90 µm and R j s ) 0.15: (A) V ) 1.5 m/s, (B) V ) 3.0 m/s.

turbulence, which occurs approximately when u∞/u* ≈ 0.78, where u* is the friction velocity (u* ) (τw/F)1/2). As the particle diameter (and u∞) increases, concentration profiles take a noticeably different shape, becoming largely dependent on local concentration and exhibiting almost no dependence on mixture velocity (or fluid turbulence levels). Numerous observations of a local maximum in particle concentration up from the bottom of the pipe have been made when the particle diameter is relatively large and the mixture velocity is high.3,11,24,32,33 Wilson and Sellgren32,33 demonstrated that this effect is the result of a near-wall lift force that occurs in certain coarse-particle slurry flows. Recently, Kaushal and Tomita24 conducted experiments with two slurries of narrowly sized glass beads (0.125 and 0.44 mm) flowing in a 55 mm pipeline loop. These results clearly demonstrated the importance of the near-wall lift force on concentration profiles and on frictional pressure gradient for slurries containing the coarse particles. These researchers confirmed many of the findings of Wilson and Sellgren: most notably, that the smaller particles are fully encapsulated in the viscous sublayer and thus are not subjected to a near-wall force. Also, Campbell et al.34 reported experimental observations of an unexpected lift-like interaction force at the center of channel containing a flowing solids-liquid mixture. This force has no apparent analog for single particles in infinite fluids and appeared to be a result of multiparticle interactions. The magnitude of this force was found to be a significant fraction (e.g., 40%) of the total weight of particles in the channel and thus may play a role in offsetting the Coulombic (contact load) friction in coarse particle slurry flows.

3. Mathematical Modeling The CFD model used in this work is based on the extended two-fluid model, which uses granular kinetic theory to describe particle-particle interactions. Particles are considered to be smooth, spherical, inelastic, and to undergo binary collisions. The fundamental equations of mass, momentum, and energy conservation are then solved for each phase. Appropriate constitutive equations have to be specified in order to describe the physical and/or rheological properties of each phase and to close the conservation equations. The solids viscosity and pressure are computed as a function of granular temperature at any time and position. A more complete discussion of the extended two-fluid model, including the implementation of granular kinetic theory, can be found in Gidaspow.35 3.1. Continuity and Momentum Equation. Each phase is described using volume-averaged, incompressible, transient Navier-Stokes equations. The volume-averaged continuity equation is given by (i ) liquid, solids): ∂ (F R ) + ∇·(FiRiui) ) 0 ∂t i i

(2)

where R is the concentration of each phase, u is the velocity vector, and F is the density. Mass exchange between the phases, for instance, due to reaction or combustion, is not considered. The momentum balance for the liquid phase is given by the Navier-Stokes equation, modified to include an interphase momentum transfer term: ∂ (F R u ) + ∇·(FlRlulul) ) -Rl∇p + FlRlg + ∇·τl + Fkm ∂t l l l (3) where g is the acceleration of gravity, p is the thermodynamic pressure, Fkm is the sum of the interfacial forces (including the

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Figure 6. Contour plots for (A) particle concentration and (B) liquid velocity taken at regularly spaced axial positions over the 10 m control volume. Obtained from numerical simulations of the following conditions: dp ) 90 µm, V ) 3.0 m/s, and R j s ) 0.19.

drag force FD, the lift force FL, the virtual mass force FVM, the wall lubrication force FWL and the turbulent dispersion force FTD). The liquid-phase stress tensor, τl, can be represented as 2 τl ) µl[∇ul + (∇ul)T] - µl(∇·ul)I 3

(4)

The solids phase momentum balance is given by ∂ (F R u ) + ∇·(FsRsusus) ) -Rs∇p + FsRsg + ∇·τs + Fkm ∂t s s s (5) The solids stress tensor, τs, can be expressed in terms of the solids pressure, Ps, bulk solids viscosity, ζs, and shear solids viscosity, µs:

{

2 τs ) (-Ps + ζs∇·us)I + µs [∇us + (∇us)T] - (∇·us)I 3

}

(6)

3.2. Kinetic Theory of Granular Flow. This is, strictly speaking, a class of models based on the kinetic theory of gases, generalized to take account of inelastic particle collisions. In these models, the constitutive elements of the solids stress are functions of the solids phase granular temperature, Θs, defined

to be proportional to the mean square fluctuating particle velocity resulting from interparticle collisions: Θs ) u′s2/3, where u′s is the solids fluctuating velocity. In most kinetic theory models, the granular temperature is determined from a transport equation. The conservation of the solids fluctuating energy balance36 can be written as 3 ∂ (R F Θ ) + ∇·(RsFsusΘs) ) τs:∇us + ∇·(ks∇Θ) - γs + 2 ∂t s s s Ωls (7)

[

]

The left-hand side of this equation represents the net change of fluctuating energy. The first term on the right-hand side represents the fluctuating energy due to solids pressure and viscous forces. The second term is the diffusion of fluctuating energy in the solids phase. The third term, γs, represents the dissipation of fluctuating energy and Ωls is the exchange of fluctuating energy between the liquid and solids phase. Although eq 7 can be solved for the granular temperature, the procedure is complex and the boundary conditions are not well understood. The procedure is also computationally expensive. A simpler and computationally cheaper method is to use an algebraic expression, where local equilibrium of generation and dissipation of fluctuating energy is assumed. Boemer et al.36 simplified eq 7 to


production ) dissipation ⇒ τsij

∂Ui ) γs ∂xj

(8)

( (

γs ) 3(1 - e2)Rs2Fsg0Θs

4 dp

Popular models for the radial distribution function are given by Gidaspow:35 g0(Rs) ) 0.6(1 - (Rs /Rsm)1/3)-1

The dissipation fluctuating energy is35

))

Θs - diVU π

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(10)

and Lun and Savage:37 (9)

where e is the coefficient of restitution for particle collisions, dp is the particle diameter, and g0 is the radial distribution function at contact. The restitution coefficient, which quantifies the elasticity of particle collisions (one for fully elastic and zero for the fully inelastic), was taken as 0.9. The radial distribution function, g0, can be seen as a measure of the probability of interparticle contact.

g0(Rs) ) (1 - (Rs /Rsm))-2.5Rsm

(11)

where Rsm is the volume fraction of a settled bed of solids. The g0 function becomes infinite when the in situ solids volume fraction approaches Rsm. A value of Rsm ) 0.64 is often assumed for a random packing of monosize spheres. In practice, though, the value of the settled bed volume fraction should be measured, as it depends primarily on particle sphericity and particle size distribution.38

Figure 7. Predicted liquid velocity profiles for (A) dp ) 90 µm, V ) 3.0 m/s and R j s ) 0.19; (B) dp ) 270 µm, V ) 5.4 m/s and R j s ) 0.20; (C) dp ) 480 µm, V ) 3.44 m/s and R j s ) 0.203. Measurements of local particle velocity shown in panel A from Gillies et al.13

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The solids pressure represents the solids phase normal forces caused by particle-particle interactions. In the kinetic theory of granular flow, both the kinetic and the collisional contributions are considered. The particle pressure consists of a kinetic term corresponding to the momentum transport caused by particle velocity fluctuations and a second term due to particle collisions:35 Ps ) FsRsΘs + 2FsRs2Θs(1 + e)g0

(12)

In eq 6, the solids bulk viscosity accounts for the resistance of the granular particles to compression and expansion and has the form39 4 ζs ) Rs2Fsdpg0(1 + e) 3

Θπ

(13)

The solids shear viscosity contains terms arising from particle momentum exchange due to collision and translation. The solids shear viscosity is expressed as a sum of the kinetic and collisional contributions: µs ) µs,col + µs,kin

4 µs,col ) Rs2Fsdpg0(1 + e) 5

Θπ

2 5√π Fsdp 4 1 + η(1 + e)g0Rs √Θ 48 (1 + e)g0 5

(

)

(16)

and Lun and Savage37

µs,kin

(

)(

)

8 1 + η(3η - 2)g0Rs 5 √Θ 2-η (17)

where η ) 1/2(1 + e) 3.3. Interfacial Forces. The interphase momentum transfer between solids and liquid due to drag force is given by 1 3 FD ) CDRsFl |ul - us |(ul - us) 4 dp

(18)

The drag coefficient CD has been modeled using the Gidaspow model,35 which employs the Wen and Yu model when Rs > 0.2 and employs the Ergun model when Rs e 0.2. The lift force can be modeled in terms of the slip velocity and the curl of the liquid phase velocity40,41 as FL ) CLRsFl(us - ul) × ∇ × ul

(19)

The lift coefficient has been assigned a value of 0.1 in the present simulation. This value is within the range suggested in the literature.41 The wall lubrication force, which is in the normal direction away from the wall and decays with distance from the wall, is expressed as42 FWL ) -RsFl

[

(

∇Rc ∇Rd 3 CD υtc R F (u - uc) 4 dp σtc d c d Rd Rc

)

(21)

Here, σtc is the turbulent Schmidt number for continuous phase volume fraction, taken here to be σtc ) 0.9. 3.4. Turbulence Equations. The turbulence model used for the liquid phase is a variant of the two-equation k-ε model, which is given in standard form as:

((

) )

((

) )

µl,tur ∂k ∂ ∂ ∂ (FlRlk) + (FlRlulk) ) Rl µ + + ∂t ∂xi ∂xi σk ∂xi Rl(G - Rlε) (22) µl,tur ∂ε ∂ ∂ ∂ (FlRlε) + (FlRlulε) ) Rl µ + + ∂t ∂xi ∂xi σ∈ ∂xi ε Rl (Cε1G - Cε2Rlε) (23) k

(15)

and the kinetic component is determined based on35

1 5√π 8 Fd ) + Rs 96 s p ηg0 5

FTD )

(14)

The collisional component of the solids viscosity is modeled as

µs,kin )

Here, ur ) ul - us is the relative velocity between phases, dp is the mean particle diameter, yw is the distance to the nearest wall, and nw is the unit normal pointing away from the wall. The wall lubrication constants C1 and C2, as suggested by Antal et al.42 are -0.01 and 0.05, respectively. The turbulent dispersion force is modeled based on the Favre average of the interphase drag force using43

(ur - (ur·nw)nw) dp max C1 + C2 , 0 dp yw

]

(20)

In these equations, G represents the generation of turbulent kinetic energy due to the mean velocity gradient. For the liquid phase, a k-ε model is applied with its standard constants: Cε1 ) 1.44, Cε2 ) 1.92, Cµ ) 0.09, σk ) 1.0, σε ) 1.3. No turbulence model is applied to the solids phase but the influence of the dispersed phase on the turbulence of the continuous phase is taken into account with Sato’s additional term.44 3.5. Boundary Conditions. At the inlet, velocities and concentrations of both phases are specified. At the outlet, the pressure is specified (atmospheric). At the wall, the liquid velocities were set to zero (no- slip condition). The velocity of the particles was also set at zero. To initiate the numerical solution, the average solids volume fraction and a parabolic velocity profile are specified as initial conditions. 4. Numerical Solution The system of equations, with the aforementioned boundary conditions, was solved using the commercial flow simulation software ANSYS CFX 10.0. Mass and momentum equations were solved using a second-order implicit method for space and a first-order implicit method for time discretization. The conservation equations were discretized using the control volume technique. The discretization of the three-dimensional domain resulted in 386 340 cells and the grid structure shown in Figure 1. The SIMPLE algorithm was employed to solve the pressurevelocity coupling in the momentum equations. The high resolution discretization scheme was used for the convective terms. Initial simulations were carried out with a coarse mesh to obtain rapid convergence and an indication of the positions where a high mesh density was needed. Grid independence was examined, but further grid refinement did not result in significant changes to the simulations results. Three dimensional transient simulations were performed. In these simulations, a constant time step of 0.001 s and pipe length of 10.0 m were used. Timeaveraged distributions of flow variables are computed over a period of 100 s.


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Figure 8. Effect of particle size on concentration profile: (A) dp ) 90 µm, V ) 3.0 m/s, R j s ) 0.19, and D ) 103 mm; (B) dp ) 125 µm, V ) 3.0 m/s, R js ) 0.20, and D ) 54.9 mm; (C) dp ) 165 µm, V ) 4.17 m/s, R j s ) 0.189, and D ) 51.5 mm. (D) dp ) 270 µm, V ) 5.4 m/s, R j s ) 0.20, and D ) 103 mm. (E) dp ) 440 µm, V ) 3.0 m/s, R j s ) 0.20, and D ) 54.9 mm. (F) dp ) 480 µm, V ) 3.44 m/s, R j s ) 0.203, and D ) 51.5 mm.

5. Results and Discussion The CFD simulations were carried out to match the experimental conditions of Roco and Shook,27 Schaan et al.,45 Gillies and Shook,12 Gillies et al.,13and Kaushal and Tomita.24 Table 1 summarizes the experimental conditions and measurement techniques employed to measure pressure drop, particle concentration profiles, and mixture velocity. Each data set described in Table 1 was simulated using the model described in the previous sections. The local, time-averaged particle concentration profiles, particle and liquid velocities, and frictional pressure drop were obtained using the model and then were compared (where possible) with the existing experimental data. A wide range of particle size (90-500 µm), solids volume concentration (8-45%), mixture velocity (1.5-5.5 m/s), and pipe diameter (50-500 mm) were considered. In the figures discussed here,

y/D is the dimensionless position along the pipe’s vertical axis, where y is the distance from the pipe bottom, the local particle concentration is Rs, the liquid velocity is ul, and the frictional pressure drop is ∆p/L. 5.1. Sensitivity Analysis. Initially, numerical simulations were conducted for three cases to demonstrate the effects that the inclusion of different forces had on the quality of the predictions. In the first case, the simulations were carried out with the k-ε model based on the kinetic theory of granular flow and drag force only. The model predictions are shown in Figure 2A, where the predicted particle concentration profile shows a peak near the bottom of the pipe. The local solids volume fraction rapidly approaches zero at a vertical position of y/D ≈ 0.85. The second simulation included drag force and the turbulent dispersion force. The predicted result is in good

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Figure 9. Effect of pipe diameter on concentration profile, dp ) 165 µm: (A) D ) 51.5 mm, R j s ) 0.0918, and V ) 3.78 m/s; (B) D ) 51.5 mm, R j s ) 0.286, and V ) 4.33 m/s; (C) D ) 263 mm, R j s ) 0.0995, and V ) 3.5 m/s; (D) D ) 263 mm, R j s ) 0.268, and V ) 3.5 m/s; (E) D ) 495 mm, R j s ) 0.104, and V ) 3.16 m/s; (F) D ) 495 mm, R j s ) 0.273, and V ) 3.16 m/s.

agreement with the experimental data. In the third case, when all the forces (drag, turbulent dispersion, lift, and wall lubrication force) were included, the results showed no significant improvement. Additionally, two radial distribution function and kinetic solids viscosity models35,37 were tested. Essentially, there is no difference between the two, as shown in Figure 2B. On the basis of these observations, all subsequent simulations were conducted using (i) k-ε turbulence model with the kinetic theory of granular flow; (ii) drag and turbulent dispersion forces; and (iii) the Gidaspow radial distribution function/kinetic solids viscosity model.35 Lift and wall lubrication forces were neglected in these simulations. 5.2. Solids Concentration Profiles. Because concentration profiles depend on many parameters, including mixture velocity,

mixture density, pipe diameter, particle size, and particle density, it is important to test the ability of a model to predict these profiles. Additionally, the knowledge of solids distribution across the pipe cross-section is essential in the evaluation or prediction of pipeline wear.46 Figure 3 shows the experimental and predicted concentration profiles for 270 µm sand slurries flowing at a constant mixture velocity (5.4 m/s) in a 100 mm pipeline. In situ solids volume concentrations of 10 to 45% were tested (and simulated). The experimental data were initially reported by Gillies et al.13 Generally, the numerical predictions show reasonable agreement with the experimental results. It can be observed from the figures that for a given velocity, increasing the in situ solids concentration reduces the asymmetry of the concentration profiles because of increased particle interactions. For these slurries, fluid turbulence is not completely effective in suspending the particles; instead, suspension results partly


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Figure 10. Effect of pipe diameter on concentration profile for slurries of fine particles, V ) 3.0 m/s: (A) D ) 54.9 mm, dp ) 125 µm, R j s ) 0.30; (B) D ) 54.9 mm, dp ) 125 µm, R j s ) 0.40; (C) D ) 103 mm, dp ) 90 µm, R j s ) 0.29; (D) D ) 103 mm, dp ) 90 µm, R j s ) 0.33; (E) D ) 150 mm, dp ) 90 µm, R j s ) 0.32; (F) D ) 150 mm, dp ) 90 µm, R j s ) 0.39.

from particle-particle interactions.13 This is a good test of the model’s ability to predict the combined importance of fluid turbulence and shear-dependent (Bagnold-like) particle-particle interactions. Overall, the results are encouraging. In Figure 3F, the experimentally determined concentration profile exhibits a reversal in local concentration near the pipe invert. This reversal is not predicted in our simulations and may be related to the existence of near-wall forces described previously. Figures 4 and 5 show experimental data and numerical predictions for 90 µm sand slurries flowing in 100 and 154 mm pipelines, respectively. The particles in these slurries are effectively suspended by fluid turbulence, as the relatively uniform concentration profiles at low in situ volume fractions attest. These concentration profiles can be accurately predicted using a Schmidt-Rouse 1D turbulent diffusion model.11,19-23

They also provide a good test of the numerical model’s ability to predict the importance of the turbulent dispersion forces. Overall, the agreement between the numerical predictions and experimental results is good. Contour plots of particle concentration and liquid velocity along the pipe cross section at axial positions separated by 1.25 m intervals are shown in Figure 6. Significant differences in particle concentration and liquid velocity can be observed between the first and fourth axial positions. The contour plots shown for axial positions 5 through 8 are nearly identical, indicating that the numerical simulations are providing results for fully developed flow. 5.3. Velocity Profiles. Velocity profiles in horizontal slurry flow are directly linked to the concentration profile; as such, they are also dependent upon particle size, in situ solids concentration,

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Figure 11. Parity plot for frictional pressure gradient (experimental data: (O) Schaan et al.;45 (∆) Gillies and Shook;12 (]) Gillies et al.;13 and (0) Kaushal and Tomita24).

and mixture velocity. Figure 7 shows three pairs of illustrations. The left-hand figure shows a traditional velocity profile, where the local time-averaged liquid velocity along the pipe’s vertical axis is plotted. The corresponding contour plot is the type that is readily attainable from CFD simulations. Figure 7A compares a measured particle velocity profile13 with predictions for a 90 µm sand slurry (R j s ) 0.19, V ) 3.0 m/s). Recall that the corresponding concentration profile, which is shown in Figure 4A, is nearly symmetric. Note also the agreement between the measured particle Velocity and the predicted fluid Velocity is excellent, confirming that the local, time-averaged slip velocity approaches zero for slurries of this type. Figure 7 panels B and C show predicted liquid velocity profiles for slurries containing coarser particles (270 and 480 µm, respectively). It can be seen that the velocity profiles become increasingly asymmetrical with increasing particle size. The maximum local velocity is found in the upper portion of the pipe and not at the centerline. This phenomenon has been demonstrated experimentally.13,27 Note also from the contour plots that the velocity distribution in a horizontal plane is symmetrical about the pipe axis. 5.4. Effect of Particle Diameter. Figure 8 shows the measured and predicted concentration profiles for four different slurries, each with a different particle size (90, 125, 165, 270, 440, and 480 µm). In situ solids volume fractions are comparable for these slurries (R j s ≈ 0.2). The experimental data shown in these figures represent a broad spectrum of fluid turbulence effects on particle suspension, from highly effective (Figure 8A) to completely ineffective (Figure 8F). Concentration profiles of the type shown in Figure 8F, for very coarse particles, depend primarily on in situ solids volume fraction, with only minimal dependence on mixture velocity or pipe diameter. Thus, the measured concentration profile of Figure 8F can be considered to be typical of one that would be found for any coarse particle slurry (dp > 300 µm). In all cases, the agreement between measured and predicted profiles is encouraging. In Figure 8E,F, a distinct reversal in the concentration profile can be seen near the pipe invert (y/D < 0.2). This is related to the near-wall lift force described previously,24,32 which occurs when the particle is large relative to the viscous sublayer thickness. The current version of the CFD model is unable to reproduce this concentration reversal. 5.5. Effect of Pipe Diameter. To investigate the effect of pipe diameter on the performance of the numerical model developed here, the flow of a number of slurries in pipes of different diameter

was considered. Figure 9 shows both experimental and predicted concentration profiles for a 165 µm sand slurry flowing in pipes that are 51.5, 103, 150, 263, and 495 mm in diameter. The measured concentration profiles were taken from Roco and Shook.27 This particular particle size was chosen for two reasons: data had been collected from experiments conducted with a wide range of pipe diameters and this sand size exhibits strong pipe diameter-dependent concentration profiles. The relative importance of fluid turbulence vis-a`-vis particle-particle interactions in determining the shape of the concentration profiles with increasing pipe diameter is clearly shown in Figure 9. The predictions are in good agreement with the experimental data for all pipe diameters. Figure 10 provides similar findings for experimental measurements made with smaller particles of differing size and shape. The mixture velocity is the same for each panel (V ) 3.0 m/s). Figure 10 panels A and B show the experimental measurements made by Kaushal and Tomita24 for slurries of 125 µm glass spheres in water flowing in a 54.9 mm pipeline loop. Figure 10 panels C-F show results and predictions for narrowly sized 90 µm sand slurries flowing in 100 and 150 mm pipelines. No noticeable effect of pipe diameter is observed for these concentration profiles, because the particles are relatively fine and the mixture velocity in each case is significantly greater than the deposition velocity. Again, the concentration profile reversal seen in Figure 10 panels A and B is not accurately reproduced with the current model. Otherwise, the model’s performance is satisfactory. 5.6. Pressure Drop. Pipeline pressure drop is one of the most important parameters in slurry pipeline design and operation. To validate the numerical results obtained with the CFD model, the simulation results were compared with the experimental data of Schaan et al.,45 Gillies and Shook,12 Gillies et al.13 and Kaushal and Tomita.24 As the information presented in Table 1 indicates, these experimental data were collected for a wide range of particle size, mixture velocity, in situ solids volume fraction, and pipe diameter. The comparison of measured and predicted frictional pressure drop results is shown in Figure 11. The predicted pressure drop is in good agreement with the experimental measurements for the wide range of slurry flow conditions represented by the data sets to which the numerical simulations were compared. 6. Conclusions A transient three-dimensional (3D) hydrodynamic model based on the kinetic theory of granular flow has been developed for horizontal slurry pipelines. Frictional pressure gradients and local, time-averaged solids concentration and liquid/particle velocities were obtained. A detailed comparison between the CFD simulation results and an expansive experimental data set (reported by Roco and Shook,27 Schaan et al.,45 Gillies and Shook,12 Gillies et al.13 and Kaushal and Tomita24) was presented. Only slurries containing narrowly sized particles were simulated in this study. Generally, excellent agreement between the predicted and the experimental data was obtained for a wide range of in situ solids volume fractions, particle diameters, mixture velocities, and pipe diameters. The degree of asymmetry in the concentration profiles depends primarily upon the particle diameter, the mixture velocity, and the in situ solids volume fraction. The CFD model described here is capable of predicting particle concentration profiles for fine particle slurries where fluid turbulence is effective at suspending the particles. It also performs satisfactorily when the particles are coarse and concentration profiles are primarily dependent upon the in situ solids volume fraction. In experimental data sets where the nearwall lift force was of sufficient magnitude to cause a reversal in the concentration profile near the pipe invert, the current CFD


model was not able to reproduce this behavior. The model described here, including the associated boundary conditions, is complete in the sense that no experimentally determined flow measurements are required as input data. In this sense, the present model can be considered to be superior to existing correlation-based, semiempirical models. Acknowledgment The authors gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and Syncrude Canada Ltd. Literature Cited (1) Durand, R. Basic Relationships of the Transportation of Solids in Pipes: Experimental Research. Proceedings of the 5th Congress, International Association of Hydraulic Research, Minneapolis, Minnesota, 1953. (2) Wasp, E. J., Aude, T. C., Kenny, J. P., Seiter, R. H., Jacques, R. B. Deposition Velocities, Transition Velocities and Spatial Distribution of Solids in Slurry Pipelines. Proceedings of the 1st International Conference on the Hydraulic Transport of Solids in Pipes (Hydrotransport 1), BHRA Fluid Engineering: Cranfield, U.K., Paper H4, 1970; pp 53-76. (3) Shook, C. A.; Daniel, S. M. Flow of Suspensions of Solids in Pipelines, Part 2: Two Mechanisms of Particle Suspension. Can. J. Chem. Eng. 1968, 46, 238–244. (4) Shook, C. A.; Daniel, S. M. A Variable Density Model of the Pipeline Flow of Suspensions. Can. J. Chem. Eng. 1969, 47, 196. (5) Oroskar, A. R.; Turian, R. M. The Critical Velocity in Pipeline Flow of Slurries. AIChE J. 1980, 26, 551–558. (6) Wilson, K. C. A Unified Physical-based Analysis of Solid-Liquid Pipeline Flow. Proceedings of the 4th International Conference of Hydraulic Transport of Solids in pipes (Hydrotransport 4), BHRA Fluid Engineering: Cranfield, U.K., Paper A1, 1976; 1-16. (7) Doron, P.; Granica, D.; Barnea, D. Slurry Flow in Horizontal PipesExperimental and Modeling. Int. J. Multiphase Flow 1987, 13, 535–547. (8) Wilson, K. C.; Pugh, F. J. Dispersive-Force Modeling of Turbulent Suspension in Heterogeneous Slurry Flow. Can. J. Chem. Eng. 1988, 66, 721– 727. (9) Nassehi, V.; Khan, A. R. A Numerical Method for the Determination of Slip Characteristics between the Layers of a Two-Layer Slurry Flow. Int. J. Numer. Method Fluids 1992, 14, 167–173. (10) Gillies, R. G.; Shook, C. A.; Wilson, K. C. An Improved Twolayer Model for Horizontal Slurry Pipeline Flow. Can. J. Chem. Eng. 1991, 69, 173–178. (11) Gillies, R. G.; Shook, C. A. Concentration Distribution of Sand Slurries in Horizontal Pipe Flow. Part. Sci. Technol 1994, 12, 45–69. (12) Gillies, R. G.; Shook, C. A. Modelling High Concentration Slurry Flows. Can. J. Chem. Eng. 2000, 78, 709–716. (13) Gillies, R. G.; Shook, C. A.; Xu, J. Modelling Heterogeneous Slurry Flows at High Velocities. Can. J. Chem. Eng. 2004, 82, 1060–1065. (14) Doron, P.; Barnea, D. A Three-Layer Model for Solid-Liquid Flow in Horizontal Pipes. Int. J. Multiphase Flow 1993, 19, 1029–1043. (15) Doron, P.; Barnea, D. Flow Pattern Maps for Solid-Liquid Flow in Pipes. Int. J. Multiphase Flow 1996, 22, 273–283. (16) Ramadan, A.; Shalle, P.; Saasen, A. Application of a Three-Layer Modeling Approach for Solids Transport in Horizontal and Inclined Channels. Chem. Eng. Sci. 2005, 60, 2557–2570. (17) Hunt, J. N. The Turbulent Transport of Suspended Sediment in Open Channels. R. Soc. London, Proc., Ser. A 1954, 224, 322–335. (18) Karabelas, A. J. Vertical Distribution of Dilute Suspensions in Turbulent Pipe Flow. AIChE J. 1977, 23, 426–434. (19) Kaushal, D. R.; Tomita, Y. Solids Concentration Profiles and Pressure Drop in Pipeline Flow of Multi-Sized Particulate Slurries. Int. J. Multiphase Flow 2002, 28, 1697–1717. (20) Kaushal, D. R.; Tomita, Y.; Dighade, R. R. Concentration at the Pipe Bottom at Deposition Velocity for Transportation of Commercial Slurries through Pipeline. Powder Technol 2002, 125, 89–101. (21) Kaushal, D. R.; Tomita, Y. Comparative Study of Pressure Drop in Multisized Particulate Slurry Flow through Pipe and Rectangular Duct. Int. J. Multiphase Flow 2003, 29, 1473–1487.

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ReceiVed for reView October 6, 2008 ReVised manuscript receiVed March 8, 2009 Accepted March 11, 2009 IE801505Z

Hydrodynamic Simulation of Horizontal Slurry Pipeline Flow Using

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