Research Developments in Pipeline Transport of Settling Slurries

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Powder Technology 1

Research developments in pipeline transport of settling slurries

Vaclav Matouxek

Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands

Received 9 November 2004; received in revised form 22 March 2005; accepted 18 May 2005

Abstract

The designing of slurry pipelines requires prediction of pipeline hydraulic performance. The preferable predictive tool is a model based on

a physical description of slurry flow behavior in a pipeline. Various mechanisms that affect pipeline flow behavior of settling slurries with a

Newtonian conveying liquid are described. Recent developments in measuring, understanding and modeling of the mechanisms are

discussed. An implementation of recent research results to current versions of a predictive model for stratified flows of settling slurries (a

two-layer model) is surveyed.

D 2005 Elsevier B.V. All rights reserved.

Keywords: Slurry flow; Solids friction; Two-layer model

1. Introduction

In the hydraulic transport of solids using pipelines the

liquid is used as a vehicle that carries solid particles to their

destination at the end of a slurry pipeline. In practice,

slurries of different liquids and solids are conveyed through

pipelines. This paper discusses the transport of ‘‘settling’’

slurries with a Newtonian carrier. Settling slurries are

mixtures in which the solid particles tend to separate from

the carrying liquid owing to the action of gravity force. If

solid particles cannot be maintained in suspension by the

turbulent diffusive action of the conveying liquid during the

transportation through horizontal or inclined slurry pipe-

lines, they settle down to the bottom of the pipe. A sand–

water mixture is typical settling slurry. Such slurry is

discrete, i.e. even the finest sand particles are too coarse to

combine with the conveying water to produce a stable

homogeneous medium. A settling-slurry flow tends to

stratify. In horizontal and inclined pipes, flows of settling

slurries at the velocities usually used during practical

operations are characterized by a non-uniform distribution

of particles across a pipe cross-section. More particles

gather near the bottom than near the top of a pipe cross-

0032-5910/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.powtec.2005.05.054

E-mail address: [email protected].

section. The flow is considered stratified if a portion of

solids forms a granular bed (stationary or sliding) at the

bottom of a pipe. The flow is fully stratified if all particles

occupy the granular bed (particles maintain a virtually

permanent contact with other particles within the bed) or

partially stratified if only a portion of particles occupies the

bed and the rest of particles maintains either sporadic

contact or no contact with other particles.

The design of slurry pipelines carrying settling slurries

requires prediction of hydraulic performance of the pipeline.

This is given by a relationship between the slurry flow rate

and the total pressure differential developed over a pipeline

at the flow rate in question. For settling slurries, the pressure

drop due to friction over the length of a pipeline is closely

associated with the internal structure of the slurry flow.

Since the early 1950s systematic research on the flow of

settling slurries in pipelines has been conducted. In the

pioneer era, the investigation was focused primarily on a

collection of basic experimental data (pressure drops at

different velocities) for flows of different pipe and particle

sizes. The collected databases served as a basis for a

construction of empirical models obtained by formulating

suitable dimensionless numbers and finding the best fit by

using the data regression. The 1970s and 1980s brought

works giving an insight into fundamental principles of

pipeline flows for settling slurries—basics of a macroscopic

56 (2005) 43 – 51

V. Matousek / Powder Technology 156 (2005) 43–5144

two-layer model by Wilson and of a microscopic slurry-flow

model by Shook and Roco. Contemporary research makes

use of the rapid development of computational and

measuring techniques permitting the examination of slurry

flow processes in still closer details and leading to better

understanding of the complex nature of the behavior of

settling-slurry flows in pipelines. Currently, practical inter-

ests focus attention on research of settling slurries that are

highly concentrated (volumetric concentrations of solids

higher than 35% approximately). Furthermore, the contem-

porary research tends to focus on analyzing settling-slurry

flow behavior under specific conditions.

A typical approach to the investigation of a pipeline flow

of settling slurries is composed of the following steps:

– observation of slurry flows under certain specific, well-

defined conditions in the closest possible detail and using

modern measuring techniques (including concentration

and/or velocity distributions)

– on the basis of the observations, determination of the

prevailing mechanisms governing the observed slurry

flow

– modeling of the processes and sub-processes in order to

simulate the slurry-flow behavior and flow parameters

important in practice.

Despite a fair degree of progress made over the years,

current understanding of the complex behavior of a pipe

flow of settling slurries is still far from complete.

2. Measuring techniques

2.1. Distribution of solids

A radiometric method has been most often used to

determine a distribution of solids across vertical cross-

sections of laboratory test pipes. At present, radiometric

density meters adapted as concentration profilers for slurry

pipes are in use in the laboratories of Saskatchewan

Research Council (SRC) and Delft University of Technol-

ogy (DUT, [1]). A radiometric density meter requires a

two-point calibration at each measuring position in a pipe

cross-section.

Delft Hydraulics uses a set of conductivity probes

mounted as pairs of electrodes along the perimeter of the

pipe wall. The temperature and salinity of the carrying

liquid strongly influences conductivity and thus the accu-

racy of the instrument. A similar electro-resistance sensor

was used by Simkhis et al. [2] to detect dunes in a slurry

pipe.

Recently, the magnetic resonance imaging (MRI) method

has been starting to gain a footing in laboratory slurry-pipe

experiments [3]. An advantage of the method is that it can

collect an entire concentration profile in one moment, a

disadvantage is a relatively complex reconstruction process

required to obtain a concentration-profile image from the

measured signal. As yet, the high price of the instrument

reduces its application to small pipes. The measuring

principle sets a maximum limit velocity at which a profile

can be measured in a pipe. Pullum and Graham [3] reported

1 m/s as the approximate maximum mean velocity for a

100-mm pipe.

2.2. Distribution of solids velocity

In general, it is difficult to measure the distribution of the

local velocity of solid particles. Usually, settling slurries are

transported at concentrations too high for the use of optical

methods like LDA and PIV.

In addition to a concentration profile, the MRI method

also provides a liquid-velocity profile of a flow. The cross-

correlation of the electrical resistance (or conductivity)

signals from two sensors located near each other is the

measuring principle most often used to determine local

solids velocity in a slurry pipe. The SRC uses an intrusive

conductivity probe as a solids velocity profiler (e.g. [4]).

3. Flow-friction mechanisms

3.1. Friction due to permanent contact of solid particles

with pipe wall

Particles traveling within a granular body in contact

with a pipe wall exert a solids stress against the pipe wall

that contributes significantly to the total friction of the

slurry flow. Wilson’s method [5,6] for the determination of

the normal solids stress at the pipe wall is based on the

simplified assumption that there is a hydrostatic-type of

distribution of the normal stress in the vertical direction

through a granular bed submerged in liquid. Furthermore,

it assumes that the normal stress at the wall, rswC,

produces the solids shear stress, sswC, at the pipe wall

according to Coulomb’s law (the normal and shear stresses

are mutually related through the friction coefficient, ls),

sswC=ls(qs�ql)gCvb(D / 2b) (sin b�b cos b).It is not easy to verify this method, since it is difficult to

eliminate other effects and measure only the effects caused

by the Coulombic solids stresses during pipe tests. Tests

with extremely high concentrated flows (plug flows) by

Wilson et al. [6] justified the proposed solids-stress theory.

The tests with extremely highly concentrated sand slurries

conducted by Korving [7] confirmed that the Wilson’s plug

model based on the solids-stress theory can successfully

interpret Korving’s data (Fig. 1) if the coefficient of

mechanical friction between the plug and the pipe wall is

considered dependent on the mean concentration of solids in

the pipe (i.e. in the plug).

Another suitable condition for the verification of the

theory is a fully stratified flow through a descending pipe

inclined at a small angle (say 25–35-), in which the velocity

Volumetric Concentration and VelocityProfiles

V = 4 [m/s]

0.5

0.60 1 2 3 4 5

Velocity [m/s]

volumetric concentration

solids velocity

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mean slurry velocity Vm [m/s]

Hyd

raul

ic g

radi

ent

[-]

Plug 1800 kg/m3Plug 1780 kg/m3Plug 1730 kg/m31800 kg/m31780 kg/m31730 kg/m3water

Fig. 1. Comparison of experiments with the plug model for the 0.103-mm sand in the 158-mm pipe (from Korving [7]).

V. Matousek / Powder Technology 156 (2005) 43–51 45

of the sliding bed is almost the same as the velocity of the

water current above the bed. Under these conditions the

solids effect on the total hydraulic gradient is exclusively

due to the acting solids stress at the pipe wall in contact with

the sliding bed. The test presented by Matousek [8] showed

that mechanical friction at the boundary between the sliding

granular bed and the pipe wall was successfully predicted by

Wilson’s friction law with the friction-coefficient value of

about 0.55 valid for all the sands and gravels that were

tested.

A problem arises in the interpretation of the verification

tests if a drop in local concentration occurs at the bottom of

the sliding bed adjacent to the pipe wall (see Fig. 2). This

phenomenon seems to occur if the velocity of the sliding

bed is high. The simulation of a fully stratified flow of

coarse particles using Discrete Element Method suggests

that this phenomenon is the result of interactions among

particles rolling over each other at the bottom of the sliding

bed [9] (Fig. 3).

Coulomb’s law assumes a constant value of the friction

coefficient relating the normal and shear stresses. However,

variations in the coefficient value for a pipe wall with a local

Fig. 2. Concentration profile and thickness of the sliding bed in the pipe

inclined to �35- (the 1.4–2.0-mm-sand mixture) (from Matousek [8]).

concentration of solids or the velocity of the bed can be

expected. Experimental evidence is rather limited and

further investigation is required to provide better under-

standing of the exact relationship between the solids normal

and shear stresses at a pipe wall.

3.2. Friction due to sporadic or zero contact of solid

particles with pipe wall

Basically, friction due to the presence of solid particles

suspended in a flow is a result of processes in a relatively

thin layer near the pipe wall. To identify interactions

between particles and the conveying liquid and interactions

0.0

0.1

0.2

0.3

0.4

0.0 0.2 0.4 0.6 0.8 10.

Volumetric Concentration [-]

y/D

[-]

Fig. 3. Distribution of solids concentration and velocity in the sliding bed at

Vm=4 m/s simulated using DEM (from Stienen et al. [9]).

Fig. 4. Pressure gradient ratio Im/I l as a function of density ratio Sm=qm/q l

at velocity 4 m/s. Parity line Im=SmI l. Index m=slurry and l= liquid. (from

Schaan et al. [14]).


among particles in the region near a pipe wall and to

describe their effect on flow friction is one of the greatest

challenges to those researching the pipeline flow of settling

slurries. As yet, available measuring techniques are not

capable of recognizing the interactive processes within a

narrow region near a pipe wall with sufficient distinction.

Crucial questions must be answered before the pipe-wall

friction of suspension flow can be determined. Friction

models use the wall shear stress as a measure of suspension

friction in a pipe flow. Basically, two wall shear stresses are

recognized: the Fliquid-like_ (or viscous) shear stress and the

shear stress caused by the contacts (impingements) of

traveling suspended particles with a pipe wall. In contrast

to the Coulombic solids stress, the solids stress due to

impingements is velocity-dependent (kinetic). For different

flow situations, it is difficult to determine whether or not

solid particles traveling near the pipe wall contribute to the

wall shear stresses, and if so to which shear stress they

contribute and with which portion of their total concen-

tration. Essentially, friction behavior of suspended coarse

particles differs from that of fine particles. The threshold for

particle diameter between the coarse-slurry behavior and the

fine-slurry behavior is rather vague but it is assumed that the

threshold particle size is related to the thickness of the

viscous sub-layer adjacent to the pipe wall.

Coarse particles in non-stratified flows (slurry flows

without Coulombic stresses) have contacts with each other

and with a pipe wall and so contribute to the total friction of

slurry flow. The contacts between particles and the pipe wall

are sporadic rather than permanent and they are a result of

turbulent dispersive action (turbulent eddies tend to disperse

solid particles to all directions) and collisional dispersive

action (particles are impelled in the direction of the pipe

wall by the collision with other particle of different velocity

in the flow). The solids stress due to particle collisions with

a pipe wall can be diminished by the hydrodynamic off-wall

lift.

Settling slurries composed of fine particles tend to

exhibit a rather different friction pattern. Near the pipe

wall, fine particles travel submerged within the viscous sub-

layer. Within this layer the effects of carrier turbulence and

hydrodynamic lift on solid particles are negligible. The

increase in the wall shear stress due to the presence of solid

particles in the near-wall region is due to increased viscous

friction rather than mechanical contact friction. Particles

locked inside the viscous sub-layer increase the density of

the carrying liquid and so increase the viscous shear stress at

the pipe wall. Nevertheless, the solids dispersive stress

seems to occur in the fine slurries if the concentration of

solids near the wall becomes high.

3.2.1. Collisional stress

In the near-wall region the velocity gradient is steep and

particles with different local velocities collide with each

other and with the pipe wall. The collisions generate the

solids dispersive stress (first observed and described by

Bagnold [10]) acting against the pipe wall. This stress is a

source of solids friction, additional to the liquid-like friction,

in a pipe. The most appropriate geometry for an observation

of the collisional stresses is a vertical pipe in which the flow

is axisymmetric and there is no Coulombic stress. On the

other hand, the need to subtract the hydrostatic part from the

measured pressure differential is a potential source of

inaccuracy.

Shook and Bartosik [11,12] observed three coarse particle

fractions (1.37-mm, 1.5-mm, 3.4-mm) in two narrow vertical

pipes (D =26-mm and 40-mm) and interpreted the measured

solids effects on the total frictional loss as being due entirely

to the Bagnold dispersive force. They adapted the original

equation for Bagnold’s stress in the inertial regime of the

sheared annular flow, ssB=KBqsd250k

2(dv / dy)2 (KB is equal

to 0.013 according to Bagnold’s measurements, dv / dy is

velocity differential) to the axisymmetric vertical-pipe flow

and calibrated the equation by using pressure-drop data,

ssw=(8.3018�107 /Re2.317)D2qsd250k

1.5(slw /ll)2 in which

Re =VmqlD /ll and slw= ( fl / 8)qlVm2. The solids effect due to

Bagnold stresses predicted by Bartosik and Shook in a

vertical pipe is virtually negligible for particles finer than

coarse sand of the diameter 1 mm. The database of very

coarse vertical flows was extended by the Ferre’s tests with

glass spheres of 1.8 and 4.6 mm in a 40.9-mm pipe and

conveying liquids of two different viscosities [13]. The

calibration gave ssw=0.0214 /Res0.36(d50 /D)0.99k1.31qsVm

2 in

which Res=Vmqsd50 /ll.

Schaan et al. [14] found pressure drops higher than those

due to viscous liquid-like friction in highly concentrated

fine slurries (particle size 0.085-mm, 0.090-mm and 0.100-

mm) in horizontal pipes of two diameters at high velocities

(the mean velocity Vm=4 m/s in Fig. 4). At this velocity any

effects of flow stratification and Coulombic friction were

Fig. 6. Pressure gradient ratio and density ratio for the 0.37-mm sand flow

in the vertical 150-mm pipe (from Matousek [17]).


very small. The estimation of the thickness of the viscous

sub-layer led the authors to an assumption that the fine-sand

particles were too large to be submerged within the layer

and to affect the carrier properties there. Instead, the solids

friction was considered to be a result of the Bagnold’s

particle-wall interactions. The total wall shear stress in

axially symmetric flow and negligible Coulombic friction

was considered as composed of two components: the

mixture shear stress (determined as viscous shear stress

for the ‘‘equivalent liquid’’ having density of slurry and

viscosity of conveying liquid) and the solids shear stress of

Bagnold type, ssw=Kqsk2V 2

m. Gillies and Shook [15]

obtained K =0.00002 from pressure-drop data for the 175-

Am sand slurries at high velocities in a horizontal 495-mm

pipe.

Korving and Matousek [16] observed the same effect as

Schaan et al. (i.e. pressure drops higher than the Fequivalent-liquid_ prediction at high velocities) in highly concentrated

slurries of the fine sand (the mass-median diameter

d50=0.102 mm) in a horizontal 158-mm pipe of the MTI

laboratory in Kinderdijk, the Netherlands. They attributed

the high friction to the slurry viscous effects.

A comparison of the MTI test results with the tests in a

pipe of the similar size (the inner pipe diameter D =150 mm,

Laboratory of Dredging Engineering of DUT) and only

slightly coarser sand (d50=0.12 mm), showed a rather

different behavior [17]. At high velocities the 0.12-mm sand

slurry exhibited pressure drops lower than those of the

equivalent-liquid friction model, Im=SmIl, in both horizon-

tal and vertical pipe sections (see Fig. 5 for the results for

vertical flows, in Fig. 5 the Im for slurry flow and Il for

water flow are both in meters water column over 1-m length

of the pipe). Furthermore, the relative hydraulic gradient

Im/ Il appeared to be relatively weakly dependent on the

mean delivered solids concentration Cvd (particularly in the

vertical flow). This indicated that the lift force might be

involved (see next paragraph). The entire solids effect

Im� Il was considered to be the result of kinetic solids

Fig. 5. Pressure gradient ratio and density ratio for the 0.12-mm sand flow

in the vertical 150-mm pipe (from Matousek [17]).

friction and this assumed a combined effect of the stress due

to collisions between particles and the pipe wall and liquid

lift force acting on solid particles in the near-wall zone.

Considering the axisymmetric flow and no Coulombic

friction ssw=(Im� Il)qlgD / 4=0.991k0.81V 0.99m in the verti-

cal flow and ssw= (Im� Il)qlgD / 4=0.499k1.36V 1.02m in the

horizontal flow at velocities equal to and higher than 3.5 m/

s. The same tests with the coarser sand (d50=0.37 mm)

exhibit even lower solids friction and weaker effects of

solids concentration on the friction at high velocities (Fig. 6)

indicating that the repelling effect of the hydrodynamic lift

is more intensive on the 0.37-mm particles than on the 0.12-

mm particles.

As mentioned above, the problem with a verification (at

least for particles of medium size) of a collision-friction

model is that pressure drops due to solids presence in non-

stratified flows are a product of a combined effect of the

Bagnold collisional force and liquid lift force acting on solid

particles in the near-wall zone of the slurry flow rather than

a product of a pure effect of collisions between particles and

the pipe wall.

3.2.2. Hydrodynamic lift

The vertical-pipe experiments discussed above suggest

that flows of particles of moderate size in slurries of low and

moderate concentrations exhibit lower friction than finer

slurries. This is a confirmation of the trend reported first by

Newitt et al. [18] who observed that vertical flows of fine

slurries behave essentially as homogeneous Fequivalent_liquids and slurries of coarser particles (but smaller than

about 1 mm in diameter) essentially as conveying liquids

(there was virtually no effect of solids on flow friction).

The test results in Figs. 5 and 6 show low slurry friction

(much lower than predicted by the Fequivalent-liquid_friction model) and only weak variation in solids concen-

tration. Horizontal flows at high velocities showed the same

trend [17]. If it is assumed that the collisional stress is

exclusively responsible for the pressure drop due to solids

presence in flow and this stress tends to increase with the

Fig. 8. Relative bed roughness, ks/d50, for the Nikuradze’s friction equation

as function of particle mobility number hb, acryl particles of w*=0.72

(from Sumer et al. [24]).


solids concentration (according to both Bagnold and

Bartosik and Shook), the data suggest that there is a lift

force repelling particles from the wall (and thus diminishing

the collisional stress at the pipe wall) and that this lift force

increases with the solids concentration (up to the concen-

tration value of about 0.43). The increase in the lift force

with the concentration seems to be of the same order as the

increase in the concentration of the normal force due to

collisional stress acting at the pipe wall.

Clear evidence of the lift effect is the presence of locus

on a concentration-profile curve near the bottom of the

horizontal pipe occupied by a heterogeneous flow without

the bed. Concentration profiles with a local drop in

concentration just above the bottom of the pipe were

measured in different test loops in fast flows of slurries of

medium to coarse solids.

The lift force is a product of the interaction between a

solid particle and liquid flow of the steep velocity gradient

near the pipe wall. The velocity gradient across the particle

causes its rotation and develops a pressure differential over

the particle. This is responsible for the hydrodynamic force

acting on the particle in the direction perpendicular to the

pipe wall (the Magnus effect).

Wilson and Sellgren [19] analyzed the criteria governing

the interaction of a solid particle with the liquid velocity

profile and leading to the repulsion of the particle from the

wall of a horizontal pipe. According to the analysis the lift is

governed by the product of the dimensionless velocity

function (actually a combination of velocity differentials), n,and the third power of the dimensionless distance from wall,

( y+)3. The n varies with the vertical position above the pipe

wall and reaches the maximum value somewhere within the

buffer layer (Fig. 7). Fig. 7 shows that the off-the-wall force

(force of inertia lift associated with carrier turbulence) is

effective only within a certain range of vertical positions

above wall, namely in a certain portion of the region of

turbulent log velocity profile and in the adjacent buffer

layer. Thus the force can act only on particles of sizes that fit

the range. For sand–water slurries, the lift force is not

Fig. 7. Dimensionless velocity function n versus dimensionless distance

from the wall y+ (from Wilson and Sellgren [19]).

effective for particles smaller than approximately 0.15 mm

and larger than of about 0.4 mm [19]. For a solid particle

interacting with the logarithmic velocity profile ( y+>30) in

a horizontal water flow, the value of the off-the-wall force,

FL, is estimated as (FL /FW)max=0.27flV2m/8(Ss�1)gd50.

Throughout the buffer layer (5<y+<30) the lift drops to a

negligible value at the top of the viscous sub-layer. Whitlock

et al. [20] discuss the method for the lift-reduction deter-

mination in the buffer layer.

It is rather difficult to create conditions in a slurry pipe

under which the quantification of the lift force would not be

ambiguous. As mentioned above in connection with the

determination of the collisional forces, it is impossible to

separate the two effects on the observed slurry friction. Since

the lift effect seems to be associated primarily with fast flows

of slurry, i.e. with velocities above those used usually during

practical operations, its impact in practice is limited.

3.3. Friction at the top of granular bed

In stratified flows of settling slurries the top of the

granular bed is an additional boundary at which friction

must be determined. If the top of the bed is not sheared-off it

can be considered a rough boundary with a roughness

related to the particle size. For the sheared-off beds,

however, the particle size is not an appropriate boundary-

roughness parameter. Instead the friction law is related to

another characteristic size of the boundary—the thickness of

the sheared portion of the bed. This thickness depends on

the bed shear stress (e.g. [4]).

The most appropriate condition for the evaluation of the

bed friction is the stratified flow with the stationary bed. In

this flow, the position of the top of the bed and the mean

velocity differential between the current and the bed can be

accurately determined.

Wilson and co-workers formulated the friction law on

basis of observations in closed conduits with various

fractions of sand and bakelite. For Im/ (Ss�1)>0.0167,

Fig. 9. Relative bed roughness, ks/d50, for the Nikuradze’s friction

equation as function of particle mobility number hb, sand particles of

w*=0.62 (+ data from Matousek [25]; line-prediction).


the proposed bed friction-law [21] reads fb=0.87(Im /

(Ss�1))0.78. The bed shear stress, sb= ( fb /8)q1V2ma.

Ribberink [22] used a rough-wall concept to determine

the bed shear stress in his generalized bed-load formula for

steady flows and unsteady oscillatory flows. Based on

Wilson’s close-conduit data in the range 1<hb<7, Ribber-

ink adapted the Wilson’s earlier relationship between the

Nikuradze particle-roughness height, ks, and the particle

mobility number for the top of the bed, hb, ((ks/d50)=5hb,

[23]) to the form (ks/d50)=1+6(hb�1).

Sumer et al. [24] also found that their bed-friction data

(e.g. for the acryl particles w*=0.72, Fig. 8) processed

by the Nikuradze’s resistance relation for a rough

boundary Vma=u4b ¼ffiffiffiffiffiffiffiffiffi8=fb

p¼ 2:46ln 14:8Rhb=ksð Þ satisfy

the relation, ks /d50= fn(hb). Data in Sumer et al. cover a

relatively narrow range of particle mobility numbers

(0.8<hb<5 approximately).

Our tests with the 0.2–0.5-mm sand [25] revealed a trend

that was very similar to that observed by Sumer et al. and

showed that there was a clear correlation between ks /d50and hb also in flow regimes with much larger hb (up to

almost 25). The data suggest the following relation in the

range 4<hb<25, ks /d50=0.13(hb +1.38)2.34 (Fig. 9).

4. Predictive models

The complex behavior of settling-slurry flows is not well

understood yet. Therefore existing tools for its simulation

and prediction are still far from accurate and versatile. For

difficult conditions (e.g. a complex structure of transported

solids), tests are recommended to determine the pressure

drop due to friction and the deposition-limit velocity in a

field pipe. If the field-pipe test is not possible, a test in a

laboratory pipe (usually smaller than the field pipe) is an

option. The laboratory results are scaled-up to a pipe of full

size using an appropriate scale-up model. If no test is

possible for the slurry in question one must rely on results

from a predictive model.

In practice, empirical models deriving from the 1950s and

1960s are still widely used. Dutch dredging companies, for

instance, use modified versions of the empirical correlations

of Durand (or of Fuhrboter or Jufin-Lopatin) to predict

settling-slurry flows in large horizontal/inclined pipes. They

calibrate the correlations to different conditions using their

own data collected during dredging works. The correlations

are simple to use and modify. However, a still better

understanding of an internal structure of settling-slurry flow

and a wider access to user-friendly computational techniques

make it more attractive for practical engineers to adopt more

complex models with a physical background.

A two-layer model (the basis of which was formulated by

Wilson in 1970s [5,26]) is now also becoming a standard

way of analyzing fully or partially stratified flows in practice.

Besides its physical background, the value of the two-layer

model as compared to empirical models is in the fact that that

it predicts more than just one slurry-flow parameter. The

model outputs are the pressure drop, the deposition-limit

velocity, slip between phases, the sliding-bed velocity and

the thickness of the bed. This makes it a suitable tool for the

prediction not only of pipeline hydraulic performance but

also of the flow pattern for an estimation of pipe-wall wear.

Various versions of a layered model are in use. Generally,

they differ in the way of modeling of friction on boundaries

and the division of solids into layers. The development of a

two-layer model has been associated with the application of

measuring techniques that make it possible to observe the

internal structure of the flow (concentration and velocity

profiles) and link it with the idealized two-layer structure

handled by the model.

The two-layer model developed in the Saskatchewan

Research Council (the SRC model) assumes the presence of

suspended particles in the contact layer (the lower layer in

the model) and the buoyancy effect of suspension on the

contact bed. This idealization is justified for flows of

particles of medium size in which the granular bed is virtual

rather than real or in flows of broad particle size distribution

with a portion of fines. The recent modification of the SRC

model to highly concentrated slurries [15] has introduced

the dispersive stress in the suspension layer (the upper layer

in the model) and modified the function for distribution of

solids into two layers (the stratification-ratio function) to

simulate increased solids friction observed in high concen-

trated settling slurries.

The model developed at the Delft University of

Technology [27] distinguishes between two suspension

mechanisms in the layer above the contact bed. The

proportion of particles contributing to suspension is found

to be primarily dependent on the ratio of the mean velocity

of slurry and the settling velocity of particle if solid particles

above the bed are suspended by the diffusive effect of

conveying-liquid turbulence. If the particles above the bed


are dispersed due primarily to the dispersive effect of

interparticle collisions, the stratification ratio varies with the

bed shear stress. However, the laboratory test results showed

that the overall exponential formula relating the stratifica-

tion ratio and the ratio of slurry velocity and particle settling

velocity (as proposed by Gillies et al. [28]) provides also

reasonable results, if calibrated with own data for horizontal

flows of different fractions of sand. As shown in Matousek

[29], a heterogeneous flow with turbulent suspension can be

successfully modeled as a two-layer system by linking a

theoretical concentration profile (the modified Rouse–

Schmidt turbulent-diffusion model) with the top of the

virtual contact bed.

A two-layer model is also a suitable tool for simulation of

a stratified flow with a non-Newtonian carrier [30].

5. Conclusions

Pipe-wall friction generated by solid particles in perma-

nent contact with the pipe wall is better understood than

friction deriving from sporadic contact (collisions) or zero

contact of particles with the wall. Recent tests with plug

flows and inclined flows with sliding beds confirmed that

Wilson’s particle-wall friction concept is appropriate to

describe the solids friction caused by permanent contact

between particles and a pipe wall.

No suitable general concept is available to describe pipe-

wall friction resulting from sporadic or zero contacts of

particles with a pipe wall when traveling in the near-wall

region of a slurry flow. Little is known about mechanisms of

particle friction at the pipe wall and particle dispersion in the

near-wall region of a non-stratified flow (flow with no

permanent contact between particles and a wall). One of the

important aspects of the mechanisms governing particle

movement in the near-wall region of non-stratified slurry

flows is the effect of the particle size on the solids friction.

Recent tests with non-stratified flows showed that fine

slurries exerted higher solids friction than coarse slurries.

The fine slurries (sand particles smaller than approximately

100 Am) exhibited hydraulic gradients higher than those

predicted by the model handling the slurries as Fequivalent-liquids_, while the coarse slurries (fine to medium- and

medium sand particles) exhibited lower hydraulic gradients

than the modeled Fequivalent-liquid_ slurries. Furthermore,

an interesting effect of the particle size and the solids

concentration on the solids friction was detected in flows of

different coarse slurries. Surprisingly low friction was

detected in highly concentrated medium-sand flows; the

friction was lower than that in flows of fine to medium sand

for the same flow conditions. The recently developed

concept of hydrodynamic off-the-wall lift suggests a

possible explanation of the observed low solids friction in

the medium-sand slurry.

Friction at the top of a granular bed has been studied

extensively in recent years. New data from several closed

conduits has opened the possibility to formulate and verify

friction laws for plane beds that together cover a broad

range of Shields numbers (0.8<hb<25).

The verification of friction and lift theories, the develop-

ment of which is currently in progress, will require

measuring techniques that can sense important parameters

of slurry flow in smaller control volumes (adjacent to a pipe

wall) than is currently possible.

Notation

cv local volumetric solids concentration in pipe cross-

section

Cv mean volumetric solids concentration in pipe

cross-section

Cvb mean volumetric solids concentration in bed

(spatial)

Cvd delivered Cv

Cvi spatial Cv

Cvmax maximum volumetric solids concentration in bed

(spatial)

d50 median particle diameter [m]

D pipe diameter [m]

fb Darcy–Weisbach friction coefficient for the top of

bed

fl Darcy–Weisbach friction coefficient for liquid

flow over pipe wall

FL hydrodynamic off-the-wall force (lift force) [N]

FW submerged weight of particle in liquid, (Ss�1)

qlgd350p / 8 [N]

g gravitational acceleration [m/s2]

Il hydraulic gradient of liquid flow

Im hydraulic gradient of slurry flow

ks characteristic grain roughness height (by Nikur-

adze) [m]

K proportional coefficient in Gillies–Shook equation

for solids shear stress

KB proportional coefficient in Bagnold’s equation for

solids shear stress

Rhb hydraulic radius of area associated with the top of

bed [m]

Re Reynolds number, VmDql /ll

Sm relative density of slurry, qm/ql

Ss relative density of solids, qs /ql

u*b shear velocity at the top of bed [m/s]

u*l shear velocity at the wall of pipe, Vm( fl / 8)0.5 [m/s]

v local velocity of liquid

vt terminal settling velocity of solid particle [m/s]

v+ dimensionless local velocity, v /u*lVm mean velocity of mixture in pipe cross-section

[m/s]

Vma mean velocity of mixture above contact bed [m/s]

w* dimensionless terminal settling velocity, v t /

((Ss�1)gd50)0.5

y vertical distance from pipe wall [m]

y+ dimensionless distance from pipe wall, yu*lql /ll

b angle defining position of top of bed [rad]


hb particle mobility number (Shields number), u*b2 /

((Ss�1)gd50)

k linear concentration of solids (by Bagnold), 1 /

((Cvmax /Cvi)0.33�1)

ll dynamic viscosity of liquid [Pa s]

ls mechanical friction coefficient of solids against

pipe wall

n lift parameter (by Wilson and Sellgren), dv+ /

dy+(�d2v+ /dy+2)

ql density of liquid (water) [kg/m3]

qs density of solid particle [kg/m3]

rswC solids normal stress exerted by sliding particles at

pipe wall [Pa]

sb shear stress at the top of bed [Pa]

slw liquid shear stress at pipe wall [Pa]

ssB solids shear stress in sheared slurry in inertial

regime (by Bagnold) [Pa]

ssw solids shear stress exerted by colliding particles at

pipe wall [Pa]

sswC solids shear stress exerted by sliding particles at

pipe wall [Pa]

Abbreviations

DEM Discrete Element Method

DUT Delft University of Technology

LDA Laser Doppler Anemometry

MRI Magnetic Resonance Imaging

PIV Particle Image Velocimetry

SRC Saskatchewan Research Council

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Research Developments in Pipeline Transport of Settling Slurries

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