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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: v.matousek@wbmt.tudelft.nl.
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]).
V. Matousek / Powder Technology 156 (2005) 43–5146
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]).
V. Matousek / Powder Technology 156 (2005) 43–51 47
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]).
V. Matousek / Powder Technology 156 (2005) 43–5148
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).
V. Matousek / Powder Technology 156 (2005) 43–51 49
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
V. Matousek / Powder Technology 156 (2005) 43–5150
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]
V. Matousek / Powder Technology 156 (2005) 43–51 51
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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