7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Analysis of the Horizontal Pipeline Flow of Settling Dense Slurries
Mario R. Rojas and A. Eduardo Saez
Department of Chemical and Environmental Engineering, The University of Arizona, Tucson, AZ 85721, USA
mrrojasc@email.arizona.edu and esaez@email.arizona.edu
Keywords: NonNewtonian Slurries, Twolayer Model, Horizontal Pipeline Flow, Solids Transport
Abstract
The steadystate flow of dense aqueous slurries in horizontal pipes with Newtonian and nonNewtonian (Casson) carrier fluids
has been analyzed using a twolayer model consisting of a top layer of flowing suspension and a settled bed of particles. The
coarse solids used have a wide range of particle density, average size and particle size distributions, and the fluids studied were
designed to simulate U.S. Department of Energy Hanford site waste slurries. The Doron and Bamea (1987) twolayer model
was modified and extended to the quantification of slurries containing wide particle size distributions, including highdensity
particles, as well as nonNewtonian fluids. The most important changes from previous models include an independent settling
analysis for different particle size fractions, effects of the shape of the particles on the settling velocity calculations, and a new
correlation to represent the turbulent particle dispersivity. To incorporate the theological properties of the fluids, the
WilsonThomas turbulent flow equation for a Casson fluid (Wilson and Thomas 1985) was used. The results indicate that the
turbulent dispersivity of settling particles is sensitive to particle size and density. A new correlation is proposed to relate
particle dispersivity to Archimedes number (Ar). The model also gives good estimation of the critical deposition velocity as the
minimum of the pressure drop vs. superficial slurry velocity relation. The existence of a stationary layer can be observed and
predicted by the model under laminar and turbulent flow conditions.
Introduction
The transport of slurries in horizontal pipes is a process with
widespread application in practice. Even though it is
generally desirable to operate at velocities that ensure
complete suspension of solids, some processes operate
under conditions at which a granular deposit occupies part
of the cross section of the pipe. In fact, for slurries
composed of dense particles, it might be economically
attractive to operate in a regime in which a granular deposit
exists. Typically, these processes involve suspension of
particles with wide particle size distributions in turbulent
flows.
Gillies et al. (1991) proposed a twolayer model to predict
the head losses for coarseparticle or settling slurries in
horizontal pipes as a modification of the Wilson (1970)
model. Gillies et al.'s model is capable of representing
solids concentration profiles over the cross section, and their
relation to the mean flow velocity and the settling particle
velocity. The same research group has used the twolayer
model in different applications, such as the study of
frictional losses in concentrated slurry flows (Gillies and
Shook 2000) and the modeling of heterogeneous slurries at
relatively high fluid velocities (Gillies et al. 2004), with
successful results.
Doron and Bamea (1993) proposed a threelayer model as
an extension of their own twolayer model (Doron et al.
1987) and other published models. They proposed that the
main limitation of the twolayer model is its inability to
predict accurately the existence of a stationary bed at low
flow rates: in some cases when a stationary bed was
observed, model results indicated flow with a moving bed.
This also leads to reduced reliability of the pressure drop
calculations for low flow rates, at which a stationary bed
can be expected. Using the threelayer model, the authors
were able to quantify the critical deposition velocity as the
limit when the stationary bed height approaches zero. The
value obtained can be viewed as an upper limit for the
critical velocity since, in practice, a bed layer can be
considered to vanish when its height is of the order of the
particle size. According to their results, model predictions
are in fairly close agreement with the Turian et al. (1987)
expression and the correlation proposed by Gillies et al.
(1991a) for critical velocity, which were derived from
semiempirical analyses.
The Doron and Bamea model incorporates a crosssectional
solids mass balance to quantify the solid distribution in the
pipe under steadystate conditions. The balance leads to a
onedimensional version of the sedimentationdispersion
equation, whose solution yields the vertical solids
concentration profile in the cross section of the moving
layer. The solids settling velocities and dispersion
coefficients are vital for the accurate performance of this
prediction.
Similar two and threelayer models have been developed
and extended to other applications (Gorji and Ghorbani,
2008). Ramadan et al. (2005) proposed an extension of the
threelayer to nonNewtonian (powerlaw) fluids to predict
solids transport in horizontal and inclined wellbore drilling
applications. The model is focused on the prediction of
transport rates for drilling applications.
In this work, we modify the twolayer model to characterize
the flow of dense/concentrated Newtonian and
nonNewtonian slurries with broad particle size
distributions in flow through horizontal pipes, in an attempt
to represent real waste slurries of the US DOE's Hanford
site. For the case of Newtonian fluids, the most important
changes from previous models include independent settling
analysis for different particle size fractions, effects of the
shape of the particles on the settling velocity calculation,
and a new correlation to represent turbulent particle
dispersivities. To extend the model to nonNewtonian fluids,
a new approach has been developed and tested. The
WilsonThomas Turbulent flow equation for Casson fluids
has been coupled with the twolayer model. The transition
between laminar and turbulent flow has been studied using
the Wasp criterion (Bingham fluids) adapted by Poloski et al.
(2008) to Casson fluids.
Nomenclature and Acronyms
a coefficient in the modified Taylor equation
A surface area (m2)
Ar Archimedes number
c particle circularity
C solid volume concentration (vol %)
CD drag Coefficient
C, slurry input volume concentration (vol %)
D pipe diameter (m)
dA/d, surfaceequivalent sphere to nominal diameter
ratio
d, particle diameter (m)
dP/dx pressure gradient (Pa/m)
ERT Electrical Resistance Tomography
f friction coefficient
Fmb dry friction force moving bed (N)
g acceleration of gravity (m/s2)
LL low density/small particle size/Newtonian slurry
LH low density/large particle size/Newtonian slurry
LH1 low density/large particle size/nonNewtonian,
low yield stress slurry
LH2 low density/large particle size/nonNewtonian,
high yield stress slurry
HL high density/small particle size/Newtonian slurry
HH high density/large particle size/Newtonian slurry
m, n adjustable parameters in the modified Taylor
equation
MM1 medium density/medium particle
size/nonNewtonian, low yield stress slurry
MM2 medium density/medium particle
size/nonNewtonian, high yield stress slurry
P pressure (Pa)
PSD particle size distribution
Re Reynolds number
s ratio of density of the particle and carrier fluid
S perimeter of layer or interface (m)
U superficial fluid velocity (m/s)
U, mean velocity for an equivalent Newtonian fluid
(m/s)
u* shear velocity (m/s)
w terminal settling velocity of particles (m/s)
Yb bed height (m)
Greek symbols
a, p constants for friction coefficient calculation
e solids dispersivity (m2/s)
S internal friction angle
Y Shear rate (s 1)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
lic Casson fluid viscosity (Pa s)
77 dry friction coefficient
p density of layers (kg/m3)
p, particle density (kg/m3)
0 angle associated with bed height
T shear stress of layers or interface (Pa)
c Casson yield stress (Pa)
, wall shear stress (Pa)
ratio of Casson yield stress to wall shear stress
Subscripts
b bed layer
h upper (moving) layer
i interface between layers
s mixture or slurry
Experimental
Materials. The suspensions (simulants) used were designed
to match specific physical properties, such as rheology and
particle size distribution (PSD), of actual waste slurries of
the Hanford site. The carrier fluid was water and all
experiments were performed at 200C. Glass (Spheriglass,
Potters Industry), alumina (Washington Mills) and stainless
steel 316 (Aemtek), were selected to represent the coarse
particles in the experiments. These materials have densities
of 2500, 3770 and 7950 kg/m3, respectively, and were
qualified as "low", "medium" and "high" particle density.
Particle size distributions range from 1 to 200 mrn. Other
properties of the simulants are listed in Tables 1 and 2.
Simulants are identified by a twoletter code: the first letter
identifies the particle density (low, medium or high as stated
above), and the second letter refers to relative particle size.
Examples of particle morphologies are shown in Figure 1.
The glass particles are spherical but the alumina and
stainless steel particles have irregular shapes. In order to
include the effect of shape in drag coefficient calculations,
we followed the analysis presented by TranCong et al.
(2004). Since their correlation for particle drag coefficient is
based on the definition of circularity and surfaceequivalent
sphere to nominal diameter ratio, these have been included
in Table 1. The definition of surfaceequivalent sphere to
nominal diameter ratio (dA/d,), is the ratio between dA, the
particle diameter defined in terms of projected area of the
sphere (Ap)
dA =4Ap / (1)
and the volumeequivalentsphere diameter or nominal
diameter, calculated from the particle volume (V),
dn = 36VI (2)
Table 1: Properties of the Newtonian simulants.
Acronym LL LH HL HH
Particle hydraulic
10.9 138.0 25.0 150.0
diameter (itm) dph
Solids content (vol %) 9.8 7.4 9.3 3.0
Particle density (kg/m3) 2500 2500 7950 7950
Particle circularity (c) 1.000 1.000 0.901 0.901
dAd, 1.000 1.000 1.151 1.151
Bed layer solids
concentration60 60 60 40
concentration Cb (vol %)
Experimental Setup. The slurries were prepared in a
400gal mixing tank connected to a flow loop system.
During the experiment, the slurry is transported through
the system by a 15hp/1800 rpm centrifugal pump (Georgia
Iron Works). The main section of the flowloop consists of
3 in schedule 40 stainless steel straight horizontal pipeline
on which different pressure ports were installed. Pressure
drops were measured with a differential pressure
transducer over a pipe length of 5.7 m.
Table 2: Properties of the nonNewtonian simulants.
Acronym
Particle hydraulic
diameter (rLm) dph
Solid vol fraction (%)
Fines vol fraction (%)
Particle density (kg/m3)
Fines particles density
(kg/m3)
Infinite shear rate
viscosity (Pa s)
Casson yield stress (Pa)
Particle Circularity (c)
Surfaceequivsphere
diameter (dAd,)
Bed Layer solids
concentration Cb (%)
LH1 LH2 MM1
116.0 152.7 83.3
8.4
7.7
2500
2500
0.0018
2.4
1.000
1.000
10.7
9.6
2500
2500
0.0026
4.4
1.000
1.000
9.5
8.9
3770
2500
0.0023
2.4
0.4472
1.1689
MM2
85.5
9.7
10.7
3770
2500
0.0024
4.7
0.4472
1.1689
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
electrical resistance tomography (ERT) probe (Industrial
Tomography Systems) that recorded crosssectional maps
of the slurry electrical conductivity in real time. Since
conductivity is a function of solids concentration, these
maps yield a representation of the solids distribution over
the cross section, including observation of the deposited
bed layer.
Model Description and Modifications
The model is based on the distribution of forces along the
flow direction over a crosssectional area of the pipe, and
mass balances. The contribution of each layer to the force
balance consists of shear stresses at the pipe wall and at the
interface, which can be calculated by means of friction
factors. More details about the formulation of the
twolayer model are provided by Doron and Barnea (1993).
Figure 2 shows the general scheme for the twolayer model,
identifying the stress distribution in the pipe
crosssectional area.
60 60 50 50
Figure 1: Micrographs of (a) LH, (b) HL, (c) MM and (d)
HH simulant particles.
The slurry is recirculated through the system until the flow
reaches steady state (approximately 3060 min). Before it
enters the main horizontal measurement section, the slurry
flows through a Coriolis flow meter (MicroMotion,
Fseries). A chiller connected to the mixingtank was used
to keep the temperature constant during the experiment
(200C).
All the tests were conducted starting with a relatively high
superficial velocity of 34 m/s, and then the velocity was
decreased in 0.15 m/s steps until a rise in differential
pressure was detected, indicating the presence of a settled
bed of particles; this point was considered as the
experimental critical deposition velocity.
After the Coriolis flow meters, the slurry entered an
Figure 2: Geometry and stress distribution in the twolayer
model.
The model assumes that vertical (y) particle transport is
governed by the convectiondispersion equation, which
balances the settling flux of particles with a turbulent
dispersion flux,
d 2C dC
e +w =0 (3)
dy2 dy
where w is the terminal settling velocity of the particles, e is
the turbulent dispersivity, and C is the solids concentration
(solids volume fraction). This equation was applied to
different particle size ranges within the particle size
distribution of the solid. Details are presented in the
Discussion section below. Integrating equation (3) over the
crosssection of the upper layer yields the mean solids
concentration in the upper layer,
CbD2 r/2
Ch 2 exp D [sin sin(Ob)] cos 2 yd2y (4)
2Ah J' ( 2c
Oh
where D is the pipe diameter, Ah is the crosssectional area
of moving layer, Ob is the angle associated with the
stationary bed location (Figure 2), and Cb is the
concentration of solids in the stationary bed, used as
boundary condition in the integration of equation (3). Mass
balances in solid and liquid phases yield
UhChAh +UbCbAb =UCA (5)
Uh(1 Ch)Ah +Ub(1 Cb)Ab = U, (1 C,)A (6)
where U, is the superficial velocity of the slurry, Ub is the
velocity of liquid moving through the stationary bed, and Ab
is the crosssectional area of stationary bed. The value of Ch
obtained from equation (4) must equal the value obtained
from the mass balances. A force balance in the flow
direction can be expressed as
ThSh + TS, Fmb + bSb TS,
Ah Ab
where r represents the shear stress acting on the surfaces of
each layer at the pipe surface (h and b subscripts) or at the
interface (i) between layers (Figure 2), S represents
perimeter, F,b is the dry friction force exerted by the bed
layer, which includes the effect of the submerged weight of
particles and the transmission of stresses from the interface,
and is given by
Fmb= b (D )[ 2) j tan (8)
where p, and pi are the densities of solid and carrier fluid,
respectively, 77 is the dry friction coefficient, yb is the bed
height, 0 is the internal friction angle and g is the
acceleration of gravity.
The angle and perimeters (Ob Sh, Sb, S,) can be expressed as
functions of yb and, therefore, for a given flow condition, it
is possible to solve the preceding set of equations having yb,
Ch, Uh and Ub as unknowns. If the system exhibits a
stationary bed (our case), the static dry friction force is no
longer equal to the maximum dry friction force. In this case,
the equations to solve are the mass balance (equation 4), and
the pressure gradient per unit length is then obtained from
dP
Ah ThSh ,S, (9)
dx
The shear stresses are expressed in terms of friction factors
as follows
1 2
S= pU, f, (10)
where the subscript i represents the interface between the
two layers. Similar definitions apply to the two layers (h and
b). The densities of each layer are given by
p, = P,C, + P,(1 C,) (11)
And the friction coefficient (/) for each layer can be
calculated using the following correlation
=aRep (12)
where a and P are constants that depend of the flow regime:
turbulent (a=0.046, p=0.2) and laminar (a=16, 1=1). The
Reynolds numbers are based on the velocity and density of
each layer. Furthermore, the friction factor in the moving
layer is given by the Colebrook correlation, using the
particle size as an equivalent roughness,
1 8 dp 2.51 (13)
S2f, 3.7Dh Re, (13)
The set of equations described above were used to model
the Newtonian simulants. The nonNewtonian simulants
used in this work were simulated using a new approach. The
Wilson and Thomas (1985) theory for turbulent flow has
been adapted and coupled to the twolayer model. This
theory considers the effect of variable fluid viscosity on the
velocity profiles of the viscous sublayers in turbulent flow.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Through the analysis of viscosity effects at the time and
length scales of dissipative micro eddies, the theory predicts
a thickening of the viscous sublayer, which tends to
increase throughput velocity and thus promotes drag
reduction. The WilsonThomas theory has been adapted to
different theological behaviors, including Casson fluids, for
which it has been shown that the superficial velocity
achieved by a Casson fluid for a given shear stress is related
to the velocity of an equivalent Newtonian fluid by
U=U +2.5u*ln 1I +
+2 3 (14)
3 3
u*[(2.5+1.254)+11.6(2 + 1)
where i is the ratio of the yield stress to the wall shear stress
(zTc/T), U,, is the superficial velocity for an equivalent
Newtonian fluid at the same shear stress, and u is the shear
velocity, calculated from
Calculation of the pressure drop of a Casson fluid for
homogeneous turbulent flow requires the use of this last
equation. For laminar flow, the equivalent expression is
SU (r 16 +_ 4 14 ,4
D 3 21 (16)
where p/ is the viscosity of a Casson fluid, whose
constitutive equation in shear flow relating shear stress, r, to
shear rate, is given by
1/2 = 1 (/2 c)1/2 (17)
We have adapted these equations to the twolayer model to
simulate the upper layer. The process starts by relating the
wall shear stress in equations (14) or (16) (depending on the
flow regime) to the shear stress in the upper flow region (Th
and z,), assuming that the lower layer is stationary. The wall
shear leads to a pressure gradient given by
dP
Ah d = (Sh +S,) (18)
dx
For a given bed height, all the geometric parameters can be
calculated. If the pressure gradient is given, the effective
wall shear stress can be obtained from equation (18), which
allows for calculation of . At this point, the effective
viscosity of the Casson fluid can be calculated from
PC
i =(1 )2 (19)
To calculate the Newtonian superficial velocity (U,,), we
solve Colebrook's equation for the friction coefficient at the
interface simultaneously with equation (9), using the friction
factors and Reynolds numbers of a Newtonian fluid. Once
the mean Newtonian fluid velocity is known, equation (14)
can be used to calculate the velocity in the upper layer (Uh),
the Reynolds number of the fluid, and the corresponding
superficial fluid velocity,
U, hAh (20)
A
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
After these calculations, the noslip condition should be
checked to insure the existence of a stationary layer,
followed by the calculation of settling velocity, dispersivity
and verification of the mass balance (equation 4). The
calculation proceeds iteratively until the concentration in the
upper layer (Ch) is the actual inlet concentration (C,).
The calculation of the dispersion coefficient used in
equation (4) and the modifications made to Taylor's
equation will be addressed in the discussion. All calculations
were performed using an iterative program developed in
MATLAB.
Results and Discussion
The twolayer model, as described above, can be used, in
principle, to predict the nonhomogeneous flow behaviour
of slurries. Given the scenario of a stable stationary bed
without a moving bed above it, according to the model
equations, one of the most important factors that control the
actual upper layer solids concentration is the balance
between particle turbulent dispersion and settling. At this
point, it is considered that the particles have a specified
particle size distribution that can be discretized into narrow
ranges, each characterized by its own dispersivity and
settling velocity. The calculation of settling velocity is one
of the most important modifications proposed here. It
includes effects of the solids concentration, turbulence
intensity and particle shape, which affect the estimation of
the drag coefficient for the particles. Setting velocities were
calculated considering narrow portions of the PSD
separately, as follows,
4(s 1)d g(21)
Wo, = (21)
3CDz
where CD, is the drag coefficient for particle size range I,
calculated using the particle Reynolds number based on wol,
and s=ps/pi. In order to account for the PSD. Values of drag
coefficient were obtained using the modified form of the
Clift et al. (1978) correlation provided by TranCong et al.
(2004), which takes into consideration the shape of the
particles. In addition, to account for turbulence effects in the
drag coefficient calculations, the correlation proposed by
Brucato et al. (1998) is used, which considers the
Kolmogorov scale of dissipative eddies.
Once the drag coefficients are known and values of w,, were
calculated for each fraction of the PSD, we calculated a
particle hindered settling velocity into the cluster for each of
the fractions using the equation developed by Cheng (1997).
Finally, the effective settling velocity of the particles to be
used in the integration of equation (4) is calculated as
follows,
S(woCh,) (22)
wV=
Ch
Calculations have shown that this approach gives a settling
velocity lower than that obtained using a simple average
particle size.
The turbulent dispersivity in the original Doron and Barnea
model is calculated from the original Taylor's equation
(Taylor 1954),
e = aDhu
where a=0.026. However, this approach does not
appropriately take into account particle characteristics
(density, size, shape, PSD), which become more relevant in
this work due to the relatively large particle size and density
of some of the simulants. Analysis of our data suggested
that the coefficient in Taylor's equation is related to
particles properties, and that there is a stronger dependence
of the dispersivity on fluid velocity for our slurries.
It is important to recall that Taylor's correlation applies to
solutes in turbulent flow and not to settling particles. Our
results indicate that turbulent transport of particles is slower
than transport of molecular species, which could be a
consequence of increase of energy dissipation in smaller
eddies (of size comparable to the solid particles).
Figure 3 shows the dependence of the dispersion coefficient
calculated by fitting our model to experimental data with
hydraulic diameter and shear velocity (u*). Different
correlations were obtained depending on particle properties,
so using preliminary data for Newtonian fluids, and based
on these results, we have postulated that the coefficient "a"
depends on Reynolds number and the Archimedes number
of the particles through an empirical correlation given by
a = kArnRem (24)
where n and m were found to be 0.411 and 0.75, respectively,
k is a constant and Ar is the Archimedes number of the
particles present in the slurry, defined by
Ar = gd3 (s 1)Ph2 (25)
hAr
105
06
1 2 3 4 5
3/4 *
Re Du
h h
Figure 3: Particle dispersivity as fitted to experimental
data. The slopes of these curves represent the different a
values that depend on Ar.
We extended the correlation to include both Newtonian
and nonNewtonian simulant data. The results are shown
in figure 4. Note that the new correlation is applicable to
all simulants. It is important to point out that particles with
very different densities and PSDs but with similar
Archimedes numbers possess similar values of a, which
supports the mathematical form of the proposed
correlation.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
3xl106
2x106
ix10 6
1 0.1 1 10 100
Figure 4: Particle dispersivity correlation.
Turbulent dispersivities have been found to be correlated in
previous works (albeit in different applications) with Ar and
Re. For instance, Wen and Yu (1966) and Reganathan et al.
(2004) used such a correlation to represent solids dispersion
coefficients in liquidsolid fluidized beds.
The twolayer model was used to predict the pressure drop
over the 5.71 m of straight horizontal pipe for different
simulants using the proposed correlation for particle
dispersivity. Figure 5 shows pressured drop vs. superficial
velocity for the lowdensity Newtonian simulants (LL, LH),
including the predictions of the modified twolayer model.
The results in Figure 5 follow the typical trends observed in
flow of concentrated slurries: at high fluid velocities, when
the solid is fully suspended, the pressure drop decreases
monotonically with a reduction in velocity. Additional
decreases in fluid velocity lead to the formation of a
stationary (or moving) layer in the bottom of the pipe,
whose growth as the velocity is reduced causes a decrease in
suspension flow area and a consequent increase in the
pressure drop. At very low fluid superficial velocities, the
pressure drop continues to rise as the bed thickness
increases uniformly.
The minimum observed in the pressure drop curves yields
the critical deposition velocity of the suspension. All the
experimental data analyzed in this work present a stable
stationary layer at low fluid velocities except the HH
stimulant, which exhibited a transition from moving bed to
stationary bed at high velocities. Figure 5 shows that, for
low particle densities, the modified model proposed here
predicts the pressure drops satisfactorily in the whole
velocity range evaluated. For both simulants, the modified
model predicts accurately the critical velocity (dashed lines).
The minimum in the predicted curves is close to the point at
which the model predicts that the stationary bed disappears
with any additional increment in fluid velocity.
Figure 6 shows results for the nonNewtonian simulants
with intermediate particle density and size (MM). The fluid
with the higher yield stress (MM2) exhibits a higher
pressure drop at low fluid velocities. Although there is an
important difference in the theological properties of these
two fluids, the critical velocities are quite similar. In this
case, the flow is always turbulent, and the model once again
gives an accurate representation of the experimental data.
0.8 .
S LL (Glass 10 ]on)
A LH (Glass 100 jon)
0.6
0.4
0.2
0.0
0.0 0.5 1.0 1.5
U (m/s)
2.0 2.5
Figure 5: Pressure drop as a function of superficial velocity
for the low density/small particle size Newtonian simulants.
Solid lines represent the model.
1 .0
0.. r > (Casson yield stress)
0.89
SAI1M2
t 0.7 Alumina 50 un
0.6
0.5 A
 0.4.
0.3
0.*2
0.5 1.0 1.5
U (m/s)
2.0 2.5
Figure 6: Pressure as a function of superficial velocity for
medium density/medium particle size fluids. Solid lines
represent the model.
Figure 7 shows the thickness of the stationary bed layer
height predicted by the model as a function of superficial
velocity. For practical purposes, it can be considered that the
bed disappears when its height approaches the particle
diameter, which usually happens at velocities in the range
0.31.5 m/s. However, for the very dense/large particle size
slurry (HH), the experimental observations and the ERT
images suggest that the stagnant bottom layer only
disappears at the maximum velocity used.
For HH, the model also predicts a high critical velocity
compared with all other simulants. There is a solids layer in
the bottom of the pipe even at velocities above the critical
velocity, which is due to a transition between a stationary
layer and a moving layer at high fluid velocities. However,
this moving layer is so small that it exists only in a narrow
range of velocity preceding the total disappearance of the
bed at high velocities. Figure 7 also shows the tomography
images (ERT) obtained for the HH simulant for different
superficial fluid velocities. These images provide a way to
study the evolution of the bed layer at the bottom of the pipe.
The images show a presence of a stationary bed over
practically all the range of fluid velocity studied. For all the
* NonNewtonanfluids
* Newtonanfluds
 Correlation
. = aDh
a= kAr Reh
k = 3.4466x107
R2 = 0.9840
other fluids, the disappearance of the bed layer predicted by
the model matches the experimental results and the ERT
images (results not shown).
50. oLL .
/ LH
mHH
40 HH
30
20
10 .
SERT Tomogram images
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
u (m/s)
Figure 7: Stationary bed height predicted by the model as a
function of superficial velocity for studied simulants. ERT
images for HH qualitatively follow the trends predicted by
the model.
Conclusions
The twolayer model of Doron and Barnea has been
modified to study slurries made up by Newtonian and
nonNewtonian fluids composed of dense particles with
broad particle size distributions and irregular shapes. To
include the nonNewtonian properties of the simulants and
their effects on the flow, the twolayer model has been
coupled with the WilsonThomas turbulence equation for
Casson fluids. The model predicts accurately pressure drops,
critical deposition velocities and the thickness of the
stationary bed when it is present.
The changes introduced in the model include a novel way to
calculate the effective settling velocity of the whole PSD,
taking into account the effect of turbulence and irregular
shape of particles in the drag coefficient calculation. A new
correlation for the solids dispersion coefficient was
developed to improve model predictions. The dispersion
coefficient depends on the particle Archimedes and
Reynolds numbers. Pressure drops, critical deposition
velocities, and thickness of the bed layers have been
successfully predicted for all the slurries, demonstrating the
potential applicability of this model to simulate the
hydrodynamics of complex slurries at the U.S. Department
of Energy's Hanford site.
Acknowledgements
This work was supported by Battelle. The authors are
grateful to Adam Poloski and Harold Adkins of Pacific
Northwest National Laboratory in Richland, WA, for helpful
discussions and for providing experimental results.
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