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Delft University of Technology
Dr.ir. Sape A. Miedema
Head of Studies
MSc Offshore & Dredging Engineering
& Marine Technology
&
Associate Professor of
Dredging Engineering
© S.A.M
DHLLDV Framework
An Overview Of
Slurry Transport
Models
Delft University of Technology
Dr.ir. Sape A. Miedema
Head of Studies
MSc Offshore & Dredging Engineering
& Marine Technology
&
Associate Professor of
Dredging Engineering
© S.A.M
DHLLDV Framework
An Overview Of
Slurry Transport
Models
Delft University of Technology – Offshore & Dredging Engineering
Dredging A Way Of Life
© S.A.M
© S.A.M
Goals & Targets
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Pump/Pipeline System
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
• Total pressure/power required
• Limit (Stationary) Deposit Velocity
• Cavitation limit of each pump
• Deposition/plugging the pipeline
Pressure/Flow Graph (Q-H Graph)
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0
1000
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3000
4000
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0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00
Pre
ss
ure
or
He
ad
H (
kP
a)
Flow Q (m3/sec)
Pump & Pipe Resistance Chart Water Q-H Curve
Mixture Q-H Curve
Water Resistance
Mixture ResistanceELM
Mixture ResistanceDHLLDV
Mixture ResistanceJufin Lopatin
Mixture ResistanceWilson
Design Point &Working PointWater-WaterWorking PointWater-Mixture
Working PointMixture-Mixture
Working PointMixture-Water
• Working points/working area in a stationary situation
Determining Slurry Transport
Behavior
Based On Known Parameters
Like:
Liquid Properties,
Pipe Diameter,
Particle Diameter,
Volumetric Concentration
As A Function Of The Flow Or
Line Speed
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Goals & Targets
The Elephant of Wilson
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
1. Small versus large pipe diameter
2. Small versus large particle diameter
3. Low versus high concentration
4. Low versus high line speed
5. Spatial versus delivered concentration
6. Uniform versus graded sands/gravels
1. Carrier liquid properties
2. Solids properties
For sands/gravels in water 64 combinations
possible
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Possibilities
The Solids Effect
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Solids Effect
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
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0.30
0.35
0.40
0 1 2 3 4 5 6 7 8 9 10
Hyd
rau
lic
gra
die
nt
i m, i l
(m w
ate
r/m
)
Line speed vls (m/sec)
Hydraulic gradient im, il vs. Line speed vls
Liquid il curve
Fixed Bed Cvs=c.
Sliding Bed Cvs=c.Lower Limit
Sliding Bed Cvs=c.Mean
Sliding Bed Cvs=c.Upper Limit
Heterogeneous FlowCvs=c.
Equivalent LiquidModel
Homogeneous FlowCvs=Cvt=c.
Resulting im curveCvs=c.
Resulting im curveCvt=c.
Limit Deposit VelocityCvs=c.
Limit Deposit VelocityCvt=c.
Limit Deposit Velocity
© S.A.M. Dp=0.1524 m, d=1.500 mm, Rsd=1.585, Cv=0.300, μ=0.420
i m-i
l
i m-i
l
i m-i
l
i m-i
l
i m-i
l
2
l lsl
l
l p
vpi
g L 2 g D
m l
r h g
s d v
i iE
R C
Hydraulic Gradient
Relative Excess H.G.
Data from Yagi et al., im-v
ls
Delft University of Technology – Offshore & Dredging Engineering
Data looks unorganized depending on the volumetric
concentration of the solids.
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0.35
0.40
0 1 2 3 4 5 6
Hyd
rau
lic
gra
die
nt
i m, i l
(m w
ate
r/m
)
Line speed vls (m/sec)
Hydraulic gradient im, il vs. Line speed vls
Liquid il curve
Sliding Bed Cvs=c.
Equivalent Liquid Model
Homogeneous FlowCvs=Cvt=c.
Resulting im curveCvs=c.
Resulting im curveCvt=c.
Limit Deposit VelocityCvs=c.
Limit Deposit Velocity
Cvt=0.225-0.275
Cvt=0.175-0.225
Cvt=0.125-0.175
Cvt=0.075-0.125
Cvt=0.025-0.075
© S.A.M. Dp=0.1552 m, d=0.910 mm, Rsd=1.585, Cv=0.150, μsf=0.550
Data from Yagi et al., Erhg
-il
© S.A.M
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Fixed Bed Cvs=c.
Sliding Bed Cvs=c.Mean
Heterogeneous FlowCvs=c.
Homogeneous FlowCvs=Cvt=c.
Resulting Erhg curveCvs=c.
Fixed Bed, Sliding Bed& Het. Flow Cvt=c.
Fixed Bed, Sliding Bed& Sliding Flow Cvt=c.
Limit Deposit Velocity
Ratio Potential/KineticEnergy
Cvs=0.225-0.275
Cvs=0.175-0.225
Cvs=0.125-0.175
Cvs=0.075-0.125
Cvs=0.025-0.075
© S.A.M. Dp=0.1552 m, d=0.91 mm, Rsd=1.59, Cv=0.150, μsf=0.800
Delft University of Technology – Offshore & Dredging Engineering
Data looks more organized not depending on the
volumetric concentration of the solids.
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Sliding Bed Cvs=c.
Equivalent Liquid Model
Homogeneous FlowCvs=Cvt=c.
Resulting Erhg curveCvs=c.
Resulting Erhg curveCvt=c.
Limit Deposit Velocity
Ratio Potential/KineticEnergy
Heterogeneous Flowwith Near Wall Lift
Homogeneous FlowMobilized
Cvt=0.225-0.275
Cvt=0.175-0.225
Cvt=0.125-0.175
Cvt=0.075-0.125
Cvt=0.025-0.075
© S.A.M. Dp=0.1552 m, d=0.910 mm, Rsd=1.585, Cv=0.150, μsf=0.550
Spatial versus Transport Concentration
& the Slip Velocity
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Spatial Volumetric Concentration is volume based.
Transport Volumetric Concentration is volume flow
based.
s l
v t v s v t v s
ls
vC 1 C C C
v
ls
v s v t
ls s l
vC C
v v
l sm l m l
r h g
ls s l s d v ss l
s d v s
ls
vi i i iE
v v R CvR 1 C
v
Relative Excess Hydraulic Gradient Erhg,
Cvt=constant.
The slip velocity here is the velocity difference
between the line speed and the particle velocity.
Flow Regimes History
Chapter 1
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Regimes History
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
The 5 Main Flow Regimes
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
The 5 main flow regimes are identified based on their
dominating behavior regarding energy dissipation.
1. The fixed bed regime is identified based on shear
stresses at the liquid-fixed bed interface (sheet flow).
2. The sliding bed regime is identified based on sliding
friction energy losses.
3. The heterogeneous flow regime is identified based on
potential and kinetic energy losses.
4. The homogeneous flow regime is identified based on
energy losses in turbulent eddies and viscous friction.
5. The sliding flow regime is identified based on sliding
friction, potential and kinetic energy losses.
At flow regime transitions, a mix of two flow regimes
will be present.
The Solids Effect Graph
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
How To Read The Graph? (Dp=6 inch)
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
2
l lsl
l
l p
vpi
g L 2 g D
m l
r h g
s d v
i iE
R C
Hydraulic Gradient
Relative Excess H.G.
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient ilFixed Bed Cvs=c.
Sliding Bed Cvs=c.Lower LimitSliding Bed Cvs=c.MeanSliding Bed Cvs=c.Upper LimitHeterogeneous FlowCvs=c.Equivalent LiquidModelHomogeneous FlowCvs=Cvt=c.Resulting Erhg curveCvs=c.Resulting Erhg curveCvt=c.Limit Deposit VelocityCvs=c.Limit Deposit VelocityCvt=c.Limit Deposit Velocity
Ratio Potential/KineticEnergyHeterogeneous Flowwith Near Wall LiftHomogeneous FlowMobilized
© S.A.M. Dp=0.1524 m, d=0.500 mm, Rsd=1.585, Cv=0.175, μsf=0.416
Cvs
d
Dp
d
Dp
d
Cvt
Different Models Fine Sand
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
4 possible models: Black heterogeneous, blue pseudo homogeneous,
light brown pseudo homogeneous & red homogeneous.
Different Models Coarse Sand & Gravel
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
5 possible models: Orange SB/He, black He, blue pseudo Ho, light
brown pseudo Ho & red Ho.
Existing Models
Chapter 6
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Zandi & Govatos, Yagi et al. & Babcock
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0.01
0.10
1.00
10.00
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10000.00
0.01 0.10 1.00 10.00 100.00 1000.00
Φ=
(im
-il)/(
i l·C
vt)
(-)
ψ=vls2/(g·Dp·Rsd)·√CD (-)
Zandi-Govatos (1967) on Durand coordinates
Zandi-Govatos A
Zandi-Govatos B
Durand & Condolios
Equivalent Liquid Sand
Lower Limit
Limit Deposit Velocity
N=0-40
N=40-310
N=310-1550
N=1550-3100
© S.A.M.
0.01
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1.00
10.00
100.00
1,000.00
10,000.00
0.01 0.10 1.00 10.00 100.00 1,000.00
(im
-il)/(
Cvt·i l)
vls2·√Cx/(g·Dp·Rsd)
Durand Gradient vs. the Durand Coordinate, Yagi et al. (1972)
DurandEquation
Yagi Sand
Yagi Gravel
Yagi SandCvt Data
© S.A.M.
0.01
0.10
1.00
10.00
100.00
1,000.00
10,000.00
0.01 0.10 1.00 10.00 100.00 1,000.00
(im
-il)/(
Cvt·i l)
vls2·√Cx/(g·Dp·Rsd)
Durand Gradient vs. the Durand Coordinate, Yagi et al. (1972)
DurandEquation
Yagi Sand
Yagi Gravel
Yagi GravelCvt Data
© S.A.M.
0.1
1.0
10.0
100.0
1 10 100
(im
-il)/(
Cv·i
l)
vls2·√Cx/(g·Dp·Rsd)
Durand Gradient vs. the Durand Coordinate, Babcock (1970)
30/45 Sand vls=5.750 fps
20/30 Sand vls=11.10 fps
20/30 Sand vls=12.72 fps
6/8 Steel Shot
Arkosic Sand
Garnet Sand
80/100 Sand
80/100 Sand, Cv>0.2
80/100 Sand, Cv<0.2
s=2.76: Limestone
s=1.98: Mine Refuse
s=1.32: Coal
s=2.14: 1"x0.5" Slag
s=2.14: 1"x4 mesh Slag
Durand Equation
Newitt Sliding Bed: y=66·√Cx/x, gravelBabcock Sliding Bed: y=60.6·√Cx/x, gravel
22 Models im-v
lsgraph
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
For small pipe diameters the models are still “close”. For
large diameter pipes the difference is much much more.
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0.25
0 1 2 3 4 5
Hyd
rau
lic
gra
die
nt
i m, i l
(m w
ate
r/m
)
Line speed vls (m/sec)
Hydraulic gradient im, il vs. Line speed vls, All Liquid il curve
Sliding Bed Cvs=c
Equivalent Liquid Model
Homogeneous Flow
Turian & Yuan Regime 0
Turian & Yuan Regime 1
Turian & Yuan Regime 2
Turian & Yuan Regime 3
Durand & Condolios
Newitt et al.
Newitt et al. Bed
Jufin & Lopatin
Fuhrboter
Zandi & Govatos Het.
Zandi & Govatos Hom.
Wilson et al. - 1.0
Wilson et al. - 1.7
Wilson et al. Bed
SRC Original Cvs=c.
SRC Weight Cvs=c.
DHLLDV Cvs=c.
DHLLDV Cvt=c.© S.A.M. Dp=0.1016 m, d=0.500 mm, Rsd=1.585, Cv=0.175, μsf=0.416
22 Models Erhg
-ilgraph
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
This graph organizes the models better, but there is still a
lot of difference between the models.
0.001
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1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Sliding Bed Cvs=c.
Equivalent Liquid Model
Homogeneous Flow
Limit Deposit Velocity
Turian & Yuan Regime 0
Turian & Yuan Regime 1
Turian & Yuan Regime 2
Turian & Yuan Regime 3
Durand & Condolios
Newitt et al.
Newitt et al. Bed
Jufin & Lopatin
Fuhrboter
Zandi & Govatos Het.
Zandi & Govatos Hom.
Wilson et al. - 1.0
Wilson et al. - 1.7
Wilson et al. Bed
SRC Original Cvs=c.
SRC Weight Cvs=c.
DHLLDV Cvs=c.
DHLLDV Cvt=c.© S.A.M. Dp=0.1016 m, d=0.500 mm, Rsd=1.585, Cv=0.175, μsf=0.416
Types of Models
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
• There are many empirical models, mainly for
heterogeneous flow, some for sliding bed and
homogeneous flow.
• Most empirical models add one term to the Darcy-
Weisbach equation, often based on Froude numbers.
• There is the equivalent liquid model (ELM) for
homogeneous flow.
• There are some 2 layer and 3 layer models for
transport with a stationary or sliding bed or sheet flow,
Wilson, Doron & Barnea, SRC Model, Matousek.
• The 2 layer and 3 layer models are closed with
empirical equations for the bed shear stress and the
concentration distribution.
Stationary/Fixed Bed Regime
Chapter 7.3 & 8.3
Wilson et al.
Doron & Barnea
SRC Model
Matousek Model
DHLLDV Framework
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Definitions
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Equilibrium of Forces
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
1 , l 1 1 2 , l 1 2 1 , l 1 2 , l
2 1
1 1
O L O L F Fp p p
A A
Kazanskij (1980), Cvs=0.17
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Sliding Bed Cvs=c.
Equivalent Liquid Model
Homogeneous FlowCvs=Cvt=c.
Resulting Erhg curveCvs=c.
Resulting Erhg curveCvt=c.
Limit Deposit Velocity
Ratio Potential/KineticEnergy
Heterogeneous Flowwith Near Wall Lift
Homogeneous FlowMobilized
Cv=0.170
Cv=0.134
Cv=0.076
Cv=0.036
© S.A.M. Dp=0.5000 m, d=1.500 mm, Rsd=1.585, Cv=0.170, μsf=0.416
Sliding Bed Regime
Chapter 7.4 & 8.4
Wilson et al.
Doron & Barnea
SRC Model
Matousek Model
DHLLDV Framework
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Definitions
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Equilibrium of Forces
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
The Models 1
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
• For the wall friction the standard Darcy Weisbach
friction coefficient is used (Moody diagram).
• The Newitt et al. Model is empirical and based on
experimental data. Newitt et al. were the first to use the
solids effect graph.
• The Wilson 2LM is based on a bed and water above.
The bed friction is the Darcy Weisbach friction
coefficient with the particle diameter as the roughness
multiplied with a factor. Televantos found a factor 2,
but Wilson also used different factors over the years
like 2.6. For the normal stress between the bed and the
wall Wilson uses a hydrostatic stress distribution,
resulting in a higher friction force compared to the
submerged weight times the sliding friction coefficient.
The Models 2
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
• The Wilson model is based on constant spatial
concentration. Constant delivered concentration curves
are constructed by interpolation.
• Doron & Barnea (2LM) basically use the Wilson
model, but extended it with suspension above the bed,
based on the standard advection diffusion equation. For
the constant delivered concentration case this always
results in a sliding bed, also at very low line speeds. So
they extended their model to a 3LM giving it the
possibility to have a fixed bed at very low line speeds.
• The SRC model is based on the Wilson model for
constant spatial concentration, but with suspension
above the bed. The fraction in suspension and the
fraction in the bed are based on an exponential power.
The Models 3
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
• This power contains the ratio of the terminal settling
velocity to the line speed. The suspended fraction
forms an adjusted carrier liquid, resulting in adjusted
liquid properties and an adjusted submerged weight of
the bed. This way there is a smooth transition of fully
stratified flow to heterogeneous flow to homogeneous
flow.
• Matousek uses a completely different method. Based
on the delivered concentration, the Shields parameter is
determined with the reversed Meyer Peter Muller
equation. Once the Shields parameter is known, the
bed friction coefficient can be determined from the
equivalent bed roughness. The method is based on
sheet flow as a transport mechanism.
The Models 4
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
• The DHLLDV Framework uses the Wilson approach,
however with the weight approach for the sliding
friction. So the sliding friction force equals the
submerged weight times the sliding friction coefficient.
Above the bed sheet flow is assumed according to
Wilson & Pugh. The method is spatial concentration
based. The delivered concentration follows from the
transport in the sheet flow layer and the transport in the
sliding bed. The method results in a solids effect
almost equal to the sliding friction coefficient.
The Submerged Weight Approach
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
s f s f l s d v b p
s in c o s
F g L R C A
The Limit of Stationary Deposit Velocity
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0 1 2 3 4 5 6 7
Hyd
rau
lic
gra
die
nt
i m(m
.w.c
./m
pip
e)
Line speed vls (m/sec)
Hydraulic gradient im vs. line speed vls, Cvr=Cvs/Cvb
LSDV HydrostaticTelevantos
Liquid
© S.A.M.© S.A.M.© S.A.M. Dp=0.7620 m, d=0.50 mm, Rsd=1.585, μsf=0.415, Cvb=0.55
Increasing Cvr
2 ,s f 2 , l 1 2 ,s f 2F F F p A
The Erhg
Value is almost μsf
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.0 0.5 1.0 1.5 2.0 2.5
Erh
g=
(im
-il)/(
Rsd·C
vs)
(-)
Relative line speed vr=vls/vsm (-)
Erhg value vs. relative line speed vr, Cvr=Cvs/Cvb
LSDV
Cvr=0.05
Cvr=0.10
Cvr=0.20
Cvr=0.30
Cvr=0.40
Cvr=0.50
Cvr=0.60
Cvr=0.70
Cvr=0.80
Cvr=0.90
Cvr=0.95
© S.A.M. Dp=0.7620 m, d=1.00 mm, Rsd=1.59, μsf=0.415, Cvb=0.55
2LM-M-W
Stationary Deposit
m l
r h g s f m l s f s d v s
s d v s
i iE a n d i i R C
R C
Resulting Hydraulic Gradient Graph, Cvt
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 1 2 3 4 5 6 7 8 9 10 11 12
Hyd
rau
lic
gra
die
nt
i m(m
.w.c
./m
pip
e)
Line speed vls (m/sec)
Hydraulic gradient im vs. line speed vls, Cvr=Cvt/Cvb
LSDV
Cvr=0.00
Cvr=0.05
Cvr=0.10
Cvr=0.20
Cvr=0.30
Cvr=0.40
Cvr=0.50
Cvr=0.60
Cvr=0.70
Cvr=0.80
Cvr=0.90
Cvr=0.95
© S.A.M.© S.A.M.© S.A.M. Dp=0.7620 m, d=1.00 mm, Rsd=1.59, μsf=0.415, Cvb=0.55
Stationary Deposit
3LM-M-W
Wiedenroth (1967), Medium Sand
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Sliding Bed Cvs=c.
Equivalent LiquidModel
Homogeneous FlowCvs=Cvt=c.
Limit Deposit Velocity
Graded Sand Cvs=c.
Graded Sand Cvt=c.
Cv=0.300
Cv=0.250
Cv=0.200
Cv=0.150
Cv=0.100
Cv=0.050
© S.A.M. Dp=0.1250 m, d=0.900 mm, Rsd=1.585, Cv=0.200, μsf=0.250
Wiedenroth (1967), Coarse Sand
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Sliding Bed Cvs=c.
Equivalent LiquidModel
Homogeneous FlowCvs=Cvt=c.
Limit Deposit Velocity
Graded Sand Cvs=c.
Graded Sand Cvt=c.
Cv=0.300
Cv=0.250
Cv=0.200
Cv=0.150
Cv=0.100
Cv=0.050
© S.A.M. Dp=0.1250 m, d=2.200 mm, Rsd=1.585, Cv=0.150, μsf=0.250
Heterogeneous Flow Regime
Chapter 7.5 & 8.5
Durand & Condolios
Newitt et al.
Jufin & Lopatin
Fuhrboter – Wilson et al.
DHLLDV Framework
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Existing Equations Depending on il
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
m l m l
m l v t
l v t l v t
i i p pp p 1 C w ith :
i C p C
3 / 22
3 / 2 ls
x
p s d
vK K C w ith : K 8 5
g D R
3
m l 1 p s d t v t 1
ls
1p p 1 K g D R v C K 1 1 0 0
v
Durand, Condolios & Gibert based on Froude numbers
Newitt et al. based on potential energy losses
3
1 / 6*m in
m l m in v t p
ls
vp p 1 2 v 5 .5 C D
v
Jufin & Lopatin empirical large diameters
Existing Equations Independent of il
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
k
m l l v s
ls
k m l k
m l v s r h g
ls s d v s s d ls
Sp p g L C
v
S i i Si i C E
v R C R v
M
s f 5 0
m l l s d v t
ls
M M
s f 5 0 s f 5 0
m l s d v t r h g
ls ls
vp p g R L C
2 v
v vi i R C E R
2 v 2 v
Fuhrboter medium diameters
Wilson heterogeneous empirical (Stratification Ratio)
DHLLDV Framework
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Energy Dissipation by:
• Turbulence Viscous Dissipation (Darcy Weisbach)
• Potential Energy Losses (Hindered Settling Velocity)
• Kinetic Energy Losses (Collisions)
s ,p o t s ,k in
m l , v is c s ,p o t s ,k in l , v is c
l , v is c l , v is c
p pp p p p p 1
p p
2
t v s C2 s lm
l l ls l s d v s l s d v s
p ls t
v 1 C / vp 1 1v g R C g R C
L D 2 v v
2
s d p t v s C s l
m l v s 2
l tls ls
2 g R D v 1 C / v1p p 1 C
vv v
Slip
Verification & Validation, Durand et al.
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Durand & Condolios (1952)
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Equivalent Liquid Model
Homogeneous
Sliding Bed Cvs=c.
d=0.20 mm, Cvt=c.
d=0.39 mm, Cvt=c.
d=0.89 mm, Cvt=c.
d=2.04 mm, Cvt=c.
d=4.20 mm, Cvt=c.
Limit Deposit Velocity
d=0.20 mm
d=0.39 mm
d=0.89 mm
d=2.04 mm
d=4.20 mm
© S.A.M. Dp=0.1524 m, Rsd=1.585, Cvt=0.050, μsf=0.416
Verification & Validation, Clift et al.
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Clift (1982)
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Sliding Bed Cvs=c.
Equivalent Liquid Model
Homogeneous FlowCvs=Cvt=c.
Resulting Erhg curveCvs=c.
Resulting Erhg curveCvt=c.
Limit Deposit Velocity
Ratio Potential/KineticEnergy
Heterogeneous Flowwith Near Wall Lift
Homogeneous FlowMobilized
Cv=0.100
© S.A.M. Dp=0.4400 m, d=0.680 mm, Rsd=1.585, Cv=0.100, μsf=0.416
Homogeneous Flow Regime
Chapter 7.6 & 8.6
Equivalent Liquid Model
Newitt et al.
Wilson et al.
Talmon
DHLLDV Framework
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
The Equivalent Liquid Model (ELM)
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
2
m l m ls
p
L 1p v
D 2
2
l lsm m
m
l l p
l s d v s
m l
r h g l
s d v s
vpi
g L 2 g D
i 1 A R C
i iE A i
R C
Phenomena
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
• Very fine particles: The liquid properties have to
be adjusted. The ELM can be used with the
adjusted liquid properties.
• Fine particles: The ELM can be used with the
original liquid properties. At high line speeds the
lubrication effect will be mobilized.
• Medium/Coarse particles: The lubrication effect is
mobilized, due to a particle poor viscous sub-layer.
This gives a reduction of the solids effect in the
ELM.
Very Fine Particles
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
v s ,x
l l p l l p
l im 0 .4
s ls , ld v s p
v s s d
x l l
v s v s
v s
v s , x v s ,r v s
v s v s
1 6 .6 C2
x l v s , x v s , x
x
x
x
S tk 9 D S tk 9 D
d = X = 1v 7 .5 D
X C R
1 C C X
X CC a n d C 1 X C
1 C C X
1 2 .5 C 1 0 .0 5 C 0 .0 0 2 7 3 e
s x
s d , x
x
a n d R
The Models
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
• Newitt et al. use a factor A=0.6 in the ELM.
• Wilson et al. Use different factors for A in the ELM.
• Talmon determined A based on a particle free viscous
sublayer in 2D channel flow.
• The DHLLDV Framework determined A based on a
concentration distribution in a circular pipe. This way
the viscous sublayer is particle poor, but not
completely particle free. The result is an equation for
A, depending on the concentration.
• The DHLLDV Framework also assumes that particles
fitting in the viscous sublayer do not result in a
particle free viscous sublayer and thus have A=1. The
larger the particles the more the particle free sublayer
is mobilised.
Fine Particles
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
v
v
2
C m l
s d v s
l vm l
r h g l 2s d v s
C m l
s d v s
l
vm l
r h g l E
s d v s
A
1 R C ln 18i i
E i 1 1 1R C dA
R C ln 18
i iE i 1 1 1
R C d
v
m l l s d v s E
v
m l l s d v s l s d v s
v
m l l s d v s E l s d v s E
i i i R C 1 1 1d
= m a x = 1 i i i R C i 1 R C E L Md
0 i i i R C i 1 R Cd
Medium/Coarse Particles
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
v
v
v
v
2
C m l
s d v s
l
r h g l E l2
C m l
s d v s
l
2
C m l
s d v s
l
m l l 2
C m l
l
C
s d v s
m l l
A
1 R C ln 18
E i i
A
R C ln 18
A
1 R C ln 18
i i i
A
ln 18
A
1 R C
p p p
v
v
2
m l
l
2
C m l
l
ln 18
A
ln 18
Lubrication Factor αE
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
λm
/λl&
αE
(-)
Volumetric concentration Cv (-)
λm/λl & αE vs. Volumetric concentration Cv
λm/λl: Law of the Wall (no damping)
λm/λl: Nikuradse (no damping)
λm/λl: Prandtl (damping)
λm/λl: Average
αE: Law of the Wall (no damping)
αE: Nikuradse (no damping)
αE: Prandtl (damping)
αE: Average
© S.A.M.
Very Fine Particles
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0.000
0.020
0.040
0.060
0.080
0.100
0.120
0.140
0.160
0.180
0.200
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Hyd
rau
lic
gra
die
nt
i m, i l
(m w
ate
r/m
)
Line speed vls (m/sec)
Hydraulic gradient im, il vs. Line speed vls
Liquid il curve
Sliding Bed Cvs=c.
Equivalent Liquid Model
Homogeneous FlowCvs=Cvt=c.
Resulting im curveCvs=c.
Resulting im curveCvt=c.
Limit Deposit VelocityCvs=c.
Limit Deposit Velocity
Cv=0.240
© S.A.M. Dp=0.1075 m, d=0.040 mm, Rsd=3.999, Cv=0.240, μsf=0.416
Thomas (1976)
Very Fine Particles, with Thomas (1965)
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Thomas (1976) Adjusted Liquid Properties
0.000
0.020
0.040
0.060
0.080
0.100
0.120
0.140
0.160
0.180
0.200
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Hyd
rau
lic
gra
die
nt
i m, i l
(m w
ate
r/m
)
Line speed vls (m/sec)
Hydraulic gradient im, il vs. Line speed vls
Liquid il curve
Sliding Bed Cvs=c.
Equivalent LiquidModel
Homogeneous FlowCvs=Cvt=c.
Uniform Sand Cvs=c.
Uniform Sand Cvt=c.
Limit Deposit VelocityCvs=c.
Limit Deposit Velocity
Cv=0.240
© S.A.M. Dp=0.1075 m, d=0.040 mm, Rsd=3.999, Cv=0.240, μsf=0.416
Fine Particles
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Whitlock (2004)
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Sliding Bed Cvs=c.Mean
Equivalent Liquid Model
Homogeneous FlowCvs=Cvt=c.
Resulting Erhg curveCvs=c.
Resulting Erhg curveCvt=c.
Limit Deposit Velocity
Ratio Potential/KineticEnergy
d=0.400 mm, Cv=0.170
d=0.085 mm, Cv=0.237
© S.A.M. Dp=0.1016 m, d=0.085 mm, Rsd=1.585, Cv=0.237, μsf=0.416
Medium/Coarse Particles
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Blythe & Czarnotta (1995)
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Sliding Bed Cvs=c.Mean
Equivalent Liquid Model
Homogeneous FlowCvs=Cvt=c.
Resulting Erhg curveCvs=c.
Resulting Erhg curveCvt=c.
Limit Deposit Velocity
Ratio Potential/KineticEnergy
Cv=0.285
Cv=0.235
Cv=0.185
Cv=0.135
Cv=0.085
© S.A.M. Dp=0.1000 m, d=0.280 mm, Rsd=1.585, Cv=0.175, μsf=0.416
Sliding Flow Regime
Chapter 7.7 & 8.8
SRC Model
DHLLDV Framework
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Phenomena
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
If the particle diameter to pipe diameter is larger than
about 0.015, the particles will not be suspended anymore,
but stay in a fast flowing sort of bed.
The behavior is a mix of the heterogeneous flow regime
and the sliding bed regime.
At d/Dp=0.015 the behavior is still heterogeneous, but
the larger the particle diameter the more it is sliding bed
behavior.
The higher the line speed the smaller the concentration of
the flowing particles at the bottom of the pipe.
Verification & Validation, Boothroyde
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Sliding Bed Cvs=c.
Equivalent Liquid Model
Homogeneous FlowCvs=Cvt=c.
Resulting Erhg curveCvs=c.
Resulting Erhg curveCvt=c.
Limit Deposit Velocity
Ratio Potential/KineticEnergy
Heterogeneous Flowwith Near Wall Lift
Homogeneous FlowMobilized
Cv=0.202
Cv=0.180
Cv=0.138
Cv=0.093
Cv=0.046
© S.A.M. Dp=0.2000 m, d=10.000 mm, Rsd=1.732, Cv=0.100, μsf=0.470
Boothroyde et al. (1979)
Verification & Validation, Wiedenroth
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Wiedenroth (1967)
0.001
0.010
0.100
1.000
0.001 0.010 0.100 1.000
Rela
tive
ex
ce
ss
hyd
rau
lic
gra
die
nt
Erh
g(-
)
Hydraulic gradient il (-)
Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il
Sliding Bed Cvs=c.
Equivalent Liquid Model
Homogeneous FlowCvs=Cvt=c.
Resulting Erhg curveCvs=c.
Resulting Erhg curveCvt=c.
Limit Deposit Velocity
Ratio Potential/KineticEnergy
Heterogeneous Flowwith Near Wall Lift
Homogeneous FlowMobilized
Cv=0.200
Cv=0.150
Cv=0.100
Cv=0.050
© S.A.M. Dp=0.1250 m, d=5.950 mm, Rsd=1.585, Cv=0.150, μsf=0.416
Main Conclusion
© S.A.M
Delft University of Technology – Offshore & Dredging Engineering
Conclusions
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Models are valid for the parameters the experiments were
carried out with.
The correct flow regime has to be identified for each (sub)
model.
For heterogeneous flow the solids effect is independent of the
Darcy Weisbach component.
Models validated with a wide range of parameters are: Jufin &
Lopatin, Wilson et al., the SRC model and the DHLLDV
Framework.
These 4 models give similar results for medium and coarse
sands over a wide range of pipe diameters.
© S.A.M
The Elephant of Wilson is our best Friend
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Questions?
The Elephant of Wilson is our best Friend
Delft University of Technology – Offshore & Dredging Engineering
© S.A.M
Questions?