dhlldv framework an overview of slurry transport models · 2016-11-25 · 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

Goals & Targets

© S.A.M


Pump/Pipeline System


© 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)


© S.A.M

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

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


Goals & Targets

The Elephant of Wilson


© 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


Possibilities

The Solids Effect

© S.A.M


Solids Effect


© S.A.M

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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


Data looks unorganized depending on the volumetric

concentration of the solids.

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0 1 2 3 4 5 6

Hyd

rau

lic

gra

die

nt

i m, i l

(m w

ate

r/m

)



Liquid il curve

Sliding Bed Cvs=c.

Equivalent Liquid Model






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

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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.


Heterogeneous FlowCvs=c.


Resulting Erhg curveCvs=c.

Fixed Bed, Sliding Bed& Het. Flow Cvt=c.

Fixed Bed, Sliding Bed& Sliding Flow Cvt=c.


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



Data looks more organized not depending on the

volumetric concentration of the solids.

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Rela

tive

ex

ce

ss

hyd

rau

lic

gra

die

nt

Erh

g(-

)



Sliding Bed Cvs=c.




Resulting Erhg curveCvt=c.



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


Spatial versus Transport Concentration

& the Slip Velocity


© 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


Regimes History


© S.A.M

The 5 Main Flow Regimes


© 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


How To Read The Graph? (Dp=6 inch)


© 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.

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1.000

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Rela

tive

ex

ce

ss

hyd

rau

lic

gra

die

nt

Erh

g(-

)


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


Cvs

d

Dp

d

Dp

d

Cvt

Different Models Fine Sand


© S.A.M

4 possible models: Black heterogeneous, blue pseudo homogeneous,

light brown pseudo homogeneous & red homogeneous.

Different Models Coarse Sand & Gravel


© 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


Zandi & Govatos, Yagi et al. & Babcock


© S.A.M

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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


N=0-40

N=40-310

N=310-1550

N=1550-3100

© S.A.M.

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10.00

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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.

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10.00

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1,000.00

10,000.00

0.01 0.10 1.00 10.00 100.00 1,000.00

(im

-il)/(

Cvt·i l)


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)


Durand Gradient vs. the Durand Coordinate, Babcock (1970)

30/45 Sand vls=5.750 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


© 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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rau

lic

gra

die

nt

i m, i l

(m w

ate

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)


Hydraulic gradient im, il vs. Line speed vls, All Liquid il curve

Sliding Bed Cvs=c


Homogeneous Flow

Turian & Yuan Regime 0




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


© S.A.M

This graph organizes the models better, but there is still a

lot of difference between the models.

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tive

ex

ce

ss

hyd

rau

lic

gra

die

nt

Erh

g(-

)



Sliding Bed Cvs=c.


Homogeneous Flow






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


© 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


Definitions


© S.A.M

Equilibrium of Forces


© 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


© S.A.M

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tive

ex

ce

ss

hyd

rau

lic

gra

die

nt

Erh

g(-

)



Sliding Bed Cvs=c.









Cv=0.170

Cv=0.134

Cv=0.076

Cv=0.036


Sliding Bed Regime

Chapter 7.4 & 8.4

Wilson et al.

Doron & Barnea

SRC Model

Matousek Model

DHLLDV Framework

© S.A.M


Definitions


© S.A.M

Equilibrium of Forces


© S.A.M

The Models 1


© 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


© 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


© 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


© 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


© 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


© S.A.M

0

0.1

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0.7

0.8

0.9

1

0.00

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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)


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


© S.A.M

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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


© S.A.M

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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)


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


© S.A.M

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gra

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nt

Erh

g(-

)



Sliding Bed Cvs=c.




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


Wiedenroth (1967), Coarse Sand


© S.A.M

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ss

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gra

die

nt

Erh

g(-

)



Sliding Bed Cvs=c.




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


Heterogeneous Flow Regime

Chapter 7.5 & 8.5

Durand & Condolios

Newitt et al.

Jufin & Lopatin

Fuhrboter – Wilson et al.

DHLLDV Framework

© S.A.M


Existing Equations Depending on il


© 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


© 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


© 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.


© S.A.M

Durand & Condolios (1952)

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Rela

tive

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ss

hyd

rau

lic

gra

die

nt

Erh

g(-

)




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.


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.


© S.A.M

Clift (1982)

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)



Sliding Bed Cvs=c.









Cv=0.100


The Equivalent Liquid Model (ELM)


© 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


© 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


© 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


© 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


© 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


© 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


© 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


© 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

)



Liquid il curve

Sliding Bed Cvs=c.







Cv=0.240


Thomas (1976)

Very Fine Particles, with Thomas (1965)


© 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

)



Liquid il curve

Sliding Bed Cvs=c.



Uniform Sand Cvs=c.

Uniform Sand Cvt=c.



Cv=0.240


Fine Particles


© 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(-

)










d=0.400 mm, Cv=0.170

d=0.085 mm, Cv=0.237


Medium/Coarse Particles


© 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(-

)










Cv=0.285

Cv=0.235

Cv=0.185

Cv=0.135

Cv=0.085


Phenomena


© 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


© 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(-

)



Sliding Bed Cvs=c.









Cv=0.202

Cv=0.180

Cv=0.138

Cv=0.093

Cv=0.046


Boothroyde et al. (1979)

Verification & Validation, Wiedenroth


© 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(-

)



Sliding Bed Cvs=c.









Cv=0.200

Cv=0.150

Cv=0.100

Cv=0.050


Conclusions


© 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.

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