THE UNIVERSITY OF TULSA Thesis/Qian...THE UNIVERSITY OF TULSA THE GRADUATE SCHOOL MEASUREMENTS AND...

THE UNIVERSITY OF TULSA

THE GRADUATE SCHOOL

MEASUREMENTS AND MODELING OF PARTICLE SEDIMENTATION RATE

AND SETTLING VELOCITY IN A VERTICAL PIPE

by Qian Li

A thesis submitted in partial fulfillment of

the requirement for the degree of Master of Science

in the Discipline of Mechanical Engineering

The Graduate School

The University of Tulsa

2003

THE UNIVERSITY OF TULSA

THE GRADUATE SCHOOL

MEASUREMENTS AND MODELING OF PARTICLE SEDIMENTATION RATE

AND SETTLING VELOCITY IN A VERTICAL PIPE

by Qian Li

A THESIS

APPROVED FOR THE DISCIPLINE OF

MECHANICAL ENGINEERING

By Thesis Committee

____________________________, Chairperson

____________________________ ____________________________

ii

ABSTRACT

Li, Qian (Master of Science in Mechanical Engineering)

(147pp.-Chapter VI)

Directed by Dr. Siamack A. Shirazi

(181 words)

In a deepwater production system, the height of the riser can be about 1 mile in

length, and around 2 miles for ultra-deepwater. When the deepwater production is shut-

in, all the sand that is entrained in the produced fluid will settle down to the riser base.

Knowing the sand settling time tells how long a shut-in can last before a sand bed is

formed. Experiments have been conducted in a vertical pipe filled with water or white oil

to measure the particle sedimentation rates. Particles used in the experiments are sand

and glass beads. A two-pipe apparatus and a one-pipe apparatus were used. Particle

bridge phenomenon was observed in the two-pipe apparatus. Experiments were also

conducted with an upward gas flow through the test section to see how gas flow affects

the sand sedimentation rate. A mechanistic model has been developed to predict settling

velocity in a vertical pipe and a finite-difference model has been developed to predict

particle sedimentation rates. The results obtained by applying the model have been

compared with experimental data. Good agreement was found between the model and

data in the one-pipe apparatus.

iii

ACKNOWLEDGEMENTS

I would like to gratefully acknowledge Dr. Shirazi for his support and guidance

throughout this work. He played a key role in all phases of research and preparation of

this thesis. I would also like to recognize Dr. McLaury, Dr. Gomez, Dr. Gene Kouba and

Dr. Song for their assistance throughout this work. Special thanks are extended to

ChevronTexaco for supporting this project.

iv

DEDICATION

I dedicate this work to my parents, Li, Ruren and Zhou, Shu and my husband, Chen,

Xianghui.

v

TABLE OF CONTENTS

Approval Page..................................................................................................................... ii

Abstract .............................................................................................................................. iii

Acknowledgements............................................................................................................ iv

Dedication ............................................................................................................................v

Table of Contents............................................................................................................... vi

List of Tables ..................................................................................................................... ix

List of Figures ......................................................................................................................x

Chapter I Introduction.......................................................................................................1

Background..................................................................................................................1

Research Goals ............................................................................................................3

Important Definitions ..................................................................................................3

Settling Velocity...................................................................................................3

Sedimentation Rate...............................................................................................4

Chapter II Literature Review ............................................................................................5

Stokes Terminal Settling Velocity...............................................................................5

Observation of Settling Velocity in Literature ............................................................6

Settling Velocity Models in Literature ......................................................................12

Chapter III Experimental Setup and Procedure ..............................................................14

Two-Pipe Apparatus and Data...................................................................................14

Experimental Apparatus .....................................................................................14

Experimental Procedure .....................................................................................15

vi

Measurements of Viscosity of White Oil ...........................................................18

Measurements of Volume Concentration of Particle Bed..................................19

Measurements of Particle Size Distribution .......................................................20

Experimental Results..........................................................................................21

Particle Bridge Phenomenon ..............................................................................28

Experimental Results Adding Gas Flow ...................................................................29

One-Pipe Apparatus and Data ...................................................................................32

Experimental Apparatus .....................................................................................32

Experimental Results..........................................................................................32

Chapter IV Mechanistic Modeling..................................................................................38

Settling Velocity Models ...........................................................................................38

TUSMP-GK Model ............................................................................................38

Corrected TUSMP-GK Model ...........................................................................40

Mechanistic Modeling of Sedimentation Rate ..........................................................43

Three-Zone Model..............................................................................................43

Finite-Difference Model .....................................................................................44

Chapter V Results ...........................................................................................................48

Results of Two-Pipe Apparatus.................................................................................48

Results of One-Pipe Apparatus .................................................................................57

Effect of Cell Height and Time Step on the Predicted Results .................................65

Chapter VI Summary, Conclusions and Recommendations...........................................67

Nomenclature.....................................................................................................................71

Bibliography ......................................................................................................................72

vii

Appendix A........................................................................................................................75

Appendix B ......................................................................................................................144

viii

LIST OF TABLES

Table III-1. Typical Measured Sedimentation Times .......................................................17

Table III-2. Experiments Conducted with Average Particle Size of 150µm in Two-Pipe

Apparatus ...........................................................................................................................17

Table III-3. Experiments Conducted with Different Particle Sizes in Two-Pipe Apparatus

...................................................................................................................................18

Table III-4. Typical Measured Sedimentation Times in One-Pipe Apparatus .................33

Table III-5 Experiments Conducted in One-Pipe Apparatus...........................................33

ix

LIST OF FIGURES

Figure I-1. Sand Accumulations in Tubing and an Elbow..................................................2

Figure I-2. Settling Velocity ...............................................................................................4

Figure I-3. Sedimentation Rate ...........................................................................................4

Figure II-1. Schematic of Return Flow...............................................................................7

Figure II-2. Experimentally Measured Settling Velocity ...................................................8

Figure II-3. Experimentally Measured Settling Velocity ...................................................9

Figure II-4. Experimentally Measured Settling Velocity .................................................10

Figure II-5. Schematic of Settling Velocity vs. Concentration.........................................11

Figure III-1. Schematic of Two-pipe Apparatus...............................................................14

Figure III-2. Photograph of Test Section ..........................................................................15

Figure III-3. Photograph of the ½ inch Graduated Pipe ...................................................16

Figure III-4. Chan 35 Viscometers ...................................................................................19

Figure III-5. Capillary Tube Viscometer ..........................................................................19

Figure III-6. Schematic of Measuring the Concentration of Particle Bed ........................20

Figure III-7. Sand Size Distribution by Weight Percent...................................................21

Figure III-8. Glass Beads Size Distribution by Weight Percent .......................................21

Figure III-9. Measured Values of Sand Level versus Time (150 µm Sand in Water,

α0=0.44%) ..........................................................................................................................22

Figure III-10. Schematic of Experimental Apparatus.......................................................23

Figure III-11. Measured Values of Sand Level versus Time (150 µm Sand in Water)....24

Figure III-12. Measured Values of Glass Beads Level versus Time (150 µm Glass Beads

in Water) ............................................................................................................................24

x

Figure III-13. Comparison of Sand Sedimentation Rate at α0=0.44% & Glass Beads

Sedimentation Rate at α0=0.47% .......................................................................................25




Sedimentation Rate at α0=3.1% .........................................................................................26

Figure III-16. Measured Values of Sand Level versus Time (150-177 µm and 177-210

µm Sand in Water, α0=0.44%)...........................................................................................27

Figure III-17. Measured Values of Sand Level versus Time (150µm Sand in Oil, α0

=0.88%)..............................................................................................................................28

Figure III-18. Schematic of Sand Bridge Phenomenon ....................................................29

Figure III-19. Photograph of Gas Injection Location ........................................................30

Figure III-20. Sand Sedimentation Rate vs. Gas Flow Rate (in water) ............................30

Figure III-21. Sand Sedimentation Rate vs. Gas Flow Rate (in oil) ..................................31

Figure III-22. Measured Values of Sand Level versus Time (150 µm Sand in Water).....34

Figure III-23. Measured Values of Sand Level versus Time (150 µm Sand in Oil) .........35

Figure III-24. Measured Values of Glass Beads Level versus Time (150 µm Glass Beads

in Oil) .................................................................................................................................35



Figure III-26. Comparison of Sand Sedimentation Rate in Oil at α0=1.8% & Glass Beads

Sedimentation Rate in Oil at α0=1.8%...............................................................................37

xi

Figure IV-1. Comparison of TUSMP-GK Models ...........................................................41

Figure IV-2. Comparison of TUSMP-GK Models with Literature Data..........................42

Figure IV-3. Comparison of Different Settling Velocity Models.....................................43

Figure IV-4. Schematic of Zones Considered in Model ...................................................44

Figure IV-5. Schematic of Experimental Apparatus.........................................................45

Figure V-1. Schematic of Particle Bridge Phenomenon...................................................48

Figure V-2. Comparison between Predicted and Measured Particle Sedimentation Rate

(150 µm Sand in Water, α0=0.44%) ..................................................................................49


(150 µm Sand in Water, α0=0.59%) ..................................................................................50


(150 µm Sand in Water, α0=1.32%) ..................................................................................50


(150 µm Glass Beads in Water, α0=0.47%) ......................................................................51







Figure V-9. Fast Recirculating Fluid Motion Observed in Oil..........................................54


(150 µm Sand in Oil, α0=0.88%).......................................................................................54

xii


(150 µm Sand in Oil, α0=0.88%).......................................................................................55

Figure V-12. Comparison between Predicted and Measured Sand Sedimentation Rate

(150 µm Sand in Oil, α01=0.38%, α02=3.45%)..................................................................55

Figure V-13. Comparison between Predicted and Measured Sand Sedimentation Rate

(150 µm Sand in Oil, α01=0.67%, α02=6.9%)....................................................................56


in One-Pipe Apparatus (150 µm Sand in water, α0=0.46%) .............................................57


in One-Pipe Apparatus (150 µm Sand in water, α0=1.38%) .............................................58


in One-Pipe Apparatus (150 µm Sand in water, α0=10%) ................................................59


in One-Pipe Apparatus (150 µm Sand in Oil, α0=0.46%) .................................................59


in One-Pipe Apparatus (150 µm Sand in Oil, α0=3.80%) .................................................60


in One-Pipe Apparatus (150 µm Sand in Oil, α0=10%) ....................................................60


in One-Pipe Apparatus (150 µm Glass Beads in Water, α0=0.49%).................................61



xiii


in One-Pipe Apparatus (150 µm Glass Beads in Water, α0=10%)....................................62




in One-Pipe Apparatus (150 µm Glass Beads in Water, α0=3.6%)...................................64


in One-Pipe Apparatus (150 µm Glass Beads in Water, α0=12%)....................................64

Figure V-26. Comparison between Predicted Particle Sedimentation Rates Using

Different dz ........................................................................................................................65


Different dt.........................................................................................................................66

Figure VI-1. Schematic of Sand Settling and Sliding in a Base of a Vertical Riser.........69

Figure VI-2. Critical Deposition Velocity vs. Pipe Angle................................................70

xiv

CHAPTER I

INTRODUCTION

Background

The critical deposition velocity is a flow velocity needed to keep sand particles in

suspension in a horizontal and inclined pipe flow and prevent accumulation of particles

on the bottom of a pipe. Often, in gas production pipelines containing sand particles, the

erosional or erosion-corrosion threshold flowstream velocity is low and could be below

the critical deposition velocity. If the production rate (flowstream velocity) is kept below

the critical deposition velocity, particles would accumulate in the pipe and create "dunes

of particles" inside the pipeline or near pipe fittings such as elbows, tees, valves, and

couplings. Accumulation of sand particles and the resulting sand dunes reduce the flow

area in these sections and may cause higher flow velocity above the dunes. Figure I-1,

for example, shows sand particle accumulation in tubing and in an elbow.

When the flow area is reduced by sand accumulation, the local flowstream velocity

above the particle dunes can be several times higher than the average flowstream velocity

in open pipe. The higher flow velocity above the sand deposits, and the resulting

deformation in the flow geometry, can cause erosion damage in tubing and pipe fittings

such as an elbow. Furthermore, the presence of sand particles and sand deposits can

prevent protective scales from forming on the pipe walls and may cause excessive

erosion-corrosion in pipelines. Therefore, it is important to determine the particle

deposition velocity and keep the flowstream velocity above the critical deposition

velocity.

1

2

Figure I-1. Sand Accumulations in Tubing and an Elbow.

In vertical pipes, if flow velocity is higher than settling velocity, then particles will

flow and will not settle down. So it is important to determine settling velocity. Also

determination of settling velocity may provide insight to understand the critical

deposition velocity in inclined pipes and has important applications in deepwater

production system as described below.

In a deepwater production system, the height of the riser that brings the production

from the seafloor to a facility, e.g., platform or ship, can be about 1 mile in length, and

around 2 miles for ultra-deepwater. When the deepwater production is shut-down, all the

sand that is entrained in the produced fluid will settle down to the riser base.

3

There may be enough sand to block the pipe at the riser base even at relatively modest

sand concentrations. Knowing the sand settling time tells how long a shut-in can last

before a sand bed is formed. Knowledge of sand bed characteristics such as the volume

and angle of repose provides a guide to estimate where the sand bed will form and

whether the pipe will be blocked. The design of the riser base can be changed to produce

a long sand bed that will not fully obstruct the pipe.

Obviously, sand settling is a complicated problem that involves many parameters.

Issues that have to be addressed include: sand settling and sedimentation in vertical,

inclined and curved pipes; sand settling in oil-water-gas environment, altering sand bed

and blockage by purging gas bubbles through the settling medium, and once the sand bed

is accumulated at the riser base, can the production system be re-started by pushing the

sand column out of the riser?

Research Goals

The objective of this work is to develop a model/program that can be used to predict sand

sedimentation rate and settling velocity in risers for deep-water production system design.

The model will help to provide guidelines that can be used to design the riser base to

minimize sand accumulation.

Important Definitions

Settling Velocity:

Settling velocity as shown in Figure I-2 is the falling velocity of particles with

respect to a fixed horizontal plane.

4

Sedimentation Rate:

Sedimentation rate as shown in Figure I-3. is change in height of sand bed with respect to

time.

Figure I-2. Settling Velocity Figure I-3. Sedimentation Rate

CHAPTER II

LITERATURE REVIEW

The behavior of solid particles settling from suspension in fluids is a very

complicated phenomenon. As far as it is known particle settling velocity depends on the

particle size, particle density, particle shape, particle concentration, fluid density and

fluid viscosity. In order to better define settling velocity of suspension, first settling

velocity of one particle is defined.

Stokes Terminal Settling Velocity

The velocity of a single spherical solid particle settling through a fluid is determined

by the balance of gravity, buoyancy and drag force acted on it:

Buoyancy+Drag Force = Gravity (II-1)

3

6dgBuoyancy f

πρ= (II-2)

220 42

1 ducF fddragπρ= (II-3)

3

6dgGravity p

πρ= (II-4)

Substituting Equations (II-2), (II-3) and (II-4) into Equation (II-1) yields,

2/1

0 3)(4

−=

fd

fp

cgd

uρρρ

(II-5)

if Re<1, du

cf

d0

24Re24

ρµ

=≈ (II-6)

Substitute Equations (II-6) into Equation (II-5), get

5

6

µρρ

18)( 2

0

gdu fp −

= (II-7)

Equation (II-7) is so called Stokes terminal settling velocity.

Where

fρ fluid density

pρ particle density

µ dynamic viscosity

d particle diameter

dc drag coefficient

0u Stokes terminal settling velocity

Observation of Settling Velocity in Literature

The settling of particles in suspension is much more complicated than that of a

single particle. McNown and Lin (1952) described how the presence of other particles

affect the movement of a given particle. The descent of each of a number of particles

creates a velocity field throughout the fluid, and hence tends to increase the velocity of all

other particles. On the other hand, in a container of finite dimensions, the downward

motion of each particle plus the downward motion of the entrained fluid must be

compensated for by an equal upward flow which tends to decrease the velocity of each

particle. Burgers (1942) has suggested that the motion of a typical single particle should

be influenced by both the motion and the presence of the other particles. The main effect

of the motion of the other particles was to cause a “return flow” (see Figure II-1) of

liquid, whilst the presence of the other particles produced and effect analogous to an

increase in the viscosity of the dispersing liquid.

7

Figure II-1. Schematic of Return Flow

Kaye and Boardman (1962) conducted some experiments in dilute suspensions. The

experiments revealed that the settling velocity exceed the Stokes terminal settling

velocity at volume concentration from 0.1% up to 3% which may be explained by the

formation of clusters. Above this concentration, up to a concentration of approximately

10%, the velocity falls rapidly as the phenomena of return flow counteracts the cluster

formation, until at 10% concentration return flow dominates and hindered settling is the

dominant feature of the suspension. Bernd and Koglin (1973) conducted experiments

with glass spheres, irregularly shaped limestone and plastic discs in motor oil at low

concentrations. They also found that at very low concentration the settling velocities of

particles are higher than their Stokes’ terminal settling velocities. In the case of glass

spheres of about 200 µm in diameter, at a volumetric solid concentration α=0.008, the

mean settling velocity reached a maximum of 1.6 times the Stokes settling velocity. For

plastic discs, a maximum of 1.4 times the Stokes’ velocity was reached at a volumetric

solid concentration α=0.0025. For limestone particles, the settling velocity reached a

maximum of 1.4 times the Stokes’ velocity at a volumetric solid concentration α=0.0008.

Terence N. Smith (1997) concluded that with increasing volume concentration from zero,

8

the settling velocity of particles first increases with volume fraction because of initial

clustering effect. But then, the settling velocity decreases as the effect of the upward

flow of fluid between the particles becomes dominant. The experimental data from the

above two papers are summarized in Figure II-2.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0% 1% 2% 3% 4% 5%

Volume Concentration of Particle

Bernd Koglin (1973), limestone

Bernd Koglin (1973), plastic discs

Bernd Koglin (1973), glass

Kaye and Boardman (1962), 850um glass spheres



Us /

Uo

Figure II-2. Experimentally Measured Settling Velocity.

But, the experimental data from other investigators like Ham and Homsy (1988),

Cheng and Schachman (1955), McNown and Lin (1952) and Oliver (1960). didn’t show a

trend of settling velocity exceeding Stokes’ terminal settling velocity at very low

concentrations. The data is summarized in Figure II-3. It is still an open question as to

why there is such a discrepancy in data from literature.

9

0.0

0.2

0.4

0.6

0.8

1.0

0% 1% 2% 3% 4% 5%


Cheng and Schachman (1955)McNown and Lin (1952)Al-Naafa and Selim (1989)Davis and Birdsell (1988)Buscall et al. (1982)Ham and Homsy (1988)Oliver (1960)

Us

/ U0


Figure II-4 shows some other data from literature that gave settling velocities from

low concentration to very high concentration. The data shows that settling velocity

decreases rapidly with the increase of concentration. At a concentration of about 35%, the

settling velocity is only 10% of u . 0

10

0.0

0.2

0.4

0.6

0.8

1.0

0% 10% 20% 30% 40% 50% 60%


Ste ino ur (1944), tapio ca

Ste ino ur (1944), glas s

Hanra tty and Bandukwala (1951)

Gurel (1951)

Olive r (1954), 53-76um Kallo do c in wa te r

Olive r (1954), 62-100um glas s in glyce ro l-water

Lewis ,Gilliland and Bauer (1949)

Olive r (1960), 161um Kallo do c in wa te r

Olive r (1960), 161um Kallo do c in 20% glycero l-wa te r

Olive r (1960), 161um Kallo do c in 25% glycero l-wa te r

Us

/ U0


The observations in the literature may be summarized as follow. When particles

settle in a fluid, they may experience three major types of settling depending on the

particle volume concentration as shown schematically in Figure II-5. The horizontal axis

in Figure II-5 is logarithmic to clearly show regions of low volumetric concentrations.

11

0 bedα

Transition to Hindered Settling

Viscous Interaction between Particles

Free Settling

Hindered

Settling

Cluster

Forming

Particle Volume Concentration

C BA

1.5

0uus

1

Figure II-5. Schematic of Settling Velocity vs. Concentration.

Region A shown in Figure II-5 is called the “free settling” region that occurs at very

low particle concentrations. When particle volume concentration is very small, particles

do not interact with each other, they settle as if other particles do not exist. The settling

velocity in this region is equal to Stokes’ velocity when particle Reynolds number is low.

0s uu =

Region B in Figure II-5 consists of a region where there is a viscous interaction

between particles and particles move faster than their terminal velocity of one particle.

As the concentration gradually increases, particles experience a drag reduction due to

their viscous interaction with neighboring particles, and also a lateral force acting on the

particles leads to clustering of particles. All these cause the particles to fall faster than

12

their Stokes velocity. Then, as a result of cluster formation and further increase in particle

concentration, an unstable and transitional region forms where the upward flow is

irregular and localized. In this report this region is called transition to hindered settling

where particle settling velocities decrease with increase in concentration.

Region C is the hindered settling region. As the concentration increases further, the

effect of resistance of the induced irregular upward-moving flow of displaced liquid

becomes dominant. The settling velocity in this region is smaller than its free settling

velocity.

Settling Velocity Models in Literature

Many investigators have proposed models for calculating settling velocities.

Steinour (1944) obtained an expression of the form:

(II-8) )()1( 20 αα fuus −=

in which the value of )(αf was shown experimentally to be 10 . α82.1−

Richardson and Zaki (1954) proposed a correlation for predicting settling velocity at

high concentration as shown in Equation (II-9):

(II-9) Ns uu )1(0 α−=

in which N is a function of the Reynolds Number of flow, u is Stokes terminal settling

velocity. This correlation is commonly used for estimation of velocities in settling and

fluidization of solids at substantial spatial concentrations.

0

Oliver (1960) developed a settling velocity model as shown in Equation (II-10):

)15.21)(75.01( 31

0 αα −−= uus (II-10)

13

Barnea (1973) has developed a correlation to predict settling velocity of particles as

shown in Equation (II-11):

( ) ( ) ( )s

f

f

f

fps u

dgd

u

21

21

31

135exp

63.08.4

1

1

363.02

−−

+

−−=

ρα

αµ

αρ

αρρ (II-11)

where pρ is the density of particles, fρ is the density of fluid, fµ is the viscosity of

fluid and α is the particle volume concentration.

However, most of the earlier studies considered high particle concentrations.

Only a few investigations of settling velocity at low concentrations have been reported in

the literature. No comprehensive model has been found in the literature to predict

settling velocity at very low particle concentrations.

CHAPTER III

EXPERIMENTAL SETUP AND PROCEDURE

Two experimental apparatuses were used for data acquisition. One was a two pipe

system and the other is a constant 2-inch diameter pipe as described below.

Two-Pipe Apparatus and Data

Experimental Apparatus

Initially the experimental apparatus was a two-pipe system where a smaller diameter

pipe was connected to the base of a 2-inch pipe to increase the height of sand

accumulated. The test section is a 50-inch long pipe, 2 inches in diameter as shown in

Figure III-1. The end section of the pipe is reduced smoothly to a ½ inch diameter pipe .

This arrangement, as described later, caused a lot of problems in data interpretation and

modeling.

119cm

166cm

43cm

Figure III-1. Schematic of Two-pipe Apparatus.

14

15

Experimental Procedure

During the experiments, the pipe was filled with liquid and sand or glass beads

were added. Two liquids, water and white oil were used. The method of measuring the

viscosity of oil is described below. The pipe was rotated several times to mix the

particles with liquid and then the pipe was brought to a vertical position. Then, particles

(sand or glass beads) sedimentation rate was recorded by recording the time that particles

fill each level on the graduated ½ inch pipe. Experimentally, the volume concentration of

sand or glass beads bed is about 60%-65%. The method of measuring the bed

concentration will be described below. Different amounts of particles are added each

time to change particle concentration for the tests that were conducted.

2-inch pipe

½ inch pipe

Figure III-2. Photograph of Test Section.

Each sedimentation test was repeated several times to increase the confidence in the

experimental results. For example, Table III-1 shows typical experimental data obtained

during an experiment with 150 µm sand with an initial volume concentration of

16

0.44% in water. (All the experimental data are given in Appendix A.) The time required

for sand to fill each of the increments (each level shown in Figure III-3) on the ½ inch

bottom pipe was recorded. Sand and glass beads with an average size of about 150 µm

are used for most experiments. The method of measuring size distribution is described

below. Particles of different narrow-ranged sizes are also used to examine the effect of

sizes. A summary of the experiments that were conducted in two-pipe apparatus is given

in Tables III-2 and Table III-3. A 95% confidence interval using statistical analysis of

some representative data is shown in Appendix B.

Level 1.31 in

Level 1.96 in

Level 2.61 in

Level 3.27 in

Figure III-3. Photograph of the ½ inch Graduated Pipe.

17

Table III-1. Typical Measured Sedimentation Times in Two-Pipe Apparatus

Pipe Arrangement: Double Pipe

Sand Size (µm) 150 1 0.5

Sand Weight (g) 30 1 43

Volume of Sand (ml) 11.32 2577 2

Density of Sand (g/cm3)

2.65 0.44% 123

Sand Level (in) Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 45 44 44 50 501.96 57 57 56 63 622.61 70 69 71 76 763.27 89 87 92 98 923.76 150 260

Measured Time to Reach Sand Level (s)

Total Volume (ml) Large Pipe Diameter (in)

Volume Concentration Large Pipe Length (cm)

Density of Water (g/cm3)

Small Pipe Diameter (in)

Viscosity of Water (cp)

Small Pipe Length (cm)

Table III-2. Experiments Conducted with Average 150 µm Particles in Two-Pipe

Apparatus

Average 150 µm Mixed Sand in Water 0.44% 0.59% 0.73% 0.88% 1.03% 1.17% 1.32% 1.46% 2.20% 2.93%Average 150 µm Mixed Glass beads in Water 0.47% 0.62% 0.78% 0.93% 1.09% 1.24% 1.40% 1.55% 2.33% 3.10%150 µm Mixed Sand in Oil 0.44% 0.88% 1.32%

Initial Volume Concentration of Particles

18

Table III-3. Experiments Conducted with Different Particle Sizes in Two-Pipe

Apparatus

Particle Size Initial Volume Concentration0.44%0.59%0.73%0.44%0.59%0.73%0.44%0.59%0.73%0.44%0.59%0.73%0.62%0.93%1.24%0.59%0.93%1.24%0.62%0.93%1.24%0.62%0.93%1.24%

Glass beads in Water

106-125 µm

125-150 µm

150-177 µm

177-210 µm

Sand in Water

106-125 µm

125-150 µm

150-177 µm

177-210 µm

Measurements of Viscosity of White Oil

A Chan 35 viscometer and capillary tube viscometer (see Figures III-4 and III-5)

were used to measure the viscosity of white oil. The measurements of viscosity from

both viscometers are about 22 cp.

19

Figure III-4. Chan 35 Viscometers Figure III-5. Capillary Tube Viscometer

Measurements of Volume Concentration of Particle Bed

The method of measuring the volume concentration of particle bed was to first add

some water to a graduated cylinder. Then read the height of the water, , and record.

Then, a certain amount of sand (glass beads) was added to the water and the water level

was raised to level . After all the particles settle down, the height of the bed, , was

recorded as shown in Figure III-6.

1h

2h 3h

Volume of Particles= Ahh )( 12 −

Total Volume of Particle Bed= h A3

3

12

3

12 )()(h

hhAh

Ahhbed

−=

−=α

Where A is the cross area of the cylinder.

20

2h1h

3h

Figure III-6. Schematic of Measuring the Concentration of Particle Bed.

The results revealed that the volume concentrations of both sand bed and glass

beads bed are about 60%.

Measurements of Particle Size Distribution

The size distribution was obtained by sorting sand or glass beads by size using a set

of sieves. The sieves were stacked in a sequence with the largest opening on the top and

the smallest opening at the bottom. 100 grams of particles were placed in the sieve with

the largest opening, and the set of sieves were shaken thoroughly. The weight of

particles retained in each sieve was then measured. The size distribution of sand is shown

in Figure III-7 and the size distribution of glass beads is shown in Figure III-8.

21

0.47% 1.13%

13.31%

20.83%22.75%

28.54%

9.51%

3.46%

0%

5%

10%

15%

20%

25%

30%

Perc

enta

ge b

y W

eigh

t

63-75µm

75-106µm

<63µm 106-125µm

125-150µm

150-177µm

177-210µm

>210µm

Figure III-7. Sand Size Distribution by Weight Percent.

0.27% 1.14%4.44%

43.50%

29.81%

20.57%

0.27%0%

10%

20%

30%

40%

50%

Perc

enta

ge b

y W

eigh

t

<75µm 75-106µm

106-125µm

125-150µm

150-177µm

177-210µm

>210µm

Figure III-8. Glass Beads Size Distribution by Weight Percent.

Experimental Results

Figure III-9 shows a typical result plotted for sand bed height (sand level) as a function of

time for 150 µm sand particles in water with an initial volume concentration

22

of 0.44%. The symbols with different shapes represent results from separate tests; the

black line is the trend line (in all figures). It is obvious from Figure III-9 that three

regions of sedimentation are observed during the majority of experiments utilizing the

two-pipe apparatus. An initial slow sedimentation rate at low concentration followed by

a nearly constant and much faster sedimentation rate at higher concentrations followed by

very slow sedimentation of very small particles at low concentration.

0.00

1.00

2.00

3.00

4.00

0 50 100 150 200 250 300

Time (s)

Sand

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

Sedimentation of Small Particles at Low

Concentration

Sedimentation at High Concentration

Initial Slow Sedimentation at Low Concentration

Trend Line

Figure III-9. Measured Values of Sand Level versus Time

(150 µm Sand in Water, α0=0.44%).

It is observed that some of this behavior is attributed to the experimental setup as

shown schematically in Figure III-10. At the beginning of experiments, the concentration

in the ½ inch pipe is very low (it is assumed to be the same as the initial concentration in

the 2-inch pipe). After some time t1, particles from the 2-inch pipe flow into the ½ inch

pipe making the concentration of the ½ inch pipe much higher (nearly 10 times higher),

and the sedimentation rate became faster. Finally, the finer particles that settle very

23

slowly cause the gradual sedimentation rate near the end of the experiment as shown in

Figure III-9.

Low Conc.

Figure III-10. Schematic of Experimental Apparatus.

A very similar trend observed in Figure III-9 was observed for all the experiments

of sand and glass beads in water. Figures III-11 and III-12, for example, show four more

typical average values of data using sand and glass beads with a high and a low initial

concentration. The results also reveal that the higher the initial concentration the faster

the sedimentation rate.

High Conc.

Low Conc. of Finer Particles

24

0.00

2.50

5.00

7.50

10.00

12.50

0 50 100 150 200 250

Time (s)

Sand

Lev

el (i

n)Average Data α0=1.32%

Average Data α0=2.93%

Trend Lines


(150 µm Sand in Water).

0.00

3.00

6.00

9.00

12.00

15.00

0 50 100 150 200 250 300

Time (s)

Gla

ss B

eads

Lev

el (i

n)



Trend Lines

Figure III-12. Measured Values of Glass Beads Level versus Time

(150 µm Glass Beads in Water).

25

The following graphs (Figure III-13 to Figure III-15) compare the sedimentation

rate of sand and glass beads for the lowest, medium and highest concentrations

considered. The figures show that glass beads settle slightly faster than sand.

0.00

1.00

2.00

3.00

4.00

5.00

0 50 100 150 200 250

Time (s)

Sand

Lev

el (i

n)

Average Data of Sand α0=0.44%

Average Data of Glass Beads α0=0.47%

Trend Lines


Sedimentation Rate at α0=0.47%.

26

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0 50 100 150 200Time (s)

Sand

Lev

el (i

n)





0.00

2.00

4.00

6.00

8.00

10.00

12.00

0 50 100 150 200Time (s)

Sand

Lev

el (i

n)

Average Data of Sand α0=2.93%Average Data of Glass Beads α0=3.1%



27

The typical results for a more limited range of particle sizes are shown in Figure

III-16. There is only a small region of slow sedimentation caused by very small particles,

because most of the particles are nearly the same size. The results also indicate that the

sedimentation rates of bigger particles are faster than that of small particles, as expected.

0.00

1.00

2.00

3.00

4.00

5.00

0 20 40 60 80 100 120

Time (s)

Sand

Lev

el (i

n)

Average Data 150-177µm

Average Data 177-210µm

Initial Slow Sedimentation at Low Concentration


Sedimentation at Low Concentration

Trend Line


(150-177 µm and 177-210 µm Sand in Water, α0=0.44%).

When experiments in oil ( 22=µ cp) were conducted, the initial slow sedimentation

at low concentration was not observed (see Figure III-17). We conducted several

experiments to investigate the reason for this phenomenon. According to our

observations, immediately after starting the test a fast re-circulating fluid motion occurred

in the ½ inch pipe causing particles in the 2-inch pipe to move to the ½ inch pipe very

quickly. Thus, a high concentration of sand was established throughout the ½ inch pipe

in a relatively short time. It appears that the sedimentation was controlled by high

28

concentration from the beginning of the experiment. All of the data for oil indicated

similar trends in the two-pipe apparatus.

0.00

2.00

4.00

6.00

8.00

0 500 1000 1500 2000 2500

Time (s)

Sand

Lev

el (i

n) Exp. #1

Exp. #2

Exp. #3


Sedimentation of Small Particles at Low Concentration

Trend Line


(150µm Sand in Oil, α0 =0.88%)

Particle Bridge Phenomenon

An interesting particle bridge phenomenon was observed at the beginning of the ½

inch pipe in the two-pipe apparatus as shown in Figure III-18. The dramatic area change

of the pipe causes the particle concentration at the beginning of ½ inch pipe (as shown in

Figure III-18) to be much higher than that in the 2 inch pipe, which makes 12 ss uu < . As

a result, the mass into the “bridged region” is more than the mass out of it and eventually

particles accumulate there and the particle concentration gets very high and may clog the

pipe. This phenomenon is called particle bridge phenomenon.

29

2su

1su

Figure III-18. Schematic of Sand

Experimental Results Adding Gas Flow

Several experiments were also conducted wit

section to see how gas flow (gas bubbling) affects

experiments, gas was injected from the bottom (s

from the top of the 2-inch pipe for all experiments

rate was measured with rotameter type flow meters

Figure III-20 shows that when the gas flow

ft3/hr, sand sedimentation rate decreases with gas f

was greater than 1.5 ft3/h, sand sedimentation rate

oil (see Figure III-21) is observed, but the transition

Low Concentration

Particle Bed

Bridged Region (very high

Bridge Phenomenon

h an upward gas flow through the test

the sand sedimentation rate. For these

ee Figure III-19) and sand is injected

reported in this section. The gas flow

.

rate was less than approximately 1.5

low rate. But, when the gas flow rate

tends to increase. A similar trend in

flow rate appears to be a little larger.

Lower Intermediate Concentration

30

on Gas Injection Locati

Figure III-19. Photograph of Gas Injection Location.

0

1

2

3

4

0 20 40 60 80 100 120 140 160

Time (s)

Sand

Lev

el (i

n)

Gas flow rate=0 SCFH

Gas flow rate=1.5 SCFH




Trend Line

Figure III-20. Sand Sedimentation Rate vs. Gas Flow Rate (in water)

31

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000 2500

Time (s)

Sand

Lev

el (i

n)






Trend Line

Figure III-21. Sand Sedimentation Rate vs. Gas Flow Rate (in oil)

32

One-Pipe Apparatus and Data

Experimental Apparatus

Due to the change in pipe area and complex behavior of sedimentation that was

observed with two-pipe experimental apparatus, it was decided to use a constant diameter

pipe to gather additional sedimentation data. The new test sections are a 79-inch long

pipe (for low concentration experiments in water) and a 35-inch long pipe (for very high

concentration of particles in water and all experiments in oil), both are 2-inch diameter

pipes. During the experiments, particles (sand and glass beads) were added to the liquid

in the pipe. The pipe is rotated several times to mix the particles and to make the particle

concentration nearly homogenous. Then, the pipe is brought to the vertical position and

sand sedimentation rate was recorded by recording the time that sand fills each

graduation mark (level) that is placed at the bottom of the pipe.

Experimental Results

Sand and glass beads with the average sizes of about 150 µm are used for the

experiments. Each test was repeated several times to increase the confidence of the

experimental results. For example, Table III-4 shows a typical result obtained during a

test with 150 µm sand with a concentration of 0.0046 (0.46%) by volume. (All the

experimental data are given in Appendix A.) The time for sand to fill each of the

increments on the bottom is recorded. A summary of the experiments that were

conducted is given in Table III-5. A 95% confidence interval using statistical analysis of

some representative data is shown in Appendix B.

33

Table III-4. Typical Measured Sedimentation Times in One-Pipe Apparatus

Pipe Arrangement: One Pipe

Sand Size (µm) 150 1 2

Sand Weight (g) 49 1 200

Volume of Sand(ml) 18.65 4053.60

Density of Sand(g/cm3)

2.65 0.46%


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 00.20 33 32 34 28 300.39 65 64 74 55 650.56 300 300 300 300 300

Total Volume (ml)

Volume Concentration


Density of Water (g/cm3)

Pipe Diameter (in)

Viscosity of Water (cp)

Pipe Length (cm)

Table III-5. Experiments Conducted in One-Pipe Apparatus

Initial Volume Concentration of Particles Average 150µm Mixed Sand in Water 0.46% 0.92% 1.38% 10% 20% Average 150µm Mixed Sand in Oil 0.46% 0.92% 1.8% 3.8% 10% Average 150µm Mixed Glass Beads in Water 0.49% 0.98% 1.46% 10% 20% Average 150µm Mixed Glass Beads in Oil 0.46% 0.92% 1.8% 3.6% 12% 20%

Figures III-22 and III-23 show some sample typical experimental results of how

sand bed height changes with time in water and oil. The lines are the trend lines. It is

observed from the graphs that the results for the water case and oil case are very similar.

34

Both show a nearly constant sedimentation rate followed by a slower sedimentation rate

near the end which is caused by lower concentration of remaining very small particles

inside the pipe.

Experimental results for glass beads are very similar to sand. A sample of

experimental results is shown in Figure III-24 for a low and a higher initial volume

concentration.

0.00

0.40

0.80

1.20

1.60

2.00

0 100 200 300 400 500

Time (s)

Sand

Lev

el (i

n)



Trend Line


(150 µm Sand in Water).

35

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 500 1000 1500 2000 2500

Time (s)

Sand

Lev

el (i

n)

Average Data α0=0.92%Average Data α0=1.8%

Trend Line


(150 µm Sand in Oil).

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 500 1000 1500 2000

Time (s)

Gla

ss B

eads

Lev

el (i

n)



Trend Line

Figure III-24. Measured Values of Glass Beads Level versus Time

(150 µm Glass Beads in Oil).

36

The following graphs (Figures III-25 to III-26) compared the sedimentation rate of

sand and glass beads, one case in water and one case in oil. Both show that glass beads

settles a little faster than sand.

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200 250 300

Time (s)

Sand

Lev

el (i

n)



Trend Line



37

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 500 1000 1500 2000 2500

Time (s)

Sand

Lev

el (i

n)

Average Data of Sand α0=1.8%Average Data of Glass Beads α0=1.8%

Trend Line

Figure III-26. Comparison of Sand Sedimentation Rate in Oil at α0=1.8% & Glass

Beads Sedimentation Rate in Oil at α0=1.8%.

CHAPTER IV

MECHANISTIC MODELING

In order to predict the sedimentation rate a model for settling velocity of particles is

needed. In addition, a procedure is required to calculate sedimentation of particle bed at

the bottom of the pipe. Two different sedimentation models are developed during this

investigation: a simplified three-zone model, and a comprehensive finite-difference

model.

Settling Velocity Models

Almost all models reported in the literature are mainly based on empirical

information. During this investigation, a mechanistic model is developed which is more

generally applicable than purely empirical models.

TUSMP-GK Model:

Based on suggestions of Dr. Gene Kouba of ChevronTexaco, a mechanistic model

was developed for the region of hindered settling (Region C in Figure II-1). This model,

as described below, is called TUSMP-GK Model (Tulsa University Sand Management

Project (TUSMP) and Dr. Gene Kouba (GK) Model).

When particles move in a fluid, an equal volume of fluid is displaced. Therefore the

upward flow rate of fluid must be equal to the flow rate of downward moving particles as

shown by Equation (IV-1).

particlefluid QQ = (IV-1)

Equation (IV-1) is written in terms of fluid and sand velocities:

38

39

AuAu sf αα =− )1( (IV-2)

where

su Settling velocity of particles with respect to a fixed horizontal plane

fu Upward velocity of displaced fluid with respect to a fixed horizontal plane

α Particle volume concentration

Particle volume concentration is defined as the volume of particles divided by the total

volume of particles and liquid.

)1

(α

α−

= sf uu (IV-3)

Therefore, the relative velocity between particles and fluid becomes:

)1

1()1

(αα

α−

=+−

=+= ssssf uuuuuu (IV-4)

where

u Relative velocity between particles and fluid

The relative velocity between fluid and particles can be assumed to be equal to the

Stokes velocity of a single particle falling in the suspension with density mρ and

viscosity of the mixture. The following equations are used to represent the mixture

density and viscosity:

mµ

)1( αραρρ −+= fpm (IV-5)

From Einstein viscosity law:

fm )5.21( µα+=µ (IV-6)

The equation for Stokes’ velocity of particles then becomes

40

+−

=

+−−

=−

=α

αα

αµρρ

µρρ

5.211

5.211

18)(

18)(

0

22

ugdgd

uf

fp

m

mp (IV-7)

Combining Equations (IV-4) and (IV-7), one obtains an expression for hindered settling

velocity of sand.

( )( )α

α5.21

1 2

0 +−

= uus (IV-8)

where is the terminal velocity of one particle falling in a pure fluid. Equation

(IV-8) is called Original TUSMP-GK model.

ou

Corrected TUSMP-GK Model

The mechanistic settling velocity model (Original TUSMP-GK model) given by

( )( )α

α5.21

1 2

0 +−

= uu

predicts that the settling velocity becomes zero at the solids concentration 1=α , while

experimentally this occurs at a maximum concentration maxα between 0.6 and 0.7.

Therefore, an ad hoc modification was added to the TUSMP-GK model to make the

settling velocity equal zero at maxαα = .

( )( )

65.0

max

2

011

5.211

−

+−

=αα

αuu (IV-9)

Equation (IV-9) is called TUSMP-GK model 1.

Also from literature, at very low concentrations, the particle settling velocity may

exceed Stokes terminal settling velocity as discussed in ChapterII, therefore, TUSMP-GK

model 2 is developed to account for this phenomena based on the original TUSMP-GK

model.

41

( )( )

( )

>+−

−

+−

−=

≤++−=

3% 5.21

103.0

2275.103.0

2275.1

3% 17.662222265.0

maxmaxmax

0

20

αα

ααα

αα

ααα

uu

uu (IV-10)

Figure IV-1 compares the three models developed during this investigation.

Figure IV-2 compares TUSMP-GK models with data from literature. The data from

literature is discussed in Chapter II.

A comparison of the TUSMP-GK models, and two selected models of Steinour

(1944) and Richardson and Zaki (1954) which are described in Chapter II are shown in

Figure IV-3.

0

0.25

0.5

0.75

1

1.25

1.5

0 0.2 0.4 0.6 0.8 1


Us/

Uo

TUSMP-GK Model 1

TUSMP-GK Model 2

Original TUSMP-GK

Figure IV-1. Comparison of TUSMP-GK Models

42

0.0

0.3

0.5

0.8

1.0

1.3

1.5

0.0 0.1 0.2 0.3 0.4 0.5 0


.6

TUSMP-GK Model 1

TUSMP-GK Model 2U

s/U 0

Figure IV-2. Comparison of TUSMP-GK Models with Literature Data

0

0.25

0.5

0.75

1

1.25

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6


Us/U

o

TUSMP-GK Model 1

TUSMP-GK Model 2

Steinour(1944)

Richardson and Zaki(1954)

Figure IV-3. Comparison of Different Settling Velocity Models

43

Mechanistic Modeling of Sedimentation Rate

Three-Zone Model

Initially, a simplified three-zone model was developed for the 2-pipe apparatus

which will not be discussed in detail here. The basic idea is to divide the two-pipe

experimental apparatus into 3 zones as shown in Figure IV-2. Zone 1 is the section of 2-

inch pipe; Zone 2 is the section of the ½-inch pipe above the solid bed; Zone 3 is the

region where the solids are being accumulated. Mass conservation is applied in each

zone.

Figure IV-4.

1pα

Schematic of Zones Considered in Model

Zone 1

Zone 2

Zone 3

44

This simple model is based on some assumptions such as the concentration in

zone 1 is equal to initial concentration all the time, which is not true; the height of zone 3

can not be longer than the height of ½ inch pipe which limited the use of the model. Also,

if the initial concentration is high, the concentration in zone 2 may even exceed xmaα . To

fix this one needs to limit the concentration to xmaα and extra mass that flows into the

zone was lost. Due to these disadvantages of the three-zone model the finite-difference

model was developed which does not have these disadvantages.

Finite-Difference Model

In the finite-difference model, the pipe is divided into n cells (as shown in Figure

(IV-5), and mass conservation is applied in each cell as described below:

n-1

1

i+1

i

i-1

n

.

.

2

Figure IV-5. Schematic of Experimental Apparatus.

First, the mass conservation is applied to the interior cells (cell 2 to cell n-1)

Rate of mass of solids from cell i-1 into cell i:

45

(IV-11) 111 )()( −−− it

it

ip Auαρ

Rate of mass of solids from cell i into cell i+1:

it

it

ip Au )()(αρ (IV-12)

Rate of change of mass of solids within cell i during time dt:

[ - ] / =iit

ip dzA)(αρ iit

ip dzA1)( −αρ dt[ ]

dtdzA ii

ti

tip

1)()( −− ααρ (IV-13)

mass in cell i mass in cell i

at time t at time t-1

Applying the conservation of mass to cell i, yields

Eq.(IV-11)-Eq.(IV-12)=Eq.(IV-13)

Then, solving for iα gives

ti

i

i

iti

ti

i

ti

ti

udzdt

AAu

dzdt

)(1

)()()()(

111

1

+

+=

−−−

− ααα (IV-14)

For the first cell (cell number 1), mass conservation is applied in a similar

manner:

Rate of mass of solids into cell 1 is zero

Rate of mass of solids from cell 1 into cell 2:

111 )()( Au ttp αρ (IV-15)

Rate of change of mass of solids within cell i:

[ ]dt

dzAttp 11

111 )()( −− ααρ

(IV-16)

Again, applying the conservation of mass to cell number 1, one can solve for α1

46

0-Eq.(IV-15)=Eq.(IV-16), which gives:

t

i

tt

udzdt )(1

)()(1

11

1

+=

−αα (IV-17)

Finally for the bottom cell, applying the mass conservation yields:

Rate of mass of solids into cell n:

1)()( −nt

nt

np Auαρ (IV-18)

Rate of mass of solids out of cell n is zero.

Rate of change of mass of solids within cell n:

[ ]dt

dzA nnt

nt

np1)()( −− ααρ

(IV-19)

Conservation of mass becomes Eq.(IV-18)-0=Eq.(IV-19), which gives:

n

ntn

tn

i

tn

tn A

Au

dzdt 1

111 )()()()( −

−−− += ααα (IV-20)

where pρ is density of particles, iα is particle volume concentration in cell I;

is particle settling velocity in cell I, is Area of outlet of cell I, is Time step, is

height of cell I and is calculated from either TUSMP-GK model 1 or TUSMP-GK

model 2.

iu

iA dt idz

u

When size distributions are considered, Equations (IV-14), (IV-17) and (IV-20) are

applied to each size independently. The settling velocity of each size is calculated using

the sum of the particle volume concentration of each size.

47

When settled sand fills cell n (i.e., particle volume concentration reaches the volume

concentration of sand bed bedα ) the calculation is terminated in cell n, and Equation (IV-

16) is applied to cell n-1. And when cell n-1 is filled, then Equation (IV-16) is applied to

cell n-2 and this procedure continues. With this finite-difference approximation and

keeping track of cells that are filled, the moving sand bed is modeled. The maximum

sand concentration (volume concentration of sand bed bedα ) that fills a cell was

experimentally determined to be 0.6 during this investigation.

CHAPTER V

RESULTS

In this chapter we will compare sedimentation rates predicted by the finite-

difference model and measured sedimentation rates for some typical cases.

Results of Two-Pipe Apparatus

For the two-pipe apparatus, due to the particle bridge phenomenon (see Figure V-1)

as described in Chapter III. Once the concentration in the particle bridge region reaches

bedα , particles there will not move anymore since the settling velocity equals zero

at bedα as predicted by the settling velocity model (TUSMP-GK model 1 and model 2).

2su

1su

Figure V-1. Schematic of Particle Bridge Phenomenon

Lower Intermediate Concentration

21 ss uu >

Low Concentration

Particle Bridged Region (αmax was limited to 35%)

Particle Bedαbed=60%

48

49

But, experimentally particles in the bridged region found a way to settle down in a

non-uniform fashion. The maximum concentration was then limited to 35% to allow

particles to flow downward from the bridged region and also to match the data better for

higher initial concentrations where sand bridge occurs.

The comparison between predicted sedimentation rates by the finite-difference

model and the experimental data of 3 typical sand cases are shown in Figures V-2 to V-4.

In Figures V-2 and V-3, the initial particle volume concentrations were very low and the

35% limit was not used because particle bridge phenomenon didn’t happen. Both models

agree with the data well and TUSMP-GK model 1 agrees better with the data at low

concentrations as shown in Figures V-2 and V-3. With the increase of the initial

concentration α0, the prediction is slightly away from data as shown in Figures V-4.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 100 200 300 400

Time (s)

Sand

Lev

el (i

n)

Exp. #1Exp. #2Exp. #3Exp. #4Exp. #5TUSMP-GK Model 1TUSMP-GK Model 2

Figure V-2. Comparison between Predicted and Measured Particle Sedimentation

Rate (150 µm Sand in Water, α0=0.44%).

50

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 100 200 300 400

Time (s)

Sand

Lev

el (i

n)Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2



0.00

2.00

4.00

6.00

8.00

10.00

12.00

0 100 200 300 400 500

Time (s)

Sand

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2



51


model and the experimental data of four typical glass beads cases are shown in Figures

V-5 to V-8. At low concentrations 35% limit was not used because particle bridge

phenomenon didn’t happen, and both models agree with the data very well as shown in

Figures V-5 and V-6. At higher concentrations, however, both models deviate from the

data due to particle bridge phenomenon as shown in Figures V-7 and V-8.

0.00

1.50

3.00

4.50

0 100 200 300 400

Time (s)

Gla

ss B

eads

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate (150 µm Glass Beads in Water, α0=0.47%).

52

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 100 200 300 400

Time (s)

Gla

ss B

eads

Lev

el (i

n)Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2



0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

0 100 200 300 400 500

Time (s)

Gla

ss B

eads

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2



53

0.00

4.00

8.00

12.00

16.00

20.00

0 100 200 300 400 500 600

Time (s)

Gla

ss Be

ads L

evel

(in)

Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2



The comparisons of two typical oil cases are shown in Figures V-10 and V-11. The

predictions do not agree with the data very well. But, if assume 11

2 2 oo AA

αα =

where

2oα Initial concentration in half-inch pipe

1oα Initial concentration in two-inch pipe

2A Area of half-inch pipe

1A Area of two-inch pipe

a good prediction is obtained as shown in Figures V-12 and V-13. This assumption

corresponds to our observation in oil (Figure V-9) that very quickly a fast re-circulating

motion occurred inside the small pipe causing particles to move into the smaller pipe

54

quickly. As a result a high concentration was obtained in a short time in the ½- inch

collection pipe.

α01

α02

Figure V-9. Fast Reticulating Fluid Motion Observed in Oil

0.00

1.00

2.00

3.00

4.00

0 1000 2000 3000 4000 5000 6000 7000 8000

Time (s)

Sand

Lev

el (i

n) Exp. #1

Exp. #2

Exp. #3

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate (150 µm Sand in Oil, α0=0.88%).

55

0.00

2.00

4.00

6.00

8.00

0 2000 4000 6000 8000 10000

Time (s)

Sand

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate (150 µm Sand in Oil, α0=0.88%).

0.00

1.00

2.00

3.00

4.00

0 2000 4000 6000 8000

Time (s)

Sand

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

TUSMP-GK Model 1

TUSMP-GK Model 2

Figure V-12. Comparison between Predicted and Measured Sand Sedimentation

Rate (150 µm Sand in Oil, α01=0.38%, α02=3.45%).

56

0.00

2.00

4.00

6.00

8.00

0 2000 4000 6000 8000Time (s)

Sand

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

TUSMP-GK Model 1

TUSMP-GK Model 2

Figure V-13. Comparison between Predicted and Measured Sand Sedimentation

Rate (150 µm Sand in Oil, α01=0.67%, α02=6.9%).

57

Results of One-Pipe Apparatus


model and the experimental data of 3 typical sand in water cases are shown in Figures V-

14 to V-16. TUSMP-GK model 2 agrees with the data very well at low concentrations.

With the increase of initial particle concentration TUSMP-GK Model 1 agrees with the

data better.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 100 200 300 400 500

Time (s)

Sand

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate in One-Pipe Apparatus (150 µm Sand in water, α0=0.46%).

58

0.00

0.40

0.80

1.20

1.60

2.00

0 100 200 300 400 500

Time (s)

Sand

Lev

el (i

n)Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate in One-Pipe Apparatus (150 µm Sand in water, α0=1.38%).

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0 50 100 150 200 250

Time (s)

Sand

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate in One-Pipe Apparatus (150 µm Sand in water, α0=10%).

59

Figures V-17 to V-19 show the comparison of 3 typical sand in oil cases. Both

models agree well with the data and TUSMP-GK model 2 agrees better with the data for

all cases.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 1000 2000 3000 4000 5000

Time (s)

Sand

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate in One-Pipe Apparatus (150 µm Sand in Oil, α0=0.46%).

In summary, predicted sedimentation rates agree with measurements for all cases.

Using TUSMP-GK Model 2 appears to slightly over predict most of the experimental

data of water cases in one- pipe apparatus.

60

0.00

0.50

1.00

1.50

2.00

2.50

0 1000 2000 3000 4000 5000

Time (s)

Sand

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate in One-Pipe Apparatus (150 µm Sand in Oil, α0=3.80%).

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0 1000 2000 3000 4000 5000

Time (s)

Sand

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate in One-Pipe Apparatus (150 µm Sand in Oil, α0=10%).

61


model and the experimental data of 3 typical glass beads in water cases are shown in the

Figures V-20 to V-22. Predictions of particle sedimentation rate using TUSMP-GK

model 2 agrees with the data very well at low initial particle concentrations. With the

increase of concentration TUSMP-GK model 1 agrees better with the data.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0 100 200 300 400 500

Time (s)

Gla

ss B

eads

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate in One-Pipe Apparatus (150 µm Glass Beads in Water, α0=0.49%).

62

0.00

0.50

1.00

1.50

2.00

2.50

0 100 200 300 400 500Time (s)

Gla

ss B

eads

Lev

el (i

n)Exp. #1

Exp. #2

Exp. #3

Exp. #4

Exp. #5

TUSMP-GK Model 1

TUSMP-GK Model 2



0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 50 100 150 200 250

Time (s)

Gla

ss B

eads

Lev

el (i

n)

Exp. #1Exp. #2Exp. #3Exp. #4Exp. #5TUSMP-GK Model 1TUSMP-GK Model 2


Rate in One-Pipe Apparatus (150 µm Glass Beads in Water, α0=10%).

63

Figures V-23 to V-25 show the comparison of 3 typical glass beads in oil cases.

Predictions of particle sedimentation rates using TUSMP-GK model 2 agree with the data

better for all cases.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 1000 2000 3000 4000 5000

Time (s)

Gla

ss B

eads

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

TUSMP-GK Model 1

TUSMP-GK Model 2



64

0.00

0.50

1.00

1.50

2.00

2.50

0 1000 2000 3000 4000 5000Time (s)

Gla

ss B

eads

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

TUSMP-GK Model 1

TUSMP-GK Model 2



0.00

2.00

4.00

6.00

8.00

0 1000 2000 3000 4000 5000

Time (s)

Gla

ss B

eads

Lev

el (i

n)

Exp. #1

Exp. #2

Exp. #3

TUSMP-GK Model 1

TUSMP-GK Model 2


Rate in One-Pipe Apparatus (150 µm Glass Beads in Water, α0=12%).

65

Effect of Cell Height and Time Step on the Predicted Results

The effect of computational cell height dz and time step dt on the predicted

sedimentation rate by the finite-difference model have been investigated. Figure V-26

compares the predicted sedimentation rate by using different dz. The results revealed that

the predictions are almost the same and the predictions using a larger dz predicts slightly

longer sedimentation times. Figure V-27 compares the predicted sedimentation rate by

using different dt. It shows that the value of dt has little effect on the results.

0

1

2

3

4

5

6

7

0 50 100 150 200 250

Time (s)

Sand

Lev

el (i

n)

dz=0.5cm, dt=0.5s

dz=1cm, dt=0.5s

dz=2cm, dt=0.5s


Different dz

66

0

1

2

3

4

5

6

7

0 50 100 150 200 250

Time (s)

Sand

Lev

el (i

n)

dz=1cm, dt=0.25s

dz=1cm, dt=0.5s

dz=1cm, dt=1s


Different dt

CHAPTER VI

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

Particle sedimentation rate experiments were carried out in a two-pipe apparatus and

a one-pipe apparatus. Sand and glass beads were used in the experiments in water and

white oil. In the data, three regions of sedimentation rate were observed for all the

experiments in water utilizing two-pipe apparatus: an initial slow sedimentation rate at

low concentration followed by a nearly constant and much faster sedimentation rate at

higher concentrations followed by very slow sedimentation of very small particles at low

concentration, while experiments in oil didn’t show the initial slow sedimentation.

Particle bridge phenomenon was observed in two-pipe apparatus at higher initial

concentrations. For one-pipe apparatus only two regions of sedimentation rate were

observed for all the experiments both in water and in oil: a nearly constant sedimentation

rate followed by a slower sedimentation rate near the end which is caused by lower

concentration of remaining very small particles inside the pipe. The data revealed that

for the particles of the same size, the higher the initial concentration the faster the

sedimentation rate. The data also revealed that at the same initial concentration, larger

particles settle faster. Data also showed that glass beads (average size of 150µm) settle

slightly faster than sand with average size of 150µm. The data for adding gas flow

revealed that particle sedimentation rate first decreases with gas flow, then increases as

the gas flow increases.

A mechanistic settling velocity model (TUSMP-GK model) has been developed.

This model predicts settling velocity to be zero at particle volume concentration equal to

100%, while experimentally this occurs at a maximum concentration maxα between 0.6

67

68

and 0.7. Therefore, TUSMP-GK model was modified to make settling velocity to be zero

at maxαα = . Two modified TUSMP-GK models were developed. One is called

TUSMP-GK model 1, the other is called TUSMP-GK model 2. TUSMP-GK model 2

was developed to account for the trend observed by some investigators that at dilute

concentrations particle settling velocity exceeded their Stokes terminal settling velocities.

A finite-difference model has been developed to predict particle sedimentation rate.

For the two-pipe apparatus, due to the particle bridge phenomenon, the maximum particle

concentration was set to 35% to allow particles to flow and also to match experimental

data better at high initial particle volume concentrations where particle bridge

phenomenon occurs. Particle sedimentation rates predicted by the finite-difference

model were compared to the experimental data. For two-pipe apparatus at low initial

concentrations, the predicted particle sedimentation rates agree with the data well but at

higher initial concentrations, predictions began to deviate from data which may due to the

particle bridge phenomenon. For one-pipe apparatus, the predicted particle sedimentation

rates agree very well with almost all of the data. The effect of time step (dt) and cell

height (dz) have been investigated on the predicted particle sedimentation rates from

finite-difference model. The results revealed that they have little effect on the numerical

solutions. A VBA computer program has been developed based on the finite-difference

model to predict particle sedimentation rates.

In order to develop a model to predict sand sedimentation rates at a base of a riser,

one needs to conduct sand sliding experiments in curved pipes as shown schematically in

Figure VI-1. The objective of these experiments is to find out how sand will slide down

in a base of a riser and where they will accumulate and whether they will block the pipe.

69

Develop a model for sand sliding in inclined and curved pipes and combine the model to

the sedimentation rate and settling velocity models that have been developed in vertical

pipes.

Sand Sedimentation

Sliding Particle Bed

Figure VI-1. Schematic of Sand Settling and Sliding in a Base of a Vertical Riser

Some preliminary experiments in inclined pipes have been conducted to observe if

the flow is capable of carrying sand. The results revealed that critical deposition velocity

decreased with the pipe angles as shown in Figure VI-2.

70

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80 90

Pipe Angle

Cri

tical

Dep

ositi

on V

eloc

ity (m

/s)

Trend Line

Figure VI-2. Critical Deposition Velocity vs. Pipe Angle

More experiments of critical deposition velocity in inclined pipes need to be

conducted. These experiments can be used to determine the effects of inclination angle

and sand rate on critical deposition velocities. Additional experiments should be

conducted to determine effects of a gas phase on critical deposition and settling velicties.

Then a model to predict critical deposition velocities in single phase and multiphase flow

can be constructed.

NOMENCLATURE

fρ fluid density

pρ particle density

mρ suspension density

mµ suspension viscosity

µ fluid dynamic viscosity

d particle diameter

dc drag coefficient

0u Stokes terminal settling velocity

su Settling velocity of particles with respect to a fixed horizontal plane

fu Upward velocity of displaced fluid with respect to a fixed horizontal plane

u Relative velocity between particles and fluid

0α Initial particle volume concentration

bedα Particle volume concentration in particle bed

α Particle volume concentration

iα Particle volume concentration in cell i

iu Particle settling velocity in cell i

iA Area of outlet of cell i

dt Time step

idz Height of cell i

71

72

REFERENCES

[1] Al-Naafa, M. A. and Selim, M. S., 1992, “Sedimentation of Monodisperse and

Bidisperse Hard-sphere Colloidal Suspensions” AIChE J. 38, 1618-1630.

[2] Batchelor, G. K., 1971, “Sedimentation in a Dilute Dispersion of Spheres” J. Fluid

Mech. Vol 52, part 2, pp. 245-268.

[3] Barnea, E. and Mizrahi,J., 1973, “A Generalized Approach to the Fluid Dynamics of

Particulate Systems Part I. General Correlation for Fluidization and Sedimentation in

Solid multiparticle Systems” The Chemical Engineering journal, 5, 171.

[4] Burgers, J. M., 1942, “On the Influence of the Concentration of a Suspension upon

the Sedimentation Velocity” Proc. Ned. Akad. Wet., Amsterdam, Vol. 44, 1045.

[5] Burger, R. and Wendland, W., 2001, “Sedimentation and Suspension flows:

Historical perspective and some recent developments” Journal of Engineering

Mathematics, 41,101.

[6] Crowley, J. M., 1971, “Viscosity-induced Instability of a One-dimensional Lattice

of Falling Spheres” J. Fluid Mech. Vol. 45 part 1, pp. 151-159.

[7] Davis, R. H. and Birdsell, K. H., 1988, “Hindered Settling of Semidilute

Monodisperse and Polydisperse Suspensions” AIChE J. 34, 123-129.

[8] Davis, R. H. and Gecol, H, 1994, “Hindered Settling Function with No Empirical

Parameters for Polydisperse Suspensions” AIChE Journal vol. 40, No. 3

[9] Felice, R. Di, 1999, “The Sedimentation Velocity of Dilute Suspensions of Nearly

Monosized Spheres” International Journal of Multiphase Flow 25, 559-574.

[10] Famularo, J. and Happel, J., 1965, “Sdeimentation of Dilute Suspensions in Creeping

Motion” Am. Inst. Chem. Engng. J. 11, 981.

73

[11] Lewis, E. W. and and Bowerman, E. W., 1952, “Fluidization of Solid Particles in

Liquids” Chem. Eng. Progr. 48, 603-609

[12] Ham. J.M., and Homsy, G.M., 1988, “Hindered Settling and Hydrodynamic

Dispersion in Quiescent Sedimenting Suspensions” Int. J. Multiphase Flow 14, 533-546

[13] Kaye, B.H. and Boardman, R.P., 1962, “Cluster Formation in Dilute Suspensions”

Interaction between Fluids & Particles, Instin. Chem. Engrs.

[14] Koglin, B., 1973, “Dynamic Equilibrium of Settling Velocity Distribution in Dilute

Suspensions of Spherical and Irregularly Shaped Particles” Proc. 1st Int. Conf. Particle

Technol. P266, IIT Research Institute, Chicago.

[15] Kynch, G. J., 1952, “A Theory of Sedimentation” Trans. Faraday Soc., 48, 166.

[16] Lewis,W. K., Gilliland, E. R. and Bauer, W. C., 1949, “Characteristics of Fluidized

Particles” Industr. Engng. Chem., 41, 1104.

[17] Lin, P. N., 1951, “Effect of Spacing and Size Distribution on the Fall Velocity of

Sediment” Doctoral dissertation, State University of Iowa.

[18] Maude, A. D. and Whitmore, R. L., 1958 “A Generalized Theory of Sedimentation”

Br. J. Appl. Phys. 9, 477.

[19] McNown, John S. and Lin, Pin-Nam, 1952, “Sediment Concentration and Fall

Velocity” Proceedings of the Second Midwestern Conference and Fluid Mechanics.

[20] Oliver, D.R., 1961, “The Sedimentation of Suspension of Closely-sized Spherical

Particles” Chemical Engineering Science, 15, 230.

[21] Richardson, J.F. and Zaki, 1954, “Sedimentation and Fluidization: Part I” Trans.

Instn chem. Engrs, 32, 35.

74

[22] Richardson, J.F., and Meikle, R.A.,1961, “Sedimentation and Fluidization: Part III”

Trans. Instn chem. Engrs, 39, 348.

[23] Smith, T.N., 1998, “A Model of Settling velocity” Chemical Engineering Scicence,

53, 315.

[24] Zigrang D. J. and Sylvester N. D., 1981, “An Explicit Equation for Particle Settling

Velocities in Solid-Liquid Systems” AIChE Journal vol. 27, no. 6.

APPENDIX A. RAW EXPERIMENTAL DATA

75

Appendix A-1. 150µm Sand in Water, α0=0.44%



Sand Weight (g) 30 1 43.18Volume of Sand (ml) 11.32 2577 2

Density of Sand (g/cm3) 2.65 0.44% 123


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 45 44 44 50 501.96 57 57 56 63 622.61 70 69 71 76 763.27 89 87 92 98 923.76 150 260




Density of Water (g/cm3) Small Pipe Diameter (in)

Viscosity of Water (cp) Small Pipe Length (cm)

76


Pipe Arrangement: Double PipeSand Size (µm) 150 1 0.5Sand Weight (g) 40 1 43.18

Volume of Sand (ml) 15.09 2577 2Density of Sand (g/cm3) 2.65 0.59% 123


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 45 46 44 45 451.96 55 58 55 56 562.61 66 69 67 66 673.27 80 81 77 78 793.92 89 95 89 90 924.57 103 110 103 102 105


Total Volume (ml) Large Pipe Diameter (in)Volume Concentration Large Pipe Length (cm)

Density of Water (g/cm3) Small Pipe Diameter (in)Viscosity of Water (cp) Small Pipe Length (cm)

77




Sand Weight (g) 50 1 43.18Volume of Sand (ml) 18.87 2577 2Density of Sand (g/cm3) 2.65 0.73% 123


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0.00 0.00 0.00 0.00 0.001.31 41.00 40.00 42.00 41.00 40.001.96 52.00 49.00 51.00 50.00 50.002.61 60.00 57.00 60.00 59.00 58.003.27 70.00 68.00 71.00 70.00 70.003.92 79.00 76.00 80.00 79.00 80.004.57 89.00 86.00 90.00 89.00 90.005.23 99.00 95.00 99.00 98.00 99.005.88 110.00 109.00 112.00 111.00 112.00




Density of Water (g/cm3) Small Pipe Diameter (in)

Viscosity of Water (cp) Small Pipe Length (cm)

78





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 38 38 34 36 361.96 47 47 42 45 462.61 54 54 49 52 543.27 62 63 58 61 623.92 70 70 66 68 704.57 78 78 73 76 775.23 84 86 81 83 855.88 94 95 90 92 946.54 103 103 99 101 1027.19 112 112 108 111 1127.84 170




79





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 35 34 37 35 371.96 44 42 45 42 452.61 52 50 52 50 533.27 61 59 61 59 623.92 68 65 69 66 694.57 76 74 76 74 775.23 84 81 83 81 845.88 92 89 91 89 926.54 100 97 99 96 987.19 109 108 108 107 1087.84 119 117 118 117 1188.50 130.00 128.00 127.00 126.00 129.00




80





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 34 31 35 35 351.96 41 39 42 43 432.61 47 46 50 50 503.27 54 54 57 58 583.92 62 60 65 65 654.57 69 66 73 72 725.23 75 73 79 79 795.88 81 80 86 86 866.54 89 87 93 94 937.19 96 96 100 101 1017.84 104 104 109 109 1108.50 111 110 116 117 1189.15 118 120 125 126 1259.80 129 128 135 136 136




81





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 36 31 33 32 321.96 44 39 41 39 392.61 50 45 48 46 463.27 57 52 56 54 543.92 64 59 63 61 614.57 70 64 69 67 685.23 77 71 76 74 755.88 83 77 82 80 826.54 88 84 89 88 897.19 95 92 96 95 967.84 104 99 104 102 1048.50 111 107 112 110 1129.15 118 114 120 119 1209.80 127 124 129 126 12911.38 200




82





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 32 30 33 33 301.96 40 37 40 41 372.61 46 43 46 47 443.27 54 51 54 54 513.92 60 57 61 61 584.57 66 64 67 67 645.23 72 69 73 74 715.88 79 75 78 80 776.54 85 82 84 87 847.19 93 88 93 93 917.84 98 95 100 104 988.50 106 104 107 108 1059.15 114 112 116 115 1129.80 122 120 123 123 121




83





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 35 30 31 26 351.96 42 37 38 32 422.61 48 43 44 38 483.27 56 51 50 45 553.92 62 57 57 51 614.57 67 63 63 57 685.23 72 68 68 62 715.88 80 75 75 67 816.54 85 80 81 73 857.19 92 87 88 80 927.84 100 94 95 87 1018.50 105 100 102 94 1049.15 112 107 108 101 1119.80 121 115 116 110 121




84





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 31 28 28 32 261.96 37 35 35 39 322.61 44 40 41 45 383.27 51 47 47 51 453.92 57 53 52 56 514.57 63 60 58 63 575.23 68 65 63 69 625.88 75 71 70 76 696.54 79 77 76 82 747.19 86 83 81 87 817.84 93 90 87 94 888.50 99 96 94 100 939.15 104 101 100 108 999.80 115 110 109 114 106




85

Appendix A-11. 150µm Glass Beads in Water, α0=0.47%

Pipe Arrangement: Double PipeGlass Beads Size (µm) 150 1 0.5Glass Beads Weight (g) 30 1 43.18

Volume of Glass Beads (ml) 12.00 2577 2Density of Glass Beads(g/cm3) 2.5 0.47% 123


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 48 48 46 48 491.96 57 57 55 57 582.61 67 67 64 66 673.27 80 79 76 78 793.86 150




86





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 48 45 46 49 471.96 56 53 53 56 552.61 63 60 60 63 633.27 71 69 69 72 713.92 79 77 78 80 804.57 87 86 86 88 88




87





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 42 42 43 45 451.96 50 50 51 52 522.61 57 57 57 59 583.27 65 64 65 66 663.92 73 71 71 73 734.57 78 77 78 79 795.23 85 83 84 86 855.88 90 90 90 92 92




88





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 43 43 43 44 421.96 50 50 49 52 482.61 56 56 55 57 543.27 63 63 62 64 603.92 69 70 68 70 664.57 74 76 72 70 715.23 80 80 78 81 775.88 85 86 84 87 836.54 90 91 90 93 897.19 98 98 96 99 94




89





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 45 45 44 43 441.96 51 52 50 49 512.61 57 58 56 55 573.27 63 65 63 62 633.92 69 70 68 67 684.57 75 76 73 72 745.23 79 81 78 77 795.88 84 86 83 82 846.54 89 91 89 87 897.19 94 96 94 93 957.84 100 102 99 98 1008.50 106 107 105 104 106




90





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 40 42 40 39 421.96 47 49 47 46 482.61 53 55 53 52 543.27 59 61 59 58 603.92 64 67 65 64 664.57 69 71 69 68 715.23 73 76 74 73 755.88 79 82 79 78 806.54 84 87 85 83 857.19 89 92 90 88 907.84 94 97 95 93 968.50 99 102 99 97 1009.15 104 106 104 102 1049.80 110 112 110 109 111




91





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 40 38 42 39 411.96 47 46 49 46 482.61 53 52 54 52 533.27 59 58 60 57 603.92 65 63 66 63 664.57 69 68 70 67 715.23 74 72 75 72 755.88 79 77 80 77 806.54 83 82 84 81 857.19 89 87 88 87 907.84 94 92 94 93 958.50 98 96 99 98 999.15 103 101 104 103 10411.74 180




92





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 41 41 40 43 421.96 48 47 47 50 482.61 53 53 52 55 533.27 60 59 58 61 603.92 65 64 64 67 654.57 71 68 68 71 695.23 75 73 72 75 745.88 80 78 77 80 786.54 84 82 81 84 837.19 89 87 86 89 887.84 94 92 91 95 928.50 98 96 96 100 979.15 102 101 100 105 102




93





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 38 37 38 38 371.96 44 44 45 45 432.61 50 49 50 51 483.27 55 55 56 56 543.92 61 60 61 61 604.57 66 65 65 66 645.23 70 69 70 70 685.88 74 73 74 75 736.54 79 78 79 79 777.19 83 82 83 83 827.84 88 87 89 88 878.50 93 91 93 93 919.15 97 96 98 98 969.80 103 101 102 102 100




94





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 37 37 35 35 361.96 42 43 40 40 422.61 49 49 45 46 483.27 55 54 50 52 533.92 59 59 55 56 584.57 64 64 59 61 635.23 68 68 63 65 675.88 73 72 67 70 726.54 77 77 72 74 767.19 82 82 76 78 817.84 86 87 81 83 868.50 91 92 85 87 909.15 96 96 89 92 95




95

Appendix A-21. 150µm Sand in Oil, α0=0.44%

Pipe Arrangement: Double PipeSand Size (µm) 150 0.89 0.5Sand Weight (g) 30 22 43.18



Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 01.31 363 388 3761.96 564 598 6012.61 831 839 8413.27 1209 1269 1250



Density of oil (g/cm3) Small Pipe Diameter (in)Viscosity of oil (cp) Small Pipe Length (cm)

96





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 01.31 292 349 3191.96 413 479 4442.61 542 619 5803.27 691 765 7203.92 831 922 8614.57 981 1051 10005.23 1114 1208 11485.88 1291 1369 13056.54 1449 1545 14777.19 1641 1751 16837.55 2300




97





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 01.31 294 287 2981.96 415 412 4202.61 536 541 5503.27 670 676 6713.92 809 829 8304.57 940 951 9485.23 1075 1084 10865.88 1224 1250 12356.54 1367 1372 13687.19 1545 1551 15367.84 1733 1734 17418.50 1944 1937 1929




98

Appendix A-24. 106-125µm Glass Beads in Water, α0=0.62%

Pipe Arrangement: Double PipeGlass Beads Size (µm) 106-125 1 0.5Glass Beads Weight (g) 40 1 43.18

Volume of Glass Beads (ml) 16.00 2577 2Density of Glass Beads (g/cm3) 2.5 0.62% 123


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 85 801.96 98 932.61 118 1063.27 125 1223.92 139 1364.57 152 1505.23 166 161




99





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 65 651.96 78 772.61 88 873.27 100 1003.92 110 1104.57 120 1205.23 130 1305.88 140 1396.54 150 1487.19 160 157




100





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 65 63 0.651.96 77 75 0.652.61 86 85 0.653.27 97 96 0.653.92 105 105 0.654.57 115 115 0.655.23 123 122 0.655.88 133 132 0.656.54 142 141 0.657.19 152 150 0.657.84 160 1608.50 170 1709.15 180 1809.80 189 18810.46 198 196




101





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 65 601.96 75 702.61 85 803.27 97 903.92 108 1004.57 119 1105.23 130 120




102





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 55 501.96 65 602.61 74 673.27 82 753.92 91 824.57 97 905.23 103 975.88 110 1046.54 120 1117.19 125 118




103





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 50 491.96 56 562.61 64 653.27 72 723.92 79 804.57 85 875.23 92 925.88 98 986.54 105 1057.19 112 1127.84 118 1208.50 123 1259.15 130 1329.80 136 13910.46 142 143




104





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 54 561.96 64 652.61 72 733.27 81 823.92 89 904.57 97 985.23 107 107




105





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 42 411.96 49 472.61 55 523.27 62 603.92 67 654.57 72 715.23 77 765.88 83 826.54 90 877.19 94 93




106





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 37 371.96 43 432.61 48 483.27 55 543.92 60 604.57 65 655.23 70 705.88 75 746.54 80 797.19 84 837.84 89 888.50 93 939.15 98 979.80 103 10210.46 108 106




107





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 39 351.96 46 422.61 53 483.27 59 563.92 65 624.57 71 695.23 78 76




108





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 31 311.96 37 362.61 42 403.27 47 463.92 51 514.57 55 555.23 60 595.88 64 626.54 67 667.19 71 717.84 75 76




109





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 30 301.96 35 352.61 40 403.27 45 453.92 50 504.57 53 555.23 57 595.88 61 636.54 65 677.19 69 717.84 73 748.50 76 789.15 80 819.80 84 8510.46 87 88




110

Appendix A-36. 106-125µm Sand in Water, α0=0.44%

Pipe Arrangement: Double PipeSand Size (µm) 106-125 1 0.5Sand Weight (g) 30 1 43.18



Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 75 77 70 72 711.96 90 92 83 85 962.61 105 107 96 97 1103.27 122 126 112 116 1283.76 150 150 135 140 148




111





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 75 65 64 68 701.96 85 78 75 79 832.61 97 90 86 89 923.27 111 102 99 104 1053.92 123 112 110 115 1144.57 135 125 121 127 1265.04 148 143 135 145 138




112





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 53 62 62 66 711.96 65 73 74 77 822.61 76 82 83 86 923.27 89 92 93 97 1023.92 99 102 104 106 1124.57 110 111 113 116 1225.23 120 121 122 123 1315.88 132 131 132 134 1406.32 145 142 143 145 148




113





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 55 60 62 60 581.96 69 73 73 72 692.61 80 85 85 85 823.27 98 101 101 101 1003.76 115 120 120 118 115




114





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 60 52 55 55 551.96 71 62 64 64 652.61 80 72 73 73 753.27 92 82 84 83 853.92 102 92 93 92 954.57 113 103 103 102 1035.04 125 115 115 111 114




115





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 01.31 50 51 50 48 471.96 60 60 59 55 552.61 68 66 65 62 623.27 76 75 74 70 703.92 84 83 81 78 784.57 92 91 88 84 845.23 99 98 97 92 925.88 107 105 103 100 1006.32 116 108 114 114 119




116





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 50 481.96 60 572.61 68 653.27 78 773.76 90 88




117





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 01.31 41 41 421.96 48 48 502.61 55 55 563.27 63 63 643.92 70 69 714.57 78 76 785.04 87 86 87




118





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 40 401.96 46 472.61 53 533.27 60 603.92 66 674.57 71 725.23 76 775.88 83 846.32 90 92




119





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 35 341.96 41 402.61 46 453.27 52 513.76 60 60




120





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 36 351.96 42 412.61 48 473.27 53 523.92 58 574.57 63 625.04 68 67




121





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 01.31 33 331.96 36 372.61 42 423.27 46 463.92 50 504.57 54 545.23 58 595.88 62 626.32 67 67




122


Pipe Arrangement: One PipeSand Size (µm) 150 1 2Sand Weight (g) 49 1 200


Density of Sand(g/cm3) 2.65 0.46%


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.20 33 32 34 28 300.39 65 64 74 55 650.56 260 270 300 250 280

Total Volume (ml)Volume Concentration


Density of Water (g/cm3) Pipe Diameter (in)Viscosity of Water (cp) Pipe Length (cm)

123






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.20 21 22 24 29 220.39 44 43 43 45 420.59 62 59 60 60 580.79 83 78 77 79 790.98 105 113 107 109 1181.13 360 360 360 360 360




124




Density of Sand (g/cm3) 2.65 1.38%


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.20 17 13 14 14 150.39 30 30 27 28 280.59 40 43 38 43 390.79 50 56 48 56 500.98 61 70 59 69 611.18 74 90 75 85 751.38 89 113 94 102 871.57 114 150 137 130 1171.69 420 420 420 420 420




125






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.79 9 9 7 8 81.57 18 17 15 16 162.36 26 26 23 24 243.15 35 36 32 34 333.94 45 45 41 45 434.72 57 57 53 57 545.51 73 73 70 75 705.83 150 120 180 180 180




126



Volume of Sand(ml) 360.59 1803.85 Cmax=0.6Density of Sand (g/cm3) 2.65 20.00%


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 00.79 6 5 3 5 51.57 12 10 10 11 112.36 18 15 15 17 173.15 24 21 20 23 233.94 30 27 26 29 294.72 36 34 32 35 355.51 43 40 37 41 416.30 49 47 44 48 487.09 55 53 51 54 547.87 62 60 57 61 618.66 68 67 63 67 689.45 76 74 74 74 7410.24 84 83 84 82 8211.02 93 92 97 91 9011.42 100 99 103 106 10511.81 150 150 150 150 150




127


Pipe Arrangement: One PipeSand Size (µm) 150 0.89 2Sand Weight (g) 22 22 89




Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.08 197 220 2070.20 500 520 5400.26 1320 1320 1320



Density of oil (g/cm3) Pipe Diameter (in)Viscosity of oil (cp) Pipe Length (cm)

128






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.08 103 118 1100.20 220 241 2350.31 388 420 4090.39 555 576 5720.47 810 850 8400.52 1500 1550 1600




129






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.08 41 38 440.31 175 168 1820.55 342 358 3550.79 546 553 5501.02 1062 1055 10711.07 1980 1980 1980




130






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 00.20 57 58 600.31 94 98 1000.55 162 165 1710.79 237 237 2481.02 315 328 3291.26 412 418 4251.50 491 516 5191.73 608 627 6391.97 772 798 8062.10 2280 2280 2280




131






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.79 110 115 1151.57 235 241 2462.36 347 369 3673.15 475 508 5053.94 653 663 6614.72 844 839 8455.51 1112 1131 11415.87 3000 3000 3000




132


Pipe Arrangement: One PipeGlass Beads Size (µm) 150 1 2Glass Beads Weight (g) 50 1 200

Volume of Glass Beads(ml) 19.86 4053.60

Density of Glass Beads(g/cm3) 2.5 0.49%


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.20 27 26 27 35 310.39 56 48 48 58 540.59 95 83 81 100 800.64 170 162 165 174 180




133






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.20 12 11 12 14 150.39 25 25 28 29 300.59 41 43 43 42 470.79 53 55 59 53 650.98 70 68 75 65 921.18 94 93 95 89 1221.29 249 230 270 275 290




134






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.20 8 8 9 12 110.39 16 19 18 25 240.59 23 33 26 33 350.79 31 43 37 43 460.98 41 53 44 54 601.18 52 64 58 62 721.38 58 73 64 75 851.57 67 81 78 85 971.77 93 110 98 112 1251.94 300 300 310 310 300




135



Volume of Glass Beads(ml) 180.39 1803.85Density of Glass Beads(g/cm3) 2.5 10.00%


Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 00.39 5 5 4 5 50.79 10 10 9 10 101.18 14 15 14 15 151.57 19 19 19 20 201.97 24 23 23 25 252.36 27 27 27 30 292.76 32 31 32 35 343.15 37 36 36 40 393.54 42 41 41 45 443.94 46 46 46 50 484.33 51 50 51 55 534.72 56 55 56 60 585.12 62 61 62 65 645.51 68 67 68 71 715.83 110 110 110 110 110




136





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 00.79 4 5 5 5 51.57 10 11 11 11 102.36 15 17 16 17 163.15 21 23 22 23 233.94 27 29 28 29 294.72 32 35 34 35 355.51 38 41 40 41 416.30 44 47 46 47 477.09 50 53 52 53 537.87 56 59 58 59 598.66 62 65 64 66 669.45 68 71 71 72 7210.24 73 78 77 78 7811.02 80 84 83 86 8511.42 87 87 87 90 9011.81 120 120 120 120 120




137

Appendix A-63. 150µm Glass Beads in Oil, α0=0.46%

Pipe Arrangement: One PipeGlass Beads Size (µm) 150 0.89 2Glass Beads Weight (g) 21 22 89




Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.08 203 210 1800.20 510 530 5000.26 1260 1260 1260




138






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.08 120 123 1300.20 240 250 2580.31 380 427 4550.39 574 582 6130.47 780 826 8400.52 1460 1430 1400




139






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.08 51 53 500.31 183 185 1920.55 346 351 3630.79 505 550 5541.02 910 930 9601.05 1620 1620 1620




140






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 00.20 51 59 540.31 94 120 1070.55 183 200 1950.79 265 299 2851.02 361 385 3751.26 463 485 4721.50 543 597 5761.73 659 712 7051.97 915 923 9022.10 1750 1750 1750




141






Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 0

0.79 109 107 1081.57 241 240 2392.36 364 365 3603.15 488 492 4863.94 617 617 6124.72 748 744 7355.51 880 882 8666.30 1029 1041 10176.93 1980 1980 1980




142





Exp. #2

Exp. #3

Exp. #4

Exp. #5

0.00 0 0 0 0 00.79 91 95 1001.57 197 211 2142.36 306 319 3273.15 418 428 4423.94 522 543 5514.72 632 653 6665.51 740 763 7756.30 846 873 8817.09 958 977 9967.87 1069 1090 11138.66 1178 1203 12219.45 1286 1312 133510.24 1403 1430 145511.02 1547 1567 159611.81 2100 2100 2100




143

APPENDIX B

95% CONFIDENCE STATISTICAL ANALYSIS OF REPRESENTATIVE DATA

A 95% confidence interval using statistical analysis of three representative

experimental data for the two-pipe apparatus is shown in Figure B-1 to Figure B-3.

Figure B-4 to Figure B-6 show a 95% confidence interval of three representative

experimental data for the one-pipe apparatus.

144

145

0.00

1.00

2.00

3.00

4.00

0 50 100 150 200 250

Time (s)

Sand

Lev

el (i

n) Trend Line

Figure B-1. 95% Confidence Interval of Experimental Data


0.00

2.00

4.00

6.00

8.00

10.00

12.00

0 50 100 150 200 250

Time (s)

Sand

Lev

el (i

n)

Trend Line



146


0.00

2.00

4.00

6.00

8.00

0 500 1000 1500 2000 2500 3000

Time (s)

Sand

Lev

el (i

n) Trend Line

(150 µm Sand in Oil, α0=0.88%).

0.00

0.40

0.80

1.20

1.60

2.00

0 100 200 300 400 500

Time (s)

Sand

Lev

el (i

n) Trend Line



147

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0 50 100 150 200

Time (s)

Sand

Lev

el (i

n)

Trend Line


(150 µm Sand in Water, α0=10%).

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 500 1000 1500 2000

Time (s)

Sand

Lev

el (i

n)

Trend Line


(150 µm Sand in Oil, α0=0.92%).

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