table of contensts - university of tulsa thesis... · 2016. 8. 9. · in horizontal flow of water...

T H E U N I V E R S I T Y O F T U L S A

THE GRADUATE SCHOOL

MODELING OF CRITICAL DEPOSITION VELOCITY OF SAND

IN HORIZONTAL AND INCLINED PIPES

by Faisal M. Almutahar

A thesis submitted in partial fulfillment of

the requirements for the degree of Master of Science

in the Discipline of Mechanical Engineering

The Graduate School

The University of Tulsa

2006

T H E U N I V E R S I T Y O F T U L S A

THE GRADUATE SCHOOL

MODELING OF CRITICAL DEPOSITION VELOCITY OF SAND

IN HORIZONTAL AND INCLINED PIPES

by

Faisal M. Almutahar

A THESIS

APPROVED FOR THE DISCIPLINE OF

MECHANICAL ENGINEERING

By Thesis Committee

, Chair Dr. Siamack Shirazi , Co-Chair Dr. Brenton McLaury Dr. Daniel Crunkleton

ii

ABSTRACT

Almutahar, Faisal (Master of Science in Mechanical Engineering) Modeling of Critical Deposition Velocity of Sand in Horizontal and Inclined Pipes Directed by Professors Siamack Shirazi and Brenton McLaury (94 pp., Chapter 6)

(165 words)

Two models are developed in this work for predicting critical deposition velocity

in two-phase (gas-solid or liquid-solid) and multiphase (gas-solid-liquid) flows. In the

first one, called “initial approach”, a horizontal two-phase model developed by Oroskar

and Turian (1980) is adopted. In the second (called the new approach), a new mechanistic

model is developed based on a force balance and turbulent theory. The initial approach is

obtained by fitting the Oroskar and Turian model to experimental data. It is extended to

multiphase flow by following a two-layer approach developed during this study. On the

other hand, the new approach is developed by assuming that particles are suspended by

turbulent velocity fluctuations generated by the flow. It is extended to multiphase flow

based on data and by accounting for the effect of the gas phase in the particles

suspension. The new approach shows satisfactory performance with two-phase and

multiphase data while the initial approach shows good performance for small particle

sizes (below 250 µm) for two-phase flow.

iii

ACKNOWLEDGEMENTS

The author would like to acknowledge Dr. Siamack Shirazi, thesis advisor, and

Dr. Brenton McLaury, thesis co-advisor, for their continuous support and guidance

throughout this work. They played a key role in all phases of research. Dr. Daniel

Crunkleton is acknowledged for serving as a member of the thesis committee and for

reviewing the thesis manuscript. He provided corrections and valuable suggestions to the

manuscript. Tulsa University Fluid Flow Project (TUFFP) is also acknowledged for

making their multiphase flow model available for this work. Special thanks are extended

to Saudi Aramco for providing the author with graduate research fellowship.

iv

TABLE OF CONTENTS

Page ABSTRACT............................................................................................................... iii ACKNOWLEDGEMENTS....................................................................................... iv TABLE OF CONTENTS........................................................................................... v LIST OF FIGURES ................................................................................................... vii CHAPTER I: INTRODUCTION Background .................................................................................................. 1 Research Goals ............................................................................................. 4 Research Approach...................................................................................... 4 CHAPTER II: LITERATURE REVIEW Modeling of Critical Deposition Velocity in Two-Phase Flow .................. 7 Modeling of Critical Deposition Velocity in Multiphase Flow.................. 15 Solids Concentration Effect on Particle Settling........................................ 18 CHAPTER III: EXPERIMENTAL DATA

Experimental Measurements of Critical Deposition Velocities in Two- Phase Flow ................................................................................................... 22 Experimental Measurements of Critical Deposition Velocities in Multi- phase Flow .................................................................................................... 26 Turbulent Velocity Fluctuation .................................................................. 27 Hot Film Anemometer........................................................................ 27 Laser Doppler Velocimeter (LDV)..................................................... 30 Experimental Measurements of Turbulent Velocity Fluctuation ....... 31

CHAPTER IV: MECHANISTIC MODELING

Initial Approach ........................................................................................... 35 Original Model (Oroskar and Turian, 1980)..................................... 36 Extension of Initial Approach to Low Concentration Slurries .......... 39 Extension of Initial Approach to Inclined Flow................................. 43 Extension of Initial Approach to Multiphase Flow............................ 45

New Approach .............................................................................................. 46 Development of New Approach for Two-phase Flow ........................ 46 Extension of New Approach to Multiphase Flow .............................. 51

CHAPTER V: RESULTS AND DISCUSSION Validation of Two-Phase Critical Deposition Velocity Models................ 57

Small Sand Size .................................................................................. 59 Medium Sand Size .............................................................................. 62

v

Coarse Sand Size................................................................................ 67 Discussion of Two-Phase Critical Deposition Velocity Results ........ 68

Validation of Multiphase Critical Deposition Velocity Models ............... 70 Initial Multiphase Approach .............................................................. 71 New Multiphase Approach................................................................. 72 Discussion of Multiphase Critical Deposition Velocity Results ........ 75

Validation of Turbulent Velocity Fluctuation Correlations .................... 77 Discussion of Turbulent Velocity Fluctuation Results....................... 79

CHAPTER VI: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Summary....................................................................................................... 81 Conclusions ................................................................................................... 83 Recommendations for Future Work .......................................................... 84 BIBLIOGRAPHY......................................................................................................... 86 APPENDIX A: Experimental Data............................................................................ 92

vi

LIST OF FIGURES

Page I-1: Sand transport regimes of sand in two-phase flow (sand-liquid,

sand-gas) .......................................................................................................................... 2 I-2: Sand deposition in tubing and fittings............................................................................. 3 II-1: Suspension forces acting on a particle........................................................................... 10 II-2: Change in settling velocity as a function of particle concentration

(Li, 2003) ...................................................................................................................... 20 III-1: Experimental measurements of critical deposition velocity of sand

in horizontal flow of water (Kokpinar and Gogus, 2001)............................................. 23 III-2: Experimental measurements of critical deposition velocity of sand

in horizontal flow of water (Parzonka, 1981) ............................................................... 23 III-3: Experimental measurements of critical deposition velocity of 360

µm sand and 78 µm fly ash in horizontal flow of water (Roco, 1991) ............................................................................................................................. 25

III-4: Experimental measurements of critical deposition velocity of 2%

and 20% (by volume) sand in inclined flow of water (Roco, 1991)............................. 25 III-5: Experimental measurements of sand concentration in horizontal

flow of (a) water and 0% gas, (b) water and 10% gas, and (c) water and 20% gas (particle distribution: 150-300 µm, pipe diameter: 70 mm) ............................................................................................................................... 26

III-6: Typical hot film anemometer measurements in slug flow ........................................... 29 III-7: Slug flow pattern .......................................................................................................... 29 III-8: Schematic diagram of Laser Doppler Velocimeter (LDV) .......................................... 31 III-9: Experimental measurements of local turbulent velocity fluctuation

(Burden, 1999) .............................................................................................................. 32 III-10: Experimental measurements of local turbulent velocity

fluctuations in the liquid phase of horizontal bubbly flow (Iskandrani, 2001) ......................................................................................................... 33

vii

III-11: Experimental measurements of local turbulent velocity fluctuations in the liquid phase of horizontal slug flow (Lewis, 2002) .......................................................................................................................... 34

IV-1: Comparison of the Oroskar and Turian original formula (Equation

(IV-1)) with experimental data (Roco, 1991) ............................................................... 40 IV-2: Comparison of the Oroskar and Turian correlation (Equation (IV-

3)) with experimental data (Roco, 1991) ...................................................................... 41 IV-3: Comparison of predicted critical deposition velocity using the

initial Two-Phase model (Equation (IV-5)) and Roco’s data ....................................... 43 IV-4: critical deposition velocity predictions by equation (IV-6) versus

experimental data at different inclination angle, sand in water .................................... 45 IV-5: Turbulent velocity fluctuation components.................................................................. 48 IV-6: Change in liquid phase turbulent velocity fluctuation at different gas

fractions as calculated from Lewis data presented in Figure III-10.............................. 53 V-1: Comparison between critical deposition velocity predictions made

by initial approach, new approach, Oroskar and Turian (1980), and Davies (1987) with experimental data gathered by Parzonka (1981) for 78 µm fly ash........................................................................................................... 60

V-2: Comparison between critical deposition velocity predictions made

by initial approach, new approach, Oroskar and Turian (1980), and Davies (1987) with experimental data gathered by Parzonka (1981) for 180 µm sand ............................................................................................................ 61




by initial approach, new approach, Oroskar and Turian (1980), and Davies (1987) with experimental data gathered by Kokpinar and Gogus (2001) for 230 µm sand ..................................................................................... 62


by initial approach, new approach, Oroskar and Turian (1980), and Davies (1987) with experimental data gathered by Roco (1991) for 360 µm sand.................................................................................................................. 63

viii








by initial approach, new approach, Oroskar and Turian (1980), and Davies (1987) with experimental data gathered by Kokpinar and Gogus (2001) for 450 µm sand..................................................................................... 65










by initial approach, new approach, Oroskar and Turian (1980), and Davies (1987) with experimental data gathered by Kokpinar and Gogus (2001) for 1150 µm sand................................................................................... 68

ix

V-15: Comparison between Oudeman’s data (1993) and critical deposition velocity predictions made by the initial approach....................................... 72

V-16: Comparison between Oudeman’s data (1993) and critical deposition

velocity predictions made by the new multiphase approach, Equation (IV-27), along with turbulent dissipation function given by Davies, (Equation IV-21)........................................................................................................... 73

V-17: Comparison between Oudeman’s data (1993) and critical deposition

velocity predictions made by the new multiphase approach, Equation (IV-27), along with modified turbulent dissipation function given by Equation IV-22.............................................................................................................. 75

V-18: Comparison between predictions made by Equation IV-19 and

experimental measurements of turbulent velocity fluctuation in the core region .................................................................................................................... 78

V-19: Comparison between predictions made by Equation IV-15 and

experimental measurements of turbulent velocity fluctuation near the bottom wall ................................................................................................................... 79

x

CHAPTER I

INTRODUCTION

Background

The oil and gas industry has developed significantly to meet the growing global

demand in energy. However, still there are many critical areas with minimal levels of

knowledge. That lack of knowledge is not necessarily attributed to low level of interest but it

could be attributed to the complexity of the issue to be investigated. Sand transport in oil and

gas pipelines is one of the complex issues that has been under investigation over the last three

decades. Oil and gas producers are highly interested in this area because inappropriate

operation of oil and gas containing sand could lead to major consequences such as production

loss and unpredicted failure. High production rates, for example, could cause sever erosion in

a very short time while low production rates could cause sand deposition that prevents

corrosion inhibitors form reaching the bottom of the pipeline causing underdeposit corrosion.

Therefore, proper velocity guidelines should be put in place to ensure reliable and safe

operation.

Sand motion in flowing pipelines is characterized by three regimes as shown in

Figure I-1. The first regime is the stationary sand bed that occurs at low flowstream

velocities, then the moving or sliding sand bed (or sand dunes) that occur at higher velocities

and finally fully suspended sand that occurs at high enough velocities. This research is to

study the transition from the sliding sand bed to fully suspend sand where the critical

deposition velocity occurs. The critical deposition velocity is defined as the minimum

1

flowstream velocity needed for keeping sand particles in suspension in pipe flow to prevent

sand deposition.

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

velocity, particles would accumulate in the pipe and create sand beds or dunes inside the

pipeline or near pipe fittings such as elbows, tees, valves, and couplings. Figure I-2, for

example, shows sand particle accumulation in tubing and in an elbow. Sand deposition can

cause serious problems such as partial flow blockage or even total blockage that may occurs

in case of production shut in. Erosion damage is also a possible problem with sand deposition

because 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.

III. FULLY SUSPENDED High Velocity

Figure I-1: Sand transport regimes of sand in two-phase flow (sand-liquid, sand-gas),

(Oudeman, 1993)

I. STATIONARY BED

II. MOVING BED Flow

Flow

Flow Low Velocity

2

Sand

Sand D une

Sand

Sand D une

b . E lbow

a. T ubing

Figure I-2: Sand deposition in tubing and fittings

Scraping is a common practice in removing deposits; however, high cost is usually

associated with the scraping process, especially for offshore applications. Scraping is hard to

be apply unless it is considered in the design stage because it requires special facilities like

launcher, receiver, and specific pipe geometry. Scraping can cause problems such as

production loss when it gets stuck in pipelines or it could cause mechanical damage to pipe

components when it hits them. Therefore more convenient and cost effective ways in either

removing or preventing sand deposition should be sought.

Increasing flowstream velocity can be used effectively to either remove or prevent

sand deposition. However, the velocity should not be increased to high limits that could

cause erosion damage. Therefore, it is important to develop a reliable prediction model for

determining critical deposition velocity. However, determining critical deposition velocity is

a challenging task because it involves many complicated issues such as sand settling,

3

particle-particle interactions, and turbulent velocity fluctuations. Each one of these issues is

complex enough to be an area of research by itself. The problem even becomes more

complicated for multiphase systems due to the variation in flow patterns.

Research Goals

The objective of this work is to develop a mechanistic model for predicting the

critical sand deposition velocity for oil and gas design applications. Specifically, the

objectives are (1) to develop a mechanistic two-phase model (solid-liquid and solid-gas) for

predicting critical deposition velocity in horizontal wells and pipelines, (2) to extend the

model to predict critical deposition velocity in inclined pipes, (3) to extend model predictions

to multiphase (gas-liquid-sand) systems, (4) and to verify the model by comparing results to

experimental data provided in the literature.

Research Approach

Developing a mechanistic model for critical deposition velocity requires a full

understanding of the physics of the problem in two-phase flow (solid-liquid, solid-gas) and

multiphase flow (solid-liquid-gas). In order to achieve this objective, a comprehensive

review of previous work conducted in this area has to be done first.

Two models have been developed in this work for predicting critical deposition

velocity. In the first one, a horizontal two-phase model developed by Oroskar and Turian

(1980) has been adopted. In the second, a new mechanistic model has been developed based

on a force balance and turbulent theory.

4

The Oroskar and Turian model has been first evaluated in this work and then it has

been modified to obtain better results at desirable particle sizes and concentrations. It has

been extended to inclined flow and multiphase flow. A two-step modeling approach has been

developed to extend it to multiphase flow.

In the second model, a force balance and turbulent velocity approach has been

followed to develop a new model for two-phase flow (gas-solid, liquid-solid). The new

model has been extended to multiphase flow by developing a relation that accounts for the

effect of the gas phase in the particle suspension. The relation is flow pattern dependent so

experimental data for different flow patterns have been used.

Both models have been evaluated by comparing predictions with a wide range of

experimental data available in the literature. However, due to lack of experimental data in

multiphase flow, the models have been compared to one set of data.

This report consists of six chapters. The next chapter, Chapter II, is a literature review

in which previous work conducted in the area of critical deposition velocity is reviewed. A

summary of all experimental data used in this study is presented in Chapter III that includes

experimental measurements of critical deposition velocities in two-phase (solid-liquid) and

multiphase (solid-liquid-gas) flows and experimental measurements of turbulent velocity

fluctuation in single-phase (liquid) and multiphase (gas-liquid). Chapter IV provides detailed

development of the two critical deposition velocity models developed in this study. In

Chapter V, the two critical deposition velocity models are validated by comparing the

models’ predictions with experimental data for two-phase (solid-liquid) and multiphase

(solid-liquid-gas) flows. The last chapter, Chapter VI, provides a summary and future work.

5

CHAPTER II

LITERATURE REVIEW

This chapter reviews previous work conducted in the area of critical deposition

velocity. The work in this area started in the early fifties when investigators tried to

determine the critical deposition velocity for hydrotransport of high concentration slurries.

Many empirical models were developed within the first twenty years but all showed low

performance. According to Thomas (1979), Carleton and Cheng (1974) reviewed over 50

correlations to conclude that hydraulic transport can not be designed with confidence from

available correlations. Later, investigators started to follow mechanistic approaches to model

critical deposition velocity. Different mechanistic models have been developed since then

such as Thomas (1979), Oroskar and Turian (1980) and Davies (1987). However, these

models were developed for horizontal two-phase (solid-liquid, solid-gas) flow of high

concentration slurries. The models have been developed for the solid hydrotransport industry

and do not show good performance for petroleum production applications. In the oil and gas

industry, sand is produced in multiphase (solid-liquid-gas) systems with much lower

concentrations and smaller particle sizes than those used in the solid hydrotransport industry.

No mechanistic models have been found in the literature for sand transport in multiphase

flow. However, some empirical correlations were developed for different flow regimes. This

could be attributed to the complexity associated with multiphase flow as the flow pattern

changes with liquid and gas rates.

The following section provides an overview of two-phase and multiphase critical

deposition velocity models found in the literature. This will be followed by an overview of

important issues related to the critical deposition velocity such as particle settling, turbulent

6

velocity fluctuations in pipe flow, and effect of concentration in particle settling and

turbulent velocity fluctuation.

Modeling of Critical Deposition Velocity in Two-Phase Flow

Three mechanistic approaches have been found in the literature for modeling critical

deposition velocity in two-phase flow (solid-liquid, solid-gas). The earliest approach reported

in the literature was based on the minimum head losses that occur at the critical deposition

velocity. In the next approach, some investigators used a theory called sliding bed theory to

theoretically justify one of the earliest critical deposition models developed by Durand

(1953). The latest approach available in the literature was based on a particle force balance

and turbulent theory. In this section, an overview of the three approaches will be provided.

Durand (1953) developed one of the earliest empirical correlations for calculating

critical deposition velocity. Durand’s correlations took into account most of the parameters

involved in the deposition process as given by

( )[ ] 211 12 −= sgDFVD (II-1)

where

VD : critical deposition velocity, m/s

F1 : empirical term accounts for particle size and concentration, dimensionless

g : gravitational acceleration, m/s2

D : pipe diameter, m

s : ratio of solid density to liquid density, dimensionless

7

Thomas (1979) was able to justify theoretically Durand’s relation (Equation II-1) by

applying a theory called sliding bed theory. The sliding bed theory was developed by Wilson

(1974) to calculate the pressure gradient required to move a stationary bed as given by

( ) εφρ bLb CsgJ 12 −= (II-2)

where

Jb : pressure gradient required to move a stationary bed, kg/m2. s2


ρL : liquid density, kg/m3


Cb : volume concentration of solids in the bed, dimensionless

ε : coefficient of sliding friction between the bed and the pipe wall, dimensionless

φ : function of solid concentration in the pipe, dimensionless

Thomas used the pressure gradient required to slide the bed as an approximation for

the pressure gradient required to suspend a particle. After that, he related pressure gradient,

Jb, to flowstream velocity that is assumed to be equal to critical deposition velocity. Thomas

used standard pressure drop equation that is given by

DV

fJ DLb

2

2 ρ= (II-3)

where

Jb : pressure gradient required to move a stationary bed, kg/m2. s2

f : friction factor, dimensionless


VD : Critical deposition velocity, m/s

8


A relation for critical deposition velocity was obtained by combining Equation (II-2)

and Equation (II-3). The relation is given by

( )[ ] 2121

⎞⎛Cbεφ 122

−⎟⎟⎠

⎜⎜⎝

= sgDf

VD (II-4)

where

itical deposition velocity, m/s

n the bed, dimensionless

wall, dimensionless

VD : cr

Cb : volume concentration of solids i

ε : coefficient of sliding friction between the bed and the pipe

φ

f

2

ity to liquid density, dimensionless

quation (II-4) is equivalent to Durand’s equation (Equation II-1) with

: function of solid concentration in the pipe, dimensionless

: friction factor, dimensionless

g : gravitational acceleration, m/s


s : ratio of solid dens

E

21⎞⎛Cbεφ

2 ⎟⎟⎠

⎜⎜⎝

=f

F (II-4)

Another approach was developed by Bain and Bonnington (1970), and Doron and

Barnea (1995). In this approach, a relation for critical deposition velocity was obtained based

on head loss in the pipe. For two-phase flow (solid-fluid), experiments have shown higher

pressure drops at velocities below the critical deposition velocity due to the formation of a

solid bed. Sand bed formation causes higher head losses due to reduction in pipe cross

9

section area and due to the increase in surface roughness. As the velocity increases the solid

bed disappears and particles are entrained in suspension at a velocity equal to the critical

deposition velocity which consequently will decrease the head loss. Continuing to increase

the flow velocity above the critical deposition velocity will cause a decrease in the pressure

drop. According to Shook and Roco (1991), the head loss is expected to reach a minimum at

the critical deposition velocity due to the removal of the sand bed. This approach has given

unsatisfactory results because the minima are often poorly defined and the correlations used

in obtaining the minimum are only approximations.

The third approach developed in the recent studies is based on a force balance and

turbulent theory. This approach has been reported by many authors such as Oroskar and

Turian (1980), Davies (1987), and Cabrejos and Klinzing (1992). In this approach, the

critical deposition velocity is calculated by considering all forces involved in keeping a

particle suspended. A particle in a horizontal flow is subjected to three forces: downward

gravity force, and upward drag force and buoyancy force as shown in Figure II-1.

Fdrag

Fgravity

Fbuoyancy

y

x

Particle

Figure II-1. Suspension forces acting on a particle

10

The main assumption made in this approach is that turbulent eddies provide the

required lifting energy, drag force, to keep a particle in suspension. Another assumption was

made to account for the concentration effect in particle settling. The change in a particle

settling velocity due to the presence of other particles is assumed by the hindered settling.

Investigators, such as Davies (1987) and Oroskar and Turian (1980), have treated these

assumptions differently which have led to different forms of critical deposition models.

Davies (1987) followed a force balance and turbulent theory approach to develop a

model for calculating critical deposition velocity of solid particles in horizontal flow of

liquid. The model was developed in two steps. In the first step, the magnitude of the turbulent

velocity fluctuation (u´) required to lift the concentration of particles is calculated. Then a

relation between turbulent velocity fluctuation (u´) and flow stream velocity (VD) is

developed. A relation for calculating the required turbulent velocity fluctuation needed to

suspend the concentration of particles is developed by applying a force balance at a

suspended particle where

Force Lifting Forceion Sedimentat = (II-5)

The sedimentation force of an isolated spherical particle is given by

( LSPbuoyancygravity gdFF ρρ )π−=−= 3

6ForceionSedimentat (II-6)

where

dP : particle diameter, m


ρS : solid density, kg/m3


11

For a group of particles with concentration (c), Davies assumed that the

sedimentation force is reduced by the hindered settling effect. A work done by Maude and

Whitmore (1958) was used to account for hindered settling that was given by

( nC−= 1 Settling Hindered ) (II-7)

where

C : particles volume concentration, dimensionless

n : function of particle Reynolds number, dimensionless

Applying hindered settling, the following equation for the sedimentation force is obtained:

( )( nLsP Cgd −−= 1

6 Forceion Sedimentat 3 ρρ )π (II-8)

The lifting force is the drag force created by a velocity equal to the turbulent velocity

fluctuation (u´) that is given by the following equation:

( ) DLPdrag CudF 22

8Force Lifting ′== ρπ (II-9)

where

u′ : turbulent velocity fluctuation, m/s



CD : drag coefficient, dimensionless

Davies developed a relation for calculating the required turbulent velocity fluctuation

magnitude to suspend a concentration of particles by combining Equation (II-8) and Equation

12

(II-9) along with assuming a drag coefficient (CD) value of 2. The relation was given by the

following equation:

( ) ( )[ ] 211182.0 −−=′ sgdCu Pn (II-10)

where

u′ : turbulent velocity fluctuation, m/s




s : ratio of solid density to liquid density

In the second step, Davies related turbulent velocity fluctuation (u´) to flowstream

velocity (VD). For single-phase (liquid), he calculated turbulent velocity fluctuation as a

function of eddy length (ℓe) and power dissipated per unit mass of fluid (Pm) as given by

(Davies 1972)

( ) emPu l=′ 3 (II-11)

The specific power dissipated per unit mass of fluid (Pm) can be calculated as a function of

Blasius friction factor and flowstream velocity (VD) as given by (Davies 1972)

4575.2413

16.02 −== DVvD

VfP DD

m (II-12)

where

Pm : specific power dissipated per unit mass of fluid, m2/s3

f : friction factor, dimensionless

VD : flowstream velocity, m/s

13


v : kinematic viscosity, m2/s

dp : particle diameter, m

Davies assumed that the magnitude of eddy length is equal to particle diameter (dp)

because eddies smaller than particle size will not have sufficient energy to carry particles and

eddies larger than particle size will be too big to reach the bottom to carry particles. Having

this assumption and combining Equation (II-11) and Equation (II-12), Davies developed the

following equation that relates turbulent velocity fluctuation (u´) to flowstream velocity (VD):

( ) 42.03192.01213116.0 −=′ DdVvu PD (II-13)

Equation (II-13) is for single-phase (liquid) which can be extended to two-phase (solid-

liquid) by adding a function that accounts for the dissipation of turbulent velocity fluctuation

due to the presence of solids which was proposed by Davies as (1/(1+3.6C)). The final form

for critical deposition velocity (Equation II-14) was obtained by introducing the turbulent

dissipation term and combining Equation (II-10) and Equation (II-13).

( ) ( ) ( )[ ] 46.054.018.009.055.009.1 12164.3108.1 DsgdCCV Pn

D −−+= −ν (II-14)

where

VD : flowstream velocity, m/s

C : particle volume concentration, dimensionless

v : kinematic viscosity, m2/s




14


The early interest in studying deposition velocity of two-phase (solid-liquid) flow has

enriched this area with many experimental studies. These studies cover wide range of particle

sizes and concentrations. Parzonka et al. (1981), for example, provided an overview of more

than 50 sets of experimental data. A summary of these data and additional data is provided in

(Chapter III).

Modeling of Critical Deposition Velocity in Multiphase Flow

Sand transportation in multiphase flow is a very complex phenomenon. One reason

for the complexity is the change in flow pattern as the gas and liquid rates vary in the pipe.

Each flow pattern may have a different sand transportation mechanism. For example, sand

transport in intermittent flow is different from sand transport in stratified or annular flow.

The complexity of multiphase flow may explain the lack of theoretical development in this

area. No mechanistic model was found in the literature for predicting critical deposition

velocity in multiphase flow. However some empirical studies were found such as the study

conducted by Peter (1971), Oudeman (1993), and Stevenson et al. (2001, 2002). In these

studies, dimensional analyses/approaches were followed to develop correlations for sand

transport in multiphase flow.

Oudeman (1993) conducted an experimental study of sand transport in an air and

water test loop. Experimental data collected by Oudeman are shown in Chapter III. Oudeman

claimed that sand transport rate is not a direct function of flow pattern so he developed a

universal correlation for all flow patterns. Oudeman’s correlation predicts delivered sand

15

concentration in multiphase flow as a function of gas and liquid superficial velocities.

Oudeman defined two dimensionless quantities: sand transport rate (Φ) and liquid flow rate

(Ψ) as given by Equations (II-15 and II-16).

( )13 −=Φ

sgd

Q

P

(II-15)

( )1

2

−=Ψ

sgdv

P

b (II-16)

where

Φ : dimensionless quantity of solid transport rate

Ψ : dimensionless quantity of liquid flow rate

Q : solid transport rate (volume per second per meter of sand bed width), m3/s.m

dP : particle size, m


s : solid to liquid density ratio, dimensionless

vb : drag velocity at sand bed (calculated as a function of liquid velocity), m/s

Oudeman used two empirical constants (m) and (n) to relate the two dimensionless

quantities as given by Equation (II-17). The constants (m) and (n) depend on the input gas

fraction. Based on experimental data, Oudeman used an average value of “m = 70” and “n =

2.7”.

nmΨ=Φ (II-17)

Another empirical correlation was developed by Stevenson et al. (2001, 2002). They

conducted experimental studies and dimensional analysis to develop flow pattern dependent

relations for stratified and intermittent flow patterns. For intermittent flow, they developed an

16

empirical relation given by Equation (II-18) to calculate average particle velocity. This

relation can be extended to estimate flow conditions at which deposition occurs by setting

particle velocity (Vp) equal to zero.

18.05.1

Re88.038.1195.0−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

⎥⎦

⎤⎢⎣

⎡+−⎥

⎦

⎤⎢⎣

⎡+=

DdFrFr

vv

vv

VV P

ffsl

sg

sl

sg

sl

p (II-18)

where

Vp : average particle velocity, m/s

vsl : superficial liquid velocity, m/s

vsg : superficial gas velocity, m/s

Ref : Reynolds number based on liquid superficial velocity (Ref = ρD vsl /µ),

dimensionless

Frf : Fraud number based on liquid superficial velocity (Frf = vsl2 /gD), dimensionless

dP : sand size, m


For stratified flow, they followed the same procedure to develop a relation for particle

velocity that was given by Equation (II-19). The effect of gas superficial velocity is indirectly

considered by using the hydraulic diameter of the liquid phase.

( )[ 330340340 1433 .f

..D Dsgvd.V −= − ] (II-19)

where

VD : critical deposition velocity of liquid film, m/s

d : particle size, m/s

ν : kinematic viscosity, m/s2

17

g : gravitational acceleration, m2/s

s : ratio of particle density to liquid density, dimensionless

Df : hydraulic diameter of liquid phase, m

Solids Concentration Effect on Particle Settling

Settling velocity of an isolated particle is given by Equation (II-20) and is known as

Stokes terminal settling velocity (Li 2003).

( )µρρ

18

2pfS

o

gdV

−= (II-20)

where

Vo : Stokes terminal settling velocity, m/s


ρs : particle density, kg/m3

ρf : fluid density, kg/m3

µ : fluid dynamic viscosity, kg/m.s

However in the presence of other particles the settling velocity becomes a function of

concentration as given by Equation (II-21).

( )CfVV oS = (II-21)

where

VS : settling velocity of particles , m/s

Vo : Stokes terminal settling velocity , m/s

f(C) : function of concentration , dimensionless

18

There is a discrepancy in the experimental data collected by investigators that studied

the effect of solid concentration in settling velocity. Some investigators like Olive (1960),

McNown and Lin (1952), and Ham and Homsy (1988) showed that the experimental

measurements of settling velocity decreases as particle concentration increases. However

others like Koglin (1973), and Kaye and Boardman (1962) showed different behavior at

concentrations below 3%. They showed that settling velocity increases with concentration in

this region.

Li (2003) summarized the change in settling velocity as a function of solid

concentration in three regions: free settling, cluster forming, and hindered settling as shown

in Figure II-2. The first occurs at very low concentrations below 0.1% where settling velocity

is not a function of concentration. At a concentration of 0.1% to almost 1.5% the second

region occurs where settling velocity increases with concentration due to the formation of

clusters. The cluster formation makes it easier and faster for particle to settle down. In the

third region, settling velocity decreases with concentrations due to the effect of hindered

settling. The hindered settling occurs as a result of return flow caused by the displaced

volume of settled particles.

19

1

B CA Free Settling

Cluster Forming

Hindered Settling

0% 3%Particle Volume Concentration

Figure II-2. Change in settling velocity as a function of particle concentration, (Li, 2003)

The change in particle settling velocity due to hindered settling effect was proposed

by Richardson and Zaki (1954), and Maude (1958) as given by the following equation:

( ) ( )nCCf −== 1Settling Hindered (II-22)

where (n) is function of particle’s Reynolds number (Rep). The change in particle settling

velocity due to cluster formation was proposed by Li (2003) as given by the following

equation.

( ) 17.662222EffectCluster 2 ++−== CCCf (II-23)

20

CHAPTER III

EXPERIMENTAL DATA

The early interest in studying particles transport (hydrotransport) enriched the area of

two-phase (solid-liquid) with a significant amount of experimental data. Parzonka et al.

(1981), for example, collected more than fifty sets of experimental measurements of critical

deposition velocity. They gathered data for different types of material such as coal, small to

coarse sand, and gravel for a wide range of particle sizes and concentrations. In contrast, the

experimental work in the area of multiphase (solid-liquid-gas) is very limited and most of it

does not provide direct measurements of critical deposition velocity. Stevenson et al. (2001),

for example, provided measurements of particle velocities in near horizontal multiphase

pipes that can be used roughly to extrapolate values of critical deposition velocity. The only

multiphase critical deposition velocity measurements found in the literature were collected

by Oudeman (1993). Oudeman measured suspended solid concentration in horizontal

multiphase (air/sand/water) test loop.

This chapter consists of three parts. In the first two parts, experimental measurements

of critical deposition velocities in two-phase (solid-liquid) and multiphase (solid-liquid-gas)

flows are provided. In the third part, experimental measurements of turbulent velocity

fluctuations are provided.

The experimental data of critical deposition velocity is used in Chapter V to validate

the initial approach and the new approach developed in this study for predicting critical

deposition velocity. The initial approach is developed by modifying an existing horizontal

21

two-phase model developed by Oroskar and Turian (1980) while the new approach is

developed based on a force balance and turbulent theory.

The turbulent velocity fluctuation data is used in Chapter V to validate correlations

developed in this study for estimating magnitude of turbulent velocity fluctuations in single-

phase (gas or liquid) or multiphase (gas-liquid) flow. Estimating magnitude of the turbulent

velocity fluctuations is an essential part of the proposed two-phase and multiphase critical

deposition velocity models. The correlations are developed in Chapter IV for predicting the

magnitude of turbulent velocity fluctuations in single-phase (gas or liquid) flow. The

multiphase turbulent velocity fluctuation data is then used to extend the single-phase

correlations to multiphase.

Experimental Measurements of Critical Deposition Velocities

In Two-Phase (Solid-Liquid) Flow

Parzonka et al. (1981) and Kokpinar and Gogus (2001) collected a wide range of

critical deposition velocity data available in the literature. They collected more than 70 sets

of data that cover a range of particle sizes from 60 µm to 5300 µm. A summary of

experimental data within the area of interest of this research is presented in this section.

Figure III-1 shows critical deposition velocities collected by Kokpinar and Gogus for

different sand sizes (230 µm - 1150 µm) in water. Figure III-2 shows additional critical

deposition velocities collected by Parzonka et al. (1981) for different sand sizes (180 µm -

600 µm) in water.

22

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0% 5% 10% 15% 20% 25%

Sand Volume Fraction

Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

dp-450um, D-102mm

dp-450um, D-152mmdp-880um, D-152mm



dp : Particle sizeD : Pipe diameter

Figure III-1. Experimental measurements of critical deposition velocity of sand in horizontal flow of water (Kokpinar and Gogus, 2001)

0.0

0.5

Figure III-2. Experimental measurements of critical deposition velocity of sand in

horizontal flow of water (Parzonka, 1981)

1.0

5

0

5

0

3.5

0% 5% 10% 15% 20% 25% 30%

Sand Volume Fraction

Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

3.

2.

2.

1.

dp-180um, D-50.8mm

dp-190um, D-76mm

dp-420um, D-206mm

dp-570um, D-51mm

dp-600um, D-13mm

dp-400um, D-103mm

dp : Particle size

D : Pipe diameter

23

The data presented in Figure III-1 and Figure III-2 indicate that critical deposition

velocity increases as particle size, particle concentration, or pipe diameter increases. At a

pipe diameter of 108 mm, the critical deposition velocities of 1150 µm sand are higher than

those for 230 µm and 585 µm sand as shown in Figure III-1. Similarly, Figure III-2 shows

that the critical deposition velocities of 570 µm sand are higher than those for 180 µm and

190µm sand. For sand size of 450 µm, the critical deposition velocities at 152 mm ID are

higher than those at 102 mm ID as shown in Figure III-1. Another observation can be made

by examining the data in Figure III-1 and Figure III-2 that the critical deposition velocity

experiences a maximum value at a particle volume concentration between 15% and 25%. A

Similar trend is shown in Figure III-3 for data collected by Roco (1991). Roco measured the

critical deposition velocity of two different particle materials in horizontal flow of water. The

materials used are 360 µm sand and 78 µm fly ash (Fly ash density is 2300 kg/m3) in 100

mm ID pipe. Roco also measured critical deposition velocities of sand at different inclination

angles as shown in Figure III-4. This experiment was conducted at a fixed sand size of 360

µm and two sand mass fractions of 2% and 20%.

24

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

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

Solid Volume Fraction

Criti

cal D

epos

ition

Vel

ocity

(ft/s

)

dp-360um, D-100mm

dp-78um, D-100mm

dp : Particle Size

D : Pipe Diameter

Figure III-3. Experimental Measurements of critical deposition velocity of 360 µm sand and 78 µm fly ash in horizontal flow of water (Roco, 1991)

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15

Inclination Ang

Crit

ical

Dep

ositi

on V

eloc

ity (f

t/s)

Figure III-4. Experimental measurements of critica

(by mass) sand in inclined flow of

25

20% Sand Mass Fraction


20 25 30

le

l deposition velocity of 2% and 20% water (Roco, 1991)

θ

4” Pipe 360 µm sand

4” Pipe

Sand-W

Experimental Measurements of Critical Deposition Velocities

In Multiphase (Solid-Liquid-Gas) Flow

Oudeman (1993) collected experimental measurements of sand concentrations in

air/water flow. The experiment was conducted in a multiphase (air/water/sand) test loop with

a 70-mm ID and 150-300 µm sand distribution. Oudeman used a full-bore sampling probe to

measure the suspended sand concentration at different water and air superficial velocities. In

order to investigate the effect of gas fraction in sand suspension, the experiment was

conducted for three cases: (a) single-phase water, (b) water and 10% by volume air, (c) water

and 20% by volume air. The experimental data for the three cases are shown in Figure III-5.

0.3

0.4

0.5

0.6

0.7

0.8

0 0.3 0.6 0.9 1.2 1.5 1.8

Sand Volume Concentration (%)

Liqu

id S

uper

ficia

l Vel

ocity

(m/s

)

(a) 0% Gas

(b) 10% Gas(c) 20% Gas

Figure III-5. Experimental measurements of sand concentration in horizontal flow of (a) water and 0% gas, (b) water and 10% gas, and (c) water and 20% gas (particle

distribution: 150-300 µm, pipe diameter: 70 mm), (Oudeman, 1993)

26

Turbulent Velocity Fluctuation

Conducting experimental measurements of turbulent velocities in either single-phase

(gas or liquid) or two-phase (liquid-gas) flows can not be done simply. Advance techniques

are needed to allow effective measurements of local velocities. Two techniques have been

reported in the literature (Iskandrani (2001), Lewis et al. (2002), and Azzopardi (1994)) for

this purpose: Hot Film Anemometer (HFA) and Laser Doppler Velocimeter (LDV). The

former has been used widely in multiphase flows over the last fifty years while the latter was

developed initially for single-phase applications; however, some have used it for multiphase

applications. This section provides an overview about the development and applications of

each technique followed by experimental measurements of turbulent velocities in single-

phase and multiphase flows.

Hot Film Anemometer

According to Lewis et al. (2002), Hsu et al. (1963) was the first to adapt HFA for

identifying flow pattern and for measuring the void fraction in upward vertical water-steam

flows. Delhaye (1969) studied the response of hot-film probes in two-phase (liquid-gas) flow.

He developed a procedure for obtaining local measurements of void fraction, liquid velocity

and turbulent intensity in two-phase (gas-liquid) flow. Since then, this technique has been

used extensively by many investigators for studying different multiphase flow patterns.

Lewis et al. (2002), for example, used this technique to study slug flow. Others like Serizawa

et al. (1975), Wang et al. (1984, 1987) and Iskandrani et al. (2001) used it to study bubbly

flow.

27

The hot film anemometry technique works by taking instantaneous measurements of

the change in heat transfer from an electrically heated sensor. As the fluid flows past the

probe, the sensor cools at different rates due to changes in local velocities or phase. These

changes in cooling rates result in voltage changes in the anemometer.

Figure III-6 shows a typical sensor output of a probe installed in horizontal slug flow.

A schematic diagram of the slug flow pattern is shown in Figure III-7. The probe was

mounted in a position that allows the sensor to encounter a slug body and a Taylor bubble.

High voltage regions of Figure III-6 represent slug bodies and low voltage regions represent

Tailor bubbles. Voltage measurements in the liquid phase are higher because the heat transfer

in the liquid phase is much larger than the heat transfer in the gas phase so the anemometer

provides higher voltages to maintain the probe at the same temperature. Within the slug

body, the anemometer was sensitive enough to measure the change in voltage due to the

presence of small gas bubbles. Close examination of Figure III-6 shows that the voltage

continues to decrease gradually within the Tailor bubble. This is attributed to the gradual

removal of liquids remaining in the sensor element as it leaves the slug body. Different

methods are available in the literatures for analyzing anemometer signals for getting local

velocities and void fraction measurements. Lewis et al. (2002) provided a detailed procedure

for calculating local velocity using the hot film anemometer technique.

28

Slug Body

Taylor Bobble Entrained

Bubbles

Vol

tage

Time, Sec

Figure III-6. Typical hot film anemometer measurements in slug flow (Lewis, 2002)

Taylor Flow Bubble Slug

Body

Figure III-7. Slug flow pattern

29

Laser-Doppler Velocimeter (LDV)

Laser Doppler Velocimeter (LDV) was developed initially for single-phase flow.

However, some have used it for multiphase flow. Theofaneous et al. (1982) and Lioumbas et

al. (2005) used LDV to study turbulence in, respectively, two-phase dispersed flow and

stratified flow. LDV was also used by Azzopardi (1994) to measure velocity fluctuations of

the gas phase in annular flow. He implemented two-color visibility technique described by

Yeoman et al. (1982). In this technique 1 µm polystyrene tracer particles are injected into the

gas phase to allow separate velocity measurements of entrained liquid and tracer particles.

This was possible as the liquid particles were much larger than tracer particles.

Process and major components of the LDV technique are shown in Figure III-8. The

LDV technique starts by generating a single beam of light produced by an argon ion laser.

The single beam is then split into two beams by a beam splitter (the Color Burst). The beams

are directed at an angle into the flow by a lens in the fiber optic probe. A measurement

volume and fringe pattern are created at the intersection of the beams. Light is refracted as

particles pass through them. Refracted lights of the particles are collected by the APV probe

and directed to the Intelligent Flow Analyzer (IFA) through the Color Link Receiver. Finally,

the data is processed by computer software that provides real time measurements.

30

Figure III-8. Schematic diagram of Laser Doppler Velocimeter (LDV)

Experimental Measurements of Turbulent Velocity Fluctuation

Burden (1999) collected experimental measurements of turbulent velocities in

horizontal flow of water. Burden used the Laser Doppler Velocimeter (LDV) technique to

measure the local axial velocities in a 50.8 mm ID pipe and a mean velocity of 0.78 m/s. The

local axial velocities were used to calculate the turbulent velocity fluctuations. Figure III-9

shows local turbulent velocity fluctuations measured from the bottom wall to the top wall.

31

0

0.02

0.04

0.06

0.08

0.1

0.12

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

Turb

ulen

t Vel

ocity

(m/s

)

1 0-1

Flowy

D = 50.8 mmV = 0.78 m/s

Figure III-9. Experimental measurements of local turbulent velocity fluctuations (Burden, 1999)

Iskandrani (2001) used the Hot Film Anemometry technique to study the flow

structure of horizontal bubbly flow. The experimental work was conducted in horizontal flow

loop of 50.3 mm ID with two-phase flow of air and water. A portable hot film probe was

installed vertically to obtain local measurements of turbulent velocity fluctuations (u′) in the

liquid phase. The experiment was conducted at a fixed liquid superficial velocity (VSL) of 5

m/s and a range of gas volume fraction that varies from 0% to 13.8%. Figure III-10 shows

the average local turbulent velocities for all gas fractions. The blue bottom curve corresponds

to the base line case of single-phase liquid and the next upper curve corresponds to the next

higher gas fraction. The experiment shows that turbulent velocity fluctuations in the liquid

phase increase as the gas fraction increases.

32

0.00

0.20

0.40

0.60

0.80

-1.00 -0.50 0.00 0.50 1.00

Probe Location

Liqu

id T

urbu

lent

Vel

ocity

(m/s

)

0% Gas4.7% Gas9.1% Gas13.8% Gas

1 0-1

Probe 1 0-1

Probe

D = 50.3 mm Vsl = 5 m/s

Flow

Figure III-10. Experimental measurements of local turbulent velocity fluctuations in the liquid phase of horizontal bubbly flow (Iskandrani, 2001)

Lewis et al. (2002) used the experimental facility mentioned above (Iskandrani 2001)

to study internal flow structure of slug flow in a horizontal pipe. Lewis et al. conducted the

experiment at a fixed liquid superficial velocity (VSL) of 1.65 m/s and range of gas volume

fractions that varies from 0% to 57%. Figure III-11 shows the average local turbulent

velocity fluctuations for all gas fractions. The bottom curve with “0% gas” corresponds to the

base line case of single-phase liquid and the next upper curve corresponds to the next higher

gas fraction. The experiment shows that turbulent velocity fluctuations in the liquid phase

increase as the gas fraction increases.

33

0.00

0.15

0.30

0.45

0.60

0.75

-1.0 -0.5 0.0 0.5 1.0

Probe Location

Turb

ulen

t Vel

ocity

(u- m

/s)

0% Gas14% Gas25% Gas40% Gas50% Gas57% Gas

10-1

Probe

Flow

D = 50.3 mmVsl = 1.65 m/s

Figure III-11. Experimental measurements of local turbulent velocity fluctuations in the liquid phase of horizontal slug flow (Lewis, 2002)

34

CHAPTER IV

MECHANISTIC MODELING

Significant experimental data and correlations for determining the critical deposition

velocity have been provided in the literature. Most of the earlier works presented in the

literature are empirical and some do not account for the effects of fluid properties and other

factors contributing to particle deposition velocity. Thus, mechanistic models are sought

which preserve some physical aspects of the problem. As an initial approach in this work, a

horizontal two-phase model developed by Oroskar and Turian (1980) has been adopted.

Some modifications are done to the original model in order to improve it and extend its

capability. A new mechanistic model has been developed in this work based on a force

balance and turbulent theory. The detailed development of the initial and new models is

provided in this chapter.

Initial Approach

An initial approach for predicting critical deposition velocity is developed based on

an existing model for horizontal two-phase (solid-liquid) flow. The original model was

developed for high concentration slurries by Oroskar and Turian (1980). To meet the

objective of this study, the model is extended to low concentration slurries, inclined flow,

and multiphase flow. An overview of the original model and all modifications made are

shown in this section.

35

Original Model (Oroskar and Turian, 1980)

This section provides an overview of the original Oroskar and Turian (1980) critical

deposition velocity model. For more detailed analysis of the derivation, the original work

should be consulted.

Oroskar and Turian, in the context of slurry transport in pipes, assumed that only

eddies having turbulent fluctuating velocities equal or greater than the settling velocities of

the particles are effective in maintaining the particles suspended in the flowstream.

The Oroskar and Turian critical deposition velocity expression is obtained from three

steps. First, the required turbulent energy necessary for keeping the particles suspended is

evaluated. Second, the turbulent energy generated by the flow is also evaluated. And third,

the required and the generated energies are considered to obtain an expression for the critical

deposition velocity. The major assumptions used to develop the model are:

1. An individual particle is kept suspended by a turbulent eddy. In this process, the

eddy energy is dissipated by the work performed by the drag force upon the particle.

2. The rate of dissipation of energy is the product of the drag force and the settling

velocity of the particles.

3. The particles are spherical and uniform in size and density.

4. The concentration of particles in the pipe cross section is uniform.

5. The presence of other particles affects the settling velocity of an individual particle.

In the Oroskar and Turian model, based on the work by Maude and Whitmore

(1958), the hindered settling velocity was calculated through multiplication of the

settling velocity by the factor (1-C)n, where “C” is the mean volumetric

concentration of particles across the open area over the particle bed and “n” is a

36

function of particle Reynolds number. However, other mechanistic models for

hindered settling velocity, such as the one developed by Li (2003) can be considered.

6. The drag force exerted by an eddy upon a particle performs work during the lifetime

of that eddy. Equivalently, the drag force acts over a distance equal to the eddy

length scale.

7. The turbulent fluctuations are nearly isotropic. The time-averaged eddy length, ℓe,

can be readily evaluated for this condition based on classical turbulent flow theories.

8. Only a portion of the total available turbulent energy is assumed to be consumed to

bring the particles into suspension, in the favorable direction (i.e., normal to the wall

and upward).

9. The root-mean square time-averaged turbulent fluctuation is estimated through the

use of the friction velocity for smooth pipes and Blasius equation for friction factor.

10. For a particle settling in an infinite, quiescent fluid, the value of the drag force is

equated to the net result of the gravitational and buoyant forces.

11. Only eddies having fluctuation velocities equal or greater than the settling velocities

of the particles are effective in maintaining the particles suspended.

With the above assumptions and simplifications, the final form of the critical

deposition velocity is given by Oroskar and Turian (1980) as:

1588/1

12 1)1()()1(5

)1( ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−=

−−

ξµ

ρ sgdD

dDCC

sgdV p

p

n

p

D (IV-1)

where


37



s : ratio of particle density to fluid density, dimensionless

C : volumetric concentration of particles, dimensionless

n : exponent of the hindered settling term, dimensionless


ρ : fluid density, kg/m3

µ : fluid viscosity, kg/m.s

ξ : defined by Equation (IV-2), ξ = 1 is used, dimensionless

ξ is the fraction of turbulent eddies having velocity greater than or equal to the

settling velocity of the particles. The term ξ in Equation (IV-1) is calculated by Oroskar and

Turian (1980) by assuming it is similar to the distribution of molecular velocities as derived

from the kinetic theory of gases, from which:

⎭⎬⎫

⎩⎨⎧ ∞+= ∫

−−

γ γγππ

ξπγπγ de22 /242 /4 e (IV-2)

where γ = V Vs D is the ratio of the particles slip velocity (flow velocity minus sand particle

velocity) and critical deposition velocity. The integral in the above equation is related to the

error function and is easy to evaluate. Oroskar and Turian (1980), based on experimental

data, found ξ close to unity (ξ>0.95). Thus a provisional value of unity is used for ξ in the

present work.

Using the form of Equation (IV-1) as a basis, Oroskar and Turian (1980) changed the

exponents and coefficients in Equation (IV-1) to fit 357 data points (for high concentration

slurries) by regression to obtain:

38

( ) 30.009.0Re

378.03564.01536.0 ~185.1

)1(ξN

Dd

CCsgd

V p

p

D−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

− (IV-3)

where

µ

ρ )s(gdDN~ p

Re

1−= (IV-4)

is a modified Reynolds number.

Extension of Initial Approach to Low Concentration Slurries

The relation obtained by Oroskar and Turian was initially evaluated in this work by

comparison to experimental data gathered by Roco (1991) for sand in water. Figure IV-1

shows a comparison between the predicted values of the critical deposition velocity using the

original “mechanistic” model developed by Oroskar and Turian (Equation (IV-1)) and

experimental data gathered by Shook and Roco. The predictions do not agree well with the

experimental data. Furthermore, the effect of the sand size on the critical deposition velocity

that is observed in the data is not reproduced by the model. In fact, a closer examination of

Equation (IV-1) indicates that the model predicts that the critical deposition velocity is

independent of the sand size. The small difference between the predicted settling critical

deposition velocities for larger and smaller particles shown in Figure IV-1 is due to the fact

that mass and volume concentration of particles are not linearly proportional. Note the

abscissa in Figure IV-1 is mass concentration, and the concentration used in Equation (IV-1)

is the volume concentration defined by:

MixtureofVolume

SandofVolumeC =

39

Predictions of the critical deposition velocity using the Oroskar and Turian

correlation, Equation (IV-3), are also compared with Roco’s data as shown in Figure IV-2. It

is observed that the predicted values are not significantly better than the original formula,

Equation (IV-1), and the effect of sand size observed from data is not predicted.

0

2

4

6

8

10

12

0 10 20 30 40 50 6

Sand Mass Fraction (%)

Crit

ical

Dep

ositi

on V

eloc

ity (f

t/s)

0

Original Oroskar & Turian360 µm sand

Original Oroskar & Turian78 µm sand

Data, 360 µm sand

Data, 78 µm Fly Ash

Pipe Diameter = 100 mm

Figure IV-1. Comparison of the Oroskar and Turian original formula (Equation (IV-1)) with experimental data (Roco, 1991)

40

0

2

4

6

8

10

12

0 10 20 30 40 50 60

Sand Mass Fraction (%)

Crit

ical

Dep

ositi

on V

eloc

ity (

ft/s

)

Oros kar & Turian correlation78 µm Fly A sh

Oros kar & Turian correlation360 µm Sand

Data, 360 µm Sand

Data, 78 µm Fly A sh


Figure IV-2. Comparison of the Oroskar and Turian correlation (Equation (IV-3))

with experimental data (Roco, 1991)

Although the Oroskar and Turian correlation is based on a mechanistic model, these

results clearly indicate that some of the assumptions and simplifications used in the model

may not accurately account for the effects of sand size on the critical deposition velocity.

Thus, in the present work as a first attempt, corrections were made to the Oroskar and Turian

model to obtain good agreement with Roco’s data. The starting point is the original relation

given by Equation (IV-1). Equation (IV-1) is modified by fitting it to the experimental data

by making two modifications. The first modification is to adjust the exponent of volumetric

concentration, C, in Equation (IV-1) to fit Roco’s data set for the 360 micron sand for less

than 20% concentration. The term (1-C)2n-1 which also involves C is not considered because

this term may not be significant for low concentration slurries considered in this analysis.

41

The second modification is an ad hoc correction factor to account for the effects of

sand size on the critical deposition velocity. This is done by multiplying the modified

relation by the factor (dp/do)b. The exponent b was chosen by fitting the equation to Roco’s

data. The corrected or modified Oroskar and Turian relation is then:

b

o

pap

pp

D

dd

CsgdD

dD

sgdV

)(1)1()(5

)1(

1588/1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

− ξµ

ρ (IV-5)

where

a = 0.31

b = 0.60

do = 360 µm

Figure IV-3 shows a comparison of the results obtained from the modified Oroskar

and Turian model, Equation (IV-5), with experimental data of Roco (1991). The agreement

between the model and data is good.

42

0

2

4

6

8

10

0 5 10 15 20Sand Mass Fraction (%)

Crit

ical

Dep

ositi

on V

eloc

ity (f

t/s)

Data

Data

Prediction

Prediction

300 µm Sand

78 µm Fly Ash


Figure IV-3. Comparison of predicted critical deposition velocity using the initial two-phase model (Equation (IV-5)) and experimental data (Roco, 1991)

Extension of Initial Approach to Inclined Flow

The Oroskar and Turian model for computing the critical deposition velocity was

developed for horizontal pipes. For extended reach pipes (inclined wells) the model is

extended to account for inclination angle. An approach used by Campos et al. (1994) is used

in the present study. The extension to inclined pipes is accomplished by modifying the

gravitational term in Equation (IV-5) to obtain:

b

o

pap

pp

D

dd

CsdSingD

dD

sdSingV

)(1)1()()(5

)1()(

1588/1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

− ξµ

θρ

θ (IV-6)

43

where


g : gravitational acceleration, m2/s

dp : particle diameter or sand size, m

s : ratio of particle density to fluid density, dimensionless




µ : fluid viscosity, kg/m.s

do : reference sand size, do = 360 µm

a : an empirical constant based on Roco’s data, a= 0.31, dimensionless

b : an empirical constant based on Roco’s data, b= 0.60, dimensionless

θ : inclination angle measured from vertical (θ=90o is horizontal), degree

ξ : defined by Equation (IV-2), ξ = 1 is used, dimensionless.

The model predictions for small inclination angles (measured from horizontal) are

shown in Figure IV-4 along with experimental data obtained by Roco (1991) for sand in

water with a sand size of 360 µm. The agreement between the model and predictions is fairly

good. Experimental data and the predictions indicate that the critical deposition velocity

slightly decreases as the inclination angle (from horizontal) is increased. The negative

inclination angle indicates a downward flow.

44

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30Inclination Angle (θ)

Crit

ical

Dep

osition

Vel

ocity

(ft/s

)

θ

4” Pipe 360 µm sand

4” Pipe

Sand-W


2% Sand Mass FractionData

Data

Prediction

Prediction

Figure IV-4. Critical deposition velocity predictions by Equation (IV-6) versus experimental data at different inclination angle, sand in water (Roco, 1991)

Extension of Initial Approach to Multiphase Flow

Sand transportation in multiphase flow is a very complex phenomenon. One reason

for the complexity is the change in flow pattern as the gas and liquid rates vary in the pipe.

Each flow pattern may have a different sand transportation mechanism. For example, the

sand transport mechanism in intermittent flow is different from sand transport in stratified or

annular flow. In order to deal with this complexity an assumption was made. It is assumed

that for multiphase flow regimes such as intermittent, stratified or bubbly, sand particles are

entrained within the liquid phase. This means that particle suspension can be calculated as a

function of liquid phase velocity. Therefore, a simplified approach has been developed based

on this major assumption. The approach consists of two steps. In the first step, the multiphase

45

flow (gas-liquid-solid) is modeled as two separate layers, namely, gas layer and liquid-solid

layer. In the second step, the initial approach is applied to the two-phase (liquid-solid) layer.

The first step is done by using a two-phase (gas-liquid) flow prediction model. In this work, a

two-phase model developed by Tulsa University Fluid Flow Project (2002) is used. The two-

phase flow model is used to calculate the liquid film velocity and hydraulic diameter. Then,

the initial approach (discussed above, see Equation (IV-5)) is used to calculate the critical

sand concentration using the calculated liquid film velocity and hydraulic diameter.

New Approach

This section describes the development of the new model for predicting critical

deposition velocity in two-phase (gas-solid or liquid-solid) and multiphase (gas-liquid-solid)

flows. The new model is developed based on force balance and turbulent theory approach

used by Davies (1987) and Oroskar and Turian (1980). The main assumption made in this

approach is that particles are lifted by the vertical turbulent velocity fluctuation, v′, generated

by the flow.

Development of New Approach for Two-Phase Flow

This model is developed in three steps. First, the required turbulent velocity

fluctuation necessary for keeping the particles suspended is evaluated. Second, the turbulent

velocity fluctuation generated by the flow is also evaluated. And third, the required and the

generated turbulent velocity fluctuation are considered to obtain an expression for the critical

deposition velocity.

46

In the first step, a relation developed by Davies (1987) is used after modification to

calculate the required turbulent velocity fluctuation necessary for keeping particles

suspended. The development of the original Davies’ relation, Equation (II-10), is provided in

Mechanistic Modeling of Two-Phase Flow of Chapter II.

( ) ( ) ( )[ ]21

Required 11820 −−=′ sgdC.v Pn (II-10)

where

v′Required : turbulent velocity fluctuation required for keeping particles in suspension, m/s





n : exponent in the hindered settling term, dimensionless

In Equation (II-10), Davies used the hindered settling effect factor of (1-C)n to

account for the change in particle settling velocity due to the presence of other particles. This

assumption is acceptable at high concentrations (>3%). However at low concentrations, <3%,

hindered settling is not the dominant effect in particle settling as observed by Li (2003). At

concentrations below 3%, the cluster effect controls the mechanism of particle settling. For

detail, refer to Concentration Effect in Particle Settling of Chapter II. Based on this

argument, Equation (II-10) is re-written as

( ) ( ) ( )[ ]21

Required 1820 −=′ fp sgdCf.v (IV-7)

where

47

f (C) = hindered settling = (1-C)n , C > 3% (Maude, 1958)

f (C) = cluster effect = -2222C2 + 66.7C + 1, C < 3% (Li, 2003)

The densities ratio S, given in Equation (II-10), is replaced with Sf to extend the

application of Equation (IV-7) to gas and liquid flows.

In the second step, a relation for calculating the turbulent velocity fluctuation

generated by the flow is developed by consulting the work of Tennekes and Lumley (1972).

Tennekes and Lumley used the frictional velocity (uτ) and experimental data for pipe flow to

approximate the turbulent velocity fluctuation components (u′, v′, and w′) in horizontal pipe.

Directions of the turbulent velocity fluctuation components are shown in Figure IV-5.

Figure IV-5. Turbulent velocity fluctuation components

Due to the big difference in turbulent velocity fluctuations between the wall region

and core region, Tennekes and Lumley (1972) made different approximations in each region.

Based on experimental data for pipe flow, the order-of-magnitude of turbulent velocity

components near the wall are (Tennekes and Lumley, 1972):

τuu 2≅′ (IV-8)

τuv 8.0≅′ (IV-9)

τuw 4.1≅′ (IV-10)

v′Flow

u′w′

48

and the order-of-magnitude of turbulent velocity components in the core region are

(Tennekes and Lumley, 1972):

τuwvu 8.0≅′=′=′ (IV-11)

The above relations for turbulent velocities can be related to mean flow velocity

through the use of frictional velocity (uτ). The frictional velocity is defined as

ρτ

τwu = (IV-12)

where

uτ : frictional velocity, m/s

τw : wall shear stress, kg/s2.m


The wall shear stress, τw, is calculated as a function of flow velocity and friction

factor as shown in Equation (IV-13):

fVw2

21 ρτ = (IV-13)

where f is the Fanning friction factor and is estimated for smooth pipe by the Blasius

equation as:

41Re0791.0 −=f (IV-14)

Considering the frictional velocity Equations (IV-12, IV-13), the turbulent fluctuation

velocities near the wall can be re-written as

81Re4.0 −=′ Vu (IV-15)

81Re16.0 −=′ Vv (IV-16)

49

81Re28.0 −=′ Vw (IV-17)

and the turbulent fluctuation velocities in the core region become:

81Re16.0 −=′=′=′ Vwvu (IV-18)

Since particles are lifted by turbulent velocity fluctuations generated in the vertical

direction, Equation (IV-16) will be used for estimating the turbulent velocity fluctuation

generated by the flow. However, Equation (IV-16) is developed for single-phase (liquid, gas)

and does not take into account the presence of solids. Solid concentration has a major effect

in turbulence generation that can not be ignored. The flow is expected to lose some of its

energy by carrying solid particles. Therefore, ignoring the effect of the solids will lead to

overestimating the generated turbulence. Based on this argument, the produced turbulence,

Equation (IV-16), is modified by adding a turbulent dissipation term (Ω) as suggested by

Davies (1987). Another modification is done to Equation (IV-16) by modifying the turbulent

generation constant from 0.16, to 0.18 as it shows better performance with data as it will be

discussed in Chapter V. Therefore, the final form of the produced turbulent velocity

fluctuation is given by:

( ) ( )Ω=′ − 81Produced 180 ReV.v (IV-19)

where,

v′Produced : turbulent velocity fluctuation generated by the flow in (y-direction), m/s

V : flowstream velocity, m/s

Re : Reynolds number, dimensionless

Ω : turbulent dissipation, dimensionless

In the last step, it is assumed that in order to keep particles in suspension, required

turbulence, (u′)Required, should be equal to produced turbulence, (u′)Produced. Therefore, the final

50

form of the critical deposition velocity can be obtained by equating Equation (IV-7) and

Equation (IV-19) then rearranging to get

( ) ( )[ ]787178 11665 ⎟

⎠⎞

⎜⎝⎛Ω⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

µρL

fpDD

sgdcf.V (IV-20)

To account for turbulent dissipation (Ω), a function that was suggested by Davies

(1987) is used for high concentrations (>> 1%):

3.64C11

+=Ω (IV-21)

However at concentration around 1% and lower, Davies function does not give good results

as will be shown in Chapter V. A new function is used to show better results at low

concentrations as given by Equation (IV-22):

( )3.64C1501+

=Ω.

(IV-22)

Extension of New Approach to Multiphase Flow

The new model is extended to multiphase flow by making two assumptions. The first

assumption is that for multiphase (gas-solid-liquid) flow, all solid particles are entrained

within the liquid phase. The second assumption is that the required turbulent velocity

fluctuation necessary for keeping the particles suspended in multiphase flow is the same as

that for two-phase (gas-solid, liquid-solid) flow. Therefore, the earlier relation developed for

calculating the required turbulent velocity fluctuation in two-phase flow, Equation (IV-7),

can be applied to the liquid phase of the multiphase flow. Therefore, the required turbulent

velocity fluctuation necessary for keeping the particles suspended in multiphase flow is given

by the following equation:

51

( ) ( ) ( ) LLsp /gdcf.v ρρρ −=′ 820Required (IV-23)

In the next step, a relation for calculating the turbulence generated by the liquid phase

in multiphase flow is developed. Developing a relation for calculating produced turbulence in

multiphase flow is much more complicated than two-phase flow because for two-phase flow

(gas-solid, liquid-solid), there is one major source of turbulence which is the carrier flow

velocity while for multiphase flow, there are at least three sources of turbulence. The

turbulence in the liquid phase of multiphase flow is generated by liquid phase velocity, gas

phase velocity and gas void fraction as shown in Equation (IV-24).

(v′L)Produced = (v′L)VL + (v′L)VG + (v′L)αG (IV-24)

Where

(v′L)Produced : produced turbulence within the liquid phase, m/s

(v′L)VL : liquid phase turbulence generated by the liquid phase velocity, m/s

(v′L)VG : liquid phase turbulence generated by the gas phase velocity, m/s

(v′L)αG : liquid phase turbulence generated by the entrained gas, m/s

To predict the overall turbulence generated in the liquid phase, the contribution of all

turbulence sources should be considered. The turbulence generated by the liquid phase

velocity can be simply predicted by the earlier developed relation (Equation (IV-19)).

However, the real challenge is to predict the turbulence generated by the gas phase velocity

and gas void fraction.

Because of the lack of development in the area of multiphase turbulence, an empirical

approach is used in this study to predict the liquid phase turbulence generated by the gas

52

phase velocity and gas void fraction. Experimental data collected by Lewis et al. (2002) is

used to develop a relation for predicting the change in liquid phase turbulence due to the

presence of the gas phase. Lewis et al. (2002) measured the liquid phase turbulence at a fixed

liquid superficial velocity (VSL) of 1.65 m/s and range of gas volume fractions that varies

from 0% gas to 57% as shown in Figure III-11. A relation for predicting the change in liquid

phase turbulence as a function of gas fraction is developed in two steps. In the first step, the

average liquid turbulent velocity fluctuation of each case is calculated, and then the change in

average turbulence of each case relative to the base line case (0% gas) is calculated. The

change in liquid phase turbulent velocity fluctuation at different gas fractions is summarized

in Figure IV-6.

0

100

200

300

400

500

0 10 20 30 40 50 60

Gas Volume Fraction (%)

Cha

nge

in tu

rbul

ence

(%)

Diameter = 50 mmVSL = 1.65 m/s

Figure IV-6. Change in liquid phase turbulent velocity fluctuation at different gas fractions as calculated from Lewis data presented in Figure III-11. (Lewis et al., 2002)

In the second step, a relation for predicting the change in turbulent velocity

fluctuation as a function of gas fraction is obtained by fitting the data shown in Figure IV-6

53

to an exponential function that is given by Equation (IV-25). Caution should be taken in

applying this relation to other conditions as it is limited to the experimental condition of slug

flow.

ΦSlug = 20.83 Exp(0.05 Gas Fraction) (IV-25)

where ΦSlug is the percentage change in average turbulent velocity fluctuation as a function

of gas fraction in percent.

Following the same procedure discussed above, another relation is developed for

bubbly flow. Equation (IV-26) predicts the change in turbulence velocity as a function of gas

fraction for bubbly flow. The equation is developed based on experimental data gathered by

Iskandrani (2001). The experimental data was shown earlier in Figure III-9.

ΦBubbly = 5.7 (Gas Fraction) (IV-26)

The next step is to add the gas effect function, Φ, to Equation (IV-26) to obtain the

final form for produced turbulence as given by Equation (IV-27)

(v′L)Produced = (v′L)VL + Φ (v′L)VL (IV-27)

where ( ) ( )Ω=′ − 81160 ReV.v SLVLL (IV-19)

The final form for predicting the minimum liquid superficial velocity needed to keep

sand in suspension is obtained by equating the required turbulence, Equation (IV-23), with

produced turbulence, Equation (IV-27) and then rearranging to obtain

( ) ( )[ ] ( )( )

787178

111665 ⎟⎟

⎠

⎞⎜⎜⎝

⎛Φ+Ω⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

L

LpSL

Dsgdcf.V

µρ

(IV-28)

where

54

VSL : liquid superficial velocity, m/s



s : ratio of sand density to liquid density, dimensionless



f(c) : concentration effect in particle settling, Equations (II-22 or II-23) , dimensionless


µL : liquid viscosity, kg/m.s

Φ : empirical function accounts for the presence of gas phase, Equation (IV-25, IV-26),

dimensionless

Ω : turbulent dissipation given by Equations (IV-21 or IV-22), dimensionless

The new approach is developed in a mechanistic manner that involves a

representation of the physics of particle suspension. The new approach is developed based on

an assumed mechanism of particle suspension. It is assumed that for particles to be

suspended, flow inside the pipe has to be turbulent and turbulent eddies tend to keep sand

particles in suspension while they are transported by the mean flow velocity. This assumption

is translated into a model by applying a force balance and turbulent theory approach. It is

shown in this chapter that this approach introduces all major parameters involved in particle

suspension, namely, particle properties (such as density, size, and concentration), fluid

properties (such as density and viscosity), flow properties (such as velocity, and turbulence

level), and pipe parameters (such diameter and friction factor). The turbulent theory accounts

55

for fluid properties, flow properties and pipe parameters while the force balance accounts for

particle properties. More thought is put toward particle concentration as the effects of particle

concentration on turbulence and in settling velocity are addressed in this approach.

56

CHAPTER V

RESULTS AND DISCUSSION

This chapter consists of three parts. In the first part, the initial and the new two-phase

models are validated by comparing model predictions with experimental data collected from

the literature. Model predictions are compared to fourteen sets of two-phase (sand-water)

experimental data. In the second part, predictions made by the initial and the new multiphase

approaches are compared to multiphase (sand-water-air) data. In the third part, the

correlations developed for predicting turbulent velocity fluctuations in single-phase is

evaluated by comparing predictions with experimental measurements of turbulent velocity

fluctuations in flows of water. Each part will be followed by a discussion.

Validation of Two-Phase Critical Deposition Velocity Models

The initial and the new models are validated in this section by comparing the models’

predictions with experimental measurements of critical deposition velocity in two-phase

(sand-water) flow. A wide range of sand size and concentration is considered in this

comparison. The comparison is divided into three sections based on sand size. The three

sections are small size (78 µm - 230 µm), medium size (360 µm – 585 µm), and coarse size

(600 µm – 1150 µm). In order to compare the performance of the models developed in this

study to existing models in the literature, two additional models are considered in this

comparison. The additional models are the original Oroskar and Turian (1980) and Davies

(1987). In summary, the following four models are considered in this comparison:

57

1. Initial Two-Phase Approach:

b

o

pap

pp

D

dd

CsgdD

dD

sgdV

)(1)1()(5

)1(

1588/1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

− ξµ

ρ (IV-5)

2. New Two-Phase Approach:

( ) ( )[ ]787178 11665 ⎟

⎠⎞

⎜⎝⎛Ω⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

µρL

fpDD

sgdcf.V (IV-20)

where,

( ) ( )nCcf −== 1settling hindered , for (C > 3%) (II-22)

( ) 17662222Effectcluster 2 ++−== C.Ccf , for (C < 3%) (II-23)

3.64C11

+=Ω , for (C >> 1) (IV-21)

( )3.64C10.51+

=Ω , for (C <~ 1) (IV-22)

3. Davies (1987)

( ) ( ) ( )[ ] 46.054.018.009.055.009.1 12164.3108.1 DsgdCCV Pn

D −−+= −ν (II-14)

4. Oroskar and Turian (1980)

( ) 30.0

09.0378.0

3564.01536.0

)1(185.1

)1(ξ

µ

ρ

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−

− sp

gdD

Dd

CCsgd

V p

p

D (IV-3)

where


g : gravitational acceleration, 9.81 m2/s


58

s, sf : ratio of sand density to water density, 2.65 (dimensionless)



do : reference sand size, do = 360 µm

a : an empirical constant, a= 0.31 (dimensionless)

b : an empirical constant, b= 0.60 (dimensionless)

n : exponent in the hindered settling term, use 0.15 for “new approach”, and 4 for

“Davies model”, dimensionless

ρ, ρL : water density, 998 kg/m3

µ : water viscosity, 0.00098 kg/m.s (or Pa.s)

ξ : defined by Equation (IV-2), ξ = 1 is used, dimensionless

Ω : turbulent dissipation, dimensionless

υ : kinematic viscosity of water, 1 x 10-6 m2/s

Small Sand Size (78 µm - 230 µm)

The smallest particle size used in this comparison is 78 µm for fly Ash (Roco 1991).

The fly ash density used for making predictions is 2300 kg/m3. At this particle size, the initial

approach shows excellent agreement with data as shown in Figure V-1. Predictions made by

the new approach show fare agreement with the experimental data while predictions made by

Davies’ and Oroskar and Turian’s models were less satisfactory. Davies’ and Oroskar and

Turian’s models overpredict the critical deposition velocity

59

0.0

0.3

0.6

0.9

1.2

1.5

1.8

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

Fly Ash Volume Concentration

Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

Exp. (Roco, 1991)Initial Approach

Oroskar & Turian, 1980New Approach

Davies, 1987

Sand Size: 78 µmPipe Size : 100 mm

Figure V-1. Comparison between critical deposition velocity predictions made by initial approach, new approach, Oroskar and Turian (1980), and Davies (1987) with

experimental data gathered by Roco (1991) for 78 µm fly ash

For sand sizes of 180 µm and 190 µm, all models perform well with the lowest

performance shown by Davies’ model as shown in Figures V-2 and V-3. Davies’ model over

predicts the critical deposition velocity. At larger sand size of 230 µm, also all models

perform well with lowest performance shown by the new approach as shown in Figure V-4.

The new approach under estimates the critical deposition velocity.

60

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0% 5% 10% 15%

Sand Volume Concentration

Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

Exp. (Parzonka, 1981)Initial Approach


Davies, 1987



experimental data gathered by Parzonka (1981) for 180 µm sand

0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

0% 5% 10% 15% 20% 25% 30%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

Exp. (Parzonka, 1981)

Initial ApproachOroskar & Turian, 1980

New ApproachDavies, 1987




61

0.0

0.5

1.0

1.5

2.0

2.5

0% 5% 10% 15% 20% 25%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

Exp. (Mehmet, 2001)Initial Approach

Oroskar & Turian, 1980 New Approach

Davies, 1987


Figure V-4. Comparison between critical deposition velocity predictions made by initial approach, new approach, Oroskar and Turian (1980), and Davies (1987) with experimental data gathered by Kokpinar and Gogus (2001) for 230 µm sand

Medium Sand Size (360 µm – 585 µm)

Figure V-5 shows a comparison between model predictions and Roco’s data for 360

µm sand that was used in the development of the initial approach. As expected, the initial

approach shows the best performance because it was modified to fit with Roco’s data. The

rest of the models show unsatisfactory performance. In order to investigate findings of Figure

V-5, models are compared to another set of data with sand size and pipe diameter close to

Roco’s data. Experimental data reported by Parzonka (1981) for sand size of 400 µm and

pipe size of 103 mm is used as shown in Figure V-6. The comparison shows reasonable

performance of all models with exception of the initial model. The same findings are found at

larger particle sizes as shown in Figures V-7, through V-12.

62

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0% 5% 10% 15% 20%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

Exp. (Roco, 1991)Initial ApproachOroskar & Turian, 1980




experimental data gathered by Roco (199) for 360 µm sand

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0% 10% 20% 30%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)


Initial ApproachOroskar & Turian, 1980





63

0.0

1.0

2.0

3.0

4.0

5.0

0% 5% 10% 15%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

Exp. (Parzonka, 1981)Initial Approach

Oroskar and Turian, 1980New approach

Davies, 1987




0.0

1.0

2.0

3.0

4.0

5.0

0% 5% 10% 15% 20%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

Exp. (Parzonka, 1981)Initial ApproachOroskar & Turian, 1980New ApproachDavies, 1987




64

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0% 2% 4% 6% 8%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

Exp. (Mehmet 2001)

Initial Approach

New Approach

O

D

Sand Pipe S

Figure V-9. Comparison between critical deposition velocity predapproach, new approach, Oroskar and Turian (1980), and Dexperimental data gathered by Kokpinar and Gogus (2001)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0% 1% 2% 3% 4% 5% 6%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

EInO

ND

Sand Pipe Si

Figure V-10. Comparison between critical deposition velocity pinitial approach, new approach, Oroskar and Turian (1980), an

experimental data gathered by Kokpinar and Gogus (2001)

65

roskar & Turian, 1980

avisOroskar & Turian, 1980

Davies, 1987

Size: 450 µmize : 102 mm

ictions made by initial avies (1987) with

for 450 µm sand

xp. (Mehmet, 2001)itial Approachroskar & Turian, 1980

ew Approachavies, 1987

Size: 450 µmze : 152 mm

redictions made by d Davies (1987) with for 450 µm sand

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0% 5% 10% 15% 20%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)


Initial Approach

Oroskar & Turian, 1980

New approach

Davies, 1987

Sand Size: 570 µmPipe Size : 50.8 mm



0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0% 5% 10% 15% 20% 25% 30%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

Exp. (Mehmet, 2001)Initial ApproachOroskar & Turian, 1980 New ApproachDavies, 1987

Sand Size: 585µmPipe Size : 108 mm


experimental data gathered by Kokpinar and Gogus (2001) for 585 µm sand

66

Coarse Sand Size (600 µm – 1150 µm)

Although the coarse sand size of (600 µm – 1150 µm) is outside the area of interest in

this research, it was included to provide an overview of the models performance for a broad

range of particle sizes. The Models continue the same performance observed at the medium

sand size where all models show reasonable performance with exception of the initial model

as shown in Figure V-13 and V-14.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0% 1% 2% 3% 4% 5% 6%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s)

Exp. (Mehmet, 2001)Initial ApproachOroskar & Turian, 1980New ApproachDavies, 1987




67

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0% 5% 10% 15% 20%


Crit

ical

Dep

ositi

on V

eloc

ity (m

/s) Exp. (Mehmet, 2001)

Initial Approach


Davies, 1987




Discussion of Two-Phase Critical Deposition Velocity Results

Three major observations can be made based on the above comparisons. First, the

unsatisfactory performance of the initial approach for medium and large particle sizes.

Second, the relatively good performance obtained by the new approach over a broad range of

particle sizes. Third, the range of sand concentrations of most of the data used in this section

is much higher than concentrations encountered in the oil and gas industry.

The satisfactory performance obtained by the new approach over a broad range of

particle sizes indicates that applying turbulent theory is a viable mechanistic approach in

modeling sand deposition.

The initial approach, Equation (IV-5), shows good performance at small particle sizes

(below 250 µm) only. At medium and large particles sizes, the initial approach shows low

68

performance that is even much lower than the performance shown by the Oroskar and Turian

correlation (Equation (IV-3)). This unsatisfactory performance could be attributed to the

experimental data used initially to refine the model. In the early stages of this study, very

limited data was available for validation. Only two sets of data gathered by Roco (1991) for

360 µm sand and 78 µm fly ash were used initially. The original Oroskar and Turian formula,

Equation (IV-1), was initially evaluated and then modified based on Roco’s data for 360 µm

sand. However, in later stages of this study more experimental data from several sources

were considered. The comparison with the available data shows that the original Oroskar and

Turian correlation ((Equation (IV-3) gives better results than the modified one developed

initially (Equation (IV-5)). It is known that measurements of critical velocity may be affected

by various experimentalist judgments and observations. Therefore, it is recommended to use

the Oroskar and Turian correlation, Equation (IV-3), over the initial approach which only

was refined with one set of experimental data. The Oroskar and Turian correlation shows

good results as good as those obtained by the new model for a variety of experimental data

and con

o far for low concentrations that fall within the area of interest in oil and gas

industr

ditions.

Particle concentration for most of the data used in this study is much higher than

those encountered in the oil and gas industry. This happen because all data used in this

section came from studies interested in high concentration slurries. No data were found in the

literature s

y.

A close examination of the models shows that they have different level of

dependency on pipe diameter. The initial approach has the lowest level of dependency with

exponent value of 0.14, while the new approach has the highest level of dependency with

69

exponent value of 0.6. Davies’s and Oroskar and Turian formulas have equivalent level of

dependency with exponent value of 0.46. With this variation on diameter’s exponent values,

it is important to conduct sensitivity analysis for the pipe diameter. However, the available

data is not comprehensive enough to meet this objective. The only two sets of data that can

be considered are shown in Figures V-9 and V-I0 with pipe diameters of, respectively, 102

mm and 152 mm. The experimental data shows that the trend of critical deposition velocity

shifts up by almost 0.5 m/s as a result of increasing the pipe diameter from 102 mm to 152

mm. Predictions made by all models show relatively close changes to the experimental data

with exception of the initial approach that show very small shift in the trend.

els

ict es are used to make the predictions:

d size, 150 x 10-6 m

2.65 (dimensionless)

Validation of Multiphase Critical Deposition Velocity Models

The experimental work in the area of multiphase (solid-liquid-gas) is very limited and

most of it does not provide direct measurements of critical deposition velocity. The only

multiphase critical deposition velocity measurements found in the literature were collected

by Oudeman (1993). Oudeman collected experimental measurements of sand concentrations

in a flow of air, water and sand. The experiments were conducted for three cases: (a) single-

phase water, (b) water and 10% by volume air, (c) water and 20% by volume air. The initial

and the new multiphase approaches are validated in this section by comparing mod

pred ions with Oudeman’s data. The following valu

dp : particle diameter or san

D : pipe diameter, 0.07 m

s : ratio of sand density to water density,

g : gravitational acceleration, 9.81 m2/s

70

do : reference sand size, do = 360 x 10

n settling term, 0.15 (new approach)

ρL

ξ defined by Equation (IV-2), ξ = 1 is used, dimensionless

Initial

, the measured

sand concentration is 1.6 % while the predicted concentration is only 0.6%.

-6 m

a : an empirical constant, a= 0.31 (dimensionless)

b : an empirical constant, b= 0.60 (dimensionless)

: exponent in the hindered

ρ, : water density, 998 kg/m3

µ : water viscosity, 0.00098 Kg/m.s (or Pa.s)

:

Multiphase Approach

Figure V-13 shows a comparison between Oudeman’s data and predictions made by

the initial approach. For case (a), single-phase (liquid), the model shows reasonable

agreement with experimental data. However, for the multiphase cases, predictions are

unsatisfactory. The initial approach significantly under estimates the effect of the gas. For a

liquid superficial velocity of 0.55 m/s, for example, and gas fraction of 20%

71

0.3

0.4

0.5

0.6

0.7

0.8

0 0.3 0.6 0.9 1.2 1.5 1.8

Sand Volume Concentration (%)

Liqu

id S

uper

ficia

l Vel

ocity

(m/s

)

20% Exp20% Model10% Exp10% Model0% Exp0% Model

dp = 150-300 µmD = 70 mm

Figure V-15. Comparison between Oudeman’s data (1993) and critical deposition velocity predictions made by the initial approach

New Multiphase Approach

The new multiphase approach is given by the following equation:

( ) ( )[ ] ( )( )

787178

111665 ⎟⎟

⎠

⎞⎜⎜⎝

⎛Φ+Ω⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

L

LpSL

Dsgdcf.V

µρ

(IV-27)

where,

( ) ( )nCcf −== 1settling hindered , for (C > 3%) (II-22)

( ) 17662222effect clustering 2 ++−== C.Ccf , for (C < 3%) ( II-23)

3.64C11

+=Ω , for (C >> 1%) (IV-21)

( )3.64C10.51+

=Ω , for (C <~ 1%) (IV-22)

72

ΦSlug = 20.83 Exp(0.05 Gas Fraction) (IV-25)

At the beginning of this study, the new multiphase approach, Equation (IV-27), was

used along with the turbulent dissipation function given by Davies, Equation (IV-21), and

cluster settling effect, Equation (II-23). The comparison between Oudeman’s data and

predictions made by Equation (IV-27) is shown in Figure V-16. Predictions were

unsatisfactory for all the three cases. The new approach was expected to show better results

at least for the two-phase (sand-water) case because it has shown good results in the previous

section for two-phase data. The only major difference between Oudeman’s data and all the

data presented earlier is sand concentration. Oudeman conducted his experiments at very low

sand concentrations that were less than 1% while all other data presented in the previous

section were conducted at much higher concentrations that reach up to 30%. This means that

the new approach does not address the effect of sand concentration properly.

0.3

0.5

0.7

0.9

1.1

1.3

1.5

1.7

0.0% 0.3% 0.6% 0.9% 1.2% 1.5% 1.8%


Liqu

id S

uper

ficia

l Vel

ocity

(m/s

)

0% Exp0% Model10% Model10% Exp20% Exp20% Model

dp = 150-300 µmD = 70 mm

73

Figure V-16. Comparison between Oudeman’s data (1993) and critical deposition velocity predictions made by the new multiphase approach, Equation (IV-27), along

with turbulent dissipation function given by Davies, (Equation IV-21)

In order to highlight the effect of sand concentration in sand suspension, the new

multiphase model is considered:

( ) ( )[ ] ( )( )

787178

111665 ⎟⎟

⎠

⎞⎜⎜⎝

⎛Φ+Ω⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

L

LpSL

Dsgdcf.V

µρ

(IV-27)

The particle concentration variable appears in two terms that account for settling

effect, f(c), and turbulent dissipation. The problem is more likely with the turbulent

dissipation term because the function used in this relation was developed by Davies (1987)

for slurry transport. Slurry transport deals with much higher particle concentrations than

those used by Oudeman (1993). So based on this observation, there is a need for a new

turbulent dissipation function that works at concentrations in the neighborhood of 1% or

lower. A new turbulent dissipation function is developed by fitting the new approach’s

predictions with Oudeman’s data. The modified turbulent dissipation function is given in

Equation (IV-22).

The new results made by the modified turbulent dissipation function are shown in

Figure V-17. The results show good agreement with data for the three cases.

74

0.3

0.4

0.5

0.6

0.7

0.8

0.0% 0.3% 0.6% 0.9% 1.2% 1.5% 1.8%


Liqu

id S

uper

ficia

l Vel

ocity

(m

/s)

0% Exp0% Model10% Model10% Exp20% Exp20% Model

dp = 150-300 µmD = 70 mm

Figure V-17. Comparison between Oudeman’s data (1993) and critical deposition velocity predictions made by the new multiphase approach, Equation (IV-27), along

with modified turbulent dissipation function given by Equation IV-22

Discussion of Multiphase Critical Deposition Velocity Results

The major deference between two-phase (solid-liquid) flow and multiphase (solid-

liquid-gas) is the presence of the gas phase. Thus, the key factor in developing a multiphase

approach is accounting properly for the effect of the gas phase in solid suspension. The

unsatisfactory results obtained by the initial multiphase approach revealed that this approach

does not account properly for the effect of the gas phase. Therefore, another approach that

involves better representation of the physics of the problem is sought. Extending the new

two-phase model (Equation (IV-20)) to multiphase flows shows satisfactory results over the

range of available data. The new multiphase approach, given by Equation (IV-27), accounts

empirically for the effect of the gas phase on the liquid phase turbulence.

75

Sand concentration plays a very important role in sand suspension and settling. Sand

concentration appears in two terms in the new multiphase approach (Equation IV-27)). It

appears in sand settling and turbulent dissipation. Both terms can not be modeled with

universal functions over a broad range of sand concentration. Sand settling at high sand

concentrations, say above 3%, is represented by hindered settling while at low sand

concentrations it is represented by the cluster effect. Li’s work (2003) is successfully used in

this work to account for sand settling at low concentrations. Similarly, two different

functions were used in this work to account for turbulent dissipation. At concentrations much

higher than 1%, a function given by Davies (1987), Equation (IV-21) is used while at

concentrations around and below 1% a new function is used (Equation (IV-22)).

There is one major effect of sand concentration that is not considered in this study.

The presence of solid particles does not only cause turbulence dissipation but also it could

cause turbulence generation. Due to slight slippage between the flow and the entrained

solids, the flow becomes more chaotic having higher level of turbulence. With two opposite

effects, there are two possible scenarios: either both effects are equal in magnitude or one

effect is more dominant than the other. If both effects are equal, each one will cancel the

other so there is no effect of concentration in turbulence. However, if one effect is more

dominant, there is a need to identify that effect. However, the data presented in the literature

is not comprehensive to examine such details regarding turbulence structure. The results

presented in this chapter may indicate that at high sand concentrations turbulent dissipation is

more dominant and this is why the turbulent dissipation term given by Davies shows good

results at high concentrations. However, at low concentrations, turbulent generation could be

76

more dominant and this is why a new function was needed to correct for the turbulent

dissipation function that was used at higher concentrations.

Validation of Turbulent Velocity Fluctuation Correlations

Modeling turbulent velocity fluctuations is a major step in the new approach. The

turbulent velocity fluctuation correlation, Equation (IV-19), is used to calculate the generated

turbulence by the flow. Satisfactory turbulent velocity fluctuation predictions should be

assured to obtain a reliable critical deposition model. The developed correlations for

predicting turbulent velocity fluctuation are validated in this section by comparing

predictions with experimental measurements of axial turbulent velocity fluctuation (u′) in the

core region and near wall region. Equation (IV-15) is used for predicting axial turbulent

velocity fluctuation (u′) in the near wall region and Equation (IV-19) is used for predicting

vertical turbulent velocity fluctuations (v′) in the core region. The turbulent dissipation term

of Equation (IV-19) is ignored in this comparison because this term accounts for the presence

of solids and the experimental data used are for single-phase (liquid).

81Re4.0 −=′ Vu (IV-15)

( ) ( )Ω=′ − 81Produced 180 ReV.v (IV-19)

In the core region, the axial turbulent velocity fluctuation (u′) equals the vertical

turbulent velocity fluctuation (v′) as estimated by Tennekes and Lumley (1972). Therefore,

predictions made by Equation (IV-19) are compared to experimental measurements of

turbulent velocity fluctuations in the axial direction. Figure V-18 shows a comparison

between predictions made by Equation (IV-19) and experimental measurements of turbulent

velocity fluctuations in the core region. Figure V-19 shows a comparison between

77

predictions made by Equation (IV-15) and experimental measurements of turbulent velocity

fluctuation near the bottom wall. Each point of the data represents a different experiment.

Three different experiments are used: Iskandrani (2001) conducted the experiment at a liquid

velocity of 5 m/s and pipe diameter of 50 mm, Lewis et al. (2002) conducted the experiment

at a liquid velocity of 1.65 m/s and pipe diameter of 50.3 mm, and Burden (1999) conducted

the experiment at a liquid velocity of 0.78 m/s and pipe diameter of 50.8 mm.

0.00

0.03

0.06

0.09

0.12

0.15

0.18

0 0.03 0.06 0.09 0.12 0.15 0.18

Experimental Turbulent Velocity (u', m/s)

Pre

dict

ed T

urbu

lent

Vel

ocity

(u',

m/s

)

Burden 1999

Iskandarani 2001

Lewis 2002

Figure V-18. Comparison between predictions made by Equation IV-19 and experimental measurements of turbulent velocity fluctuation in the core region

78

0.0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Experimental Turbulent Velocity (u', m/s)

Pre

dict

ed T

urbu

lent

Vel

ocity

(u',

m/s

)

Burden 1999Iskandarani 2001Lewis 2002

Figure V-19. Comparison between predictions made by Equation IV-15 and experimental measurements of turbulent velocity fluctuation near the bottom wall

Discussion of Turbulent Velocity Fluctuation Results

For single-phase flow, the developed relations for predicting turbulent velocity

fluctuation show good results in the near wall region and in the core region. The results show

that the developed relations are able to predict turbulent velocity fluctuation for different

liquid velocities.

For multiphase flow, relations for predicting turbulent velocity fluctuation in bubbly

and slug flow patterns are developed empirically based on experimental data. The developed

relations are not validated further in this work because there are no experimental data

available for comparison. At this stage of work, it is not intended to develop similar relations

for annular or mist flow patterns because critical deposition velocity occurs at low flow

79

velocities where annular or mist flow does not occur. However, stratified flow is to be

considered because sand settling is very likely to occur at low velocities.

80

CHAPTER VI

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

FOR FUTURE WORK

Summary

The critical deposition velocity is the flow velocity required to prevent accumulation

of sand particles on the bottom of a pipe. 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 below

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

particles" or particle bed inside the pipeline or near pipe fittings such as elbows, tees, valves,

and couplings. This accumulation of sand particles can accelerate erosion and corrosion

damage beyond safe operating conditions.

Two models are developed in this work for predicting critical deposition velocity in

two-phase (gas-solid or liquid-solid) and multiphase (gas-solid-liquid) flows. In the first one,

a horizontal two-phase (solid-liquid) model developed by Oroskar and Turian (1980) is

adopted. In the second, a new mechanistic model is developed based on a force balance and

turbulent theory. Both models are evaluated by comparing predictions to experimental data

available in the literature.

As an initial approach in this work, an existing model developed by Oroskar and

Turian (1980) for predicting critical deposition velocity is adopted. Oroskar and Turian

developed their model for horizontal two-phase flow of high concentration slurries. Using the

original model as a basis, Oroskar and Turian (1980) modified the model to fit 357 data

81

points for high concentration slurries. The initial evaluation of the original model and the

modified correlation show that they do not account for particle size and they do not give

good results for low concentration slurries. In order to meet the objective of this study, the

original Oroskar and Turian model is extended to low concentration slurries, inclined flow,

and multiphase flow to obtain, what is called in this study, the Initial Approach. The Initial

Approach is extended to low concentration slurries by fitting the model with Roco’s data

(1991) for 360 µm sand. The extension to multiphase flow is achieved in two steps. In the

first step, the multiphase flow (gas-liquid-solid) is treated as two separate layers, namely, gas

layer and liquid-solid layer. In the second step, the initial approach is applied to the two-

phase (liquid-solid) layer.

The evaluation of the initial approach shows good performance at small particle sizes

(below 250 µm). At medium and large particles sizes, the initial approach shows low

performance that is even much lower than the performance shown by the Oroskar and Turian

correlation. The low performance could be attributed to the experimental data used to

develop the initial approach. At the beginning of this study, very limited data was considered.

Only two sets of data gathered by Roco (1991) for 360 µm sand and 78 µm fly ash were

used. The original Oroskar and Turian formula was initially evaluated and then modified

based on Roco’s data. However, in a later stage of this study more data were considered. The

comparison with the available data shows that the original Oroskar and Turian correlation

gives better results than the modified one in this study. It is known that measurements of

critical velocity may be affected by various experimentalist judgments and observations.

Therefore, it is recommended to use Oroskar and Turian correlation over the initial approach

82

which only was refined with one set of experimental data. The Oroskar and Turian

correlation shows good results over a broad range of particle sizes.

The other mechanistic model developed in this study is referred as the New

Approach. It is developed based on force balance and turbulent theory. The New Approach

for two-phase flow is developed in three steps. First, force balance is applied to develop an

equation for sizing turbulent velocity fluctuation needed to suspend particles. Then, an

equation for sizing the turbulent velocity fluctuation generated by the flow is developed by

consulting some relations available in the literature. In the last step, the required turbulence

and the produced turbulence are combined to develop a relation for critical deposition

velocity. The new two-phase approach is extended to multiphase flow by developing

empirical correlations that account for the effect of the gas phase in turbulence and particles

suspension.

The New Approach shows better results than the Initial Approach. In the area of two-

phase (solid-liquid) flow, it shows satisfactory performance over a broad range of particle

sizes. Similarly, for multiphase (sand-air-water) flow, the New Approach shows good

agreement with available data.

Conclusions

1. In the area of two-phase flow, the initial approach shows good performance at small

particle sizes (below 250 µm) while the new approach shows satisfactory

performance over a broad range of particle sizes.

2. In the area of multiphase flow, the initial approach shows unsatisfactory results while

the new approach shows satisfactory results as compared to the available data.

83

3. The satisfactory performance obtained by the new approach indicates that turbulent

theory is a viable mechanistic approach in modeling sand deposition in two-phase and

multiphase.

4. The Oroskar and Turian correlation (1980) shows good results over a broad range of

particle sizes.

5. Sand concentration plays a very important role in sand suspension. Sand

concentration has a pronounced effect on sand settling, turbulence dissipation, and

turbulence generation. All effects have to be considered in modeling critical

deposition velocity.

6. The developed correlations for predicting the magnitude of turbulent velocity

fluctuation in single-phase (gas or liquid) show good results in the near wall region

and in the core region. However, for two-phase flow (gas-liquid), the developed

correlations for bubbly and slug flow patterns are not validated further in this work

because there are no experimental data available for comparison.

Recommendations for Future Work

This section highlights some areas of improvement for two-phase (solid-liquid and

solid-gas) flow, multiphase (solid-liquid-gas) flow, and common areas that apply to both

two-phase and multiphase. The new approach only is recommended to be considered for

future work since it shows more promising results than the initial approach.

For two-phase flows, there is a need to validate the new approach with experimental

data for gas-sand flow because all the experimental data used in this study are based on

water-sand flow.

84

For multiphase (solid-liquid-gas) flow, there is a major area of improvement. There is

a need to develop a model for predicting the turbulent velocity fluctuation in the liquid phase.

In the current approach, the liquid phase turbulence is estimated empirically based on

experimental data of slug and bubbly flow patterns.

In addition to the above mentioned areas of improvement, there are three common

areas of improvement that apply to two-phase and multiphase flows. The first area is

extending the critical deposition velocity models to non-uniform sizes of particles as they are

originally developed for uniform sizes of particle. In this area, the work conducted by Wani

(1986) and Chien (1994) can be consulted. The second area of improvement is developing a

function that predicts the magnitude of turbulent velocity fluctuation generated by solids. It is

shown in this study that turbulent dissipation does not accommodate all the effects generated

by the presence of solids. A turbulent generation term should be added to the critical

deposition velocity model. For better understanding of turbulent dissipation and turbulent

generation due to the presence of solids, a literature review should be conducted in the area

of low solid concentrations (below 1%). The third area is studying the effect of the presence

of fine particles in sand deposition. In oil production, fine sand particles usually coexist with

larger sand particles and it could lead to some effects that are not addressed in this study. The

data collected by Parzonka (1981) can be considered in this area. Parzonka collected some

experimental measurements of critical deposition velocity of sand in the presence of fine

particles. The last area is the range of sand concentration considered in this study. The

available data used in this study has higher sand concentrations than what is usually

experienced in oil and gas industry. The developed models should be compared to additional

experimental data with sand concentrations below 1%.

85

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91

APPENDIX A

Experimental Data Table A-1: Experimental measurements of critical deposition velocities in horizontal two-phase (sand-water) flow

Reference Origin Particle

Size

(µm)

Pipe

Size

(mm)

Sand

Volume

Conc.

(%)

Critical

Deposition

Velocity

(m/s)

0.7 1.551.0 1.713.0 1.9

450 102

7.0 1.980.8 1.781.9 2.122.5 2.27

450

152

5.4 2.420.8 1.951.1 2.043.0 2.2

Graph, 1971

880 152

5.0 2.255 1.83

10 1.9415 2.08

230

108

20 2.355 1.99

10 2.1220 2.96

585

108

25 2.445 2.52

10 2.32

Yotsukura,

1961

1,150

108

15 2.675 2.47

10 2.65

Kokpinar and

Gogus, 2001

Durand, 1953

440

150

15 2.713.8 0.9488.4 1.190

Hayden,

1971

180 50.8

13.9 1.220

Parzonka, 2001

Smith, 1955 190 76.2 1.9 0.960

92

3.5 1.0206.5 1.4209.7 1.280

0.17 1.41017.7 1.50020.4 1.42022.5 1.41023.1 1.3400.26 1.28020.4 1.27026.1 1.50024.1 1.600

13 1.7103.1 2.4006.0 2.560

Silin, 1962 420 206

12.2 2.8900.04 1.5308.7 1.630

Hayden,

1971

570 50.8

14.3 1.760

0.9 0.4873.1 0.6223.9 0.6485.0 0.6806.1 0.6997.1 0.7188.1 0.737

Sinclair,

1962

600 12.7

10.2 0.7693.1 1.9906.1 2.190

14.3 2.30024.2 2.300

Silin, 1973 400 103

3.1 1.9900.04 0.7500.75 1.3500.85 1.7503.64 2.2000.07 2.66010.2 3.000

13.10 3.125

360

100

18.50 3.1500.44 0.612.23 0.754.6 0.88

Roco, 1991 Roco, 1972

78 100

9.78 0.94

93

15.68 0.9827.38 0.9139.42 0.83

94

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