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

    THE GRADUATE SCHOOL

    A MECHANISTIC MODEL FOR LIQUID

    HYDROCYCLONES (LHC)

    by

    Juan Carlos Caldentey

    A Thesis Submitted in Partial Fulfillment of

    the Requirements for the Degree of Master of Science

    in the Discipline of Petroleum Engineering

    The Graduate School

    The University of Tulsa

    2000

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    ABSTRACT

    Caldentey, Juan Carlos (Master of Science in Petroleum Engineering)

    A Mechanistic Model for Liquid Hydrocyclones (LHC)

    (98 pp. Chapter V)

    Directed by Professor Ovadia Shoham and Professor Ram S. Mohan

    (180words)

    Hydrocyclones provide economical and effective means for liquid-liquid

    separation in the petroleum as well as other industries. This study is focused on the

    deoiling of produced water utilizing a liquid hydrocyclone, LHC.

    A simple mechanistic model is developed for the LHC. The model is capable of

    predicting the hydrodynamic flow field of the continuous phase within the LHC. The

    separation efficiency is determined based on droplet trajectories, and the inlet-underflow

    pressure drop is predicted using an energy balance analysis.

    The predictions of the proposed model are compared with elaborate published

    experimental data sets. Good agreement is obtained between the model predictions and

    the experimental data with respect to both separation efficiency and pressure drop. The

    underflow separation efficiency is predicted with an average relative absolute error of

    4%, while the pressure drop is predicted with an average relative absolute error of 11%.

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    A user friendly computer code is developed in Excel-Visual Basic platform based

    on the proposed model. The code provides easy access to the input data and very fast

    output, and can be used for design of LHCs by the industry.

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    v

    ACKNOWLEDGEMENTS

    I would like to thank my co-advisors Dr. Ovadia Shoham and Dr. Ram Mohan for

    their continued support, guidance and for the freedom they gave me to work

    independently, which allowed me to explore several alternatives during my research. I

    wish to acknowledge Dr. Charles Petty of Michigan State University for his valuable

    assistance in the recompilation of literature.

    I want to express gratitude to my colleagues within the TUSTP group, from whom

    I learned invaluable knowledge. I am especially grateful to Luis Gomez and Carlos

    Oropeza for the many helpful suggestions and assistance. Also, Ferhat Erdal, Shoubo

    Wang and Carlos Gomez with whom I held many helpful discussions, and to Judy Teal

    whose collaboration made this project a reality.

    The research was made possible by the financial support of the TUSTP member

    companies.

    It is also important to acknowledge the Petroleum Engineering Staff of The

    University of Tulsa, outstanding full time professors who share their time and experience

    with the alumni and make this Department one of the top in the nation.

    Finally, I would like to thank my family and friends who are the source of my

    inspiration and motivation. This work is dedicated to my beloved wife, Ana, who not

    only encouraged me to pursue my Masters studies but also helped me complete this

    thesis.

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    vi

    TABLE OF CONTENTS

    TITLE PAGE i

    APPROVAL PAGE ii

    ABSTRACT iii

    ACKNOWLEDGEMENTS v

    TABLE OF CONTENTS vi

    LIST OF FIGURES viii

    LIST OF TABLES xi

    CHAPTER I

    INTRODUCTION 1

    1.1 Motivation and Objective 1

    1.2 LHC Hydrodynamic Flow Behavior 2

    1.3 LHC Geometry 4

    1.4 Thesis Structure 6CHAPTER IILITERATURE REVIEW 8

    2.1 Solid Hydrocyclones 9

    2.2 Liquid Hydrocyclones, LHC 13

    2.2.1 LHC Modeling 14

    2.2.2 Field Applications 15

    2.2.3 Experimental Studies 16

    2.2.4 Velocity Field measurements 19

    CHAPTER IIILHC MECHANISTIC MODEL 23

    3.1 Overview 23

    3.2 Swirl Intensity 26

    3.3 Velocity Field 29

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    vii

    3.3.1 Tangential Velocity 29

    3.3.2 Axial Velocity 31

    3.3.3 Radial Velocity 33

    3.4 Droplet Trajectories 34

    3.5 Separation Efficiency 37

    3.6 Pressure Drop 40

    3.7 LHC Mechanistic Model Code 42

    CHAPTER IVRESULTS AND DISCUSSION 44

    4.1 Swirl Intensity Prediction 44

    4.1.1 Experimental Data Sets 44

    4.1.2 Results 474.1.3 Discussion 50

    4.2 Velocity Field Prediction 50

    4.2.1 Experimental Data Sets 50

    4.2.2 Tangential and Axial Velocity Results 51

    4.2.3 Discussion 51

    4.3 Droplet Trajectory Prediction 67

    4.4 Separation Efficiency Prediction 69

    4.4.1 Experimental Data Sets 69

    4.4.2 Migration Probability and Underflow Purity Results 70

    4.4.3 Discussion 77

    4.5 Pressure Drop Prediction 78

    4.5.1 Experimental Data Sets 78

    4.5.2 Results 79

    4.5.3 Discussion 81

    CHAPTER VSUMMARY, CONCLUSIONS AND RECOMMENDATIONS 82

    5.1 Summary and Conclusions 82

    5.2 Recommendations 85

    NOMENCLATURE 88

    REFERENCES 91

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    viii

    LIST OF FIGURES

    Figure 1.1 LHC Hydrodynamic Flow Behavior............................................................... 3

    Figure 1.2 Colman and Thews Hydrocyclone Design..................................................... 5

    Figure 1.3 LHC Inlet Design ........................................................................................... 6

    Figure 2.1 Tangential Velocity Diagram........................................................................ 11

    Figure 2.2 Axial Velocity Profile From Colman (1984)................................................. 21

    Figure 3.1 LHC Mechanistic Model Structure ............................................................... 24

    Figure 3.2 LHC Characteristic Diameter....................................................................... 28

    Figure 3.3 Rankine Vortex ............................................................................................ 30

    Figure 3.4 Axial Velocity Diagram................................................................................ 32

    Figure 3.5 Droplet Velocities ........................................................................................ 34

    Figure 3.6 Forces Acting on a Droplet........................................................................... 35

    Figure 3.7 Droplet Trajectory and Migration Probability ............................................... 38

    Figure 3.8 Migration Probability Curve ......................................................................... 39

    Figure 3.9 LHC Mechanistic Model Code .......................................................................43

    Figure 4.1 Colmans Designs (1981) ............................................................................. 45

    Figure 4.2 Swirl Intensity Prediction - Case 1................................................................ 48

    Figure 4.3 Swirl Intensity Prediction - Case 2................................................................ 48

    Figure 4.4 Swirl Intensity Prediction - Case 3................................................................ 49

    Figure 4.5 Swirl Intensity Prediction - Cases 4 and 5 ..................................................... 49

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    Figure 4.6 Tangential Velocity Prediction - Case 1 ........................................................ 53

    Figure 4.7 Tangential Velocity Prediction - Case 2 ........................................................ 56

    Figure 4.8 Tangential Velocity Prediction - Case 3 ........................................................ 56

    Figure 4.9 Tangential Velocity Prediction - Case 4 ........................................................ 57

    Figure 4.10 Tangential Velocity Prediction - Case 5 ...................................................... 58

    Figure 4.11 Axial Velocity Prediction - Case 1.............................................................. 59

    Figure 4.12 Axial Velocity Prediction - Case 2.............................................................. 62

    Figure 4.13 Axial Velocity Prediction - Case 3.............................................................. 62

    Figure 4.14 Axial Velocity Prediction - Case 4.............................................................. 63

    Figure 4.15 Axial Velocity Prediction - Case 5.............................................................. 64

    Figure 4.16 Axial Velocity Prediction - Case 6.............................................................. 65

    Figure 4.17 Predicted Droplets Trajectories Case 7..................................................... 67

    Figure 4.18 Trajectories of a 15 Microns Droplet Case 7 ........................................... 68

    Figure 4.19 Migration Probability Curve - Case 7 ......................................................... 71

    Figure 4.20 Underflow Purity, u - Case 7 ..................................................................... 71

    Figure 4.21 Migration Probability Curve - Case 8 ......................................................... 72

    Figure 4.22 Underflow Purity, u - Case 8 ..................................................................... 72

    Figure 4.23 Droplet Size Distributions for Kuwait Oil (Colman et al., 1980) .................73

    Figure 4.24 Droplet Size Distribution for Forties Oil (Colman et al., 1980) ................... 74

    Figure 4.25 Migration Probability Curve Cases 16, 18 and 20 .................................... 75

    Figure 4.26 Migration Probability Curve Case 23 ....................................................... 76

    Figure 4.27 Migration Probability Curve Case 24 ....................................................... 76

    Figure 4.28 Comparison of Model Underflow Purity and Experimental Data Set ..........77

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    Figure 4.29 Pressure Drop Prediction Case 25 ............................................................ 79

    Figure 4.30 Pressure Drop Prediction Case 26 ............................................................ 80

    Figure 4.31 Comparison Between Pressure Drop Model and All Experimental Data .....81

    Figure 5.1 Hypothetical Swirl Intensity Decay .............................................................. 87

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    LIST OF TABLES

    Table 3-1 Drag Coefficient Constants............................................................................ 36

    Table 4-1 Geometrical Parameters of Colmans Designs (1981) .................................... 46

    Table 4-2 Operational Conditions of Colmans Designs (1981) .................................... 46

    Table 4-3 Geometrical Parameters of Hargreaves (1990)............................................... 47

    Table 4-4 Geometrical Parameters, Wolbert et al. (1995) .............................................. 70

    Table 4-5 Underflow Purity Results Cases 7 to 24......................................................... 75

    Table 4-6 Geometrical Parameters, Young et al. (1990) ................................................ 78

    Table 4-7 Pressure Drop Cases 27 to 35 ..................................................................... 80

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    1

    CHAPTER I

    INTRODUCTION

    1.1 Motivation and Objective

    The petroleum industry has traditionally relied on conventional gravity based

    vessels to separate multiphase flow. They are bulky, heavy, expensive and have large

    residence time. The growth of the offshore oil industry, where platform costs to

    accommodate these separation facilities are critical, has provided the incentive for the

    development of compact separation technology. Hydrocyclones have emerged as an

    economical and effective alternative for produced water deoiling and other applications.

    The hydrocyclone is inexpensive, simple in design with no moving parts, easy to install

    and operate, and has low maintenance cost.

    In the past, hydrocyclones have been used to separate solid/liquid, gas/liquid and

    liquid/liquid mixtures. For the liquid/liquid case, both dewatering and deoiling have been

    used in the oil industry. This study focuses only on the latter case, using the liquid

    hydrocyclones (LHC) to remove dispersed oil from a water continuous stream.

    In general, oil is produced with significant amount of water and gas. Typically, a

    set of conventional gravity based vessels are used to separate most of the multiphase

    mixture. The small amount of oil remaining in the water stream, after the primary

    separation, has to be reduced to a legally allowable minimum level for offshore disposal.

    Hydrocyclones have been used successfully to achieve this environmental regulation.

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    There is a large quantity of literature available on the LHC, including

    experimental data sets and computational fluid dynamic simulations. However, no simple

    and overall mechanistic model has been developed to date for the LHC. The objective of

    the current work is to develop a mechanistic model for the LHC to predict the separation

    efficiency and the flow capacity (pressure drop flow rate relationship).

    The developed model will allow the performance prediction for a given geometry

    and operating conditions, that can be utilized for LHC design. Also, it will permit the

    design of alternative geometries under similar conditions for optimization purposes.

    1.2 LHC Hydrodynamic Flow Behavior

    The hydrocyclone, as shown in Figure 1.1, utilizes the centrifugal force to

    separate the dispersed phase from the continuous fluid. The swirling motion is produced

    by the tangential injection of pressurized fluid into the cyclone body. The flow pattern

    consists of a spiral within another spiral moving in the same circular direction (Seyda and

    Petty, 1991). There is a forced vortex in the region close to the LHC axis and a free-like

    vortex in the outer region. The outer vortex moves downward to the underflow outlet

    while the inner vortex flows in reverse direction to the overflow outlet. Moreover, there

    are some recirculation zones associated with the high swirl intensity at the inlet. These

    zones, with a long residence time and very low axial velocity, have been found to be

    diminished as the flow enters the low angle tapered section (see Figure 1.1).

    An explanation of the characteristic reverse flow in the LHC is well described by

    Hargreaves (1990). With high swirl at the inlet, the pressure is high near the wall region

    and very low toward the center. As a result of the pressure gradient profile across the

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    diameter, decreasing with downstream position, the pressure at the downstream end of

    the core is greater than at the upstream, causing flow reversal.

    As the fluid moves to the underflow outlet, the narrowing diameter increases the

    fluid angular velocity and the centrifugal force. It is due to this force and the difference in

    density between the oil and the water, that the oil moves to the center where it is caught

    by the reverse flow and separated flowing into the overflow outlet. Instead, if the

    dispersed phase is the heaviest, like solid particles, it will migrate to the wall and exit

    through the underflow.

    Figure 1.1 LHC Hydrodynamic Flow Behavior

    The amount of fluid going through the different outlets differs with heavy and

    light dispersion. That means that for these two different separation cases, two different

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    geometries are necessary (Seyda and Petty, 1991). In the deoiling case, usually between 1

    to 10 percent of the feed flow rate goes to the overflow.

    Another phenomenon that may occur in a hydrocyclone is the formation of a gas

    core. As Thew (1986) explained, dissolved gas may come out of solution because of the

    pressure drop, migrating very fast to the axis, and eventually emerging through the

    overflow outlet. A significant amount of gas can be tolerated but excessive amounts will

    disturb the vortex. An experimental study on this topic is found in Smyth and Thew

    (1996).

    1.3 LHC Geometry

    The deoiling LHC consists of a set of cylindrical and conical sections. Colman

    and Thews (1988) design has four sections, as shown in Figure 1.1. The inlet chamber

    and the reducing section are designed to achieve the higher tangential acceleration of the

    fluid, reducing the pressure drop and the shear stress to an acceptable level. The latter has

    to be minimized to avoid droplet breakup leading to reduction in separation efficiency.

    The tapered section is where most of the separation is achieved. The low angle of this

    segment keeps the swirl intensity with high residence time. An integrated part of the

    design is a long tail pipe cylindrical section in which the smallest droplets migrate to the

    reversed core at the axis. This configuration gives a very stable small diameter reversed

    flow core, utilizing a very small overflow port.

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    Figure 1.1 Colman and Thews Hydrocyclone Design

    Young et al. (1990) achieved similar results to Colman-Thews LHC, in terms of

    separation efficiency, with a different hydrocyclone configuration. Three sections were

    used instead of four. The reducing section was eliminated and the angle of the tapered

    section was changed from 1.5 to 6. Later, Young et al. (1993) developed a new LHC

    design, which resulted in an improvement in the separation performance. The principal

    modification of the enhanced design was a small change in the tail pipe section. A minute

    angle conical section was used rather than the cylindrical pipe.

    Another important parameter in the LHC geometry is the inlet design (Figure 1.2).

    Rectangular and circular, single and twin inlets have been most frequently used by

    different researchers. The main goal is to inject the fluid with higher tangential velocity

    avoiding the rupture of the droplets. The twin inlets have been thought to maintain better

    symmetry and for this reason maintain a more stable reverse core (Colman et al., 1984;

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    Thew et al., 1984). Good results have also been achieved with the involute single inlet

    design.

    Figure 1.2 LHC Inlet Design

    The last element of the LHC is the overflow outlet. This is a very small diameter

    orifice that plays a major role in the split ratio, defined as the relationship between the

    overflow rate and the inlet flow rate. Most of the commercial LHC permit changing the

    diameter of this orifice depending on the range of operating conditions.

    1.4 Thesis Structure

    Current chapter is a brief preface to the study. It begins with a statement of the

    incentive to develop this project from the oil industry point of view. It is followed by the

    objective and the scope of the thesis. Then, the principles of operation of a hydrocyclone

    is discussed, focusing in the hydrodynamic behavior. The last section contains the typical

    geometry of a deoiling hydrocyclone including two different patent designs.

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    The second chapter is a review of some works pertinent to solid and liquid

    hydrocyclones. A general review of solid hydrocyclones is presented in the first section

    while a more detailed review is done for LHC. The LHC section is divided into

    theoretical, experimental and applications studies. The last topic of the LHC literature is

    related to the velocity field measurements and CFD simulations.

    Chapter III consists of description of the developed LHC mechanistic model. This

    chapter is divided into sections presented in the same order as the calculations that the

    model follows. The first topic is the swirl intensity, an important parameter for defining

    the velocity field which is the next subject. The velocity field allows the calculation of

    the droplet trajectories which define the separation efficiency. These two are discussed in

    the following sections. The LHC pressure drop - flow rate relationship is covered next

    and the last topic of the chapter is related to the developed LHC mechanistic model code.

    The accuracy of the mechanistic model is evaluated in Chapter IV through

    comparison with available data from other researchers. The results include the swirl

    intensity, the velocity profile, the separation efficiency and the pressure drop as well. The

    conclusions and suggestions for further work are covered in Chapter V.

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    CHAPTER II

    LITERATURE REVIEW

    For many years hydrocyclones have been used in different industries such as pulp

    and paper production, food processing, chemical industries, power generation,

    metalworking, and oil and mining industries. Both solid-liquid and liquid-liquid

    separation are possible with this technology. Most of the available literature on

    hydrocyclones is related to solid-liquid separation. Since the 1980s, liquid-liquid

    separation has become popular due to the relevant application area in the oil industry.

    This work is focused on liquid-liquid separation, specifically for a lighter

    dispersed phase. However, a brief review of solid hydrocyclones is imperative for

    understanding the principle of operation of this device and the evolution of the different

    models. It is important to stress up-front what are the main differences between these two

    types of separation processes.

    The density difference is much smaller for liquid-liquid mixtures, making

    the separation more difficult and creating the necessity of operating with

    higher centrifugal forces.

    The solid particles can be considered rigid unlike the liquid droplets which

    deform with the interaction of external forces. If high shear stress is

    present, this may cause droplet break up, reducing the probability of the

    smaller droplets to be separated. In the opposite case, if two droplets get

    close enough, a coalescence effect can occur, whereby the larger droplets

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    can be separated more easily.

    As mentioned in the previous chapter, the amount of fluid going through

    the different outlets differs with either heavy or light dispersion. For solid-

    liquid separation, more than 90% of the fluid exits from the top of the

    hydrocyclone while a similar quantity goes to the underflow outlet in the

    liquid-liquid case. This characteristic may suggests that the velocity field

    of the continuous flow differs for the two different cases.

    Because of the centrifugal force, the solid particles move outward until

    they reach the wall and fall to the underflow outlet. Therefore, the

    boundary layer is an important zone for this case and should be considered

    in any modeling (Bloor et al., 1980). In the LHC for lighter dispersed

    phase, more attention has to be centered on the region away from the wall

    where the separation occurs.

    More information about the differences between solid and liquid hydrocyclones

    can be found in Thew (1986).

    Two textbooks that condense pioneering works in hydrocylones and fundamental

    theories, including experimental data, design, and performance aspects, are Bradley

    (1965) and Svarovsky (1984). Both refer in most of the chapters to solid hydrocyclones

    with only a small section in liquid-liquid separation and other application areas.

    2.1 Solid Hydrocyclones

    The hydrocyclone was introduced after World War II by the Dutch State Mines as

    a new tool to separate dispersed solid material from a liquid of lower density (Rietema,

    1961). Although widely used nowadays, the selection and design of hydrocyclones are

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    still empirical and experience based. Even though quite a few hydrocyclone models are

    available, the validity of these models for practical applications has still not been

    established (Kraipech et al., 2000). A thorough review of the different available models

    can be found in Chakraborti and Miller (1992) and Kraipech et al. (2000).

    The models can be divided into empirical and semi-empirical, analytical solutions

    and numerical modeling (Chakraborti and Miller, 1992). The empirical approaches are

    based on correlations of the key parameters, considering the separator as a black box. The

    semi-empirical approach is focused on the prediction of the velocity field in the main

    flow using existing data as a major support. The analytical and numerical solutions solve

    the non-linear Navier-Stokes Equation. The first one is a mathematical solution, which is

    achieved neglecting some of the terms of the momentum balance equation. The

    numerical solution uses the power of computational fluid dynamics to develop a

    numerical simulation of the flow. As Svarovsky (1996) comments, it seems that the

    analytical flow models have been abandoned in favor of numerical simulations.

    Based on the experimental data taken by Kelsall (1952) using an optical method,

    many researchers have attempted to correlate the velocity field inside the hydrocyclone,

    especially the tangential velocity. It can be determined using the following relationship

    (Kelsall, 1952, see also Bradley and Pulling, 1959):

    Constant=nWr (2.1)

    This implies that the tangential velocity (W) increases as the radius (r) decreases

    for positive values of the empirical exponent (n). The exponent, n is usually between 0.5

    and 0.9 (Svarovsky, 1984) in the outer vortex, while in the core region it is close to -1

    (see Figure 2.1). If n = 1 a free vortex is obtained where a complete conservation of

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    angular momentum is implied or no viscous effect is considered. However, if n = -1 a

    forced vortex or a solid body rotation type is expected. Also, Kelsall's results are an

    evidence of the low dependence of the tangential velocity on the axial position.

    Figure 2.1 Tangential Velocity Diagram

    Analytical flow models have been pursued by Bloor and Ingham for many years

    (Bloor and Ingham, 1973, 1984 and 1987 and Bloor, 1987). The momentum and

    conservation of mass equations are mathematically solved for an incompressible and

    inviscid fluid using the stream function concept in an axi-symmetric flow. Kang (1984)

    and Kang and Hayatdavoudi (1985) follow this approach. But unlike the Bloor and

    Ingham's model, a cylindrical coordinate system was used instead of spherical. In this

    work, it was assumed that the velocities do not depend on the axial position. Kang

    considered that the axial and radial velocity obtained from this inviscid model can be

    applied without serious error. However, the addition of the turbulence effect had to be

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    included for the tangential component. A constant eddy viscosity was considered to

    account for the turbulence fluctuation following a procedure similar to Rietema (1961).

    Presently, numerical simulations or CFD are used widely to investigate flow

    hydrodynamics. As expressed by Hubred et al. (2000), the solution of the Navier Stokes

    Equations for simple or complex geometry for non-turbulent flow is feasible nowadays.

    But current computational resources are unable to attain the instantaneous velocity and

    pressure fields at large Reynolds numbers even for simple geometries. The reason is that

    traditional turbulence models, such as k-, are not suitable for this complex flow

    behavior. On the other hand, more realistic and complicated turbulence models increase

    the computational times to inconvenient limits.

    The flow inside hydrocyclones has been numerically simulated by Rhodes et al.

    (1987). A commercial computer code, PHOENICS, was used to solve the required partial

    differential equations which govern the flow. Prandtl mixing-length model was used to

    account for the viscous momentum transfer effect. In further work, Hsieh and Rajamani

    (1991) (see also Rajamani and Hsieh, 1988; Rajamani and Devulapalli, 1994) used a

    modified Prandtl mixing-length model with a stream function-vorticity version of the

    equation of motion. Good agreement with experimental data was observed in this study.

    The authors mention that the key for success is choosing the appropriate turbulence

    model and numerical solution scheme. In 1997, He et al. used a fully three dimensional

    model with a cylindrical coordinate system and curvilinear grid for the calculation of the

    flow field. A modified k- turbulence model was proved to achieve good results.

    In most of the work reviewed in the previous paragraph, excluding Rhodes et al.

    (1987), the models were evaluated through comparison with laser-doppler anemometry

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    2.2.1 LHC Modeling

    From extensive experimental tests, Colman and Thew (1983) developed some

    correlations to predict the migration probability curve, which defines the separation

    efficiency for a particular droplet size in a similar way that the grade efficiency does for

    solid particles (see 2.1.6 Grade efficiency in Svarovsky, 1984). Later it was found that the

    optimized Stokes Number vs. Reynolds Number correlation used in this work was

    erroneous (Nezhati in Thew and Smyth, 1997). However, relevant conclusions can be

    extracted from this study, such as that the separation efficiency is independent of the split

    ratio in the range 0.5 to 10%.

    Seyda and Petty (1991) evaluated the separation potential of the cylindrical tail

    pipe section. A semi-empirical model to predict the velocity field in a cylindrical

    chamber was developed to calculate the particle trajectories, and hence, the grade

    efficiency. In the model, the axial velocity was assumed to be independent of the axial

    location and a constant eddy viscosity was considered. The theoretical results showed an

    optimum split ratio, as opposed to previously reported results, and an increment in the

    efficiency when the feed flow rate was increased.

    Estimation of LHC efficiency based on a droplet trajectory was the target of

    Wolbert et al. (1995) work. The velocity distribution in the tapered section of Colman

    and Thew's design was modeled. This was achieved using a modified Helmholtz law for

    the tangential velocity, a polynomial correlation for the axial component, and the

    continuity equation and wall condition (Kelsall, 1952) for the radial velocity. The

    importance of the tail pipe section to the LHC separation efficiency was confirmed by

    comparing the model with experimental results.

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    An extension of Bloor and Ingham (1973) model (see section 2.1 Solid

    Hydrocyclones) for LHC was elaborated by Moraes et al. (1996). The modification takes

    into account the difference in the split ratio for liquid and solid hydrocyclones. Although,

    this model is sophisticated, results shown by the authors, where no reverse flow is

    achieved in the parallel section, disagree with existing data.

    2.2.2 Field Applications

    Field trials of deoiling units began in 1983-84, with the first permanent

    installation in the North Sea and Bass Strait, Tasmania, in 1985. By the end of 1985, a

    North Sea installation of 42 units in parallel was handling nearly 15 m3/min (135,860

    bpd) successfully (Thew and Smyth, 1997). The field tests conducted by Serck Baker

    have demonstrated that a single deoiling unit can maintain effluent oil content below 300

    ppm in spite of the large fluctuations in oil content of the inflow up to 2000 ppm. A

    comparison of this field data with laboratory measurements showed that two or three

    units of hydrocyclones in series can provide substantial improvements (Colman et al.,

    1984).

    Meldrum (1988) discussed the operational performance of the four-in-one

    hydrocyclone concept on the Murchison platform. It was found that the separation

    efficiency falls for low flow rates as well as for high flow rates. This was attributed to the

    low swirl generated for the lower flow rate limit and due to droplet break up in the higher

    flow rate case. Meldrum also found that the efficiency increases as the split ratio

    increases until it gets to a point where it remains constant.

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    Usually, hydrocyclones have been used, where adequate system feed pressure for

    satisfactory operation is present. Flanigan et al. (1992) revealed successful field trials

    with a low shear progressive cavity pump that overcame this limitation.

    LHC have been successfully applied not only offshore but also in standard

    oilfields. Stroder and Wolfenberger (1994) showed how this technology can be applied to

    high water cut electric submergible pump (ESP) wells, as a much more economical

    alternative instead of expanding the conventional water separation facilities. Also, good

    results in application of hydrocyclones for heavy oil treatment were achieved by Hashmi

    et al. (1996). A two stage hydrocyclones system accomplished similar performance to

    that of the free water knockout (FWKO) vessels.

    The oil industry has realized the benefits of downhole separation. High water cut

    wells are produced re-injecting the water and pumping the oil to the surface. Field trials

    and description of this application using LHC with an ESP system are found in Bowers et

    al. (1996). As it is expressed by the authors, this technology has the potential to become

    as significant a revolution in oilfield production as the LHC itself was to the oil industry

    in the 1980's.

    2.2.3 Experimental Studies

    Before reviewing the experimental studies, let us consider the definition of

    underflow efficiency as given below (Young, 1990):

    FeedtheatesentedPrOil

    DiscardedOil= (2.2)

    Utilizing continuity equation:

    uuiiooQkQkQk = (2.3)

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    where k is the concentration, Q is the flowrate and the subscripts o, i, and u are for the

    overflow, inlet and underflow streams.

    Rewriting Equation (2.2), yields:

    ii

    uu

    ii

    oo

    Qk

    Qk1

    Qk

    Qk== (2.4)

    Since very small amount of flow is taken out of the overflow, Qu/Qi is almost

    equal to one. This is the basis used by many authors and also in the current study when

    considering oil/water separation. This efficiency is called underflow purity, defined as

    follows:

    i

    u

    u k

    k1= (2.5)

    If ku tends to 0, u becomes 1. On the other hand, if ku is equal to ki, u is 0. In the

    latter case, the hydrocyclone splits the flow without achieving any separation.

    Colman et al. (1980) examined oil/water separation efficiency of a series of LHC.

    In this work, the performance criterion used was the underflow purity, u. Important

    observations can be made from this study. The u is independent of the ki within a range

    of 100 to 1000 ppm. The authors concluded that this is a sign of no interaction between

    the oil droplets. Also, constant values of uwere found for different split ratios. However,

    for split ratio values less than 2.5% it was found that ubegins to decrease. It was also

    observed that under these conditions, breakdown of the flow structure finally leads to a

    complete loss of separation.

    Nezhati and Thew (1987) investigated the effect of variation in the inlet area and

    other parameters, such as temperature, on the performance of the LHC. The principal

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    objective of this work was to see the variation of dimensionless groups, namely, Stokes,

    Euler and Hydrocyclone Number with the flow conditions. Relevant conclusions from the

    experimental results are that the separation increases as the cylindrical length (tail pipe

    section) is increased and that the pressure drop increases following a simple exponential

    relationship with flowrate, given by

    n

    iQP (2.6)

    As mentioned in section 1.3 LHC Geometry, Young et al. (1990) searched

    through a broad set of experiments for optimum dimensions of the LHC. Similar to

    Nezhati and collaborators, Young found that the separation efficiency increases with the

    underflow length until it gets to a point where no additional separation occurs. Contrary

    to this, as the inlet chamber length is increased the separation efficiency reduces. Other

    variables studied were the angle of the conical section, the overflow outlet diameter, the

    underflow diameter and the oil properties. In this work, it is noted that the oil droplet size

    distribution at the inlet of the LHC has the greatest impact on the separation, namely, that

    the bigger the droplets are, the better the separation will be.

    In 1991, Weispfennig and Petty explored the flow structure in a LHC using a

    visualization technique (laser induced fluorescence). Different types of inlets were

    studied including an annular entry. A parameter that measures the strength of the swirling

    flow, the Swirl Number, was used to characterize most of the results. This is defined as

    the ratio of the axial flux of the axial component of angular momentum to the axial flux

    of the tangential component of angular momentum. Vortex instability and recirculation

    zones were strongly dependent on the Swirl Number and a characteristic Reynolds

    Number.

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    The performance of a small hydrocyclone was summarized by Ali et al. (1994).

    Deoiling hydrocyclones of 10 mm-diameter have achieved high performance with cut

    size as low as 4 microns. The cut size (d50) is the particle size which has a 50% chance of

    being separated.

    Experiments have also been carried out in dewatering hydrocyclones where the

    dispersed phase is water and the continuous phase is oil (Smyth et al., 1980, 1984; Smyth

    and Thew, 1987; Smyth, 1988; Young, 1993; Sinker and Thew, 1996). Due to the high

    viscosity of the oil, this type of separation is more difficult than the deoiling case but with

    an adequate geometry good results may be achieved.

    2.2.4 Velocity Field Measurements

    Flow field measurements within hydrocyclones can be obtained using

    photographic, optical, Pitot tubes and Laser Doppler Anemometry (LDA) (Chakraborti

    and Miller, 1992). LDA has been the preferred method in the last two decades because it

    permits high speed data acquisition and also because it is a non intrusive technique that

    consequently does not cause any flow perturbation.

    Thew et al. (1980), used a Residence Time Distribution (RTD) technique to

    complement the information gathered from LDA measurements. The authors used a

    tracer method in zones where the LDA is limited, i.e. near the boundary wall. Later,

    Thew et al. (1984) showed results using the same technique in an improved hydrocyclone

    design. The effect of changes in operational parameters, such as split ratio and inlet flow

    rate, on the RTD were studied. Good separation efficiency was achieved using split ratios

    down to 1%, as opposed to previously reported results.

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    Colman (1981) measured the velocity field in four different hydrocyclone

    configurations. Axial and tangential velocities at different axial locations were acquired

    using LDA technique. Detailed information of the flow structure was used to improve the

    LHC geometry. The observed tangential velocity was a combination of a semi-free vortex

    in the outer region and a forced vortex in the core region. In most of the cases, the axial

    velocity measured has a reverse flow region in the LHC axis which is inside the forced

    vortex region. Both axial and tangential velocities are reduced close to the cylindrical

    section wall as the flow moves downward, due to the frictional losses. The reverse flow

    begins to decrease as the swirl intensity decays.

    Analyzing the LDA axial velocities (Figure 2.1), Colman et al. (1984) concluded

    that 85% of the fluid that exits the overflow outlet comes from the reverse flow contained

    in the tapered and tail pipe section, while the rest (15%) is made up from the radially

    inward moving fluid at the top wall boundary layer. This effect is known as a short circuit

    where part of the feed flow rate goes directly to the reject orifice.

    Similar results to Colmans (1981) were obtained by Hargreaves (1990) in a

    single LHC. Several flow rates and split ratios were explored, measuring the velocity and

    the turbulence quantities. It was confirmed that the reverse flow is a body rotation type

    and the magnitude of the axial velocity was found to be four times greater than that of the

    mean axial velocity in the cylindrical section, and six times more than in the tapered

    region. In this study a CFD simulation flow field was compared with LDA

    measurements. The turbulence model used in the numerical solution was the Four

    Equation Algebraic Stress Model. A review of this modeling and CFD applied to

    deoiling hydrocyclones is found in Hargreaves and Silvester (1990).

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    Figure 2.1 Axial Velocity Profile From Colman (1984)

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    The rankine vortex behavior of the tangential velocity was confirmed by

    Weispfennig et al. (1995). LDA measurements in the cylindrical section of the

    hydrocyclone show how the angular momentum flux ratio decreases with the axial

    distance. A flow control was used in order to increase the angular momentum. Parks and

    Petty (1995) solved a numerical model using a constant eddy viscosity Boussinesq

    approximation to predict the angular momentum distribution in the LHC.

    After a critical analysis of the extensive literature available on LHC, it becomes

    evident that researchers have directed great efforts toward understanding not only the

    separation mechanisms but also the highly complex velocity and pressure field within

    hydrocyclone. A large number of experiments have permitted the design of successful

    configurations for specific industrial applications. However, it is the authors impression

    that an optimum design can be achieved for each range of operational conditions. A

    robust mechanistic model, in which the swirling motion and the collateral mechanisms

    that influence the LHC efficiency are predicted, is developed in this study as described in

    chapter III. This model has the potential to become an excellent tool to predict

    performance of existing design, or even to design optimum LHC over a broad range of

    operational conditions.

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    CHAPTER III

    LHC MECHANISTIC MODEL

    3.1 Overview

    The mechanistic model is an intermediate approach between the empirical

    approach and the exact solution. In this approach a simplified physical model is built,

    which attempts to describe closely the nature of the physical phenomenon. This physical

    model is then expressed mathematically to provide a tool for prediction and design

    purposes. The closer the physical model is to the real phenomenon, the better the

    mathematical model is, as well as its prediction. One must remember that a mechanistic

    model is not a rigorous solution as the physical model is approximated by taking into

    consideration the most important processes, neglecting other less important effects that

    can complicate the problem without considerably adding to the accuracy of the solution.

    The present work focuses on the development of a simple mechanistic model for

    deoiling hydrocyclones, which captures the nature of the hydrodynamic flow behavior in

    the LHC. The model is capable of predicting the separation efficiency and the pressure

    dropflow rate relationship. In order to obtain these desired output, the LHC Mechanistic

    Model needs as input variables, the geometry of the LHC and the operational conditions

    such as fluid properties, flow rate and feed droplet size distribution.

    The structure of the model can be seen in Figure 3.1. The first step of the model is

    the prediction of a parameter known as swirl intensity. With the swirl intensity

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    the velocity field of the continuous phase is determined within the LHC, in terms of axial,

    tangential and radial velocities. The model then predicts the separation efficiency and the

    pressure drop from the inlet to the underflow outlet of the LHC.

    Migration Probability Underflow Purity

    Separation

    Efficiency

    Droplet Trajectories Pressure Drop

    Velocity Field

    Swirl Intensity

    Figure 3.1 LHC Mechanistic Model Structure

    The separation efficiency is computed based on droplet trajectories of the

    dispersed phase and can be expressed in two modes: the migration probability curve and

    the underflow purity (see section 2.2.3 Experimental Studies). The former yields the

    separation probability for a specific droplet size, while the latter gives the ratio of the oil

    concentration at the clean stream to the one at the inlet, for a given feed droplet size

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    distribution. All these concepts will be explained in more detail in the subsequent

    sections.

    There are some key parameters that must be considered in the LHC Model. The

    angle of the tapered section is one of them. As the angle is increased, the tangential

    velocity will be increased and with this the centrifugal force. This may suggest that better

    separation can be achieved. However, as the angle is increased the resulting smaller cross

    sectional area increases the axial velocity of the fluid which will give less residence time

    for the droplets to separate. Thus, there is a compromise between the centrifugal force

    and the residence time. The model considers this relationship with the prediction of the

    swirl intensity, which is a parameter that defines the strength of the swirling motion

    compared to the mean axial velocity.

    Another geometrical parameter to be considered is the inlet configuration. The

    swirl intensity, as well as the velocity field, is strongly dependent on how the swirling

    motion is promoted at the inlet. In the present model the two most commonly used inlet

    configurations are included, the involute single inlet and the twin inlets (see Figure 1.2).

    The model does not consider the shape of the inlet at all, as rectangular and circular inlets

    are treated in the same manner. Only the cross sectional area is considered crucial in the

    calculations.

    As mentioned before, some effects have to be neglected in order to arrive at a

    sufficiently simple solution. The main assumptions that reduce the numerical effort are

    listed as follows: 1) axisymmetric flow is considered where there is no variation in the

    tangential component; 2) no deformation or interactions within the droplets, namely, no

    coalescence or droplet break up occur; 3) no presence of a gas core and 4) no turbulence

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    effect on the droplet trajectory. Some of these assumptions as well as others will be

    explained in greater detail in later sections.

    3.2 Swirl Intensity

    Diverse definitions of swirl intensity have been used by researchers in different

    fields. The importance of this parameter, characterizing the swirling flow in liquid-liquid

    hydrocyclones, is recognized by Weispfenning and Petty (1991) (see section 2.2.3

    Experimental Studies) and also by Thew and Smith (1997). In both cases the swirl

    number concept was used. For deoilers the swirl number is commonly within the range of

    8-10 which is high enough to achieve a good separation, but low enough to avoid droplet

    break up and vortex core instability (Thew and Smyth, 1997).

    In the current model the swirl intensity, , is defined as the ratio of the rate of

    tangential to total momentum flux at a specific axial location, given by (Mantilla, 1998

    and Chang and Dhir, 1994):

    avz22

    zc

    R

    0c

    UR

    uwrdr2z

    =

    (3.1)

    where u and w are the axial and tangential velocities of the continuous fluid, respectively,

    r is the radial position, c is the continuous phase density and R is the radius. Uav is the

    bulk axial velocity and the subscript z is for a given axial position.

    Several published data sets on cylindrical cyclones indicate that the swirl intensity

    decays exponentially with the axial position (Mantilla, 1998). Mantilla also developed a

    modification of Chang and Dhir (1994) correlation to account for fluid properties and

    inlet effects. In the current study, based on analysis of experimental data sets, a modified

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    correlation of Mantillas model is developed. The modified correlation takes into account

    the semi-angle, , of a tapered section, resulting in:

    ))tan(2.11(IM

    M

    48.115.0

    93.0

    2

    T

    t

    +

    =

    ( )

    +

    12.07.016.0

    z

    35.0

    4

    T

    t)tan(21

    Dc

    z

    Re

    1I

    M

    M

    2

    1EXP (3.2)

    This correlation was developed using experimental data for small semi-angles, ,

    from 0 to 0.75. However, a good prediction has also been obtained for 3 case (see

    section 4.5 Pressure Drop Prediction). Due to this limitation and lack of experimental

    data for larger angles, this equation is mainly valid for the tapered and tail pipe sections

    of Colman and Thews LHC Design (Figure 1.1).

    In the above equation Dc, also shown in Figure 3.1, is the characteristic diameter

    of the LHC, measured where the angle changes from the reducing section to the tapered

    section in the Colman and Thews Design, and at the top diameter of the 3 tapered

    section of the Youngs Design (see section 1.3 LHC Geometry); z is the axial position

    starting from Dc.

    T

    t

    M

    Mis the ratio of the momentum flux at the inlet slot to the axial momentum flux

    at the characteristic diameter position, calculated as follows:

    is

    c

    cc

    isc

    avc

    is

    T

    t

    A

    A

    A/m

    A/m

    Um

    Vm

    M

    M=

    ==&

    &

    &

    &

    (3.3)

    where Vis is the velocity at the inlet, Uavc is the average axial velocity at Dc, m& is the

    mass flow rate, Ac is the cross sectional area at Dc and Ais is the inlet cross sectional area.

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    Figure 3.1 LHC Characteristic Diameter

    The Reynolds number is defined in the same way as for pipe flow with the

    caution that it refers to a given axial position, yielding:

    c

    zavzc

    z

    DURe

    = (3.4)

    where c is the viscosity of the continuous fluid.

    The inlet factor, I, which was modified from Mantilla (1998), is defined as:

    ( )nEXP1I = (3.5)

    where n = 1.5 for twin inlets and n = 1 for involute single inlet.

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    The LHC Mechanistic Model considers only the separation occurring at the

    tapered and the tail pipe sections. This is a good assumption for the following reasons:

    1) Several researchers have reported that most of the separation is achieved in the low

    angle tapered section. 2) It can be expected that the biggest droplets that may separate

    close to the inlet section will be separated anyway in the consecutive sections of the LHC

    and 3) the length of the inlet and reducing sections is usually less than 10% of the total

    length of the LHC.

    3.3 Velocity Field

    The swirl intensity is related, by definition, to the local axial and tangential

    velocities. Therefore, it is assumed that once the swirl intensity is predicted for a specific

    axial location, it can be used to predict the velocity profiles (Mantilla, 1998). Both

    tangential and axial velocities are calculated following a similar procedure as proposed

    by Mantilla (1998). The radial velocity, which is the smallest in magnitude, is computed

    considering the continuity equation and the wall effect.

    3.3.1 Tangential Velocity

    It has been experimentally confirmed that the tangential velocity is a combination

    of forced vortex near the hydrocyclone axis and free-like vortex in the outer wall region,

    neglecting the effect of the wall boundary layer (Figure 3.1). This type of behavior is

    known as a Rankine Vortex.

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    Figure 3.1 Rankine Vortex

    Algifri et al. (1988) proposed the following equation for the tangential velocity

    profile:

    =2

    c

    c

    m

    avcR

    rBEXP1

    R

    r

    T

    U

    w(3.6)

    where w is the local tangential velocity, which is normalized with the average axial

    velocity, Uavc, at the characteristic diameter; r is the radial location and Rc is the radius at

    the characteristic location.

    Tm represents the maximum momentum of the tangential velocity at the section

    and B determines the radial location at which the maximum tangential velocity occurs.

    The following expressions were obtained by curve-fitting several sets of the experimental

    data.

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    =m

    T (3.7)

    Involute Single Inlet:7.1

    7.55B= (3.8)

    Twin Inlets:35.2

    8.245B= (3.9)

    It can be seen that the above equations are only functions of the swirl intensity, .

    Thus, for a given axial position, the tangential velocity is only function of the radial

    location and the swirl intensity.

    3.3.2 Axial Velocity

    In swirling flow the tangential motion gives rise to centrifugal forces which in

    turn tend to move the fluid toward the outer region (Algifri 1988). Such a radial shift of

    the fluid will result in a reduction of the axial velocity near the axis, and when the swirl

    intensity is sufficiently high, reverse flows can occur near the axis. This phenomenon

    causes a characteristic reverse flow in the LHC axis, which allows the separation of the

    different density fluids.

    A typical axial velocity profile for LHC is illustrated in Figure 3.1. Here, the

    positive values represent the downward flow near the wall, which is the main flow

    direction, and the negatives values represent the upward reverse flow near the LHC axis.

    The reverse radius, rrev, is the radial position where the axial velocity is equal to zero.

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    Figure 3.1 Axial Velocity Diagram

    To predict the axial velocity profile, a third-order polynomial equation is used

    with the proper boundary conditions. The general form is as follows:

    43

    2

    2

    3

    1 ararara)r(u +++= (3.10)

    where a1, a2, a3 and a4 are constants. The boundary conditions considered are:

    1. 0dr

    )Rr(du z ==

    the velocity is maximum at the wall,

    2. 0)rr(u rev == zero velocity at the location of reverse flow, rrev,

    3. 0dr

    )0r(du=

    =the velocity is symmetric about the LHC axis and

    4. 2zc

    R

    0avzc RUrdr)r(u2

    z

    = Mass Conservation.

    Substituting the boundary conditions in Equation (3.10), yields the axial velocity

    profile, which is a function of the swirl intensity, only:

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    1C

    7.0

    R

    r

    C

    3

    R

    r

    C

    2

    U

    u2

    z

    3

    zavz

    ++

    = (3.11)

    7.0Rr23

    RrC

    z

    rev

    2

    z

    rev

    = (3.12)

    358.0

    z

    rev293.0

    R

    r= (3.13)

    Several assumptions are implicit in these equations. First, an axisymetric

    geometry is imposed. Then, the effects of the boundary layer are neglected, and finally

    the mass conservation balance does not consider the split ratio. The last assumption can

    be considered a good approximation for small values of split ratios used in the LHC,

    usually less than 10%.

    3.3.3 Radial Velocity

    The radial velocity, v, of the continuous phase is very small, and has been

    neglected in many studies. In our case, in order to track the position of the droplets in

    cylindrical and conical sections, the continuity equation and wall conditions suggested by

    Kelsall (1952) and Wolbert, (1995) are used for the radial velocity profile, yielding:

    )tan(uR

    rv

    z

    = (3.14)

    The radial velocity is a function of the axial velocity and geometrical parameters.

    In the particular case of cylindrical sections, where tan( ) = 0, the radial velocity, v, is

    equal to 0.

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    3.4 Droplet Trajectories

    The droplet trajectory model is developed using a Lagrangian approach in which

    single droplets are traced in a continuous liquid phase. The droplet trajectory model

    utilizes the flow field presented in the previous section.

    Figure 3.1 presents the physical model. A droplet is shown at two different time

    instances, t and t + dt. The droplet moves radially with a velocity V r and axially with Vz.

    It is assumed that in the tangential direction the droplet velocity is the same as the

    continuous fluid velocity, as no force acts on the droplet in this direction. Therefore, the

    trajectory of the droplet is presented only in two dimensions, namely r and z.

    Figure 3.1 Droplet Velocities

    During a differential time dt, the droplet moves at velocity Vr = dr/dt in the radial

    direction and Vz = dz/dt in the axial direction. Combining these two equations and

    solving for the axial distance yields the governing equation for the droplet displacement:

    === drV

    Vz

    V

    V

    dtdr

    dtdz

    dr

    dz

    r

    z

    r

    z(3.15)

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    2

    1

    D

    2

    c

    dc

    sr C

    d

    r

    w

    3

    4V

    = (3.18)

    where d is the droplet diameter; d is the density of the dispersed phase and CD is the

    drag coefficient calculated using the following relationship (Morsi and Alexander, 1971

    and Hargreaves, 1990):

    2

    d

    3

    d

    2

    1D Re

    b

    Re

    bbC ++= (3.19)

    The coefficients b are dependent on the Reynolds Number of the droplets,

    defined as:

    c

    src

    D

    VdRe

    = (3.20)

    The values for the b coefficients as function of the range of ReD are shown in

    the Table 3-1.

    Table 3-1 Drag Coefficient Constants

    Range b1 b2 b3

    ReD < 0.1 0 24 0

    0.1 < ReD < 1 3.69 22.73 0.0903

    1 < ReD < 10 1.222 29.1667 -3.8889

    10 < ReD < 100 0.6167 46.5 -116.67

    Finally, a numerical integration of Equation (3.16) determines the axial location

    of the droplet as a function of the radial position. The trajectory of a droplet of a given

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    size is mainly a function of the LHC velocity field and the physical properties of the

    dispersed and continuous phases.

    3.5 Separation Efficiency

    The separation efficiency of the LHC can be determined based on the droplet

    trajectory analysis presented above. Starting from the cross sectional area corresponding

    to the LHC characteristic diameter, it is possible to follow the trajectory of a specific

    droplet, and determine if it is either able to reach the reverse flow region and be

    separated, or if it reaches the LHC underflow outlet, dragged by the continuous fluid and

    carried under.

    As illustrated by Figure 3.1, the droplet that starts its trajectory from the wall

    (r = Rc) is not separated, but rather carried under. However, if the starting location is at

    r < Rc, the chance of this droplet to be separated increases. When the starting point of the

    droplet trajectory is the critical radius, rcrit, the droplet reaches the reverse radius, rrev, and

    is carried up by the reverse flow and is separated.

    Therefore, assuming homogeneous distribution of the droplets, the efficiency for a

    droplet of a given diameter, (d), can be expressed by the ratio of the area for which the

    droplet is separated, defined by rcrit, over the total area for flow. This assumption has also

    been applied by other researchers (Seyda and Petty, 1991; Wolbert et al., 1995 and

    Moraes et al., 1996).

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    Figure 3.1 Droplet Trajectory and Migration Probability

    As proposed by Moraes et al. (1996), the efficiency is given by:

    =

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    have an efficiency very close to zero and as the droplet size is increased, (d) increases

    sharply until it reaches d100, which is the smallest droplet size with a 100% probability to

    be separated.

    Figure 3.2 Migration Probability Curve

    The migration probability curve is the characteristic curve of a particular LHC for

    a given flow rate and fluid properties. This curve is independent of the feed droplet size

    distribution and it is used in many cases to compare the separation of a given LHC

    configuration.

    Using the information derived from the migration probability curve and the feed

    droplet size distribution, the underflow purity, u, can be determined as follows:

    =i i

    iii

    u V

    V)d(

    (3.22)

    where u is expressed in %, and Vi is the percentage volumetric fraction of the oil

    droplets of diameter di.

    The underflow purity is the parameter that quantifies the LHC capacity to separate

    the dispersed phase from the continuous one (see section 2.2.3 Experimental Studies).

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    3.6 Pressure Drop

    The pressure drop from the inlet to the underflow outlet is calculated using a

    modification of the Bernoullis Equation:

    ( ) LsinghhU2

    1PV

    2

    1P

    cfcfc

    2

    UcU

    2

    iscis++++=+ (3.23)

    where c is the density of the continuous phase; Pis and Pu are the inlet and outlet

    pressures respectively; Vis is the average inlet velocity and Uu is the underflow average

    axial velocity; L is the hydrocyclone length, is the angle of the LHC axis with the

    horizontal; hcfcorresponds to the centrifugal force losses and hfis the frictional losses.

    The frictional losses are calculated similar to pipe flow:

    2

    )z(V

    )z(D

    z)z(f)z(h

    2

    R

    f

    = (3.24)

    where f is the friction factor and VRis the resultant velocity.

    In the case of conical sections, all parameters in Equation (3.24) change with the

    axial position, z. The conical section is divided into m segments and assuming

    cylindrical geometry in each segment, the frictional losses can be considered as the sum

    of the losses in all the m segments, as follows.

    )2

    z)1n2((atV

    2

    DD

    z

    2

    )z(fh 2

    R

    m

    1n n1n)conical(f

    +

    =

    = (3.25)

    The resultant velocity, VR, is calculated as the vector sum of the average axial and

    tangential velocities, The annular downward flow region is only considered, as presented

    in the following set of equations:

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    2

    Z

    2

    Z

    2

    R WU)z(V += (3.26)

    = 20 Rr

    2

    0

    R

    r

    z z

    rev

    z

    rev

    rdrd

    WrdrdW

    (3.27)

    For simplification purposes, the average axial velocity in Equation (3.26), Uz, is

    calculated assuming plug flow, namely, Uz is equal to the total flow rate over the annular

    area from the wall to the reverse radius, rrev. The Moody friction factor is calculated using

    Halls Correlation (Hall, 1957).

    +

    +=

    3/16

    4

    )zRe(

    10

    )z(D10x210055.0)z(f (3.28)

    where is the pipe roughness and Re is the Reynolds Number, calculated based on the

    resultant velocity computed in Equation (3.26).

    The centrifugal losses are the most important one in Equation (3.23), and account

    for most of the total pressure drop in the LHC. They are calculated using the following

    expression:

    ( )= urev

    R

    r

    2

    u

    cfdr

    r

    )r(nWh (3.29)

    where Wu is calculated from Equation (3.27) at the underflow outlet and the centrifugal

    force correction factor, n = 2 for twin inlets, and n = 3.2 for involute single inlet.

    The centrifugal force correction factor compensates for the use of Bernoullis

    Equation under a high rotational flow condition. Its meaning is similar to the kinetic

    energy coefficient used to compensate for the non-uniformity of the velocity profile in

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    pipe flow (Munson et al., 1994). Rigorously, the Bernoullis Equation is valid for a

    streamline and the summation of the pressure, the hydrostatic and the kinetic terms can

    only be considered constant in all the flow field if the vorticity is equal to zero.

    3.7 LHC Mechanistic Model Code

    In order to validate and compare the model with published experimental data, a

    computer code was built, which includes the equations shown in this chapter. The

    program was developed in Visual Basic for Excel Application. Excel/VBA platform can

    provide great advantages such as user-friendly interface forms and easiness to manipulate

    the output data.

    Figure 3.9 presents the multipage form used in the computer code where the user

    can interact with the program. All the input such as geometry, operating conditions, fluid

    properties and feed droplet size distribution are located in this form as separate folders.

    Buttons to run the program, as well as save and open input cases are also included. All

    the results of the program are presented in the worksheets of the Excel Application.

    The code uses mainly two different numerical methods to obtain the results. The

    tangential velocity, given by Equation (3.27), is solved using the Trapezoidal Rule, and

    for the droplet trajectory, a fourth-order Runge-Kuttta method is used to solve Equation

    (3.16). Also, a commercial program (Mathematica 4.0) was used to verify the resulting

    numerical values given by the computer code.

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    Figure 3.9 LHC Mechanistic Model Code

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    CHAPTER IV

    RESULTS AND DISCUSSION

    This chapter presents a comparison between 35 published experimental data sets

    and the prediction of the LHC mechanistic model developed in the present study. The

    outline is similar to the previous chapter, starting from the swirl intensity prediction and

    ending with the pressure drop prediction. In each section, the source of the experimental

    data is described followed by the results and discussion. The only section that differs

    from this structure is the droplet trajectory, where only the model predictions are shown.

    4.1 Swirl Intensity Prediction

    4.1.1 Experimental Data Sets

    The swirl intensity, which is the ratio of the local tangential momentum flux to

    the total momentum flux, can be obtained from the numerical integration of Equation

    (3.1). The numerical method employed for this purpose was the trapezoidal rule. The

    experimental data sets used to compare with the swirl intensity predicted by the model,

    are described next:

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    Colmans (1981) work, where the flow field was measured using a

    Laser Doppler Anemometer. In this study four different hydrocyclone

    designs were used. However, only designs II, III and IV, as named in

    the original work, are used here. The configurations of these

    hydrocyclones are shown in Figure 4.1 and the geometrical and

    operational conditions of each study case are detailed in Table 4-1 and

    Table 4-2, respectively.

    Figure 4.1 Colmans Designs (1981)

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    Table 4-1 Geometrical Parameters of Colmans Designs (1981)

    Case Design Dc(mm) L As 1 2 D2 L2 Ds Ls Du

    1 II 58 30Dc 0.125 - 90 0.5Dc 22Dc - - 0.14Dc

    2 III 30 20Dc 0.0625 10 10 0.5Dc 21Dc 2Dc 3Dc 0.14Dc

    3 IV 30 - 0.0625 10 0.67 0.5Dc 21Dc 2Dc 3Dc 0.14Dc

    Table 4-2 Operational Conditions of Colmans Designs (1981)

    Case Design Dc(mm) Flowrate (lpm) T (C) F (%)

    1 II 58 175 25 10

    2 III 30 60 25 10

    3 IV 30 60 25 10

    where F(%) is the split ratio.

    Hargreaves data (1990) were taken with a LDA in a similar LHC

    configuration as that of Colmans Design IV, but with a single involute

    inlet instead of the twin inlets. The cross sectional area of the inlet is 644

    mm2. Figure 4.1 is used as a reference for the rest of the geometrical

    parameters expressed in Table 4-3. The flow rate used in these cases is

    180 lpm and two different split ratios, F, were tested. Case 4 with F =

    10%, which corresponds to Du = 0.117 Dc, and the Case 5 with F = 1%,

    which corresponds to Du = 0.04 Dc.

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    Table 4-3 Geometrical Parameters of Hargreaves (1990)

    Case Design Dc(mm) 1 2 D2 L2 Ds Ls

    4 and 5 IV 60 10 0.6365 0.5Dc 15Dc 1.5Dc 1.67Dc

    The amount of experimental data, presented for each case in the next section,

    depends on the availability of axial and tangential velocity measurements at specific axial

    locations published by the above mentioned authors.

    4.1.2 Results

    The experimental data shown in Figure 4.1 to Figure 4.4 correspond to cases 1, 2,

    3, 4 and 5 mentioned in the previous section. All the data used from design II and III

    (Figure 4.1) correspond to the cylindrical sections with Dc diameter, while the data

    shown for design IV correspond to the small angle tapered section.

    The results display the swirl intensity versus the dimensionless axial position,

    where z is the axial distance from the characteristic diameter (see Figure 3.1).

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    0

    1

    2

    3

    4

    5

    6

    0 5 10 15 20 25 30 35

    z / Dc

    SwirlIntensity,

    Experimental Data

    LHC Mechanistic Model

    Figure 4.1 Swirl Intensity Prediction - Case 1

    0

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    4

    0 5 10 15 20 25

    z / Dc

    SwirlIntensity,

    Experimental Data

    LHC Mechanistic Model

    Figure 4.2 Swirl Intensity Prediction - Case 2

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    0

    1

    2

    3

    4

    5

    6

    0 5 10 15 20 25 30 35 40 45

    z / Dc

    SwirlIntensity,

    Experimental Data

    LHC Mechanistic Model

    Figure 4.3 Swirl Intensity Prediction - Case 3

    0

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    4

    4.5

    0 5 10 15 20 25 30 35 40

    z / Dc

    SwirlIntensity,

    Experimental Data F=10% (Case 4)

    Experimental Data F=1% (Case 5)

    LHC Mechanistic Model

    Figure 4.4 Swirl Intensity Prediction - Cases 4 and 5

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    4.1.3 Discussion

    It has been experimentally proved by several researchers that the swirl intensity

    decays exponentially with axial position in cylindrical pipes due to the wall frictional

    losses (Chang and Dhir, 1994 and Mantilla, 1998). The results in Figure 4.1 show that the

    swirl intensity in a cylindrical hydrocyclone predicted by the modified model of Mantilla

    (1998) agrees very accurately with the experimental data.

    The model also shows good agreement for the low angle conical sections (Case 3,

    4 and 5). However, there are not sufficient experimental data to ensure the reliability of

    the modified correlation to predict the swirl intensity as a function of a variety of angles.

    In this sense, a question that remains open is to what extent can the correlation predict the

    swirl intensity for larger angles of the conical sections.

    4.2 Velocity Field Prediction

    4.2.1 Experimental Data Sets

    The velocity field predicted by the proposed mechanistic model is compared with

    the same experimental data sets used for the swirl intensity prediction, namely, cases 1, 2,

    3, 4 and 5. The only new sets of data incorporated are from Case 6 (Hargreaves, 1990),

    which is for the same geometry of Case 5 but with a change of the flow rate from 180

    lpm to 150 lpm. In the latter case only the axial velocity was published for the lower

    angle conical section.

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    4.2.2 Tangential and Axial Velocity Results

    Figure 4.1 to Figure 4.5 present the comparison between the data and model

    prediction for the tangential velocity at different axial positions. The y-axis of each chart

    corresponds to the axis of the LHC, and the x-axis represents the radial position. The

    units used originally were conserved, namely, millimeters per second for the tangential

    velocity, and millimeters for the radial position.

    Next, the axial velocities predicted by the model are compared with the

    experimental data at different axial locations in Figure 4.6 to Figure 4.11. In all the charts

    the positive values of axial velocities correspond to downward flow, which is the

    direction of the main flow, while the negative values represent the reverse flow.

    Figure 4.11 shows the axial velocity in the entire cross sectional area of LHC. In

    the rest of the figures, the axial velocity is only shown from the center line to the wall of

    the LHC.

    4.2.3 Discussion

    In general, the model predictions are close to the experimental data. The closer

    the prediction of the velocities is to the real phenomenon, the better will be the prediction

    of the separation efficiency and the pressure drop.

    Tangential Velocity

    The experimental data of case 1 (Design II) cannot be predicted accurately by an

    axy-symmetric model. It seems that if the data are shifted to the right, as if to get a zero

    velocity value in the LHC axis, the model predictions will get much closer to the

    experimental data. The non-symmetry of this case compared with the others is one of the

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    reasons that this design had an inferior separation performance to the Design III and IV

    (Colman, 1981) Cases. Except for Case 1, the model predicts with acceptable accuracy

    the tangential velocity at the wall, the peak velocity and the radius where it occurs. The

    experimental data and the model display a Rankine Vortex shape, namely, a combination

    of forced vortex near to the LHC axis and a free like vortex in the outer region.

    It can also be seen from Case 1 that the experimental data and the model

    predictions follow the same tendency as the fluid moves downward. The tangential

    velocity at the wall decreases and the peak velocity value increases approaching the LHC

    axis.

    Axial Velocity

    The mechanistic model performance is excellent with respect to the axial velocity,

    in the downward flow region, and not so good in the reverse flow. Considering the

    calculations that the model follows to compute the separation efficiency, the prediction of

    the reverse flow velocity profile is not so important. What is really important is the

    prediction of the radius of zero velocity since beyond this point the droplet is assumed to

    be separated.

    Further, it can be observed how the model and the experimental data show a

    reduction in the reverse radius as the axial position increases and the swirl intensity

    decreases.

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    z / Dc = 9

    0

    2000

    4000

    6000

    8000

    10000

    12000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelo

    city(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 10.5

    0

    2000

    4000

    6000

    8000

    10000

    12000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 12

    0

    2000

    4000

    6000

    8000

    10000

    12000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    Figure 4.1 Tangential Velocity Prediction - Case 1

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    z / Dc = 15

    0

    2000

    4000

    6000

    8000

    10000

    12000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 18

    0

    2000

    4000

    6000

    8000

    10000

    12000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 21

    0

    2000

    4000

    6000

    8000

    10000

    12000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    Figure 4.1 Tangential Velocity Prediction - Case 1 (Contd.)

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    z / Dc = 24

    0

    2000

    4000

    6000

    8000

    10000

    12000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 27

    0

    2000

    4000

    6000

    8000

    10000

    12000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 28

    0

    2000

    4000

    6000

    8000

    10000

    12000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    Figure 4.1 Tangential Velocity Prediction - Case 1 (Contd.)

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    z / Dc = 10.5

    0

    4000

    8000

    12000

    16000

    0 4 8 12 16

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    Figure 4.2 Tangential Velocity Prediction - Case 2

    z / Dc = 10.5

    0

    4000

    8000

    12000

    16000

    0 4 8 12

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    Figure 4.3 Tangential Velocity Prediction - Case 3

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    z / Dc = 3.75F = 10%

    0

    2000

    4000

    6000

    8000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 7.5

    F = 10%

    0

    2000

    4000

    6000

    8000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 15

    F = 10%

    0

    2000

    4000

    6000

    8000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    Figure 4.4 Tangential Velocity Prediction - Case 4

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    z / Dc = 3.75

    F = 1%

    0

    2000

    4000

    6000

    8000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelo

    city(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 7.5F = 1%

    0

    2000

    4000

    6000

    8000

    0 5 10 15 20 25 30

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 15

    F = 1%

    0

    2000

    4000

    6000

    8000

    0 5 10 15 20 25

    Radius (mm)

    TangentialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    Figure 4.5 Tangential Velocity Prediction - Case 5

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    z / Dc = 10.5

    -6500

    0

    6500

    -30 0 30

    Radius (mm)

    AxialVelocity

    (mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 12

    -6500

    0

    6500

    -30 0 30

    Radius (mm)

    AxialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 15

    -6500

    0

    6500

    -30 0 30

    Radius (mm)

    AxialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    Figure 4.6 Axial Velocity Prediction - Case 1

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    z / Dc = 18

    -6500

    0

    6500

    -30 0 30

    Radius (mm)

    AxialVelocity

    (mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z / Dc = 21

    -6500

    0

    6500

    -30 0 30

    Radius (mm)

    AxialVelocity(mm/sec)

    Experimental Data

    LHC Mechanistic Model

    z /


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