CALDENTEY, C., J.; A Mechanistic Model for Liquid Hydrocyclones (LHC)

THE UNIVERSITY OF TULSA 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

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 (180 words) 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%.

iii

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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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. v

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

6

CHAPTER II LITERATURE 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 III LHC MECHANISTIC MODEL

23

3.1 Overview

23

3.2 Swirl Intensity

26

3.3 Velocity Field

29 vi

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 IV RESULTS AND DISCUSSION

44

4.1 Swirl Intensity Prediction

44

4.1.1 Experimental Data Sets

44

4.1.2 Results

47

4.1.3 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 V SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

82

5.1 Summary and Conclusions

82

5.2 Recommendations

85

NOMENCLATURE

88

REFERENCES

91 vii

LIST OF FIGURES

Figure 1.1 LHC Hydrodynamic Flow Behavior ............................................................... 3 Figure 1.2 Colman and Thew’s 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 Colman’s 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 viii

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, å u - Case 7 ..................................................................... 71 Figure 4.21 Migration Probability Curve - Case 8 ......................................................... 72 Figure 4.22 Underflow Purity, å 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

ix

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

x

LIST OF TABLES

Table 3-1 Drag Coefficient Constants............................................................................ 36 Table 4-1 Geometrical Parameters of Colman’s Designs (1981).................................... 46 Table 4-2 Operational Conditions of Colman’s 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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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. 1

2 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

3 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

4 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 Thew’s (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.

5

Figure 1.1 Colman and Thew’s Hydrocyclone Design

Young et al. (1990) achieved similar results to Colman-Thew’s 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 1.5º to 6º. 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;

6 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.

7 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.

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 8

9 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 solidliquid 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

10 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):

Wr n = Constant

(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

11 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

12 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

13 (LDA) data. LDA has many advantages over other techniques. It is not as tedious as the optical technique and does not cause flow distortion like the Pitot tubes (Chakraborti and Miller, 1992). Many researchers have used this technique to measure the velocity field and the turbulence intensities (Dabir, 1983; Fanglu and Wenzhen, 1987; Jirun et al., 1990; Fraser and Abdullah, 1995).

2.2

Liquid Hydrocyclones, LHC

The modern renaissance in deoiling hydrocyclones was instigated by Martin Thew and Derek Colman at the University of Southampton (Young et al., 1990). Works published in the First International Conferences on Hydrocyclones (Colman et al., 1980; Colman and Thew, 1980; Thew et al., 1980) were the motivators that generated the application of this technology in the field a few years later. Today, the deoiling hydrocyclones have become the standard world-wide equipment for cleaning produced water offshore and have extensive use on-shore (Thew and Smyth, 1997). A thorough review of LHC is found in Thew and Smyth (1997). As mentioned in this work, in spite of the steady advances in analytical theories, and more recently CFD simulations, hydrocyclones have principally been designed based on experimentation. A comprehensive review on LHC is given below in four sections. Starting from the modeling, field applications and experimental studies and finally, the velocity field measurements and CFD simulations. These topics are treated separately because of their significance to the present study.

14 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.

15 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.

16 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):

ε=

Oil Discarded Oil Pr esented at the Feed

(2.2)

Utilizing continuity equation:

k oQ o = k i Q i − k u Q u

(2.3)

17 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:

ε=

k o Qo kQ = 1− u u k i Qi k i Qi

(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:

ε u = 1−

ku ki

(2.5)

If ku tends to 0, å u becomes 1. On the other hand, if ku is equal to ki, åu 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, åu. Important observations can be made from this study. The å 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 å u were found for different split ratios. However, for split ratio values less than 2.5% it was found that åu begins 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

18 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

∆ P ∝ Q in

(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.

19 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.

20 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 Colman’s (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).

21

Figure 2.1 Axial Velocity Profile From Colman (1984)

22 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 author’s 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.

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 drop–flow 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 23

24 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.

Swirl Intensity

Velocity Field

Droplet Trajectories

Pressure Drop

Separation Efficiency

Migration Probability

Underflow Purity

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

25 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

26 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):

2πρc ∫0 uwrdr Ω= πρc R 2z U 2 avz Rz

(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

27 correlation of Mantilla’s model is developed. The modified correlation takes into account the semi-angle, , of a tapered section, resulting in:

M  Ω = 1.48 t I 2  (1 + 1.2 tan( β) 0.15 )  MT  0.93

0.35 0.16   1  M    1   z 0.7 t 4 0.12       (1 + 2 tan( β ) ) I EXP −     2  M T   Re z   Dc  

(3.2)

This correlation was developed using experimental data for small semi-angles, , from 0º to 0.75º. However, a good prediction has also been obtained for 3º 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 Thew’s 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 Thew’s Design, and at the top diameter of the 3º tapered section of the Young’s Design (see section 1.3 LHC Geometry); z is the axial position starting from Dc. Mt is the ratio of the momentum flux at the inlet slot to the axial momentum flux MT at the characteristic diameter position, calculated as follows:

& Vis & / ρc A is A c Mt m m = = = & U avc m & / ρc A c A is MT m

(3.3)

& is the where Vis is the velocity at the inlet, Uavc is the average axial velocity at Dc, m mass flow rate, Ac is the cross sectional area at Dc and Ais is the inlet cross sectional area.

28

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:

Re z =

where

c

ρc U avz D z µc

(3.4)

is the viscosity of the continuous fluid.

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

I = 1 − EXP( − n ) where n = 1.5 for twin inlets and n = 1 for involute single inlet.

(3.5)

29 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.

30

Figure 3.1 Rankine Vortex

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

  r 2  w Tm  1 − EXP − B   = U avc  r     R c      Rc 

(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.

31

Involute Single Inlet:

Twin Inlets:

Tm = Ω

(3.7)

B = 55.7 Ω−1.7

(3.8)

B = 245.8 Ω−2.35

(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.

32

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:

u ( r ) = a 1r 3 + a 2 r 2 + a 3 r + a 4

(3.10)

where a1, a2, a3 and a4 are constants. The boundary conditions considered are: 1.

du ( r = R z ) =0 dr

2. u ( r = rrev ) = 0 3.

du ( r = 0) =0 dr

the velocity is maximum at the wall, zero velocity at the location of reverse flow, rrev, the velocity is symmetric about the LHC axis and

4. 2πρ c ∫ u ( r ) rdr = U avz ρ πR 2z Rz

0

c

Mass Conservation.

Substituting the boundary conditions in Equation (3.10), yields the axial velocity profile, which is a function of the swirl intensity, only:

33

u 2  r  3  r  0.7 =   −   + +1 U avz C  R z  C  R z  C 3

r  C =  rev  Rz 

2

2

   3 − 2 rrev  − 0.7  R z  

rrev = 0.293 Ω0.358 Rz

(3.11)

(3.12)

(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:

v=−

r u tan( β) Rz

(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.

34 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 Vr 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:

dz dz V = dt = z dr dr Vr dt

⇒ z=∫

Vz dr Vr

(3.15)

35 Neglecting the axial buoyancy force (no-slip condition), the droplet axial velocity, Vz becomes the axial velocity of the fluid, u. This simplification is reasonable when the acceleration due to the centrifugal force in the radial direction is thousand times larger than the aceleration of gravity. Due to this, the LHC is not sensitive to external movements and it can be installed either horizontally or vertically. The droplet velocity in the radial direction is equal to the fluid radial velocity, v, plus the slip velocity, Vsr. Rearranging Equation (3.15) yields the total trajectory of the droplet, namely:

u  r= r  ∆ r z = ∑r=r   v + Vsr  2

1

(3.16)

The only unknown parameter in Equation (3.16) is the slip velocity, which can be solved from a force balance on the droplet in the radial direction, as shown Figure 3.2.

Figure 3.2 Forces Acting on a Droplet Assuming a local equilibrium momentum yields: 2 w 2 πd 3 1 2 πd ( ρc − ρd ) = C D ρc Vsr r 6 2 4

(3.17)

where the left side of the equation is the centripetal force, and the right side is the drag force. Solving for the radial slip velocity, results in:

36 1

 4  ρc − ρd  w 2 d 2   Vsr =   3 ρ r C    c D  where d is the droplet diameter;

d

(3.18)

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):

C D = b1 +

b2 b + 32 Red Re d

(3.19)

The coefficients “b” are dependent on the Reynolds Number of the droplets, defined as:

Re D =

ρc d Vsr µc

(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

37 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).

38

Figure 3.1 Droplet Trajectory and Migration Probability

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

ε( d )

=

 0, if rcrit = rrev  2 2  πrcrit − πrrev  2 if rrev < rcrit < R c 2 ,  πR c − πrrev 1, if rcrit = R c

(3.21)

Repeating this procedure for different droplet sizes, the migration probability curve is obtained (Figure 3.2). This function has an “S” shape and represents the separation efficiency,

(d), vs. the droplet diameter, d. It can be seen that small droplets

39 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, å u , can be determined as follows:

εu

∑ ε (d ) V = ∑V i

i

i

i

(3.22)

i

where å 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).

40 3.6

Pressure Drop

The pressure drop from the inlet to the underflow outlet is calculated using a modification of the Bernoulli’s Equation:

1 1 Pis + ρc Vis2 = PU + ρc U 2U + ρc (h cf + h f ) + ρc g sin θ L 2 2 where

c

(3.23)

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; hcf corresponds to the centrifugal force losses and hf is the frictional losses. The frictional losses are calculated similar to pipe flow:

∆ z VR2 ( z ) h f ( z ) = f (z ) D( z ) 2

(3.24)

where f is the friction factor and VR is 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.

f (z ) ∆z ∆z VR2 at (( 2 n − 1) ) 2 D n−1 + D n 2 n=1 2

=∑ m

h f ( conical )

(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:

41

VR2 ( z ) = U 2Z + WZ2

Wz

(3.26)

∫ ∫ Wrdrdφ = ∫ ∫ rdrdφ 2π

Rz

0

rrev

2π

Rz

0

rrev

(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 Hall’s Correlation (Hall, 1957). 1/ 3    ε  10 6   4 + f ( z ) = 0.00551 +  2 x10     D( z )  Re( z )    

where

(3.28)

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:

h cf = ∫r

Ru rev

(nW ) u

r

2

(r)

dr

(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 Bernoulli’s 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

42 pipe flow (Munson et al., 1994). Rigorously, the Bernoulli’s 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.

43

Figure 3.9 LHC Mechanistic Model Code

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:

44

45 §

Colman’s (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 Colman’s Designs (1981)

46 Table 4-1 Geometrical Parameters of Colman’s Designs (1981) Case

Design

Dc(mm)

1

II

58

2

III

3

IV

L

As

1

D2

L2

30Dc

0.125

-

90º

0.5Dc

22Dc

-

-

0.14Dc

30

20Dc

0.0625

10º

10º

0.5Dc

21Dc

2Dc

3Dc

0.14Dc

30

-

0.0625

10º

0.67º

0.5Dc

21Dc

2Dc

3Dc

0.14Dc

2

Ds

Ls

Du

Table 4-2 Operational Conditions of Colman’s Designs (1981) Case

Design

Dc(mm)

Flowrate (lpm)

T (ºC)

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 Colman’s 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.

47 Table 4-3 Geometrical Parameters of Hargreaves (1990) Case 4 and 5

Design IV

Dc(mm) 60

1

10º

D2

L2

Ds

Ls

0.5Dc

15Dc

1.5Dc

1.67Dc

2

0.6365º

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).

48

6

Experimental Data

Swirl Intensity, Ù

5

LHC Mechanistic Model

4

3

2

1

0 0

5

10

15

20

25

30

35

z / Dc

Figure 4.1 Swirl Intensity Prediction - Case 1

4

Experimental Data

3.5

LHC Mechanistic Model

Swirl Intensity, Ù

3 2.5 2 1.5 1 0.5 0 0

5

10

15

20

z / Dc

Figure 4.2 Swirl Intensity Prediction - Case 2

25

49

6

Experimental Data

Swirl Intensity, Ù

5

LHC Mechanistic Model

4

3

2

1

0 0

5

10

15

20

25

30

35

40

45

z / Dc

Figure 4.3 Swirl Intensity Prediction - Case 3

4.5

Experimental Data F=10% (Case 4)

4

Experimental Data F=1% (Case 5) 3.5

Swirl Intensity, Ù

LHC Mechanistic Model 3 2.5 2 1.5 1 0.5 0 0

5

10

15

20

25

30

35

z / Dc

Figure 4.4 Swirl Intensity Prediction - Cases 4 and 5

40

50 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.

51 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

52 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.

53

12000

z / Dc = 9

Experimental Data

Tangential Velocity (mm/sec)

10000

LHC Mechanistic Model

8000

6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

12000

z / Dc = 10.5

Experimental Data LHC Mechanistic Model

8000

6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

12000

z / Dc = 12

Experimental Data

10000

Tangential Velocity (mm/sec)

Tangential Velocity (mm/sec)

10000

LHC Mechanistic Model

8000

6000

4000

2000

0 0

5

10

15

20

25

Radius (mm)

Figure 4.1 Tangential Velocity Prediction - Case 1

30

54

12000

z / Dc = 15

Experimental Data

Tangential Velocity (mm/sec)

10000

LHC Mechanistic Model

8000

6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

12000

z / Dc = 18

Experimental Data

Tangential Velocity (mm/sec)

10000

LHC Mechanistic Model

8000

6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

12000

z / Dc = 21

Experimental Data

Tangential Velocity (mm/sec)

10000

LHC Mechanistic Model

8000

6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

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

55

12000

z / Dc = 24

Experimental Data

Tangential Velocity (mm/sec)

10000

LHC Mechanistic Model

8000

6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

12000

z / Dc = 27

Experimental Data LHC Mechanistic Model

8000

6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

12000

z / Dc = 28

Experimental Data

10000

Tangential Velocity (mm/sec)

Tangential Velocity (mm/sec)

10000

LHC Mechanistic Model

8000

6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

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

56

16000

z / Dc = 10.5

Experimental Data

Tangential Velocity (mm/sec)

LHC Mechanistic Model 12000

8000

4000

0 0

4

8

12

16

Radius (mm)

Figure 4.2 Tangential Velocity Prediction - Case 2

16000

z / Dc = 10.5

Experimental Data

Tangential Velocity (mm/sec)

LHC Mechanistic Model 12000

8000

4000

0 0

4

8

Radius (mm)

Figure 4.3 Tangential Velocity Prediction - Case 3

12

57

8000

z / Dc = 3.75 F = 10%

Experimental Data

Tangential Velocity (mm/sec)

LHC Mechanistic Model 6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

8000

Tangential Velocity (mm/sec)

z / Dc = 7.5 F = 10%

Experimental Data LHC Mechanistic Model

6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

8000

z / Dc = 15 F = 10%

Experimental Data

Tangential Velocity (mm/sec)

LHC Mechanistic Model 6000

4000

2000

0 0

5

10

15

20

25

Radius (mm)

Figure 4.4 Tangential Velocity Prediction - Case 4

30

58

8000

z / Dc = 3.75 F = 1%

Experimental Data

Tangential Velocity (mm/sec)

LHC Mechanistic Model 6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

8000

z / Dc = 7.5 F = 1%

Experimental Data

Tangential Velocity (mm/sec)

LHC Mechanistic Model 6000

4000

2000

0 0

5

10

15

20

25

30

Radius (mm)

8000

z / Dc = 15 F = 1%

Experimental Data

Tangential Velocity (mm/sec)

LHC Mechanistic Model 6000

4000

2000

0 0

5

10

15

20

Radius (mm)

Figure 4.5 Tangential Velocity Prediction - Case 5

25

59

6500

z / Dc = 10.5

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-6500 -30

0

30

Radius (mm)

6500

z / Dc = 12

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-6500 -30

0

30

Radius (mm)

6500

z / Dc = 15

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-6500 -30

0

Radius (mm)

Figure 4.6 Axial Velocity Prediction - Case 1

30

60

6500

z / Dc = 18

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-6500 -30

0

30

Radius (mm)

6500

z / Dc = 21

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-6500 -30

0

30

Radius (mm)

6500

z / Dc = 24

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-6500 -30

0

30

Radius (mm)

Figure 4.6 Axial Velocity Prediction - Case 1 (Contd.)

61

6500

z / Dc = 27

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-6500 -30

0

30

Radius (mm)

6500

z / Dc = 28.5

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-6500 -30

0

Radius (mm)

Figure 4.6 Axial Velocity Prediction - Case 1 (Contd.)

30

62

7000

z / Dc = 10.5

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-7000 0

4

8

12

16

Radius (mm)

Figure 4.7 Axial Velocity Prediction - Case 2

8000

z / Dc = 10.5

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-8000 0

4

8

Radius (mm)

Figure 4.8 Axial Velocity Prediction - Case 3

12

63

6000

Experimental Data

Axial Velocity (mm/sec)

z / Dc = 3.75 F = 10%

LHC Mechanistic Model

0

-6000 0

5

10

15

20

25

30

Radius (mm)

6000

Experimental Data LHC Mechanistic Model

Axial Velocity (mm/sec)

z / Dc = 7.5 F = 10%

0

-6000 0

5

10

15

20

25

30

Radius (mm)

6000

Experimental Data LHC Mechanistic Model

Axial Velocity (mm/sec)

z / Dc = 15 F = 10%

0

-6000 0

5

10

15

20

25

Radius (mm)

Figure 4.9 Axial Velocity Prediction - Case 4

30

64

6000

Axial Velocity (mm/sec)

z / Dc = 3.75 F = 1%

Experimental Data LHC Mechanistic Model

0

-6000 0

5

10

15

20

25

30

Radius (mm)

6000

Axial Velocity (mm/sec)

z / Dc = 7.5 F = 1%

Experimental Data LHC Mechanistic Model

0

-6000 0

5

10

15

20

25

30

Radius (mm)

6000

Experimental Data LHC Mechanistic Model

Axial Velocity (mm/sec)

z / Dc = 15 F = 1%

0

-6000 0

5

10

15

20

Radius (mm)

Figure 4.10 Axial Velocity Prediction - Case 5

25

65

5000

z / Dc = 3.75

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-5000 0

5

10

15

20

25

30

Radius (mm)

5000

z / Dc = 7.5

Experimental Data

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-5000 0

5

10

15

20

Radius (mm)

Figure 4.11 Axial Velocity Prediction - Case 6

25

30

66

5000

Experimental Data

z / Dc = 11.25

Axial Velocity (mm/sec)

LHC Mechanistic Model

0

-5000 0

5

10

15

20

25

30

Radius (mm)

5000

Experimental Data LHC Mechanistic Model

Axial Velocity (mm/sec)

z / Dc = 15

0

-5000 0

5

10

15

20

25

Radius (mm)

Figure 4.11 Axial Velocity Prediction - Case 6 (Contd.)

30

67 4.3

Droplet Trajectory Prediction

The prediction of the separation efficiency is based on the droplets trajectories. Once the velocity field of the continuous phase is known, the slip velocity of the droplets can be calculated and consequently, the trajectory can be determined. Figure 4.1 and Figure 4.2 present the model prediction for the droplet trajectory for a case study. The geometry and operational conditions selected for this analysis are from Case 7, detailed in the experimental data sets of the Separation Efficiency Prediction (Section 4.4.1). Only model predictions are shown for droplet trajectory due to non-availability of data. In Figure 4.1, the path that the different droplet sizes follow within the LHC are shown starting from the radius at Rc only. On the other hand, Figure 4.2 describes the trajectory of a 15 microns droplet only, with different starting trajectory radial positions.

Radius (mm) 0

5

10

0

Axial Position (mm)

Reverse Flow Radius

d=60ìm 350

700

d=45ìm

d=35ìm

1050 d=29ìm d=25ìm

d=15ìm

1400

Figure 4.1 Predicted Droplets Trajectories – Case 7

15

68

Radius (mm) 0

Axial Position (mm)

0

5

10

15

Reverse Flow Radius

350

700

1050

Droplet Size Diameter 15 ìm

1400

Figure 4.2 Trajectories of a 15 Microns Droplet – Case 7

As can be seen from Figure 4.1, the larger droplets are separated easily, but the smallest droplets of 25 and 15 microns diameter are not able to reach the reverse flow region, even when their starting trajectory point is at the wall. Figure 4.2 demonstrates the concept in which the separation efficiency is calculated by the model. A 15 microns droplet is dragged by the fluid when it enters the top cross sectional area near the wall. However, as the droplet is introduced closer to the LHC axis the probability of this droplet to be separated is increased. It can be observed that this droplet reaches the reverse flow region when it is released at a radius equal to 11 mm, which is close to the critical radius, rcrit for this case. If we consider that the tangential velocity increases considerably as it approaches the LHC center, and that the axial velocity behaves in an opposite manner, once the

69 droplet is able to reach these zones of very high centrifugal forces and moderate resident times, it migrates very fast to the center, reaching the reverse flow region and is separated. This is the reason why the case study droplet (15m diameter) passes suddenly from a starting radius where it cannot be separated, to one where it is easily separated.

4.4

Separation Efficiency Prediction

The main objective of the LHC is to produce very clean water through the underflow outlet, while minimizing the amount of water loss that exits with the oil core through the overflow outlet. An expression that helps to evaluate the separation performance is the underflow purity given by Equation (2.2). This variable expresses the ratio of the difference in concentration from the feed to the underflow stream over the feed concentration. Another common way of looking at the efficiency of the hydrocyclone is through the migration probability curve. This kind of chart represents the separation efficiency of a particular droplet size. Both, the underflow purity and the migration probability curve predicted by the model are evaluated through comparisons with published experimental data. In order to calculate the underflow purity, the LHC model needs as an input the feed droplet size distribution which is also shown in the results section.

4.4.1 Experimental Data Sets The experimental data sets used for the separation efficiency comparisons are from four sources:

70 Case 7 is from Colman’s (1981) study. This case is the same as Case 3 but the experimental temperature was 50ºC, which corresponds to a water viscosity of 0.55 cp. The dispersed phase used were solid particles (polypropylene) instead of oil droplets. The density of these particles at the operational temperature is 0.89 g/cc. Case 8 includes the geometry and operational conditions reported by Wolbert et al. (1995). Using Figure 4.1 as a reference, the following table gives the details of the geometrical parameters for this case. The flow rate used in this case was 32 lpm with an oil dispersed phase density of 0.902 g/cc. The feed droplet size distribution for Case 7 and 8 are shown in Figure 4.20 and Figure 4.4 respectively. Table 4-1 Geometrical Parameters, Wolbert et al. (1995) Case

Design 8

IV

Dc(mm) 20

1

10º

2

0.75º

D2

L2

0.5Dc

30Dc

Ds 2Dc

Ls 2Dc

Di 0.35Dc

where Di is the inlet diameter of one of the twin inlets (mm)

Cases 9 to 23 are part of the set of experiments published by Colman et al. (1980) and Case 24 by Colman et al. (1984). These experimental data sets are for the same configuration as the one shown in the Table 4-1 and the characteristic diameter and operational conditions are reported together with the results in Table 4-1.

4.4.2 Migration Probability and Underflow Purity Results The results of Cases 7 and 8 can be seen in Figure 4.1 to Figure 4.4. Initially the migration probability curve is illustrated, followed by a chart that contains the underflow purity and the experimental feed droplet size distribution, and the underflow droplet size distribution, as calculated by the model. Next, the feed droplet size distribution of Cases

71 9 to 24 are plotted for two different oils, from Kuwait and Forties, as shown in Figure 4.5 and Figure 4.6.

100

Separation Efficiency (%)

90 80 70 60 50 40 30 20

Experimental Data 10

LHC Mechanistic Model

0 0

8

16

24

32

40

48

56

64

Droplet Diameter (microns)

Figure 4.1 Migration Probability Curve - Case 7

Volumetric Fraction (%)

40

Experimental å u = 91% LHC Mechanistic Model å u = 93%

30

20

10

0 12.75

15.9

20

25

31.4

39.6

50

62.6

79.8

100

129

Droplet Diameter (microns) Feed (Experimental)

Underflow (LHC Mechanistic Model)

Figure 4.2 Underflow Purity, å u - Case 7

72

100

Separation Efficiency (%)

90 80 70 60 50 40 30 20

Experimental Data

10

LHC Mechanistic Model

0 0

8

16

24

32

40

48

56

64

Droplet Diameter (microns)

Figure 4.3 Migration Probability Curve - Case 8

Volumetric Fraction (%)

50

Experimental å u = 81% LHC Mechanistic Model å u = 76%

40

30

20

10

0 1.5

5

7

10

14

20

28

40

56

Droplet Diameter (microns) Feed (Experimental)

Underflow (LHC Mechanistic Model)

Figure 4.4 Underflow Purity, å u - Case 8

73

25

Mean Drop Size = 17 ìm Oil Density = 0.84 g/cc Volumetric Fraction (%)

20

15

10

5

0 3.5

5

6.4

8.12

10

12.75 16.25

20

25.15 31.62 39.75

50

81.23 81.23

Droplet Size (microns) 25

Mean Drop Size = 35 ìm Oil Density = 0.84 g/cc

15

10

5

0 3.5

5

6.4

8.12

10

12.8 16.3

20

25.2 31.6 39.8

50

63.7 81.2 100

115

Droplet Size (microns) 30

Mean Drop Size = 70 ìm Oil Density = 0.84 g/cc

25

Volumetric Fraction (%)

Volumetric Fraction (%)

20

20

15

10

5

0 5

6

8

10

13

16

20

25

32

40

50

64

81

100

115

135

Droplet Size (microns)

Figure 4.5 Droplet Size Distributions for Kuwait Oil (Colman et al., 1980)

74

25

Volumetric Fraction (%)

Mean Drop S ize = 41 ìm Oil Density = 0.87 g/cc 20

15

10

5

0 6.4

8.12

10

12.75 16.25

20

25.15 31.62 39.75

50

63.73 81.23 100

115

Droplet Size (microns)

Figure 4.6 Droplet Size Distribution for Forties Oil (Colman et al., 1980)

The experimental underflow purity results and the one computed by the LHC model for cases 7 to 24 are described in Table 4-1. The migration probability curve for the Cases 16, 18, 20, 23 and 24 are reported in Figure 4.7 to Figure 4.9 as examples.

75 Table 4-1 Underflow Purity Results Cases 7 to 24

Case 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Dc (mm)

Flowrate (lpm)

30 20 30 30 30 30 30 58 58 58 58 58 58 58 58 58 58 51

60 32 60 40 50 60 70 160 190 220 250 220 250 220 250 220 250 150

Oil Mean Experimental Model Density Drop Size Underflow Underflow (g/cc) Purity (%) Purity (%) (ìc) 0.89 0.902 0.87 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.87 0.87 0.84

91 81 88 78 82 84 88 72 74 78 81 43 47 96 97 80 -

41 35 35 35 35 35 35 35 35 17 17 70 70 41 -

93 76 86 73 79 83 86 65 69 74 79 42 47 92 93 76 -

100

Separation Efficiency (%)

90 80 70 60 50 40 30 20

Experimental Data 10

LHC Mechanistic Model

0 0

8

16

24

32

40

48

56

Droplet Diameter (microns)

Figure 4.7 Migration Probability Curve – Cases 16, 18 and 20

64

76

100 90

Separation Efficiency (%)

80 70 60 50 40 30 20

Experimental Data

10

LHC Mechanistic Model

0 0

8

16

24

32

40

48

56

64

72

80

Droplet Diameter (microns)

Figure 4.8 Migration Probability Curve – Case 23

100

Separation Efficiency (%)

90 80 70 60 50 40 30 20

Experimental Data LHC Mechanistic Model

10 0 0

8

16

24

32

40

48

56

Droplet Diameter (microns)

Figure 4.9 Migration Probability Curve – Case 24

64

72

77

LHC Mechanistic Model Underflow Purity (%)

100

Average Relative Error (%) = -3.7 Average Relative Absolute Error (%)= 4

90

80

70

60

50

40 40

50

60

70

80

90

100

Experimental Underflow Purity (%)

Figure 4.10 Comparison of Model Underflow Purity and Experimental Data Set

4.4.3 Discussion Figure 4.10, incorporates all the data sets, showing how well the model predicts the underflow purity for such broad range of conditions. The characteristic diameter of the hydrocyclone varied from 20 to 58 mm, and the flowrate studied from 32 to 250 lpm. The average relative absolute error is 4% and the average relative error is –3.7%, which confirms what can be seen in the chart, that the model is consistently under predicting the underflow purity. This difference stems from the discrepancy between the theoretical and experimental values of the migration probability. In most of the cases the model estimates lower separation efficiency for droplet diameters less than 25 microns. On the other hand the biggest droplets that have experimental efficiencies around 95%, are predicted by the model to be completely separated. Both effects may compensate each other and give more realistic results.

78 It is interesting to note that the only case in which the model predicts higher efficiency is Case 7, which is the case with a solid particle dispersion. This may suggest that the under-prediction of the separation efficiency by the model may be due to the effect of droplet coalescence. From Table 4-1 it can be concluded that the model and the experimental data follow the same trends: §

Higher droplet diameters produced better underflow purities (Case 17, d35, å u =81% and Case 19, d17, å u =43%).

§

The separation efficiency increases at higher flow rates (see Cases 10 to 13).

4.5

Pressure Drop Prediction

In this section the pressure drop from the inlet to the underflow outlet is calculated by the model and compared with the experimental data described as follows.

4.5.1 Experimental Data Sets Young et al. (1990) conducted experiments using Colman and Thew’s (1988) design. The configuration that the authors used consisted of an involute inlet of area equal to 197 mm2. Referring to design IV of Figure 4.1, the following table gives the details of the geometrical parameters. Table 4-1 Geometrical Parameters, Young et al. (1990) Case 25

Design IV

Dc(mm) 35

1

10º

2

0.75º

D2

L2

0.73Dc

26Dc

Ds 2Dc

Ls 2Dc

79 The source of Case 26 is from the specification of a commercial 2 inch hydrocyclone called MQ HYDRO-SWIRL manufactured by MPE/NATCO. This is the only experimental data collected for a design different from Colman and Thew’s and is the only case in which the model is evaluated against a LHC with a semi-angle greater than 0.75º. This 3º semi-angle LHC was designed by Young et al. (1993) and the differences with Colman and Thew’s (1988) design is explained in section 1.3 LHC Geometry. The last experimental data set used for the pressure drop is from Colman and Thew (1983). Cases 27 to 35 are defined with the geometry of Table 4-1 and the experimental results are provided in Table 4-1. In this study the effect of the viscosity of water on the pressure drop was studied by varying the temperature.

4.5.2 Results The pressure drop comparison between the data and model predictions are given in Figure 4.29, Figure 4.30 and Table 4-7.

160

Flowrate (lpm)

120

80

40

Experimental Data LHC Mechanistic Model 0 0

40

80

Pressure Drop (psi)

Figure 4.1 Pressure Drop Prediction – Case 25

120

80

300

Flowrate (lpm)

250

200

150

100

50

Experimental Data LHC Mechanistic Model

0 0

50

100

150

200

250

Pressure Drop (psi)

Figure 4.2 Pressure Drop Prediction – Case 26

Table 4-1 Pressure Drop – Cases 27 to 35

Case

Dc (mm)

Water Density (g/cc)

Flowrate (lpm)

Water Viscosity (cp)

T (ºC)

Experimental LHC Mechanistc Model Pressure Drop Pressure Drop (Psi) (psi)

27

58

250

1.00

1.23

12

32.63

31.37

28 29

58 58

250 250

1.00 0.99

1.27 0.55

11 50

32.49 36.98

31.08 38.87

30

58

220

1.05

1.32

14

35.24

24.40

31

58

220

1.00

1.14

15

27.56

23.93

32

30

60

1.00

1.10

16

19.14

20.86

33 34

30 30

40 60

1.00 0.99

1.10 0.55

16 50

7.83 26.83

8.17 25.42

35

30

50

0.99

0.55

50

17.55

16.72

The model is now evaluated with the results of Cases 25 to 35 in Figure 4.3.

81

LHC Mechanistic Model Pressure Drop (psi)

210

Average Relative Error (%) = -7.9 Average Relative Absolute Error (%)= 11.1

180

150

120

90

60

30

0 0

30

60

90

120

150

180

210

Experimental Pressure Drop (psi)

Figure 4.3 Comparison Between Pressure Drop Model and All Experimental Data

4.5.3 Discussion Satisfactory results for pressure drop prediction were achieved by the model for the Colman and Thew’s Design (Case 25) and in Young’s Design (Case 26). Figure 4.3 shows the evaluation of the model for all the data gathered. The average relative absolute error equal to 11.1% and average relative error equal to –7.9% are evidences of good performance of the model. The pressure drop model exhibits correct sensitivity to the fluid properties. Table 4-1 shows that the experimental data and the model predictions have the same trend with respect to the variation of the water viscosity.

CHAPTER V

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

5.1

Summary and Conclusions

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. A user friendly computer code is developed based on the proposed model. The code provides easy access to the input data and very fast output, and can be used for the design of LHC by the industry. The prediction of the proposed model are compared with elaborated 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. A summary of the tasks performed during this study and the most important conclusions are described as follows. §

A set of correlations are developed to predict the hydrodynamic flow behavior of the continuous phase in the LHC. The swirl intensity, which is the ratio of tangential momentum flux to the average axial momentum flux, can be predicted using a modified form of Mantilla (1998) model, incorporating the semi-angle of a conical section and adjusting the inlet factor for LHC geometry. Good agreement with experimental data is 82

83 observed for a small angle range, from 0º to 0.75º. These are the range of values used in the Colman and Thew (1988) Design. §

It is confirmed that the swirl intensity defines the velocity field within the LHC. The tangential velocity exhibited a forced vortex near the axis and free-like vortex in the outer region. This behavior and the order of magnitude are well predicted by the model, utilizing some of the parameters of the Rankine Vortex Equation used by Mantilla (1998) and Algifri et al (1988). The axial velocity, which shows a reverse flow in the core region, is predicted by a third order polynomial equation, as suggested by Mantilla (1998). A modification of the relationship between the reverse flow radius and the swirl intensity is proposed. The prediction of the downward flow by the model is excellent as opposed to the reverse upward flow. No attempt to correct this is done in this study mainly because the critical parameter considered by the model is only the downward flow, where the separation is achieved. The radial velocity is predicted using the continuity equation and wall conditions suggested by Wolbert et al. (1995). There are no data available for this velocity component.

§

A droplet trajectory analysis is developed assuming local momentum equilibrium. The only forces acting on the droplet are the centripetal and drag forces in the radial direction. For simplification it is assumed that the droplet moves at the fluid velocity in the axial and tangential directions.

84 §

Based on the droplet trajectory the separation efficiency of the LHC is determined using a similar procedure proposed by Wolbert et al. (1995). The underflow purity can be computed for a given feed droplet size distribution. Through comparison with 17 cases, where the characteristic diameter of the hydrocyclone, Dc, varies from 20 to 58 mm and the flowrate ranges from 32 to 250 lpm, the model predicts the underflow purity with an average relative absolute error of 4%. One of these cases is the study by Wolbert et al. (1995), where their model predicted 90% of underflow efficiency, while the experimental results reported 81%. However, the proposed LHC model predicted 76% underflow efficiency for this same case. This may suggest that in general the present model predicts a more realistic velocity field within the LHC.

§

Based on the velocity field of the continuous phase and using an energy balance equation, the pressure drop is predicted by the model. Comparison with 20 data points reveals an average relative absolute error of 11.1% and an average relative error equal to –7.9%. The pressure drop is compared not only with the Colman and Thew’s Design but also with the Young’s (1993) Design, and good results are also achieved for the latter. It is important to mention that Young’s design has a conical section of a 3º semi-angle, what goes beyond the range for which the velocity correlation was developed.

85 After a critical analysis of several experimental data available for the LHC, it is possible to conclude that the LHC mechanistic model predicts with a good confidence level the performance of liquid hydrocyclones with geometrical proportions similar to the Colman and Thew’s (1988) Design.

5.2

Recommendations

The developed mechanistic model has proven to be a good tool to predict the performance of various LHC sizes, for Colman and Thew’s Design. Unfortunately, most of the experimental data published to date comes from this LHC Design. In order to use the current model as a design tool, further comparisons with experimental data from different designs are needed. Some recommendations that may improve the performance of the model and help to understand the limitations of its application are as follows. §

Acquire local velocity measurements for the axial and tangential velocity distribution at different tapered section angles, from 0º to 10º semi-angle section. These data can be used to improve the set of correlation that defines the LHC flow field.

§

The axial velocity profile needs to be further investigated, since under high values of swirl intensity double reversal may occur, for which the equation that the model uses will no longer be valid.

§

The model assumes a stable core. However, vortex instability may occur under certain conditions, as confirmed by Weispfenning and Petty (1991). They found that this phenomena is strongly dependent on the swirl

86 intensity and a characteristic Reynolds Number. Knowledge of the swirl intensity values where these undesirable conditions occur will provide a realistic range of applicability of the model. §

This proposed model does not consider recirculation zones or short circuits at the inlet. These two phenomena cause either the return to the main flow of some of the fluid that goes with the oil core to the overflow outlet, or cause the feed to go directly to the reject orifice. These conditions may affect to some degree the separation efficiency, and they have to be included in order to have a more robust model.

§

The model does not consider the overflow to underflow split ratio. This parameter is crucial for a desirable operation of the LHC but does not affect considerably the LHC flow field. At this point the model assumes that the split ratio is sufficient to accommodate the volume of oil that is separated and that the efficiency does not change with the split ratio, as many researchers have reported. This assumption may be true in the typical range of operation of the LHC, namely, 1 to 10 %, but outside this range a change of the velocity field may occur, and that must be accounted for.

§

There is a relationship between the swirl intensity and the reverse flow radius. As shown by the experimental data and followed by the model’s prediction, the reverse flow radius is reduced as the swirl decays. But there is a point where there is no longer reverse flow and still some swirling motion can occur. Under this condition the model will still consider a

87 reverse flow. A proper improvement will be to know for which small swirl intensity values the flow will not exhibit reverse flow and incorporate this aspect in the model. Finally, to use this model as a design tool, a good prediction of the swirl intensity with the axial position for different taper sections is crucial. It is believed that in the small angle tapered section the swirl intensity decreases at a slower rate as compared to a cylindrical section. Nevertheless, a point can be reached at the conical section where lower values of swirl intensity are generated, as illustrated in the next hypothetical diagram.

Figure 5.1 Hypothetical Swirl Intensity Decay

At this point, not sufficient information is available to confirm this notion, but it is important to note that the swirl intensity is crucial for the LHC performance and also for design purposes. If a model is able to predict accurately the swirl intensity, this can be used as a design parameter, where an optimum design will be the one with the highest possible swirl intensity.

NOMENCLATURE

A = cross sectional area B = factor that determines the peak tangential velocity radius CD = drag coefficient d = droplet diameter D = diameter Dc = LHC characteristic diameter f = friction factor h = losses I = inlet factor k = concentration L = length & = mass flow rate m Mt = momentum flux at the inlet slot MT = axial momentum flux at the characteristic diameter position n = centrifugal force correction factor P = pressure Q = volumetric flow rate r = radial position R = LHC radius 88

89 Re = Reynolds Number t = time Tm = maximum momentum of the tangential velocity at the section u = continuous phase local axial velocity U = bulk axial velocity v = continuous phase local radial velocity V = volumetric fraction / velocity Vr = droplet radial velocity Vsr = droplet slip velocity in the radial direction Vz = droplet axial velocity w = continuous phase local tangential velocity W = mean tangential velocity

Greek Letters: = swirl intensity = taper section semi-angle = efficiency / purity / pipe roughness = axis – horizontal angle = viscosity = density

90 Subscripts: av = average c = characteristic diameter location / continuous phase cf = centrifugal crit = critical d = dispersed phase / droplet f = frictional g = gravity acceleration i = inlet is = inlet section o = overflow r = resultant rev = reverse flow u = underflow z = axial position

Abbreviations: CFD = Computational Fluid Dynamics ESP = Electric Submergible Pump LDA = Laser Doppler Anemometry LHC = Liquid Hydrocyclones

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