PI I IIIIIIIIIIIIIIIIIIII “Ila T \ f . f h ‘5'" R " J. {’0’ I '63.}; g r 5 Li P '3'“? 1 . q. : buss-fig I” ‘ 95”!” i ‘ ' ~ I I ' 5:7\‘-' 2:... M J. )1 ' '” “W““M -—--..'.......4 This is to certify that the dissertation entitled MEAN VELOCITY MEASUREMENTS IN A 3"-HYDROCYCLONE USING LASER DOPPLER ANEMOMETRY presented by Bahram Dabir has been accepted towards fulfillment of the requirements for Ph .D. degreein Chemical Engineering 6Zwsum Major professor Date 2:- 2‘0 —- ‘63 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 NASLJ LIBRARIES w... RETURNING MATERIALS: Piace in book drop to remove this checkout from your record. FINES wiii be charged if book is returned after the date stamped beiow. ' Fill “if“? {-3 MI i" 11' ”a? (H 5%.!ng it""17 :‘1 k‘l i“ \I‘i'v Ldzhii . MEAN VELOCITY MEASUREMENTS IN A 3"-HYDROCYCLONE USING LASER DOPPLER ANEMOMETRY BY Bahram Dabir A DISSERTATION Submitted to Michigan State University . in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1983 ABSTRACT MEAN VELOCITY MEASUREMENTS IN A 3"-HYDROCYCLONE USING LASER DOPPLER ANEMOMETRY BY Bahram Dabir Within the class of inertial separators, the hydrocyclone is especially noteworthy because of its simplicity of operation and its many diverse applications. Although a good qualitative under- standing of solid-liquid separations in hydrocyclones is presently available, the relative importance of the various secondary motions on the separation efficiency of a hydrocyclone has not been quanti- tatively considered. . 'In the absence of a particulate phase and an air core, a quan- titative study of the axial and tangential components of the mean velocity in a glass 3"-hydrocyclone using laser doppler anemometry has revealed multiple reverse flows in the vortex core. Flow visuali- zation by dye injection shows that these flows are coherent over a significant portion of the hydrocyclone and that little radial mixing occurs between these secondary flows and the outer helical flow, al though the Reynolds number basedon the entrance diameter and feed flow rate exceeded 10,000. Several different vortex finder designs were studied, including one which satisfies standard optimal scaling rules. It was discovered that a 2:1 contraction in the vortex finder causes four distinct simultaneous countercurrent flows in the conical section of the hydrocyclone. Moreover, for low backpressure opera- tion (< 2 psig), a recirculation zone was detected in the upper part of the hydrocyclone with as many as six reverse flows. For some hydrocyclone designs and operating conditions, a jet-like flow on the axis occurs from the apex region to the vortex finder. This phenomenon partially explains why the design criteria of Rietema are optimal. The measured tangential velocity profiles were inde- pendent of axial location, and showed a characteristic "forced" and “free" vortex behavior in the inner and outer regions, respec- tively. The addition of 100 wppm Separan AP-30 decreased the tan- ‘gential velocity and reduced the magnitude of the reverse flows in the core region. For the hydrocyclone studied, a split ratio from zero to infinity and a feed rate from 150-500 cmZ/sec have little effect on the vortex diameter (= 3 mm) and the conical locus of zero axial velocity in the outer flow domain. Mixing Patterns in a 3"-Hydrocyclone To my family ii ACKNOWLEDGMENTS I wish to express my deepest gratitude to Dr. C.A. Petty for his guidance and assistance throughout this study and in preparation of this manuscript. I also wish to thank Professors R.E. Falco, J.D. Schuur, and P-E. Hood for their ideas, guidance, and assistance during the author's graduate studies. Special thanks are also given to Professor D.R. Nobds of McMaster University who kindly loaned us his 3"-hydrocyclone for this study. The interest in this research by Dr. C.A. Nillus 01’ Dorr-Oliver, Inc. is also appreciated. This research was supported by the National Science Foundation u"tier grant NSF 71-1642 and by the Division of Engineering Research at Michigan State University. This assistance is warmly acknowledged Finally, I would like to thank my friend Dr. Manochehr Hessary for his help and moral support. iii TABLE OF CONTENTS LIST OF TABLES ........................ LIST OF FIGURES ....................... CHAPTER 1. INTRODUCTION ..................... 1.1. The Physical Problem ............. i . 1.2. Background, ................... 1.2.1. Confined vortex flows ........ ’. . 1.2.2. Previous velocity measurements in hydrocyclones . . . ........... 1.2.3. Effect of polymer additives on secondary flows ............. 1.3. Objectives and Scope of This Research ...... EXPERIMENTAL APPARATUS AND PROCEDURES ......... 2.1. Flow Loop .................... 2.2. The Hydrocyclone ................ 2.3. The Laser-Doppler System ............ 2.4. Alignment Procedure for the Laser Doppler Anemometer ................... 2.5. Procedure for Obtaining Good Doppler Signals 2.6. Data Analysis and Consistency Checks ...... FLON VISUALIZATION STUDIES .............. iv Page vii viii (”010‘ 18 23 25 34 39 45 47 55 59 65 CHAPTER ‘1. VELOCITY MEASUREMENTS IN A 3"-HYDROCYCLONE WITH A 2:1 CONTRACTION IN THE VORTEX FINDER ......... 4.1. Axial and Tangential Profiles .......... 4.2. The Effect of Split Ratio on the Velocity Profiles . . . . ................ 4.3. The Effect of Reynolds Number and Backpressure on the Velocity Profiles ............ 4.4. The Effect of Separan AP-3O on the Velocity Profiles .................... VELOCITY MEASUREMENTS FOR DIFFERENT VORTEX FINDER CONFIGURATIONS .................... 5.1. Rietema's "Optimal" Design ........... 5.2. Equal Overflow and Underflow Areas ....... 5.3. The Effect of Vortex Finder Design on Flow Reversals .................... DISCUSSION AND CONCLUSIONS .............. 6.1. Summary Discussion of Results .......... .6.2. Conclusions Based on this Research ....... 6.3. Indirect Conclusions Suggested by the Results of this Research ................ RECOMMENDATION FOR FURTHER RESEARCH ....... -. . . 7.1. Experiments Needed to Complete This Study . . . . 7.2. Problems Which Would Provide New Fundamental Understanding .................. 7.3. Problems Which Could Lead to New Applications . . Page 76 76 85 91 96 100 100 116 125 132 132 136 141 148 148 150 151 vi Page APPENDIX A. APPLICATION OF LASER DOPPLER ANEMOMETRY T0 VELOCITY MEASUREMENTS IN A HYDROCYCLONE ............ 154 A.1. Basic Principles ................ 154 A.2. Frequency Shifting ............... 169 A.3. Corrections Due to Refraction Phenomena . . . . 170 A.3.1. The cylindrical section ........ 172 A.3.2. The conical section .......... 181 A.4. Additional Remarks on Experimental Apparatus . . 193 A.4.1. Light source .............. 193 A.4.2. Optical componénts . .' ......... 194 A.4.3. Data processing ............ 197 A.5. Example Problem: Laminar Flow in a Circular Tube ...................... 201 A.6. Example Problem: Axial and Tangential Velocity Profiles in a Hydrocyclone ........... 204 B. DRAG REDUCTION CHARACTERISTICS OF SEPARAN AP-30 . . . 208 8.1. Viscosity Measurements ............. 208 8.2. Drag Reduction Results ............. 213 C. FLOW CHARACTERISTICS FOR THE 3“-HYDROCYCLONE ..... 218 0. DATA FOR THE 3"-HYOROCYCLONE WHICH SATISFIES . RIETEMA'S DESIGN CRITERIA .............. 225 L IST OF REFERENCES ...................... 242 IVAESLE 1..1. 1.2. 1-3. 1...£5. LIST OF TABLES Summary of problems studied for this research ..... Velocity profiles measured in the hydrocyclone with a 2:1 contraction in the vortex finder Velocity profiles measured in the hydrocyclone with a 2:1 contraction in the vortex finder for 100 wppm Separan AP-30 ..................... Velocity profiles measured in the hydrocyclone which satisfy Rietema's optimal design criteria ....... Velocity profiles measured in the hydrocyclone with equal overflow and underflow areas . . Velocity profiles measured in the hydrocyclone for other vortex finder geometries Components of the T51 laser doppler anemometer used in this study ..................... Scope of flow visualization studies .......... Change in flow topology with average circulation . . . Mean axial velocity data measured in the hydrocyclone which satisfies Rietema's optimal design criteria . . . Mean tangential velocity data measured in the hydrocyclone which satisfies Rietema's optimal design criteria Values of exponent 'm' for various size hydrocyclones Mean velocity data measured in the hydrocyclone which satisfy Rietema's optimal design criteria ....... vii Page 26 29 30 31 32 32 48 66 109 205 207 222 226 FEIGURE 1.1. 1.2. 22.1, 2-2. £2..3. 2-9. 2-10. 2 - 11a-b 2 ~ llc-d LIST OF FIGURES Page Main flow structures in a hydrocyclone ........ 2 Flow reversal observed by Nuttal [1953] for swirling flow in a circular pipe ............... 8 Secondary flows in the core region and on the boundaries in confined vortex flow (from Rosenzweig et al., 1962) .................... 14 Flow loop (nominal lengths given in cm) ....... 35 Flow loop for polymer studies . ........... 37 Dimensions of the hydrocyclone used in this study (nominal lengths given in mm) ............ 40 Side and front view of the test section ....... 41 Vortex finder configurations studied (all dimensions given in mm) ..................... 42 Schematic diagram of test section for laser doppler and flow visualization setting ............ 44 Schematic of laser doppler anemometer for forward mode of operation .................. 46 Shifted and unshifted beam location as seen on the alignment mask .................... 50 Projection of beams to check the crossing of the beams 52 Test section set-up to locate center of the hydrocyclone ..................... 54 Typical doppler signals obtained from the signal processor ...................... 56 Typical doppler signals obtained from the signal processor ...................... 58 FIGURE 2.12. 2.13. 2.14. 3.1. 3.2. 23.3. 3.4. 3 .5. 4-2. 4-3. ix Output of the photomultiplier tube when the beam crossing intersects the hydrocyclone wall ...... Instantaneous axial velocity in the hydrocyclone (illustrative purposes only) ............. Symmetry and Reproducibility of velocity profiles Flow visualization of the core region for Series I . . Flow visualization of composite flows in the core region for 00/0u = 4.0 (Series I) .......... Flow patterns observed in the 3"-hydrocyclone using dye injection . . . . . . . . . . . . . . . . . Flow visualization of the core region for Series II and III ....................... Composite results of flow visualization study for Series II and III .................. Mean axial velocity profiles for 00/0u - 4 and ReF a 30,000 ....... Mean tangential velocity profiles for 00/0u - 4 , and ReF = 30,000 .................. Deviation from free vortex flow in the outer region for a split ratio of 4 and an inlet Reynolds number of 30,000 ..................... Mean axial and tangential velocity profiles at z = 32 cm for Q IQu = 4 and ReF-= 30,000 (0.0 present 5% Average radial velocity profiles for 00/0u = 4 and Resz 30,000 . . ................ The effect of split ratio on at z = 32 cm and ReF a 30,000 . . . . . . . E .......... The effect of split ratio on at z = 32 cm and 8 ReF a 30,000 .................... Mean axial velocity profiles for 00/0u = 0.25 and ReF a 30,000 .................... udy; o,lKnowles, 1971) ...... Page 61 63 67 69 72 73 75 77 79 80 82 84 86 87 89 FIGURE 4.9. 4.10. 4.11. 41.12. 4.13. The effect of Reynolds nunber and backpressure on for 00/0u = 4 at z = 30 cm ........... The effect of Reynolds number and back pressure on for 00/0u = 4 at 20 cm ............. The effect of low back pressure on flow reversals for 00/0u = 4 and ReF a 30,000 at z = 32 cm ....... Mean axial velocity profiles for 100 wppm Sepa an AP-3O with QO/Qu =4 and Rer 24,300 (QF 3 500 cm/sec, T = 12°C) ...................... Mean tangential velocity profiles for 100 wppm eparan AP-BO with Qo/Qu' 4 and Rep = 24,300 (QF a 500 c /sec, 8 12°C ....................... The effect of split ratio on for ReF a 24,200 at z = 20 cm ..................... 24.200 The effect of split ratio on for ReF a at z - 20 cm ......... The effect of Reynolds number on for 00/0u = 4 at z - 20 cm ........ . ............ The effect of Reynolds number on for 00/0u = 4 at z = 20 cm ..................... Mean axial velocity profiles for 00/0u = 0.25 and ReF = 24,200 ..................... Mean tangential velocity profiles for 00/0u = 0.25 and ReF a 24,200 ................... Mean axial velocity profiles for 00/0u = 4 and ReF = 24,300 ..................... Mean tangential velocity profiles for Qo/Qu = 4 an ReF = 24,300 .................. . . . Mean axial velocity profiles for 00/0u = w and ReFe= 24,100 ..................... Mean tangential velocity profiles for QO/Qu = w and ReF = 24,100 ..................... Page 92 93 95 97 98 101 102 105 106 110 111 112 113 114 115 FIGURE 5.11. 5.12. 5.13. 5.14. 5.15. 5.16. 5.17. 5.18. 5.20. 5.21. 5.22. 6.1. 6.2. 7.1. A.1i. xi Average radial velocity profiles for 00/0u = 4 and ReF = 24,300 ...................... Mean axial velocity profiles for 00/0u 8 0.25 and ReF a 24,300 ..................... Mean axial velocity profiles for 00/0u 8 4 and ReF = 24,300 ..................... Mean axial velocity profiles for 00/0u 8 a:and ReF = 24,300 ..................... The effect of split ratio on for ReF 3 24,300 at z 8 32 cm ...................... The effect of Reynolds number on for 00/0u 8 4 at z 8 20 cm . ........... . ........ The effect of Reynolds number on for 00/0u 8 4 at z 8 30 cm ..................... The effect of the degree of contraction in the vortex finder on for 0 l0 8 4 and Re a 24,300 at z 8 32 cm .2. . . .0. V ..... I ......... The effect of the degree of contraction on for 00/0u 8 4 and ReF = 24,300 at z 8 32 cm ....... The effect of the degree of contraction on for 00/0u 8 aland ReF = 24,300 at z 8 32 cm ....... The effect of vortex finder configuration on for QO/Qu 8 4 at z 8 32 cm ................ The effect of vortex finder configuration on for 00/0u 8 4 at z 8 32 cm ................ The effect of polymer (100 wppm Separan AP-30) on for 00/0u 8 4 and ReF a 24,300 ......... Mean axial velocity profiles for 00/0u 8 4, ReF = 24,300, and at z-= 2 cm ....... ' ..... Formation of oil core in a hydrocyclone with a contraction ..................... Dual beam mode .................... Page 117 118 119 120 124 126 127 128 129 131 138 FIGURE A.2. A.3. A.4. A.5. A.6. A.7. A.8. A.9. A.10. A.11. A.12. 8.1. 8.2. A 8.3. 8.4. 8.5. C.1. xii Vector diagram of light propagation ......... Cross section of two beams from the same source The coordinate system used for a fringe system . . . . Beam intersection region in the real fringe system. The inner ellipse represents Eq. (A.17) and the outer ellipse is Ehe locus of points at which the signal falls to e‘ of its maximum ............. Typical signal from the photodetector ........ Refraction of laser beams during axial velocity mea- surement in the cylindrical section ......... Refraction of laser beams during tangential velocity measurements in the cylindrical section ....... Refraction corrections for the cylindrical section . . Refraction of laser beams during axial velocity mea- surements in the conical section ........... Geometry of test ring for LDA measurements of, parabolic profiles .................. Mean velocity profile for fully developed laminar flow in a circular tube ............... Chemical structure of Separan AP-30, a co-polymer of polyacrylamide and polyacrylic acid .......... The effect of polymer concentration on the apparent viscosity of aqueous AP-30 mixtures at different temperatures .................. .. . . The effect of polymer concentration and PH on the apparent viscosity of aqueous AP-30 mixture ..... The effect of PH on drag reduction characteristic of Separan AP-30 (100 wppm) ............... The effect of shear on the % drag reduction for 100 wppm Separan AP-3O ................ Operating characteristics of a 3"-hydrocyclone for Qo/Qu = 0.25 ..................... Page 162 164 183 202 211 215 217 219 xiii FIGURE Page C.2. Operating characteristics of a 3"-hydrocyclone for 00/0u = 4 ...................... 220 0.3. Operating characteristic of a 3"-hydrocyclone for 00/0u 8 w ...................... 221 C.4. Effect of inlet pressure on the Q 0u of a 3"- hydrocyclone with a 2:1 contraction above the vortex finder ........................ 224 CHAPTER 1 INTRODUCTION 1.1. The Physical Problem Solids dispersed in a continuous fluid phase can be separated by several complementary methods which exploit different fundamental principles. Within the class of inertial separators, the hydro- cyclone is especially noteworthy because of its simplicity of opera- tion and its many diverse applications. In a hydrocyclone a cen- trifugal force field is created by the swirling motion of a con- tinuous fluid phase. One of the main flow features in all hydro- cyclones (see Figure 1.1) is the occurrence of a downward helical flow in the outer region and an upward helical flow in the core. Solids having a larger density than the fluid migrate to the outer boundary layer and are removed with the underflow fluid; however, some solids short-circuit to the overflow in a boundary layer which runs across the roof of the hydrocyclone to the vortex finder. This inward radial flow is caused by a pressure gradient induced by the tangential flow. Other secondary flows occur in hydrocyclones as anply illustrated by the frontespiece of this dissertation and the results of a flow visualization study summarized in Chapter 3. The basic ideas currently being used to design hydrocyclones can be traced to the pioneering studies of Kelsall [1952], Bradley 80d l’ulling [1959], and Rietema [1961a-d]. An excellent summary Overflow 00’ PO, AO Vortex finder To: ml Wm n IOIIIOI Inward short-circuit boundary layer across the roof Inner helical flow .._ Outer helical flow and boundary layer ii 0: volumetric flow 1'1 P: pressure (It l" A: cross sectional area I— TIIIII!‘ Apex Underflow 0U. Pu’ Au Figure 1.1. Main flow structures in a hydrocyclone. and analysis of these and other works can be found in Bradley [1965]. Poole and Doyle [1968] and, more recently, Svarovsky [1977] also give good, but brief, reviews of design strategies. Also see Rietema and Verver [1961]. The work of Rietema has led to the general acceptance of an "optimal" design for hydrocyclones. The geometric proportions of the 3"8hydrocyclone used in this study satisfy his criteria. Presumably, such a design minimizes the adverse effects of turbulent mixing on separation efficiency. Most hydrocyclones are presently designed based on experience rather than fundamental principles. The basic interactions between the outer swirling flows and the boundary layer flows have not been exploited systematically. The separation efficiency is certainly affected by multiple recirculation zones and flow reversals on the core axis, yet quantitative understanding of these and other phenomena is still lacking. Perhaps a better understanding would provide some clues as to why highly efficient hydrocyclones for small par- ticle separations cannot be scaled-up. Although the interactions between the various mixing patterns within a hydrocyclone can be influenced significantly by changing geometric scales, another and potentially more flexible possibility exists. For instance, it is widely known (see, especially, Metzner et.al., 1969; and Virk, 1975) that small amounts of a high molecular "91th polymer can have a dramatic effect on certain turbulent and non-turbulent flow structures of a Newtonian fluid. This suggests that: certain combinations of hydrocyclone geometry and fluid rheology may also improve the performance of hydrocyclones. The feasibility of this idea in a 3"-hydrocyclone partially motivated this research. Some initial findings are given in Section 4.4. Several years ago, a preliminary study was initiated in our laboratory to determine if the efficiency of a 10 mm hydrocyclone could be affected by adding a drag reducing polymer to the feed stream. The results of this study (see Dabir et al., 1980) con- firmed the earlier discovery by Wallace [1980] that the addition of a small amount of Separan AP-30 to a 2 wt. % water-clay suspen- sion of a 1n size particles results in significant changes in the separation efficiency. Depending on the sepcific details of how the polymer-water-clay mixtures were prepared, increases or decreases in efficiency by as much as 70% were observed. Thomas [1982] has recently completed another study which basically verifies this para- doxical mixing effect (also see Wallace et al., 1980). Because of the complex nature of the flow patterns within a hydrocyclone, it would be premature to propose a mechanism for the foregoing phenomenon without a quantitative study of the mean velocity field with and without polymer additive. Furthermore, the distinct possibility that Separan AP-30 could also promote stable flocculation of clay particles in a high shear field makes any indirect conclusions to the contrary speculative. Mean velocity measurements using a laser-doppl er technique even without polymer additives should provide new information regarding internal flow Pattierns in hydrocyclones. Earlier measurements are still incomplete and contradictory, so a complete set of velocity data would be wel- comed. Chapters 4 and 5 summarize the behavior of the tangential and axial components of the mean velocity in a 3F-hydrocyclone over a wide range of operating conditions. 1.2. Background Swirling flows have numerous applied applications and the hydrocyclone is only one of many practical devices which exploit the unique features of confined vortex flows. In what follows, a brief review of some flow phenomena characteristic of confined vortex flows relevant to the performance of hydrocyclones is developed (Section 1.2.1). The more fundamental aspects of rotating flows in general are lucidly presented by Greenspan [1968] in his now classical monograph on the topic. Lewellen [1971] gives an excellent discussion of the relevant fluid mechanics related to several impor- tant applications including the cyclone separator and advanced nuclear rocket concepts using vortex confinement. In Section 1.2.2, previous quantitative studies of swirling flows are examined with an emphasis on those specific areas where detailed knowledge is lacking or where inconclusive results need additional clarification. The known effects of polymer additives on flow structures are reviewed briefly in Section 1.2.3. From this discussion, we anticipate that Separan AP-30 could alter some of the secondary flow patterns illustrated in Figure 1.1. 1.2.1. Confined vortex flows Several fundamental properties of confined vortex flows are worth noting. First, because the swirl velocity is large, the mean radial pressure distribution is primarily determined by the centri- fugal force field per unit volume (i.e., p BSEZ.~ ._JL__ 3'. s p r o (101) Furthermore, in rotational flows there is a strong tendency toward two dimensional behavior (see p. 2 in Greenspan, 1968). Within the hydrocyclone, this translates into the important observation that the flow field quickly becomes axisymmetric, even if the fluid is introduced asymmetrically as shown in Figure 1.1. Moreover, the tangential component of the velocity to a good approximation is independent of both the e- and the z-coordinate directions. Of course, this feature is altered dramatically near the boundaries of the hydrocyclone, especially deep in the apex region. The important qualitative conclusion from the foregoing dis- cussion is that the axial variation of the pressure is determined largely by the secondary flows within the central core region and within the boundary layer on the cone wall. The radial distribution of

between the inner vortex core and the outer boundary layer is determined basically by the swirling velocity of the primary f1CMN. If the primary flow tries to conserve the axial component of angular momentum, then r a rapidly from the outer to the inner region of the flow field. Indeed, the inner core pressure could become subatmospheric causing an air core to be sucked into the hydrocyclone through the apex or vortex finder. This phenomenon is well known for hydrocyclones and is clearly documented in the early flow visualization study of Bradley and Pulling [1959]. A similar entrainment of external fluid also occurs for gas cyclones (see Ter Linden, 1949; and Stairmand, 1951). This is a general phenomenon of rotating flows caused by a pressure distribution satisfying, in part, Eq. (1.1). In a comprehensive experimental study of flow reversal in a cylindrical vortex chamber closed at one end, Donaldson and Snedeker [1962] showed that entrainment of surrounding fluid was primarily govered by the relative magnitudes of a characteristic swirl velocity and a characteristic radial velocity. Transition from a single celled vortex structure (no entrainment) to a two- celled structure (entrainment) occurred when the swirl component was approximately three times larger than the radial component. Nuttal [1953] showed that reverse flows are also characteristic of swirling flows in circular pipes. For Reynolds numbers in the range 1 x 104 to 3 x 104, the three types of flow patterns illus- trated in Figure 1.2 were observed. Transition from a one-celled to a two- and, finally, a three-celled vortex structure occurs as the swirl component of the velocity increases. Nuttal [1953] gave no explanation for this phenomenon but,as clearly discussed by Donaldson and Snedeker [1962], the basic underlying physical effect is contained in Eq. (1.1). O— |—.. .._..+.__“_....__ l one cell two cells three cells low swirl medium swirl high swirl Figure 1.2. Flow reversal observed by Nuttal [1953] for swirling flow in a circular pipe. #7’7’77’ Binnie [1957] also investigated the nature of the flow patterns when swirling water passes through a long, circular tube. Several different vortex chambers were studied at low Reynolds numbers (jL3OO). Transition from a single-celled vortex structure to a three-celled structure depends on the Reynolds number and the ratio of swirl to axial flow. For fixed Re, reverse flow was induced by increasing the swirl component of the flow. Nissan and Bresan [1961] examined some of the underlying factors governing flow reversals at high Reynolds numbers in cylinders by measuring pressure and velocity profiles. Because of the appearance of a large stationary bubble on the core axis, they also concluded that a three-celled structure was possible at high Reynolds numbers. Escudier et al. [1980] studied confined turbulent vortex flows by using a laser doppler anemometer and a flow visualization tech- nique. They showed that the size of the exit hole affects the loca- tion and magnitude of the swirl velocity and, thereby, the number of flow reversals possible. For the conditions studied, transition from a two-celled structure to a three-celled structure occurs as the exit hole diameter to vortex chamber diameter increases. The critical ratio was about two. Binnie and Teare [1956] showed that reverse flows were not limited to cylinders by observing a two-celled vortex structure at very high swirl velocities for flow through a conical nozzle; a three-celled structure was not reported. In cyclone separators, Ter Linden [1949], Kelsall [1953], and Ohasi and Maeda [1958] all 10 obtained velocity measurements showing a core flow in the opposite direction to the flow along the conical wall. Kelsall [1953] noted that under certain conditions a three-celled vortex structure was possible at high inlet Reynolds numbers within a 3"-hydrocyclone. Smith [1962] also demonstrated multiple cell phenomena (as many as three) in a gas cyclone separator. Exact solutions to the Navier-Stokes equation exist which show multiple cell behavior (see, esp., Donaldson, 1956; Sullivan, 1959; and Donaldson and Sullivan, 1960). Unfortunately, because important physical effects due to the boundaries have been neglected, these solutions do not provide a quantitative basis for understanding flow reversals; however, some insight from the Navier-Stokes equa- tion does follow by making the equation dimensionless (see, esp., p. 7 in Greenspan, 1968). Basically, two characteristic velocity scales can be identified in confined vortex flows: one related to the swirl component of the velocity and the other to either the radial or the axial flow. For hydrocyclone applications, a characteristic swirl velocity is simply the feed volumetric flow rate divided by the cross sec- tional area of the inlet (see Figure 1.1): Another natural velocity scale would be one related to the overflow rate» Thus, with UZ :-: QolAO , (1.3) 11 the following two dimensionless groups determine the behavior of the velocity field, N1 5 U25 (1.4) e and U Maui. (La 6 The characteristic length scale 0 is the diameter of the hydrocyclone and v is the kinematic viscosity of the fluid. U6 and U2 are simply the bulk average velocities in the feed and overflow lines, respec- tively. N1 is a characteristic Ekman number for hydrocyclones and N2 is a characteristic Rossby number (cf. Greenspan, p. 7). The relevant observation here is that the Ekman number (or, equivalently, the inverse Reynolds number) is extremely small in most cases of interest where the primary effects of rotation are important. In this study, NlReF = DF/D = 0.28. So with ReF a 28,000, N1 a 10's. The Rossby number, however, varies from zero to unity depending on the split ratio 00/0“. The relationship between N2 and Qo/Qu is AF Qo/Qu N = -————-———- . 1.6 2 swan- ‘ ’ For the hydrocyclone used in this research, AF 2 A0. The Ekman number multiplies the highest spatial derivatives it! the dimensionless Navier-Stokes equation. Because N1 is so small, boundary layers must exist somewhere within the flow domain. Thin 12 shear layers are certainly located along the conical surface of the hydrocyclone and these have been analyzed mathematically by 6.1. Taylor [1950]; Binnie and Harris [1950]; Wilks [1968]; Bloor and Ingham [1976]; and, recently, by Sam and Mukherjee [1980]. Vis- cous layers may also occur in the interior to offset the development of very steep velocity gradients in the core region. Thus, with the viscous action concentrated into narrow layers near the solid boundaries (and, perhaps, near the vortex core), the primary flow could easily be dominated by inviscid effects. The major theoretical consequence of the foregoing observation follows by examining the e-component of the Navier-Stokes equation (or, if turbulence is considered, the Reynolds equation). If vis- cous stresses and turbulent stresses are unimportant, then it is easy to show that (see Donaldson and Snedeker, 1962) D ( _ O R r) - , (1.7) where D/Dt s a/at +

-V. This means that the axial_component of the angular momentum is constant in a frame of reference which moves with the local mean velocity. A vortex flow which satisfies Eq. (1.7) is usually referred to as a "free" vortex. Obviously, this result would be rare for a confined vortex but the extent to which deviates from this simple behavior provides some measure Of'the importance of turbulent mixing and viscous transport of momen- tunlacross the primary flow. 13 Heuristic thinking about swirling flows often centers around various mechanisms which attempt to explain how the flow.tries to preserve angular momentum. The interactions between the secondary boundary layer structures and the primary flow are subtle and crucial. This is amply illustrated by Figure 1.3 which shows the results of a flow visualization study by Rosenzweig et al. [1962] in a jet- driven vortex tube. An insightful analysis of this flow by Rosenzweig et al. [1964] concluded that substantial turbulence levels must be assumed to explain these experimental results; boundary layer inter- actions alone were insufficient. However, mass ejection from the boundary layer is still an important process which supports the angular momentum distribution and retards the decaying process. For large Rossby numbers, Lewellen [1965] showed theoretically that axial decay of angular momentum retards the axial velocity on the axis and that the motion illustrated by Figure 1.3 is forced to be two-dimensional except in thin shear regions. Lewellen concluded that flow with a large swirl component offers a high resistance to radial flow. If axial variations in occur, then this happens in a very thin shear layer (Ekman layer) which offers little resis- tance to radial flow. Thus, the physical picture of the flow pattern, which emerges from Lewellen's analysis and which is supported by .experimental observation, is that the fluid seeks regions in the physical domain having a relatively low centrifugal force field. In this way, the circuitous secondary flows (see Figure 1.3) support the basic swirl pattern by transporting angular momentum from the c; /11‘T all LAI‘AIIJ‘L‘LIIA‘A‘AgkIIII“ 1 Figure 1.3. 14 111111111“:;; Region of low resistance to radial flow 1‘7 L Region of high k resistance to , radial flow I '! [ mam k WW4” , i : 3-celled vortex flow \ N I N N Mass ejection from the 7 \ Ekman layer Region of low resistance to radial flow Secondary flows in the core region and on the boundaries in confined vortex flow (from Rosenzweig et al., 1962). 15 outer region to the inner core by radial convection. Apparently,- the radial velocity distributes itself so the swirling motion is approximately two dimensional. Within hydrocyclones, three regions of low centrifugal force exist: the conical boundary layer, the roof of the hydrocyclone, and the central core region. As suggested by Figure 1.1, fluid is injected tangentially at the outer wall and moves radially to the overflow and underflow through these three regions. Based on the analysis of Lewellen, it should not be surprising that the under- flow fluid is carried to the apex region by the boundary layer. This is generally understood in the literature and is supported also by Taylor's earlier analysis of spray atomizers. However, what is not generally recognized is that a significant fraction of the overflow fluid arises from an ejection of mass from the conical boundary layer deep in the apex region (see, esp., Bhattacharyya, 1980a and b) rather than by a weak radial flux counter to the cen- trifugal force field across the primary swirling motion. The basic physical mechanism which causes this in an Ekman layer (see Fig- ure 1.3) is also operative here. For solid-liquid separations, the ejection of fluid containing a high concentration of solids from the conical boundary layer into an upward directed flow toward the vortex finder could easily degrade the performance of a hydrocyclone. Indeed, this phenomenon may be more significant than the usual short- circuit effect on the roof of the hydrocyclone. 16 Bradley and Pulling [1959] studied the flow patterns inside a 3"-hydrocyclone using a flow visualization technique for different size vortex finders and cone angles. At a flow rate of 500 cm3/sec and a high split ratio, the flow patterns observed were similar to those reported earlier by Kelsall [1953], viz., an outer downward helical flow surrounding an inner upward helical flow. A stagnant zone surrounding the core indicated a region of negligible axial and radial velocities. Such a region, which has been observed by many researchers, is commonly referred to as the "mantle". Bradley [1959] observed that the “mantle" is basically unaffected by changes in the vortex finder dimensions, provided the overflow diameter is Smaller than the "mantle" diameter. For large overflow diameters, the radial velocity is too large for a "mantle“ to exist. The “mantle“ diameter increases as the cone angle increases (Bradley and Pulling, 1959) and, for large cone angles, double "mantles" can occur. Earlier, Fontein and Dijksman [1953] suggested that such multiple recirculation zones exist below the roof of the hydrocyclone. These may result from an ejection of mass by the radially directed Ekman layer on the roof, or from a Taylor-like instability of the primary swirling motion. A specific mechanism for multiple circulation patterns in hydrocyclones has not been identified. The swirling flow in the core region of a vortex tube (or hydrocyclone) can undergo sudden and dramatic changes involving flow reversals. This phenomenon, termed "vortex breakdown", is 17 often characterized by a region of closed circulation with a free stagnation point and surface of zero axial velocity perpendicular to the flow axis. The recirculation region appears as a hemispherical zone of almost stationary fluid relative to the surrounding flow field; As the swirl component of the velocity increases, the iso- lated fluid region stretches along the axis of the vortex tube. Three types of breakdown phenomena have been classified in the litera- ture: double-helix, spiral form, and axisymmetric form (see, esp., Sarpkaya, 1971). The axisymmetric vortex breakdown may be a result of the first two types, or it may form directly by the swelling of the vortex core. The vortex breakdown phenomenon involves very sensitive flow structures and should be studied by a nonintrusive technique such as laser doppler anemometry together with a passive flow visualiza- tion technique (see, e.g., Harvey, 1962;Cass.idy, 1970; Sarpkaya, 1971; and Faler and Leibovich, 1978). Some summary conclusions regarding this phenomenon are as follows: (1) The primary parameter which determines vortex breakdown is the ratio of the swirl velocity to the axial velocity (i.e., an inverse Rossby number). Breakdown occurs at a critical Rossby number or, equivalently, swirl angle. (2) The location and occurrence of a vortex breakdown is closely associated with the degree of contraction of the vortex tube and the adverse pressure gradient along the axis. 18 (3) The transition from non-reversal to reversal flow occurs smoothly rather than abruptly as in an instability (see, esp., Harvey, 1962). (4) The swirl angle and adverse pressure gradient are not the only parameters controlling vortex breakdown. The boundary layers along the divergent walls (see the paper by Sarpkaya, 1974) play an important and unpredictable role in the break- down process. Vortex breakdown due to an amplification of small disturbances (wave phenomenon) was first proposed by Squire [1960] and developed by Benjamin [1962] in a sequence of papers. Another approach, developed by Hall [1966], argues that small axial variations in the swirl velocity (due to decay or contraction) can produce sig- nificant changes in the axial velocity. Thus, for a given combina- tion of swirl and axial velocity, a critical swirl angle exists which induces an infinitesimal expansion of the vortex core. This makes a 172—‘0 and gig)”; which leads to stagnation on the axis and ultimate breakdown of the vortex flow. 1.2.2. Previous velocity measurements in hydrocyclones With the exception of the entry region, the flow pattern in '3 NYdrocyclone is axisymmetric. Therefore, it is natural to resolve 19 the mean velocity field into radial, tangential, and axial components: SE? = (r,z)§r + (r,z)ge + (r,z)_(=_:_z . (1.7) Here gr, 96’ ands.z are the unit base vectors for cylindrical geometry. Eq. (1.7) states explicitly that the mean velocity com- ponents do not depend on the e-coordinate (axisymmetry) and time (statistically stationary flows). The instantaneous velocity field satisfies the full Navier-Stokes equation and the continuity equa- tion for incompressible fluids (see p. 85 in Bird, Stewart, and Lightfoot, 1960). The mean velocity satisfies the so-called Reynolds equation for turbulent flows; for a specific application of Reynolds equation to gas cyclones see Boysan et al. [1982]. Based on the discussion in Section 1.2.1, we anticipate that experiments will verify some additional kinematic properties of guy. For instance, if the swirling flow tries to maintain a two dimensional behavior, then the magnitude of the radial component in the primary flow field should be much smaller than either the tangential or axial components. Furthermore, it is expected that 3 ~ in the primary flow field and that the radial distribution of (r,z) is determined by Eq. (1.1). Understanding the interplay among all the velocity components where three dimensional behavior is. expected is an important and challenging task for the design 20 engineer. Unfortunately, little attention has been given to this aspect of the problem. Standard reviews of the velocity field (see, for example, Chapter 3 in Rietema and Verver, 1961; Chapters 3 and 4 in Bradley, 1965; and Chapter 6 in Svarovsky, 1977) emphasize the behavior of SE? within the primary flow (see Figure 1.1) measured by Kelsall [1952] using an optical technique based on flow visualization. Kelsall's study has had a major impact on the development of hydrocyclones over the past thirty years. His velocity data has been the major touchstone for theoretical studies on hydrocyclones (see, e.g., the analyses of Bloor and Ingham over the last ten years and the recent work of Bhattacharyya, 1980a and b). Two other quan- titative velocity studies in hydrocyclones using a non-intrusive flow visualization technique have been reported (Ohasi and Maeda, 1958; and Knowles et al., 1973). These experimental studies also emphasized the primary flow domain and little insight regarding the boundary layer flows or the central core region can be developed from these data. The 3"-hydrocyclone operating with an air core studied by Kelsall [1952] unfortunately had an unusually long vortex finder with a diameter much smaller than the "optimal" one recommended later by Rietema [1961]. The motion of small alumina particles (5.9. 8 2.7) was followed to determine and . The radial component of the velocity was calculated by using the continuity equation. 21 Knowles et al. [1973] studied a 3"-hydrocyclone operating without an air core. Their design, unlike Kelsall's, satisfied the scaling rules of Rietema with one unfortunate exception in the vortex finder (see Figure 2.3 and Section 5.3 of this dissertation). The experimental methodology of visualizing the motion of a seed particle is similar to Kelsall's. Here high speed cine photography was used to follow anisole particles (5.9. a 0.993, particle size a 50-300 u). The basic results obtained earlier by Kelsall and, more recently, by Knowles are similar. The main flow feature observed consists of a strong downward velocity in the outer region close to the conical wall and an upward flow close to the core. A surface of zero axial velocity is located quantitatively in both studies. Along the rim of the vortex finder a strong downward flow was quan- tified. Presumably, this flow originated as an ejection from the short-circuit Ekman layer on the roof of the hydrocyclone.‘ Below the vortex finder and close to the central axis, the tangential velocity profiles reported by Kelsall increase with the radial coordinate; Knowles' data are inconclusive. Thus, for a hydrocyclone operating with an air core, a region exists where the fluid rotates as a "forced" vortex. Quantification of this behavior for a hydro- cyclone operating without an air core is one of the main objectives of this research. At larger radial positions both Kelsall and Knowles observed a decrease in with r. However, the "free" vortex behavior 6 22 which follows from Eq. (1.7) does not occur. Instead, a r'n where n a 0.8. with an air core and n a 0.3 without an air core. The difference here is significant and should be studied carefully. Based on this result, large differences in pressure losses under similar operating conditions could be anticipated between air core and no air core operations. Furthermore, and perhaps more to the point, if this difference in "n" holds upon further examination, then the ejection processes discussed in Section 1.2.1 would be quantitatively altered. ' The magnitude of in the primary flow domain determined r by Kelsall and by Knowles turns out, as expected, to be small com- pared with the axial and tangential components. However, these two authors differ on the sign of , which suggests another sig- nificant difference between the two operations (air core vs. no air core). Kelsall's data seems to imply that ("r> is negative and decreases monotonically with increasing r; on the other hand, Knowles' data shows an outward radial flow in the inner region and an inwardly directed radial flow in the outer regions. This dif- ference could be an important factor in solid-liquid separations. The laser doppler anemometer is rapidly being employed to study swirling flows quantitatively. Orloff et al. [1973] studied some problems related to the vortex breakdown phenomenon (see Sec- tion 1.2.1) in a rotating tube and, more recently, Escudier et al. [1980] studied a confined vortex flow at high Reynolds numbers using LDA. The quantification of the core region using this technique is 23 exemplified in this study and led to the important conclusions regarding flow reversals previously mentioned in Section 1.2.1. Thew and his colleagues at the University of Southampton have been using LDA to study the mean velocity and turbulent intensities in a variety of swirling flows, including the hydrocyclone (see Bedi and Thew, 1973; Loader and Thew, 1975; and Thew et al., 1980). A recent report by Grabek et al. [1980] also illustrates the utility of using LDA for gas cyclones. Loader and Thew [1975] measured instantaneous axial and tan- gential velocities in a vortex chamber using LDA. The exit? tube had a mild contraction followed by a honeycomb flow straightener. The tangential velocity measured within the vortex chamber showed an unusual “kink' or shoulder in the outer ”free" vortex region. The complementary axial velocity measurements indicated that this "kink" was caused by a complex multiple circulation zone induced by the exit design. -The LDA measurements further showed that the axial and tangential turbulent intensities are about the same, even in regions where was an order of magnitude larger than the mean axial velocity. For swirling flows, what mechanism keeps the turbulence isotropic? The question and answer go beyond the scope of this research. 1.2.3. Effect of polymer additives on secondary flows Vortex formation for Newtonian fluids in draining tanks is a fanfiliar phenomenon and can be inhibited by using mechanical devices (Hayduk and Neale, 1978) or by adding a high molecular weight polymer 24 (Balakrishnan and Gordon, 1971). Chiou and Gordon [1976] showed that only 3 wppm of Separan AP-273 reduces the maximum tangential velocity and the centerline axial velocity of a draining tank by a factor of 10. Chiou and Gordon attribute this phenomenon to the presumably large elongational viscosity of dilute polymer solutions. Coincidently, inhibition of vortex stretching by polymers also plays a significant role in some explanations of drag reduction (see, e.g., Seyer and Metzner, 1969). Polymer additives, such as poly- ethylene oxide and guar gum, can also suppress flow induced cavita- tion (Ting, 1974). Both vortex inhibition and cavitation suppression by polymers could have important implications on hydrocyclone per- formance in which the secondary separation by the inner helical flow is significant. The presence of a small amount of polymer can also have a dramatic effect on expanding and contracting motions. This phenomenon, exemplified in the work of Metzner et al. [1969] on converging flows. of viscoelastic fluids (esp. Separan AP-30) at low Reynolds numbers, could also be important in hydrocyclones near the vortex finder or in the conical apex region. Even for fully developed turbulent flows in a square duct, Reischman and Tiederman [1975] have argued that the secondary motion in the corner of the duct for dilute polymer solutions is so strong that it can affect turbulent measurements in the buffer zone far from the corner. The separation efficiency and secondary flow structures within a hydrocyclone may similarly be altered by the presence of a high molecular weight polymer. 25 1.3. Objectives and Scope of This Research Although a good qualitative understanding of the various mechanisms for solid-liquid separations in hydrocyclones is presently available, quantitative understanding is lacking. Several distinct flow structures such as helical flow, turbulent boundary layers, converging flows, and secondary circulation in the upper part of the hydrocyclone all have a direct influence on performance. The interactions among these (and other) flow structures are currently not understood--either experimentally or theoretically. One of the major objectives of this research is to provide an experimental basis from which important physical phenomena important for solid- liquid separations can be delineated. Toward this goal ten major tasks were completed during the course of this study. These are listed in Table 1.1 with references to specific sections of this dissertation where the results are presented. Laser doppler anemometry (LDA) was the major experimental tool used. This non-intrusive technique permitted a quantitative study of the central core region of the hydrocyclone. These experi- ments are fig! and complement the earlier studies of Kelsall [1952] and Knowles et al. [1973]. One of the novel features of this work is the use of a high molecular weight polymer (Separan AP-30) to alter secondary flow structures. LDA is probably the only way this type of information could be developed accurately. A systematic set of experiments was conducted to investigate the effect of feed pressure, split ratio, and vortex finder geometry 26 .~.o no?» -omm :F :o>_m m? mag» o» seamen =< ~mcoguo op o>Fuupoc gpaepuno= an =o_mme mpg» uF=Oem sex .covomc ogoo ozu :. mgsuoo corp_n*;:_ xmuco> a? were -gouou on ma: mew; o>vuuonno ugh .mcowmoc xmuco> Facucou as» :9 mpnm igm>og :o—e may xwppceac on me: one; mo>wpuonao spas ogu ea one .oums on upaosm mucosoczmmme zuvuopo> Ppmuov mews: devasouou op mccouama mcvxvs wen we once one: wepae ace mgamcmoucga maoeoszz .covuoucommwv mpzu cw cmueoaoc gucmomoa we» nopo>oc cu corum>Paos opgsu nouv>oca xusum mega c? peace mapsmoc ugh ~.m ccwaumm ¢.v cowuumm a teenage m teenage ~.H =o_oumm ecu mommag ._a pa cFeao mam «Foouwcu acvpmum =pus.uno= mcwaumwuem «copuxuogv»: u=m a c? acosmczmmoz new: on» so omla< :ecmaom mo uumwmm we» :owpuucuccu a gap: ocopozuocu»: u=m a c? mucosmczmmoz zopa ago—uxuogu»:.ssuoH a cw mcopumcmaom cwacwuuvrpom co omum< :mceaom ma uuomwu och mxcusom mu—smoa mo eowpmucu game .cucoomos meg» cow vo.uzum msopnoca mo xgmeezm .H.H «pack 27 .mcopuzpom omega gee cm:.5cmuou mm: zepe on_a nonopo>ov appae c? cowuusumc most a use .covumN -vcouomemgu meow cocoa: a xmmp com non: meowpapom coexpoa och .mowusum oea Empnoca oaauoucca mesh ~mccouuma gape m>wuopwucmsa ecu w>pump uppmaa on» omcmgo Love.» xoaeo> mg» cw copaumeucou a mo mogmou on» ecu «cosmumpa on» woos .mpomcm>mc zap; com meowuvvcou xgommouo: mg» ocpucmmoc mugmwmcp meow ovp> -oca zuaum mpg» mo mupzmmc ugh m xwucoqa< u xvccoag< m.< covuumm .< xvucmng< m.» eoppuam ~.m cowuuom omua< casuaom we mupumwgmuumgosu covauauom amen new zupmoomw> .ofi ocopozuocux: ugm a com mnmsmcowampmm magnumezmmmcm .mamxuzopm .m maze empsocpu a cw zap; cmcwsmq nonopm>oauzppsu god xupuo—m> powx< ea mucosmczmmoz o¢ :o—m new u—mpm xuvuo—m> cam: on» :o :mvmoo govern xouco> mo uumema as» .N ocopuxuoccxr omcogumvo ooc< szom no see mucosocammoz (r.z) + 2(2: PF: 00/0“) . (1.8) For the mean tangential profiles, (r.z) + e (2; PF; 00/0") . (1-9) For example, with z 8 8 cm, PF 8 15 psig, and 0°10u==4, the profiles for and from the axis to the outer wall are given by z 0 Z(8;15;4) and 0(8;15;4), respectively. If the reader would like to examine the actual data of one of the profiles listed in Tables 1.2- 1.5, use of the above convention when making the request would be helpful. CHAPTER 2 EXPERIMENTAL APPARATUS AND PROCEDURES 2.1. Flow Loop A schematic of the flow loop used in this study is shown in Figure 2.1. The overflow and underflow streams from the hydro- cyclone discharge freely into a large holding tank (200 liters) containing a cooling coil for temperature control. Some back pres- sure develops at the vortex and apex of the hydrocyclone because of the resistance to flow through the two copper pipes 170 cm in length. The inlet and outlet pressures around the hydrocyclone were measured manometrically. In all the studies reported herein, the hydrocyclone operated without an air core. The flow loop was designed so that a portion of the flow could be recycled from the high pressure side of the centrifugal pump (Myer, OF 30-3) back into the holding tank. A surge tank (38 liters) in the feed line helped to reduce pressure fluctuations. The feed line (20 mm 1.0., copper tubing) to the hydrocyclone contained a single gate valve, a manometer tap, and a pressure gauge. The nominal location of each is indicated in Figure 2.1. Flexible tygon tubing (19 mm 1.0.) was used to connect the flow loop to the glass hydrocyclone. This permitted some movement in the test section which was useful for making velocity traverses. 34 Pump .~ 35 (N2 cylinder with regulator Surge tank (38 liter) ' L18118. 30 , ._ 2.5 (on) ' l I . -. J _ I 2 (on) Y r- — ‘ Li “ i 297 . L I : % ' . (33 '30I I | , . l By-pass Hydrocyclone : : I | I l I .....J «C- /1'1 (OD) N r A ‘- L_ ‘ + #:g i . .. r14 8 46 ‘15" Coolingcoil 200 liter -N— Gate valve T . Manometer —.i- Supply tank — *1/8— Tygon tube —H— Gas line ? Pressure gauge ‘— $ Drain Figure 2.1. Flow loop (nominal lengths given in cm). 36 The 200 liter reservoir shown in Figure 2.1 was filled with tap water. The water quality was inspected for color and unusually large particles. Enough small scattering particles were present in the water so artificial seeding for the laser doppler measure- ments was unnecessary. The cooling water flow rate was adjusted to keep the water temperature in the supply tank constant at 20°C. The inlet and outlet valves were adjusted to fix the flow rate, split ratio, and pressure at the desired values. The flow rate was measured by recording the time needed to fill a precalibrated plastic container. Sane modifications in the flow loopwere necessary so fresh polymer solutions could be injected into the feed line (Section4c4). Figure 2.2 shows that an 87 liter tank (galvanized iron) was used to hold a concentrated polymer solution (2000 ppm) which was con- tinuously injected into the suction of the centrifigual pump. The polymer reservoir was pressurized by connecting it to the same nitro- gen cylinder as the surge tank. Pressure balance on the reservoir was maintained by a regulator. The concentrated polymer solution passed through 300 cm of 3.55 mm stainless steel tubing before being injected. A rotameter and small plastic valve were used to regulate the flow rate. This technique was followed to stretch the polymer before backmixing to the feed concentration (100 wppm). A plastic container was used to collect the overflow and underflow stream from the hydrocyclone so the extent of drag reduction (see Appen- dix B) of the spent liquid could be measured. H2 cylinder 37 Figure 2.2. i 200 liter Cooling coil upply tank 4— Polymer Err—L. injection e¢¢—— Drain Flow loop for polymer studies. 100 liter T‘é‘d‘ Spent tank Surge Polymer tank holding tank 38 liter (87 liter) , = 229 25 I ‘1 LA -' $5 —_’———-J W ‘— 131 r I M "" Hydrocyclone q) Pressuregauge —N— Valve T Manometer ' "\"8"Tygon tube "”" Air line 38 To study the effect of a high molecular weight polymer on the flow structures inside the hydrocyclone, a concentrated polymer solution (2000 wppm) was prepared by dissolving Separan AP-30 in tap water and leaving it for twenty-four hours. The solution was examined under light for homogeneity before pouring it into the pressurized tank. The polymer flow rate to the suction of the cen- trifugal pump was adjusted by Valve 5 shown in Figure 2.2 (Valve 4 was fully open). The flow rate was adjusted to give 100 wppm polymer concentration when mixed with fresh water inside the centrifugal pump. After the desired polymer concentration was obtained at the hydrocyclone inlet, Valve 4 was closed. The operation of the flow loop followed the procedure as previously outlined. Valve 4 was gradually opened and the final polymer flow rate was adjusted by using Valve 7. i ' When polymer was introduced into the system, changes in the hydrocyclone flow rates and split ratio occurred. Final adjustments to the hydrocyclone flow and split ratio were made by fine tuning Valves 1, 2, and 3. The rotameter was monitored during the course of the experiment. Because of the small capacity of the pressurized tank used for polymer injection, these experiments were completed in about 40-50 minutes. After the velocity measurements were com- pleted, the flow loop used for polymer injection was cleaned with fresh water and prepared for drag reduction measurements. The over- flow and underflow streams from the hydrocyclone were mixed together 39 and the drag reduction characteristics of this composite solution were determined according to the procedure outlined in Appendix B. 2.2. The Hydrocyclone The hydrocyclone* used in this study was made from pyrex (index of refraction 8 1.5) with dimensions as indicated in Figure 2.3. The optimal scale ratios recommended by Rietema are also listed in Figure 2.3 for comparison. To reduce optical distor- tion and minimize curvature effects, the hydrocyclOne was surrounded with a plexiglass box (index of refraction 8 1.47) having flat sur- faces. Glycerol (index of refraction 8 1.46) was placed in the space between the hydrocyclone and the plexiglass box. The hydro- cyclone was attached to a flat plate (see Figure 2.4) mounted on a milling table which allowed horizontal movement in two directions with a precision of 0.0254 mm. Vertical adjustments were less pre- cise and made with a hydraulic jack. For this study movement in the vertical was unnecessary. Once the test section was leveled in a horizontal plane, it remained fixed in this plane during the experiment. Figure 2.5 shows the dimensions and designs of the various vortex finders studied in Chapter 5. A cylindrical aluminum rod was machined to slide inside the vortex finder. This allowed'four additional configurations to be studied in addition to the one *Professor Donald Woods of McMaster University kindly loaned us his 3"-hydrocyclone for this study. Velocity measurements for this equipment have been reported by Knowles et al. [1973] using a flow visualization technique. Figure 2.3. 40 0 © 102 C23 1 FT 136 T 02 l 57 L28 1460 v. :67 rvf [7 i 1 l i i ‘ ‘LT__ it: 1 I 13 01 I H2 I 320 .— g ' ‘Ii’ 1 1 H1 _ Rietema i Dimension This Stud . [1961.] ‘ 0 76 mm 76 Inn 200 —— ‘, I 9, D ~0.28 D 0.28 D l I ,/ 02 0.34 0 0.34 0 4< D3 0.16 D -- 30 I H] 5 0 5 D i l Hz 0.67 D -- -0 __ .11. H3 0.4 0 0.4 0 2' ’ll 0 " 3 _ I78 I 0 .ll.3° -- .._L_ [D Dimensions of the Hydrocyclone used in this Study (nominal lengths given in mm). 41 33> .<< / 833 2:2: III! MW T xumn 3 sea»: — _ W 4 \ l. .4111. x 4. . _ I x x/ i \ \ PO>UJ \ .copuuom umwa as» ea zov> peace ecu poem .¢.~ wean?“ n xoa co>oo iiii l 42 4:2. 5 2338.3 :3 3:03pm 2.8523328 gong... xmto> .m.~ 2:3“. E. .. 2; 4|: mm _ . . . . e: e: . . 21L T. . z . 3 o .3 u _. _l .2 r S _ 3 . A .. mm, _m a: N2 . . . ‘ e LL m fl 43 defined by Figure 2.3. Cases (b) and (c) compare the effect of an abrupt and gentle contraction a significant distance from the entrance of the vortex finder, whereas Cases (d) and (e) place these contractions at the entrance. Case (c), which contains a 2:1 con- traction, was originally studied by Knowles [1971]. Case (a) satis- fies the original design criteria of Rietema. An interesting result of this research is the discovery that the core flows of Cases (a) and (c) are very different. The results of a flow visualization study are given in Chap- ter 3. The specific connections between the hydrocyclone and the flow loop for these experiments and those discussed in Chapter 4 are slightly different. Figure 2.6 shows that a hypodermic neEdle (0.0. a 0.7 mm) with a flexible joint was fixed in the overflow and underflow exits.. By inserting the long needle (8 30 cm) through the flexible joints, different places within the flow field could be marked with either Rodumine B dye (red) or green food coloring. The density of these dyes is approximately the same as water. The 3“-hydrocyclone obtained from Professor Woods originally contained a 2:1 contraction in the vortex finder so it could be interfaced with a standard fitting (see p. 20 in Knowles, 1971). Preliminary flow visualization studies (see Chapter 3) showed that the contrac- tion in the vortex finder (see Figure 2.5c) causes a flow reversal which affects the flow patterns throughout the hydrocyclone. This surprising discovery prompted the study summarized by Chapter 4. Unfortunately, our experiments were not designed for simultaneous 44 8-44 r- "'l l | / I —ll— Hydrocycl one Tygon I | tube I I . I l Needle (00 8 0.7 mm) l ' l L- Flow visualization setting 4 I Joint Support needle Overflow .L_——————- Hydrocyclone Tygon tube Cover box Laser doppler setting Underflow Figure 2.6. Schematic diagram of test section for laser doppler and flow visualization setting. 45 flow visualization and mean velocity measurements, so Chapter 3 only gives a visual map of the secondary flows for the original hydrocyclone on loan from McMaster University. Quantitative mean velocity data obtained using LDA are summarized in Chapters 4 and 5. 2.3. The Laser-Doppler System Figure 2.7 shows the relative position of each part of the TSI laser-doppler anemometer used in this study. All experiments were performed in the dual-beam mode with forward scattering. A 35 mW He-Ne laser was used as a light source. Frequency shifting was necessary in this study to define the direction of flow and to reduce fringe bias. The system consists of an optics module and a down- mix circuit. The optics module contains an acousto-optic cell (Bragg cell) which shifts the frequency of one of the laser beams issuing from the beam splitter by 40 MHz. This is too high for the range of velocities encountered in this study, so a down-mix circuit was used to shift the frequency in twelve unequal increments from 2 KHz to 10 MHz. The counter processor used in this study was specifically designed for processing signals from a photomultiplier of a laser- doppler anemometer. The data display gives an analog voltage as output which is directly proportional to either the mean velocity or the mean doppler frequency. During each experiment, the mean frequency of the doppler signal at each location in a traverse was read directly off the display panel and recorded. The mean velocity 46 .cowuocmao we once mcwcouuoum neoxeom opzuoe «accept oncomoppwumo mp 8F Lommwuoga emuczoo toxwe-:zon F— m_ I o_ oo—_.,_ Apaasm Luzon a [\— Lom Louososmco ewpanoc mewp mo uwposmsum mp toe_aeo_=eooo;a mzop mcvmzuou pesos mcwm Loumwzm accosamed ewauwpam zoom Axmosv mqua xoom Lounge; cowpm~_eopoa Lomop mz-o: :2 mm ~~ _ Faazm Logan .N.N mesmea .o. r—NMQ’LD‘ONM 47 was then calculated from the expression (see Appendix A for a brief tutorial of this technique) 8 df (2.1) taking into account proper corrections for the effect of refraction on the fringe spacing df. The above expression gives the magnitude of the meanvelocity normal to the bisector of the beam intersec- tion; represents the mean doppler frequency.l Table 2.1 lists each component in the set-up along with a few summary remarks regarding its use in this research. Appendix A contains a more detailed description of the function of each com- ponent in Figure 2.7 as well as a discussion of the important steps in processing the doppler signal from the photomultiplier tube. 2.4. Alignment Procedure for the Laser Doppler Anemometer To achieve good results, the step-by-step procedure outlined in the TSI instruction manual should be followed(see TSI, 19828). Here only a few brief observations are made regarding the methodology developed during this study. Basically the following five steps were followed. (1) Set up the polarization rotator and the beam splitter (see Figure 2.7). The orientation of the beam splitter (and the Bragg cell) depends on which velocity component will be measured (see Appendix A). When the dual-beam mode is used, the beam intensity 48 Table 2.1. Components of the TSI laser doppler anemometer used in this study. Item TSI (see model Figure 2.7) number Description Remarks 1 9126-150A He-Ne laser, 35 mW --- 2 9101-2 Polarization rotator Only for He-Ne laser 3 --- Back plate Mask 4 9115-2 Beam splitter Splits the beam intensity 50-50% or 99.5-O.5% 5 9180-2A Frequency shift assembly 2 KHz-4O MHz 6 9176 Ring mount --- 7' 9118 Focusing lens (Achromat) 83 mm 0.0., 250 mm F.L. 8 9140 Receiving optics assembly --- 9 9160A Photomultiplier 200 MHz response 10 9165 Power supply --- 11 1990 Counter processor 100 MHz clock 12 9186 Down mixer --- 13 1992A Readout module Direct velocity display 14 465 M Portable oscilloscope 100 MHz two channels (2) 49 should be equally divided. The beam splitter is designed to do this provided the direction of polarization is perpendicular to the plane spanned by the two issuing beams. The polariza- tion rotator, which is mounted on the front of the laser. is marked for the proper orientation but fine adjustments were made by simply turning the rotator until the two beams were visually of equal intensity. Set up the Bragg cell for frequency shifting. The laser beam should be parallel to the mounting base for the Bragg cell to function properly, so care should be exer- cised in leveling the laser (see TSI manual, 1982b). Because the direction of the fringe movement (see Appendix A) depends on which beam enteerthe Bragg cell, this should be decided before mounting this optical component. For axial velocity measurements the Bragg cell was set up so the fringes moved in the opposite direction as the fluid in the outer region. Likewise, for tangential velocity measurements the cell was set up so the fringes moved in the opposite direction of the fluid. Frequency shifting causes the laser beam to split into several (sometimes four) closely spaced, parallel beams. The only one of interest, however, is the one which strikes the alignment mask at the 50 "I'll circle. Figure 2.8 ‘shows a sche- matic of what is typically observed.. The tilt in the Bragg cell is adjusted until the intensity of the beam on the 50 mm circle is about 75% of the original. The other minor beams ‘ Unshifted beam Figure 2.8. 50 50 mm Shifted beam Row of spots Shifted and unshifted beam location as seen 0n the alignment mask. (3) 51 are blocked at the test section with a beam blocker. To avoid damage to the Bragg cell, the frequency shifter should not be turned on unless the Bragg cell has been connected to the A/O cell jack on the back of the frequency shifter module. This and other precautions listed in the instruction manual should be carefully observed. Install the transmitting optics and adjust for polarity and beam crossing. The transmitting optics assembly is positioned so that the optical axis is perpendicular to the test section. The axis of the test section, which is not in place during this adjust- ment, should be approximately 250 mm from the focusing lens (see Figure 2.7). The beam crossing in air is projected onto the wall by placing a microscope objective (eyepiece) near the focal point of the transmitting lens. By moving the eye- piece back and forth across the focal point, the horizontal or vertical alignment of the beam can be checked (see Fig- ure 2.9). Fine adjustments can be made by turning the com- pensating wedge in the Bragg cell (see TSI instruction manual). It is important that this adjustment be made carefully since the refraction corrections (see Appendix A) assume that the beams enter the test section either in a horizontal plane (axial velocity measurements) or in a vertical plane (tangen- tial velocity measurements). 52 Incorrect Correct . Beam crossing (:) Incident beam (59 Incident beam Figure 2.9. Projection of beams to check the crossing of the beams. 53 (4) Align the test section. A small diameter wire (0.0. 8 0.44 mm) is used to locate the center of the hydrocyclone. The wire is coated with an anti- reflective paint. Both ends of the wire are fixed in the center of two small fittings. These fittings are machined from aluminum and designed to fit the outlets of the hydroclyclone. By tightening two thumbscrews, the coated wire can be placed firmly in the center of the hydrocyclone. By slowly opening the inlet valve, the hydrocyclone is filled with water, and any trapped air in the hydrocyclone is removed. Both ends of the overflow and underflow are sealed by tightening the ends of the wire. The test section is raised until the laser beams strike the wire. By moving the test section horizontally, the beam crossing is centered on the wife. The minor laser beams induced by the Bragg cell are blocked by simply mounting on a magnet two small pieces of stainless steel plates which have been coated with an antireflective paint. The axial location of the beam crossing on the wire is located by moving the milling table forward and backward while viewing through the pinhole of a calibrated ruler positioned on the top surface of the hydrocyclone cover box. The axial reference point is the end of the apex region (see Figure 2.10). When the wire and the calibrated ruler are positioned correctly on the test section, the center wire can be viewed from all pinholes on the rule along the axis of the hydrocyclone. Once the beam crossing is located on the 54 \\\\\\\\\——_.. Cover box Wire_/ ' \\\\\\\‘\,___.. Ruler with pin hole l l I l I l | | L Figure 2.10. Test section set-up to locate center of the hydrocyclone. 55 center of the hydrocyclone at the desired axial location, the wire is removed from the test section and the hydrocyclone is gently connected to the flow loop without misaligning the test section. The cover box of the hydrocyclone should be perpendicular to the two laser beams. This is checkéd by using a mirror to reflect the laser light back onto a transparent sheet of paper. Final adjustments are made by either rotating the top of the milling table or by "jimmying" the test section with small pieces of paper. ' (5) Set up the receiving optics and photomultiplier tube. The receiving optics assembly is positioned along the optical axis, allowing approximately 250 mm between the center of the hydrocyclone and the collecting lens. The best focus of the scattered light from the beam crossing is achieved by moving the receiving optic lens back and forth until the image viewed by an eyepiece is as small and clear as possible. The x-y adjustment thumbscrews on the photodetector mounting plate are used to move the image to the center of the eyepiece. The eyepiece is then removed from the mounting plate and the photo- detector is attached. 2.5. Procedure for Obtaining Good Doppler Signals The photomultiplier tube is connected to the oscilloscope. The x-y thumbscrews on the photomultiplier are adjusted until the output of the scope appears as shown in Figure 2.11a. Each peak 56 Volts (50 mv/div) Time (1 mS/div) (a) Output of the photomultiplier Volts (10 mv/div) Time (100 uS/div) (b) Output from the signal processor at a low sweep speed Figure 2.11a-b. Typical doppler signals obtained from the signal processor. . 57 contains velocity information. At high sweep speeds these peaks can be expanded to show a doppler signal (see Appendix A for a dis- cussion). After proper alignment, the photomultiplier output is connected to the signal processor. The filtered output from the signal processor is displayed on the oscilloscope. As oscilloscope sweep speed is increased, the pedestal component and the doppler frequency component of the signal appear. The high pass filter is positioned so the pedestal component and any Signal asymmetry are removed without losing the doppler signal. . Figure 2.11b shows several doppler bursts at a low sweep speed. If the sweep speed is increased, the signal expands on the oscillo- scope screen from which the doppler signal is easily obtained. Figure 2.11c is a complete doppler burst at a high sweep speed which has been properly filtered. The amplitude of the signal varies periodically from a minimum to a maximum. The maximum amplitude corresponds to scattered light coming from the central region of the probe volume with high light intensity. Figure 2.11d is a properly filtered signal which shows multiple bursts as a result of more than one particle passing simultaneously through the probe volume. To obtain these high quality signals several parameters related to the signal processor (see Figure 2.7) must be selected. A brief summary of these details is presented in Appendix A after the equipment shown in Figure 2.7 has been discussed. When the internal wall of the hydrocyclone is struck by the beam crossing, the signal at a high sweep speed will appear as shown .II‘I'I'Q.!IIU Volts (10 mv/div) Time (20 ps/div) (c) Properly filtered doppler signal Volts (10 mv/div) Time (10 us/div) (d) Multiple doppler signals due to several particles Figure 2.11c-d. Typical doppler signals obtained from the signal processor. 59 in Figure 2.12. This position of the beam crossing on the wall can also be checked by visual inspection. The frequency of this signal should be the same as the shifted frequency of the beam. For a frequency shift of 1 MHz, this corresponds to a peak to peak distance of 1 us/div. on Figure 2.12. The analog signals proportional to the instantaneous axial velocity are shown in Figure 2.13. Figure 2.13a is the analog output of the signal from the core region near the center of the hydro- cyclone, and Figure 2.13b shows the signal from the outer region at the same axial location. The flow conditions and signal pro- cessor settings are similar for both measurements. This illustrates that axial velocity fluctuations in the core region are higher than those in the outer region. In this research, only the long time average (8 10 sec) of these signals are recorded and analyzed. 2.6. Data Analysis and Consistency Checks In this study, the axial and tangential components of the mean velocity were measured using LDA. The radial component was estimated by using a material balance (see Section 4.1). Each axial velocity profile was checked for consistency by computing the under- flow rate Qu from (r,z) and comparing the result with the mea- sured rate. The difference was always less than 5%. Near the axis of the hydrocyclone, the axial velocity is very large for the con- ditions studied (8 ZOO-400 cm/s) and, because of a reverse flow, drops off rapidly to zero over distances 8 5 mm. Although the 3 velocity gradients were very steep (8 10 .s'l), the mean velocity 60 Volts (20 mv/ div) Time (1 115/div.) Figure 2.12. Output of the photomultiplier tube when the beam crossing intersects the hydrocyclone wall. Volts . (500 mv/div; Time (500 us/div) (a) Core region Volts (500 mv/div W“ Time (500 us/div) (b) Outer regions Figure 2.13. Instantaneous axial velocity in the hydrocyclone (illustrative purposes only). 62 profiles measured in the core region were very stable and reproducible. Figure 2.14 shows typical traverses obtained on different days. The symmetry in the axial velocity should also be noted. Velocity data for many of the studies presented in Chapters 4 and 5 were taken only between the center of the core region and the hydrocyclone wall. It is noteworthy that the asymmetry in the velocity field, which is surely very strong in the upper protion of the hydrocyclone near the feed entry, has essentially disappeared in the conical section. The velocity profiles were all corrected for index of refrac- tion effects. The details of this procedure are delegated to Appen- dix A. Generally the most significant correction was in the radial position. Some experimentally determined points had to be adjusted by as much as 40% for the axial velocity and by 80% for the tangen- tial velocity. Thus, the location of the axis of symmetry is crucial. In this study, we assumed that the flow was axisymmetric and thus made the final alignment between the test section and the light source by defining the center of the hydrocyclone as coinciding with a local extremum in the mean axial velocity. This procedure was convenient and agreed closely with using the wire previously discussed (see Figure 2.10). When the center of the core was located, the position was marked as zero radius. The mean velocity distribution was obtained by moving the test section by small increments along the radius of the hydrocyclone. An increment advancement of the milling tables moves 63 + o Run l i 0 Run 2 , cm/sec Characteristics of Experiment vortex finder Figure 2.5a PF l5.4 psig P0 ~ l3.6 P519 u l3.8 psig 00/0u - Figure 2.l4. Symmetry and Reproducibility of Velocity Profiles. 64 the test section 0.024 mm. The traverse of the milling table and the mean velocity (or, frequency) were recorded on the data sheets (see Appendix D). To carry out a consistent policy in data record- ing, all information on the counter processor was recorded. During the course of the experiments, the flow conditions were menitored continuously. No significant changes occurred during any of the experiments. CHAPTER 3 FLOW VISUALIZATION STUDIES The qualitative flow patterns in a 3"-hydrocyclone were examined by dye injection for three different split ratios. Table 3.1 defines the parameters for each series of experiments and Figure 2.3 gives the dimensions of the hydrocyclone used. Rodu- mine B and a green food color were used to visualize the flow and the results were photographed. This information was valuable in“ planning the quantitative study of the velocity profiles using the laser doppler anemometer. Figure 3.1a shows the result of injecting a green dye into the apex region of the hydrocyclone operating at a split ratio of 4.0. The hypodermic needle, which is slightly off-center and inserted some distance into the apex region, rotates freely with the flow. The stability and coherence of the central core region is remarkable. Little mixing with the outer swirling flow is evident, but this does not mean the absence of turbulent transport in the flow or radial directions. The existence of this flow structure presumably depends on an interaction between the downward moving boundary layer structure and the geometry of the apex region. For the flow conditions here (see Table 3.1) the reversal of the outer helical .flow (see Figure 1.1) apparently occurs deep within the conical region, 8 4 cm from the bottom exit. 65 66 Table 3.1. Scope of flow visualization studies. Pressure Split Flow Rates Reynolds psig Ratio cc/sec Number . Series PF Qo/Qu QO Qu Rep 1 5.2 4.0 400 100 24,850 11 5.2 0.24 96 400 24,651 111 5.2 99 590 0 25,297 67 Qo/Qu = 4.0 (a) Injection from apex only (b) Green dye injected from apex; red dye from vortex finder Figure 3.1. Flow visualization of the core region for Series I. 68 An interesting feature of Figure 3.1a is the appearance of a lightly colored sheath of upward moving fluid which surrounds a more darkly colored core of fluid moving at a much higher velocity. The green dye seems to persist around the core region after the dye supply has been interrupted. This nearly stationary layer, or ”mantle", of fluid is due to a reverse flow which starts inside the vortex finder and surrounds the upward moving core of fluid. Figure 3.10 shows an interesting discovery. A flow reversal within the vortex finder, which has a 2:1 contraction, causes an annular flow countercurrent to the upward flow on the axis. This secondary flow, which is surprisingly stable, is only 3 mm thick, yet extends over more than half of the length of the hydrocyclone. The diameter of the core fluid is also about 3 mm, so its length to diameter ratio is approximately 102. The flow patterns shown in Figure 3.1 were unexpected and have raised many fundamental ques- tions related to the performance of hydrocyclones. Clearly, the small secondary flows can have an important effect over the entire flow field. Figure 3.1b illustrates the possibility that two boundary layers separated by a significant distance (one deep in the apex region and another deep within the vortex finder) can still interact by ejecting two fluid streams which pass countercurrent to each other. This intriguing phenomenon could perhaps be exploited in the design of a novel classifier or separator. Figure 3.2a illustrates the same flow structures as Figure 3.1, but now the red dye is introduced below the vortex finder with the (a) Green dye injected from (b) Green dye from apex; red apex; red dye below the dye from vortex finder vortex finder Figure 3.2. Flow visualization of composite flows in the core region for 00/0u 8 4.0 (Series I). 7O injection needle stabilized against the inner wall of the vortex finder. The swirling pattern of the flow is clearly evident. From this photograph a rough estimate of the pitch angle is 20°. For helical flow with zero radial velocity this implies that the ratio of axial to tangential velocity in this region is 8 0.36, which should be viewed only as a crude guess. The mixing outside the core region is very efficient as indi- cated by the rapid spread of the red dye in Figure 3.2a. Still, this is mainly confined to an upward swirling flow which runs counter- current to a downward outer helical flow (see Figure 1.1 again). Furthermore, the red dye is unable to penetrate the jet-like core flow (marked green) issuing from the apex region. Note also the faint trace of red dye surrounding the green core, which is the reverse flow illustrated in Figure 3.1b. The important conclusion from Figure 3.2a is that the annular reverse flow unambiguously comes from the outer flow inside the vortex finder tube rather than the axial core.flow. Figure 3.2b shows the result of injecting green dye into the upward swirling flow which begins in the apex region and red dye into the reverse flow from the vortex finder. The dark green annular flow is part of the same upward flow marked initially by the red dye in Figure 3.2a. Note that the movement of the green fluid to larger radial positions is very restrictive. The central core in Figure 3.2b is entirely marked by red dye and extends deep into the apex region. The jet-like upward core flow, shown in the previous photographs, does not appear here, probably 71 because the dye needle has been extended into the apex region beyond the point where the jet forms. This disturbance has changed the typological structure of the flow. The sequence of flow visualization photographs from Series I has shown the existence of four distinct, simultaneous, countercurrent flows in the conical section of a 3"-hydrocyclone. This four cell structure is very sensitive to end conditions in the vortex finder and in the apex. Earlier research (see, esp., Binnie, 1957) showed the possibility of three cells, but as far aswe know this is the first report of four cells. Many additional photographs and films were taken and analyzed. Figure 3.3 portrays the composite results and conclusions of this analysis. The axial and tangential velocity profiles across the four cells discovered here were measured quan- titatively using LDA; the results are reported in Chapter 4. Reducing the split ratio eliminates the gpggrg_jet-like core flow shown in Figures 3.1a-c. For a split ratio of 0.25, a downward jet-like core flow issues from the vortex finder as shown by Fig- ure 3.4a. No upward flow on the axis was ever observed for Series II, even when the needle shown in the apex was withdrawn as in Figure 3.18. In a certain sense, Series II is just Series I turned upside down. The details of the vortex finder contraction and the convergence angle of the apex region are certainly important features in determining which jet flow will appear, but the split ratio may be the ultimate factor. It is tempting to speculate on the possibility that a jet flow could issue simultaneously from 72 Overflow I \\\\\\\ Flow reversal inside the vortex finder Tangential f d ee —> \J _l Short-circuit Strong axial core flow Downward helical flow Annual reverse flow Downward annular flow vanishes Upward core flow from apex Flow reversal 1 Underflow Figure 3.3. Flow patterns observed in the 3"—hydrocyclone using dye injection. 73 (a) “0% 8 0.25 (b) 00/0u . .. Figure 3.4. Flaw visualization of the core region for Series II an 111. 74 both the vortex finder and the apex region with a stagnation plane separating them somewhere in the hydrocyclone body. In Chapter 5 data are presented which support this conjecture (see Figure 5.5). Figure 3.4b implies that the outer helical flow (see Figure 1.1 once again) penetrates deep into the apex region even for an infinite split ratio. Even though the feed is introduced near the vortex finder at the top of the hydrocyclone, it does not simply short- circuit to the vortex finder. The swirling outer fluid still drives the boundary layer down into the conical region until the pressure builds up too much and causes the fluid to reverse direction. Figure 3.5 shows a composite of the flow patterns observed in Series II and III. An interesting distinction between these results and the ones summarized by Figure 3.3 for Series I is the reduction in the number of cellular structures. For QO/Qu 8 4.0, four cells were observed. However, for 00/0u 8 0.25 only three cells were detected, and for 00/0u 8 8; only two cells. Thus, the split ratio has an important influence on the secondary flow patterns in a hydrocyclone operating without an air core. 75 A Overflow I Overflow l / p Flow reversal i i inside the vortex Tangential feed Short circuit i Strong axial f Upward down flow flow Downward helical flow - Reverse .\ [[1 flow Flow reversal 1 Underflow ‘ 1 Series II Series III 00/0u 8 0.25 » 00/0u = ” Figure 3.5. Composite results of flow visualization study for Series II and III. CHAPTER 4 VELOCITY MEASUREMENTS IN A 3"-HYDROCYCLONE WITH A 2:1 CONTRACTION IN THE VORTEX FINDER 4.1. Axial and Tangential Profiles Figure 4.1 shows how the axial component of the mean velocity varies with radial and axial position. These quantitative profiles are consistent with the flow visualization studies previously men- tioned. The traverses at z 8 20 and 32 cm definitely show a reverse flow over a very thin annular region surrounding an inner core. The volumetric flow of the central core is about 5% of the overflow rate and the flow rate of the reverse flow is much less than 5%. The reverse flow has disappeared at z 8 8 cm but the velocity still goes through a mininum in this region. This type of behavior has also been reported by Nissan and Bresan [1961] for swirling flow in cylinders. Knowles et al. [1973] using the same hydrocyclone reported some undulations in the axial profile above 2 8 32 cm, but no reverse flows. Further from the axis a strong upflow occurs with a slow tran- sition to an outer downward flow. Because of no slip at the hydro- cyclone wall, decreases to zero across the boundary layer. 2 The surface of zero axial velocity in the outer region conforms to the conical geometry of the hydrocyclone whereas the diameter of the central core region seems to be bounded above by the size 76 '77 Vortex finder 400 wall 150. 1 U 3 E U I? N 3 80. 0.0 -80 I I 0.0 0.8 1.6 2.4 3.2 4.0 ' r, cm Figure 4.1. Mean axial velocity profiles for Q /Q 8 4 and ReF 8 30,000.= 0 " 78 of the contraction in the vortex finder. At 2 = 32 cm, the results show a fairly large annular region between the vortex finder and the hydrocyclone wall where z 0. This may correspond to a recirculation zone. Figure 4.2 shows the mean tangential velocity at three dif- ferent axial positions for the same flow conditions as before. in the central core region is approximately proportional to r and independent of 2. For non-turbulent flows, this feature would imply that the radial velocity is zero withinthe core region. It is noteworthy that the maximum angular velocity and the flow reversal. due to the contraction in the vortex finder, occur at the. same radial position. Thus, with a 400 cm/s at r = 0.5 cm, the centrifugal force changes from zero on the axis to nearly four orders of magni- tude larger than the force due to gravity. This happens uniformly over the core from 2 - 8 cm to z = 32 cm. This means that With negligible radial velocity in the central core region, an upward moving particle with a larger density than the fluid could easily migrate to the outer region given enough time. Apparently, the location of the maximum tangential velocity together with the length of the core region and its axial velocity account for some of the unique classifying characteristics of hydrocyclones. ‘ In the outer region the tangential velocity decreases with radial position. A "gaze vantex“ would be inversely proportional to r but Figure 4.3 shows a much weaker behavior. The smaller exponent observed experimentally can be attributed to the viscous , cm/sec 79 480-1 r\ 0’ 400' \ Vortex l \ finder wall 320 . 2404 . 160~ - PF = 15.4 psig \\\ ‘ P0 = 11.8 \\ Pu = 12.4 \ 30. 0.0T I I I I 1 0.0 0.8 1.6 2.4 3.2 4.0 r, cm Figure 4.2. Mean tangential velocity profiles for QO/Qu = 4 and ReF = 30,000. 80 1000. d I \ 600 . 32 \ -1 o z = cm "'1 500 . \\ rn = constant \. aue>i o 20 300.. . 8 200.. 100 U I y l v r I 1 I T I T I I T 1 fi 0.1 1.2 0.3 0.40.5 0.6 1 2 3 4 5 10 r. cm n Type of flow 1 Free vortex = +0.8 Air core (Kelsall, 1952) 0.62 1 No air core (this research) a 0.3 No air core (Knowles, 1971) Figure 4.3. Deviation from free vortex flow in the outer region for a split ratio of 4 and an inlet Reynolds number of 30.000. 81 or turbulent transport of momentum. Note that the dependence on r found here is comparable to the one reported by Kelsall [1952] fora 3"- hydrocyclone operating with an air core but significantly lorger than the exponent found by Knowles et al. [1973]. The significant result illustrated by Figure 4.2 is that does not change much 8 with axial position even in the outer region. Of course, no slip at the hydrocyclone wall reduces to zero across the boundary layer. ‘ Figure 4.4 compares the axial and tangential velocities at z = 32 cm measured in this study with the ones reported by Knowles et al. [1973] using the same hydrocyclone. The flow rates, split ratio. and pressures are all approximately the same for the two studies. The mean axial data used for comparison are the locally averaged ones which close the internal mass balances (see p. 37 in Knowles, 1971). Axial velocities measured using laser doppler anemometry always yielded internal material balances which closed to within 5% (see Section 2.5). In the outer region, the results reported here for both and are in good agreement with the earlier experiments from McMaster; however, the reverse flows and the position of the maximum tangential velocity are both obscured by the measuring technique employed by Knowles. . The radial component of the velocity was not measured. How- ever, from the three traverses of the mean axial velocity, an esti- mate of the mean radial velocity can be obtained. For instance, if the flow is axially symmetric (this was shown to be a good Mean velocity. cm/sec 82 520 f \/ Vortex finder walls 360.1 _. 32 280. z = 0 cm Tangential. 2004 VE1OC1t‘Y r, cm Figure 4.4. Mean axial and tangential velocity profiles at z = 32 cm for QO/Q = 4 and REF 3 30,000 (a,cgpresent study; OJ Know es, 1971) . 83 approximation for this hydrocyclone), then a material balance between two level planes located at 21 and 22 implies that z 2 r r]:1 (r,z) dz =.}£ [(r,zl) - (r,zz)] r dr . From this expression and the data from Figure 4.1, two characteristic radial prOfiles can be calculated. Figure 4.5 shows 22 ' (r,z) dz 21 722 - 21) ilr) 5 for the lower and upper part of the hydrocyclone. In the upper part, the radial flow is outward in the inner core region and inward for r > 2 cm. From Figure 4.1 the surface of zero axial velocity is located at radial positions greater than 2 cm for z > 20 cm. This is interesting because it suggests that an inwardly directed viscous force may counter the centrifugal force in this region and actually drag solid particles out of the downward moving boundary layer and into the upward swirling flow. This phenomenon would be even more pronounced at smaller axial positions. Finally, the outwardly directed radial flow in the upper part of the hydrocyclone would act in concert with the centringal force to clarify the core region of solid particles with densities larger than the continuous phase. Just the opposite would occur for lighter particles or oil droplets. Because Elr) changes sign in the lower - 84. Vortex finder walls * I \\ 4.. E. 3 . . Upper Lower z = 0 cm a -1'- \ jg \. Upper . -2- ‘- ,‘ \ L . e, \ I: _3- \. I" \ f \ ,' Lower '4‘ \ f \. I. \ .. . / -5. \. I. \1' -5 , r i ' 0.0 0.8 1.6 2.4 3.2 4.0 r, cm Figure 4.5. Average radial velocity profiles for Qo/Qu = 4 and ReF = 30,000. 85 part of the hydrocyclone, this mechanism of classification may not be applicable for z < 20 cm. 4.2. The Effect of Split Ratio on the Velocity Profiles Figures 4.6 and 4.7 respectively summarize the measurements of the axial and tangential velocities for three different split ratios at z = 32 cm. The striking conclusion from these data is that the split ratio has little noticeable effect on the outer flow. The position of the outer mantle (i.e., where = 0), for instance, seems oblivious to changes in QO/Qu. This means that the split ratio would not be a good choice for a control variable for those applications, if any, where the outer structure determines the per- formance of a hydrocyclone. For some situations, however, the objec- tive may be to alter the central core structure, and Figure 4.6 shows that this region changes dramatically with split ratio. As illustrated by Figure 4.6, reverse flows are found as 00/0u decreases from at For 00/0.u = as the axial velocity has an inter- esting minimum value at the same radial position where the tangen- tial profile is a maximum. Apparently, the transport of mean axial momentum into the velocity defect region makes a contribution to the kinetic energy of swirl rather than speeding up the axial com- ponent of velocity. Thus, the centrifugal force is balanced by other factors in addition to the radial pressure gradient. This undulatory type profile has also been observed by Knowles [1971] at a finite Qo/Qu in this hydrocyclone and by Thew et al. [1980] in a cylindrical hydrocyclone. , cm/sec 240 - 160 .' ’-——-._. __.._..—. 80 -' n \ - \.,//\\\ 0.0 ./ "\ M- \\ Vortex finder walls ____. 32 z = 0 cm A U/ -30_ ,/ 00/0”l . Pr- p0 Pu (psig) / 0.25 15.5 13.4 11.5 ----- / 4 15.4 12.2 12.8— -160- II .. 15.4 11.9 13.0 ————— / l -240- I I I I -320-I I -40 , . . . 0.0 0.8 1.6 . 2.4 3.2 3.0 r, cm Figure 4.6. The effect of split ratio on at z = 32 cm and ReF = 30,000. 87 560 1 480 1 I\. \\ Vortex finder walls , cm/sec 400 ‘ 320 ' 240.. ' 160 . Qo/Qu PF P0 Pu (9519) 80 0.25 15.4 13.2 11.5 ----- 4 15.4 11.8 12.4 --—- 0° 15.4 11.9 13.0 —'-°— 0.0 I I ' U I ' 0.0 0.8 1.6 2.4 3.2 4.0 r, cm Figure 4.7. The effect of split ratio on at z = 32 cm and ReF = 30,000. 88 The axial profile for 00/0u = .25 is qualitatively similar to the three cell reversals reported earlier by Nuttal [1953] and by Binnie [1957] for confined vortex flows in cylinders. These earlier results were produced at a fixed flow rate by increasing the initial swirl of the feed (see Section 1.2). The new result here is that the split ratio also controls the morphology of the central core region. At a fixed flow rate, the transition from a two cell reversal to a three cell reversal has as an intermediate flow structure four cells. Data also show that this four cell reversal at QOIQu = 4.0 becomes a six cell structure by decreas- ‘ ing the backpressure. The additional countercurrent flow appears in the outer region where = 0. The axial profile at z - 8 cm in Figure 4.8 shows that the three cell reversal at z - 32 cm becomes a two cell structure further down the conical section. This phenomenon is consistent with the study of Nissan and Bresan [1961], who showed that transition from three to two cells in.a cylinder occurs simultaneously with a reduc- tion in the swirl velocity. For this geometry, this is equivalent to either a reduction in the average angular momentum per unit volume in the cross section or to the circulation r(z). With _. R(z) P(z) a Zn (r) r dr (4.7a) 0 . and with the data shown in Figures 4.2 and 4.3, it follows that Fm . 02(2)]2 (4.75) , cm/sec 89 80 - .‘\ A 0.0 ./v ¥V \ - \, ~ \ a. \\ “‘~\ \ \-—' \.——’ -80 4 Vortex finder ‘—""":/ walls -160 - -240 .. F I 15.4 psig] —— 32 . P a 13.4 P0 z 11 5 —- 2° u 0 -32‘0 - 8 I. z = 0 cm I I 'i I- ,1 -400 -II I! I .480 I I T 1 l I 0.0 0.8 1.6 2.4 3.2 4.0 r, cm Figure 4.8. Mean axial velocity profiles for Qo/Qu = 0.25 and ReF a 30,000. 90 for the hydrocyclone exmployed in this study. Actually, the tan- gential profiles for 00/0u = 0.25 were measured at different axial levels and the results, not reported here, are very similar to those displayed in Figure 4.2 for QO/Qu = 4.0. Therefore, the propor- tionality expressed by (4.7b) is a good approximation for B< z.< 32 cm. The radius of the hydrocyclone decreases by about a half from 2 = 32 cm to z = 8 cm.(see Figure 2.3). Therefore. the circulation is reduced by one-fourth so the results summarized by Figure 4.8 are qualitatively consistent with the previous findings of Nuttal [1953], Binnie [1957]. and Nissan and Bresan [1961]. It is inter- esting that the design of this hydrocyclone maintains about the same centrifugal force distribution over a significant portion of the conical section. The reverse flows discussed so far apparently play a subtle, if not explicit, role in maintaining this feature. Dye injection into the apex region for-QO/Qu = 0.25 picks up the reverse flow at z = 8 cm. Figure 4.8 shows that this annular upflow at z = 20 cm is approximately 1 cm thick and continues to grow with increasing 2. The data also show that the maximum axial velocity in the outer downward flow increases as 2 decreases. which is probably due to a decrease in the cross sectional area. The profile at z = 8 cm has an undulating character like the one in Figure 4.6 for QO/Qu =‘m. Similarly, the tangential velocity is a maximum in the region where a/ar = 0. Therefore. the steady state velocity defect which occurs at z = 8 cm implies once again that the mean axial momentum transported into this zone probably 91 makes an important contribution to the kinetic energy of the swirl velocity. It would be interesting to study, both theoretically and experimentally, the transition which occurs between 2 = 8 cm and z = 20 cm. Presumably, the wall boundary layer has grown suf- ficiently large at z = 8 cm that it begins to interact with the core vortex and ejects_fluid upward (see discussion in Section 1.2). As illustrated by Figure 3.1, this phenomenon also caused a flow reversal in the vortex finder to form the core fluid in Figure 4.8. 4.3. The Effect of Re nolds Number and Backpressure on the Vel0city rofiles Increasing the Reynolds number (actually the flow rate) has a large quantitative effect on the axial and tangential profiles, but for the two comparisons made in Figures 4.9 and 4.10 the apparent qualitative behavior is unchanged. In the next chapter, results over a broader range of flow rates for this hydrocyclone without the 2:1 contraction in the vortex finder are presented. The results shown here clearly show that the magnitude of the reverse flow around the vortex core increases with Reynolds number. The average cir- culation at z . 20 cm for the second traverse in Figure 4.10 is obviously larger than '5 for traverse 1. Thus, the data presented for this case are consistent with the previous observations and with the criteria for reverse flows given by Binnie [1957]. Fig- ure 4.9 suggests that transition from a two-cell to a four-cell vortex at z = 20 cm occurs for ?'= 6818 if the split ratio is 4.0. 92 560 II I I I 480 . I I 400 . : Vortex finder walls I I / n ' : . | ~ I 320 'l . I : ~ I I t I ' I~ I I § 240« , 20 ‘~ I E ° I “N I z = 0 cm 3 I 160 « | I Pressure in psig : ReF PF Po Pu I 29900 15.4 12.2 12.8 —————-1 80- I 45500 10.5 1.9 3.2 ----- 2 | 30900 5.2 1.2 1.8 -—«—«— 3 0.0. Iv \ \J ‘\‘\ \ . \ \‘/ ‘80 ‘ I l l . 1 0.0 0.8 1.6 2.4 3.2 ' 4.0 I", cm Figure 4.9. The effect of Reynolds nunber and back pressure on for 00/0u = 4 at z = 20 cm. 93 720 ,1 I\ I\ 640 .. I \ ““‘;::>\ I \ Vortex finder 560 _ | I I 480 . I :- -——————- 20 400-: I z = 0 cm a I :1 320 . I 5 I g' I 5‘" I V 240.| I I I 160 “I I I aol 0.0 . 0.8 4.0 r, cm Figure 4.10. The effect of Reynolds number and back pressure on 4at20' cm. for 00/0u = 94 Lowering the back pressure has little effect on the mean velocity profiles provided the pressure drop between the feed and the two exit streams remains balanced (see traverses 1 and 3 in Figures 4.9 and 4.10). The characteristic four cells are clearly present and the tangential velocity again peaks in the region of flow reversal around the inner core. The small differences in the outer and the reserve flow regions may be due to slight differences in the flow rates and unevenness in the actual pressure drops (see data listed in Figure 4.9). It seems noteworthy, however, that lowering the backpressure on the vortex finder did not remove the reverse flow at the 2:1 contraction. Figure 4.9 does suggest that increasing the backpressure, rather than decreasing it, may remove the flow reversal induced by the 2:1 contraction. This experiment could not be run because of the pressure limitations on the glass hydrocyclone. Figure 4.11 shows the axial profile close to the vortex finder for approximately the same conditions as traverse 3 in Figure 4.9. Another set of flow reversals in addition to the four already shown in Figure 4.1 seems to have been triggered by reducing the back- pressure. This broad region of small axial velocities suggests that a recirculation zone occurs in the outer part of the hydro- cyclone. Fontein [1951] has previously discussed the existence of multiple eddies and their significance in separation performance. 95 .250 q '777’7 210 .\\\ /// Vortex finder walls 32 160 - z = 0 cm 110 - , cmVsec 60 . I l I I 1 0.0 0.8 1.6 2.4 3.2 4.0 I“, cm Figure 4.11. The effect of low back pressure on flow reversals for 00/0u = 4 and ReF a 30,000 at z = 32 cm. 96 4.4. The Effect of Separan AP-30 on the Velocitngrofiles ' Some preliminary measurements of the velocity profiles with 100 wppm Separan AP-3O in water show little, if any, differences with the results already reported. Experiments for 00/0u = m, 4.0, and 0.25 are qualitatively the same with only some minor quantita- tive differences. Because the polymer was prepared and used imme# diately, its temperature was only 12°C compared with 20°C for the water studies. This difference has a significant effect on the viscosity of the solvent, not to mention the polymer-water mixture. Therefore, any quantitative comparisons with the results in Sec- tions 4.1 and 4.2 should be cautious. Figures 4.12 and 4.13 summarize the results obtained with polymer additive. The experiments with water alone are shown for conparison and. although the temperatures are different between these two studies, it is nevertheless useful to speculate a little on some of the potentially significant differences. Note that the addition of polymer seems to have eliminated the flow reversal sur- rounding the central core vortex. At the same flow rate, only two cells were observed, but the undulating axial profile remains. A decrease in viscosity would perhaps give a similar result but this conclusion requires additional experiments. Figure 4.13 shows that the polymer may (excluding viscosity effects) decrease the tangential velocity with the largest effect occurring in the axial velocity defect region. The central core still has a tangential velocity proportional to "r" and a center-line 97 320.3 Vortex finder walls 15.4 psig 12.1 12.7 , cm/sec V, cm Figure 4.12. Mean axial velocity profiles for 100 wppm Separan AP-30 with 09/0 = 4 and ReF = 24,300 (QF = 500 cm lsec, T= 12 C)‘.I 98 400 I 320 J 240 U I 3: I; 0 AG 3a: - 'v 160 _ P a 15.4 psig so F P 3 12.3 o 0 Hater at 20 C for Pu=12.8 ,8§_z_<_32cm 0.0 ' l I | ‘ 1 0.0 0.8 1.6 2.4 3.2 4.0 Figure 4.13. Y‘, Clll Mean tangential velocity profiles for 100 wppm Separan #9-30 with 00/0u = 4 and ReF 2 24,300 (0F 5 500 cm3/sec, = 12°C . 99 axial velocity which changes with "2", but not nearly as much as before. The axial velocity at r = 0 also shows a maximum in the range 8l§.z.§_32 cm. The deceleration on the axis may indicate that a fluid particle sees an adverse pressure gradient as it approaches the vortex finder. Once again, a combination of x r and 0 3/az f 0 within the central core implies that turbulent transport of momentum may play a key role in balancing the convective radial transport into this region. Because the axial velocity in Fig- ure 4.12 does not change as much as the water case shown in Fig- ure 4.1, a tentative indirect conclusion is that the addition of Separan AP-30 has reduced turbulent transport within the core region (see Section 6.3). CHAPTER 5 VELOCITY MEASUREMENTS FOR DIFFERENT VORTEX FINDER CONFIGURATIONS 5.1. Rietema's "Optimal" Design In the 3"-hydrocyclone without the contraction in the vortex finder, a three cellular structure was observed at z = 20 cm for split ratios between 0 ande. The inlet Reynolds number was = 24,200 (feed rate of 500 cm3/sec and T = 12°C) for all runs. For QO/Qu = a, only a two-cell structure was observed at z a 20 cm, but the axial profile has a velocity defect around the axis. This was also observed at z = 8 and 32 cm for Oo/Qu = a» The results for both the axial and tangential components of the mean velocity are summarized in Figures 5.1 and 5.2. The most remarkable, and obviously apparent, difference in the results reported here without the contraction in the vortex finder and the results of Chapter 4 is the absence of a jet-like core on the axis running from the apex and out the vortex finder (also see Figure 3.2). Figure 4.12, which is at the same Reynolds number as the study summarized by Figure 5.1, suggests that the 2:1 contraction is responsible for the jet-like vortex on the center line. 100 101 .56 cm u N «a oo~.¢m m uma Low A~3v co opumg.uwpam po uomwmm ugh .H.m manor; EU I» o.¢ m.m ¢.~ o.“ w.o .ugu m.o o.H ¢.~ ~.m o.e b b n p p p b p - m.o xxxxx e - 85 I--- o ..... =o\oo oce- omn- cem- omHI 0.0 cm oas/wo ‘ 102 560 a Vortex finder wall 480 l QO/Qu PF P0 Pu (psig) O 15.3 14.0 10.4 """ 0.25 15.3 14.0 11.9 ----- 400 f 4 15.3 13.7 13.4 —-———- '3 .. 15.3 13.7 13.9 —-—-— , cm/sec 1 ' ' I' 1 1 0.0 0.8 1.6 2.4 3.2 4.0 l", cm Figure 5.2. 'The effect of split ratio on for Re a 24.200 at z = 20 cm. 6 F 103 The size of the core region, as measured by the location of the maximum tangential velocity, is a more sensitive function of split ratio at this lower Reynolds number (c.f. Figure 4.7). As seen from Figure 5.1, the strength of the downward flow in the core is increased when the split ratio is decreased. Obviously, this causes the core diameter to increase. The mass flux of the upward flow is less for Qo/Qu . 0 compared with other split ratios. Thus, for low split ratios, a high inward radial flow is expected in the core region. However, a significant fraction of the inward flux of momentum is transferred to the tangential component inasmuch as the maximum value of changes by more than a factor of 2 when QO/Qu decreases from 0.25 to 0 (see Figure 5.2). Note that the increase in at r = O is less than 20% for the same change in split ratio (see Figure 5.1). In the outer region, the mean radial velocity is inward and independentof the split ratio, so it. is expected that the swirl velocity closer to the axis should increase with a decrease in split ratio to conserve angular momentum. Because the axial velocity in the outer region is independent of split ratio for z = 20 cm (see Figure 5.1) and for z = 32 cm as well as 2 = 8 cm (not reported here), it follows that (Ur) is also independent of 00/0u in the outer region, provided the flow is axisymetric. This implies that should have a similar behavior, which is confirmed by the data shown in Figure 5.2 in the outer region. 104 It is noteworthy that a reverse annular flow still occurs for QO/Qu = O at z = 20 cm. For QO/Qu = w, a reverse flow on the core probably happens within the vortex finder. Evidence for this is given later. Figures 5.3 and 5.4 show the effect of feed rate (i.e., Reynolds number) on the tangential and axial profile at z = 20 cm and a split ratio of 4.0. All of the experiments were conducted at T = 12°C. Changes in QF have a pronounced effect, as expected, on the tangen- tial profile. These results underscore one of the characteristic difficulties in operating a hydrocyclone compared to a centrifuge. A small reduction in QF will decrease the centrifugal force field in the inner and outer region. Figure 5.1. shows that a reduction in split ratio, which will dilute the underflow suspension, may partly compensate for this in the inner region where the secondary separation of solids and liquid occurs. As QF increases, in the core region tends to increase with a slight increase in the diameter of the downward core flow. At the same time, the upward flow around the core and the downward flow near the wall also increase. This suggests that a large internal recirculation flow occurs for large QF (also see the velocity pro- files in Figure 5.7 at z = 8 cm). Thus two factors seem important as the Reynolds number increases: (1) the centrifugal force field obviously increases (see Figure 5.4); and (2) the increase 105 , cm/sec O O -80 r I I i l ‘ ‘ 3.2 2 4 1 GI 0 8 0 0 8 1 5 2 4 3 2 r g cm Figure 5.3. The effect of Reynolds number on for QO/Qu = 4 at z = 20 cm. 106 — 25500 18.6 15.5 15 ----- 24300 15.3 13.7 13.4 _._._ . 20100 10.3 9.2 9.0 ----- 14300 5.1 4.57 4.63 Vortex finder wall 240 . , cm/sec 160 . _____ 20 80- z e 0 cm 0.0 . u 0.0 0.8 1.6 2.4 3.2 4.0 I“, cm Figure 5.4. The effect of Reynolds number on for QO/Qu = 4 at z = 20 cm. 107 recirculation of the fluid from the vortex finder increases the importance of secondary separation by the inner swirling flow. As seen from Figures 5.1 and 5.3, the locus of zero axial velocity in the outer region is insensitive to operating condition. These results support the previous speculation by many researchers that this parameter is determined mainly by the geometry (see p. 15 in Bradley. 1965). Note also that the size of the core region for fixed split ratio is also insensitive to the flow rate (see Fig- ure 5.4). Apparently, the diameter of the vortex finder determines the size of the core region, which is a new result with this research. The data in Figures 5.2 and 5.4 show that decreases with r in the outer core. The index n, defined in Figure 4.3, for this set of data is approximately 0.5 at z = 20 cm for all_operating conditions. This result is smaller than the one observed with a hydrocyclone having a 2:1 contraction in its vortex finder (see Figure 4.3) and with a 3"-hydrocyclone operating with an air core (see Kelsall, 1953), yet larger than the one reported by Knowles et al. [1973] for this hydrocyclone. The value n z 0.5 is universally reported for the operation of a gas cyclone (see, e.g., Ter Linden, 1949). Here n varies slightly from 2 = 32 cm to z = 8 cm, which shows some retardation of swirl due to viscous effects. Figures 5.2 and 5.4 show another important feature of this hydrocyclone. From other data, Bradley [p, 39, 1965] has shown that the reduction in the swirl velocity at the inlet is given by a = 3.7 01/0 1 I (5.1) 108 where a is simply the ratio of the peripheral tangential velocity to the inlet average velocity, QF/AF' From Figure 2.3, D1/D z .28 for this hydrocyclone, so according to Eq. (5.1) a = 1.04. This is very consistent with the data reported in Figures 5.2 and 5.4 and elsewhere in this dissertation. Figures 5.5-5.10 show the axial and tangential profiles mea- sured at different axial levels in the hydrocyclone for QO/Qu==O.25, 4.0, and «a respectively. The results are qualitatively similar to what has been discussed before, but a few features are worth noting. Figure 5.5 shows that the three cell structure at z = 32 and 20 cm has disappeared at z = 8 and 2 cm. Table 5.1 shows that this correlates with a reduction in the circulation (i.e., loss of angular momentum) as suggested by the previous work of Nuttal [1953], Binnie [1956], and Nissan and Bresan [1961]. Note that a small velocity defect at r = 0.8 remains at z = 2 cm. Figure 5.9 shows a stagnation point on the axis just below the entrance to the vortex finder for Qo/Qu = «n Apparently, a flow reversal occurs in the vortex finder even for this extreme case. This observation yields the interesting, and most remarkable, conclusion that two stable_vortex flows aligned with the axis of symmetry can exist with a surface of zero axial velocity between them! This phenomenon has been reported before for confined swirling flows (see Chanaud, "1965), and is one of the consequences of an instability known as "vortex breakdown" (see, esp., Lewellen, 1971). Table 5.1. Change in flow topology with average circulation. Volumetric Position .Qirculation flow rate, 2, cm r, cm3/sec c /sec Type of flow 0 -- 4OO -- 8 1683 401 1-cell with velocity defect 20 3673 383 3 cells 32 6878 325 3 cells 110 80 . ~~\ \ \ 0.0 \ \\ \/ \'- \,\ ./° \‘ -80 . vortex finder wall ' -160 . R U N a I- .- : 5 -240. - : ‘ x“ . .' N . 5’ : I [3 -320. I ./ .5 l .' -400. ; PF -- 15.4 psig ' P0 = 14.0 Pu -- 11.7 -480- 5 -520 , - - 0.0 0.3 1.5 2.4 3.2 730 r, cm Figure 5.5. Mean axial velocity profiles for QO/Qu = 0.25 and ReF = 24,200. 320 . 240 - 160 . . , chsec 80 - 0.0 111 .___.- vortex finder wall * ____.32 -———— 20 ----- 3 _____ z = 0 cm 15.3 psig O = 14.0 Pu = 11.8 1 ' u I I I 0.0 0.8 1.6 2.4 3.2 4.0 Figure 5.6. 7‘, cm Mean tangential velocity profiles for QO/Qu = 0.25 and ReF a 24,200. , cm/sec 112 Vortex finder wall . \\\ p . 20 ..... 160 . . I I _-O _._._.+ I I 1 l ”*1 0.0 0.8 1.6 2.4 3.2 4.0 Figure 5.7. Mean axial velocity profiles for QO/Qu = 4 and ReF a 24,300. 320 . , cm/sec 113 vortex finder wall / —‘I' Figure 5.8. r, Clll Mean tangential velocity profiles for Qo/Qu = 4 and ReF = 24,300. 114 320. vortex finder wall / 240. 150 1 .- 80. 3 E U A. N 3 V 0.0 '80 ' fi r T I 0.0 0.5 1.5 2.4 3.2 4.0 r, cm Figure 5.9. Mean axial velocity profiles for QO/Qu = «land , ReF a 24,100. 115 , cm/sec vortex finder ---3 '--- 320 7 wall _._20 ----- 3 _____ 240 z = 0 cm 160 80 0.0 , , ' , . . 0.0 0.8 1.6 2.4 3.2 4.0 P, cm Figure 5.10. Mean tangential velocity profiles for QO/Qu = w and ReF = 24,100. 116 As far as we know this is the first time that such a structure has been observed in hydrocyclone flows. The average radial velocity was determined by applying Eq.(4.1) to the data shown in Figure 5.7. Figure 5.11 shows the average radial velocity profiles which.are similar in the outer region to the results shown in Figure 4.5. In the upper part of the hydro- cyclone, U(r) is still slightly positive and, thereby - would work in concert with the centrifugal force to remove solids from the central core region by convection and viscous drag. 5(r) in the lower portion also looks like the results found earlier with the exception of the core flow. This was expected since the jet-like core with its characteristic negative radial velocity (see Fig- ures 4.1 and 4.5) is absent in these experiments. 5.2. Equal Overflow and Underflow Areas The predominant three cell reversal structures characteristic of the large vortex finder with and without a contraction are essen- tially equal overflow and underflow areas; the exception occurs for a split ratio of 0.25 as illustrated by Figures 5.12-5.14. These qualitative findings were also observed by Escudier et al. [1980] for converging, swirling flows with only a single exit. They reported transition from a three-cell to a two-cell structure for a 2:1 contraction. These results provide some understanding of why the design criteriaof Rietema (see Figure 2.3) are optimal. With a three-cell structure like the one shown in Figure 5.7, entrainment of solids from the apex region should be less than what 117 Vortex finder wall Figure 5.11. Average radial velocity profiles for QO/Qu = 4 and ' ReF 8 24,300. , cm/sec 118 vortex finder wall 40 0.0 -40 -120 -200 -280 I I I I I —I 0.0 0.8 1.6 2.4 3.2 4.0 r, cm Figure 5.12. Mean axial velocity profiles for QO/Qu = 0.25 and ReF = 24,300. 119 480. 400 vortex finder wall / 320- 0 g 240.. \ E U AN IF = 15.3 psig 3 P0 . 11.1 150 - P . 12.7 U 30 _ 0.0 -80 ”-fi. , - . 1 0.0 0.8 1.5 2.4 3.2 4.0 r, cm Figure 5.13. Mean axial velocity profiles for QO/Qu = 4 and ReF 3 24,300. 400 q 320 - 240. 120 \ vortex finder wall U 35 15°— PF 5 15.3 psig 5 P0 = 10.5 A. - ' 12.9 :N \ V o 80- .0. . «. \. ./ ‘-\‘ \ _. \. -80 A l l l I *1 0.0 0 8 1.5 2.4 3 2 4 o r, Clll Figure 5.14. Mean axial velocity profiles for Qo/Qu = w and ReF a 24,300. . 121 would occur with the two-cell structure shown in Figure 5.13. This important conclusion is discussed further in Chapter 6. The velocity defect at r - O and z = 8 and 20 cm in Figure 5.4 may be the result of some small gas bubbles in the core region when these experiments were conducted; therefore, the physical signifi- cance of these profiles for r < 3 mm may be questionable. The tangential profiles at z = 32 for the three split ratios studied are shown in Figure 5.15.. Once again the outer core does not change much with 00/0u but the tangential velocity in the forced vortex region changes significantly. The results of Figure 5.13, which shows that the mean axial velocity for r = O is essentially constant for 2 cm 3 z 5.32 cm, and Figure 5.15, which shows that a r for r §_4 mm, imply that the mean radial flow in the forced 8 vortex region is negligible (c.f. Section 4.1). A corollary to this conclusion is that the turbulent transport of momentum in the vortex core is also negligible. This could be an important feature for the equal area discharge design but the evidence for this (and the reason) is minimal. Figures 5.16 and 5.17 show, respectively, the effect of capacity (i.e., Reynolds number) on the axial and tangential velocities. It) is interesting that the location of the zero axial velocity is insen- sitive to the Reynolds number and that it occurs at about the same radial location as the outer "mantle" in the Rietema design (see Figure 5.2). The tangential profiles shown in Figure 5.12 have a much broader shoulder and a sharper maximum than those inFigure 5.2. , cm/sec 122 I'\_ 560- l" I \ .8-_I I vortex finder wall 400 _ . 320 _ 'I -I 2409' II 150-“ _I II II 0/0 P P p 80..- 0 U F 0 u 0.25 15.3 ' 12.2 11.0 -—-— 4 15.3 11.0 12.4 ----- W 15.3 10.5 12.8 -"-w- 0.0 1 - T . - 0.0 0.8 1.6 2.4 3.2 4.0 r, cm Figure 5.15. The effect of split ratio on for ReF s 24,300 at z = 32 cm. T400 320- 160. , cm/sec 80. 0.0 123 vortex finder wall _._._ 20 z = 0 cm psig ReF PF P0 Pu 24.300 15.4 11.1 12.7-————— 14.600 5.1 4.0 4.6 ..... -80 0.0 1 I I I fit 0.8 1.6 2.4 3.2 4.0 I". cm Figure 5.16. The effect of Reynolds number on for Qo/Qu = 4 at z = 20 cm. ‘ , cm/sec 124 Vortex finder wall Tin 400- -——~ ———20 320 - I \ \ = I \ z 0 cm 240. I \ P519 '- \ p. p. p- I! ‘\ \ 24.300 15.3 11.0 12.5 —— f" '\ \ 20.230 10.2' 7.5 8.5 ----- 15°— '\ \ 14,450 5.1 3.5 4.4 ————— '\. \\ 9.590 2.0 1.5 1.9 °°°°° \ . \ \\ \. \\ ..' '.. \. \ 80 1 : 2... -..~__ .-_. ‘K ‘-\.\ ................. \.\ ..... -.,\.\ I 1 ' ‘ ' j 0.0 0.8 1.5 2.4 3.2 4.0 r, cm Figure 5.17. The effect of Reynolds number on for Qo/Qu = 4 at z = 20 cm. 125 For PF = 15.3 psig. the maximum tangential velocity observed for the hydrocyclone fitted with the smaller vortex finder is about 33% larger than the results for the optimal design. The values of in the outer region for these two cases are nearly equal. 6 The reason for this difference is probably due to the reverse flow phenomenon not seen in the equal area discharge case. 5.3. The Effect of Vortex Finder Design on Flow Reversals During the course of this research. it was discovered that the geometry of the vortex finder plays an important role in deter- mining the flow patterns in the core region. A series of experi- ments was conducted under similar operating conditions to catalogue some of the possibilities. Figure 2.5 shows the various vortex finders studied. The axial velocity is unaffected by the shape of a contraction within the vortex finder (see Figure 5.18). However. Figure 5.19 shows that for a finite split ratio (here QOIQu = 4) the axial velocity profile obtained with a smooth contraction at the inlet has a "defect" of r a 3 mm. If the split ratio is increased to a. then the qualitative behavior for the two entries is the same. as shown in Figure 5.20. Figure 5.21 shows that the detail design of a vortex finder does not influence the outer swirl velocity. but has a significant effect. as expected. on the inner region. For the same split ratio and capacity. the tangential velocity for the small diameter vortex finder is higher and the core diameter is smaller. This occurs 126 320 vortex finder wall 1 DC 240 __ . ' ¥ ‘ \ J — PF 2' 15.4 psig ‘ P0 3 11.9 | Pu a 13.0 160..| . 1 JE ‘” I m 3 l ———1 o , . A“ 80. l j J "’T N l 3 \ l 0.0 V —\_.J_ '80 l 1 I l I 0.0 0.8 1.6 2.4 3.2 4.0 r. cm Figure 5.18. The effect of the degree of contraction in the vortex finder on for QO/Qu = 4 and ReF = 24,300 at z = 32 cm. 127 vortex finder \ l :7 7 § ‘ PF = 15.3 psig E 160. ‘ -P0 = 10.6 0 o A. l P = 12.9 N l H "‘ u 3 l \ -8“ I - r I I i 0.0 0.8 1.6 2.4 3.2 4.0 r, cm Figure 5.19. The effect of the degree of contraction on for QOIQu = 4 and ReF a 24,300 at z = 32 cm. . cm/sec 128 480 . \ vortex finder wall 240- 160. 80- 0.0 .3Q | l I 1 l 0.0 0.8 1.6 2.4 . 3.2 4.0 r, cm Figure 5.20. The effect of the degree of contraction on for QO/Qu = w and ReF = 24,300 at z = 32 cm. 560 480 _ 400 - 320 - , cm/sec 240 - 129 AI I I I I I l ' I ¥$:§:: wall (psig) I \ PF 15.4 i '\ P0 2 11.8 I . I H Pu a 12.4 —\ . PF a 15.3 . P0 = 13.5 I Pu a 13.3 E 15.3 0 = 11.0 3 12.4 '\ 80 I I I I 0.0 l l l l I 0.0 0.8 1.6 2.4 3.2 4.0 r, cm Figure 5.21. The effect of vortex finder configuration on for Qo/Qu = 4 at z = 32 cm. 130 because the acceleration of the fluid, caused by the contraction, tends to preserve angular momentum by increasing the swirl to compen- sate for the reduction in the cross sectional area. Figure 5.21 also illustrates that the "size" of the core region scales with the smallest diameter in the contraction. Bradley [p.115. 1965] mentions that the vortex finder radius is usually specified to make a maximum at a radial position within the vortex finder. 6 He argues that this would give a very high and desirable centrifugal force field near the mouth of the vortex finder. However. the results of Figure 5.21 imply that this occurs for any vortex finder diameter. so this is really a tenuous design criterion. Perhaps the vortex finder diameter should be related to the flow reversal phenomenon previously discussed. Figure 5.22 sumnarizes how the nunber of flow reversals in the core region changes'with design. For the design specified by Rietema. three cells occur with a core flow toward the apex. By decreasing the diameter. the morphology of the flow changes from a three to a two cell structure with a core flow toward the vortex finder tube. Finally. with a 2:1 contraction in the vortex finder tube. a four cell flow pattern occurs with a jet-like core flow toward the vortex finder. These results are currently not applied to design hydrocyclones..but a better understanding of their quantita- tive importance may lead to significant improvements in some applications. 131 . cm/sec 400 . . TI Rietema \‘ (psig) I —\ _ _ __ PF :- 15.3 320 . \ \\\\\ // Po 6 11.2 I . Pu a 12.5 I vortex finder ‘ ' ‘ wall a [I 240 I l I * .. I PF a 15.4 ‘ \ 0 = 11.9 I = 13.0 160 . \ 11 F \ - U _.«__._ P = 15.4 F 50 _ I.‘ ,4 I J P0 2 13.5 13.3 -80 1 , , , ._1 0.0 0.8 1.6 2.4 3.2 4.0 r. cm Figure 5.22. The effect of vortex finder configuration on for QO/Qu = 4 at z = 32 cm. CHAPTER 6 DISCUSSION AND CONCLUSIONS 6.1. Summary Discussion of Results A systematic set of experiments were conducted in order to investigate the effect of operating conditions and design variables of a 3"-hydrocyclone on the mean velocity field. The conclusions developed from these experiments resulted in new findings as well as confirmation of some previous results obtained by other researchers. Mean velocity measurements for the hydrocyclone with a 2:1 contraction resulted in very complex and interesting flow patterns inside the core region. For a split ratio of 4. four helical reverse flows were measured quantatively and confirmed by flow visualization (see Figure 4.1). The central core region for this configuration is a stable and coherent high velocity region. which extends over the entire length of the hydrocyclone. A reverse flow is initiated from the contraction zone within the vortex finder and tends to fade close to the apex region (see Figures 3.1 and 3.2). For this flow condition. the locus of zero axial velocity is located on a conical surface in the outer region and on a cylindrical surface in the inner region. This shows that the surfaces of zero axial velocity are determined by the geometry rather than the operating .conditions. 132 133 The axial location of the contraction inside the vortex finder was found to be an important factor in creating multiple helical flows. However. the degree of contraction has little effect (see Section 5.3). The morphology of the fTow field is also affected by the split ratio. As the split ratio changes from w>to 4. the flow pattern increases from a two-celled to a four-celled vortex flow. With a further decrease in the split ratio. the number of cells drops from four to three. The velocity field in a hydrocyclone based on Rietema's "optimal" design criteria was studied quantitatively using LDA. The results of this study i1lustrate the effect of the split ratio and the inlet flow rate on the tangential and axial velocities. The split ratio completely altered the flow field qualitatively whereas the inlet pressure affected the flow quantitatively. A region of reverse flow was studied for a split ratio of 4. This reverse flow started inside the vortex finder and its magnitude decreased as it moved toward the apex region. For a split ratio of 0.25. the high velocity core region tends to increase as it moves downward (see Figures 5.5. 5.7. and 5.9). The flow in the outer region was unaffected by variations in the split ratio. . The complete flow pattern for a hydrocyclone with equal over- flow and underflow areas revealed only two regions of flow: an upward helical flow in the core region and a downward helical flow surrounding the core flow (see Figure 5.13). 134 A comparison study of the axial velocity for different vortex finder configurations showed that the vortex finder design influences the core region but not the outer region. The velocity profiles measured for all vortex finder configurations discussed in this study revealed no significant change in outer region as a result of change in the split ratio; however. the effect of the vortex finder on the core flow is important. Depending on the specific design. a three-celled. a two-celled. or a one-celled vortex flow is obtained (see Figure 5.22). Mean tangential velocity measurements showed two regions of flow: force.and free vortex. In the force vortex region. the tan- gential velocity changes linearly with radius and. in the 'free' vortex region. the relationship is described by orn = constant where the index n ranges from 0.5 - 0.7 for the majority of experi- ments. For the hydrocyclone investigated. the tangential velocity profiles are approximately independent of axial location. The split ratio affects the maximum tangential velocity and the core diameter only slightly; however, for smaller vortex finder diameters. the maximum tangential velocity and the core diameter are affected sig- nificantly (see Figure 5.21). The inlet flow rate affects the flow pattern inside the hydro- cyclone quantitatively. not qualitatively. The locus of zero axial 135 velocity and the core diameter are unaffected by changes in the inlet flow rate. Over the operating range of the hydrocyclone with a contrac- tion. the addition of 100 wppm Separan AP-30. which showed 50% drag reduction in fully developed pipe flow. eliminated the flow reversal and suppressed the tangential velocity. Figures 4.12 and 4.13 illus- trate this quantitative effect.* The effect of polymer on specific flow structures in the hydrocyclone at higher Reynolds numbers remains unclear. The effective viscosity of fresh Separan AP-30 is a function of chemical composition of the solvent (pH and salt content). Results obtained for drag reduction in a capillary tube for fresh polymer solution with pH 3 10.5 exceeded Virk's maximum drag reduc- tion asymptote (see Figure 8.4). This is a new discovery which merits additional attention. The flow-rate. pressure drop relationship for the 3"-hydro- eyclone used in this research was found to be in good agreement with other investigators (see Table 8.1). The theoretical result. QF “ (PF ' P0)% 9 *The mean axial velocity profiles measured at 12° and 20° showed only minor differences in the viscous core region. Therefore. the above effect is probably due to the viscoelastic nature of the fluid rather than to the 24% difference in the viscosity. This should. however, be studied further. 136 is consistant with the experimental findings of this study. The small difference between the experimentally determined exponent of 0.42 on AP and the theoretical result is due to entrance effects. 6.2 Conclusions Based on this Research FlouIRevenaala A 2:1 contraction in the vortex finder plays an important role in the formation of four distinct. simultaneous. countercurrent 'axial flows in the conical section of thehydrocyclone. The con- traction causes an annular flow reversal. without the contraction a flow reversal is still possibTe. but it contains the axis. .Ihgjififiect 05 SpiLtIZatio The morphology of the flow field is affected by the split ratio. With a 2:1 contraction in the vortex finder. as the split ratio decreases from alto 4. transition from a two-celled structure to a four-celled structure implies that a critical Rossby number exists between 1 .and 0.8 (see Eq. (1.7)). With further decreases in split ratio from 4 to 0.25. transition from a four-celled struc- ture to a three-celled structure occurs (critical Rossby number between 0.8 and 0.2). In the upper part of the hydrocyclone a six- celled vortex flow was detected by reducing the backpressure. This could result in a multiple recirculation zone. For Rietema's 'optimal' design as the split ratio is changed from «'to 0. transition from a two-celled to three-celled vortex flow occurs at QO/Qu= 6.8. Four cells were not encountered for this 137 hydrocyclone configuration. For this hydrocyclone design. a stag- nation point on the axis just below the entrance to the vortex finder for 00/0u a «’was observed. Apparently. two stable vortex flows aligned with the axis of symmetry can exist with a surface of zero axial velocity between them. The outer region of the hydrocyclone is unaffected by the split ratio. The [556th ofi Vouex Pinata. and Lu Contmetéon Change in the design of the vortex finder only affected the core region. The locus of zero axial velocity remained unchanged. Results of this research show that the location of the maximum tan- gential velocity is unaffected by the size of the vortex finder. This conclusion is in sharp contrast to the current understanding of vortex finder design discussed by Bradley [p. 115. 1961]. The flow field is unaffected by the degree of contraction in the vortex finder. but its morphology is strongly dependent on the location of the contraction. The.Efifiect ofiPoLymen onnthe.F£ouIFi££4 A dilute polymer solution (100 wppm Separan AP-30) eliminated the flow reversal and suppressed the tangential velocity in a 3"- hydrocyclone with a 2:1 contraction. Figure 6.1 shows the axial profiles obtained with and without polymer additive at the same Reynolds number (different temperature including viscosity effect). Note that the addition of polymer seems to have eliminated the flow reversal surrounding the central core vortex due to viscoelastic 138 , cm/sec 400 Vortex finJcr wall 320 t \ t I I 20 ‘ 5 240. z 8 0 cm I I I , . 150 | 9519 C . | . : PF Po Pu T I -————- 15.4 12.1 12.7 20° I | ----- 15.4 9.8 10.5 12° 30. l . i I | I I I 0.0 -80Ir_ . ' . - 0.0 0.8 1.5 2.4 3.2 4.0 Figure 6.1. The effect of polymer (100 wppm Separan AP-30) on for 00/0u = 4 and ReF = 30.000. 139 property of the dilute polymer solution. Drag reduction experiments in a capillary tube for this polymer at pH 2 10.5 exceeded Virk's maximum drag reduction asymptote. This enhancement in drag reduc- tion can be interpreted as a result of molecular expansion due to electrostatic repulsion between like charges in the polymer chain. The £562.01 06 Inlet onwRaLéo The morphology and the locus of zero axial velocity are unaffected by the inlet flow rate; however. the magnitude of the velocity profiles is changed to compensate for flow displacement inside the hydrocyclone. Compatéaon 06 Rietmna’a 'OpLOnaC' 02.55293}- with othem-ngnoegggone‘Debégné Comparison of Rietema's optimal design with other hydrocyclone configurations revealed low swirl and a smaller outward radial velocity for the optimal design. A lower centrifugal field and a low outward radial flux create an unfavorable condition for separa- tion. Although a high centrifugal field is desirable. the entrain- ment of solid particles from the conical boundary-layer into an upward flow deep in the apex makes other hydrocyclone designs inferior to Rietema's optimal design. This conclusion stems from Figure 6.2 which shows a comparison between the axial velocities in the two different hydrocyclones in the apex region. Note the absence of upward jet-like flow in the Rietema design. 140 Rietema's optimal design . oI - Equal overflow and under- 0 cm/sec flow areas 01 ".‘. 1r* . m I 1.0 . ' 9. sir. 1.0 .0. o 0’. o "O I Figure 6.2. Mean axial velocity profiles for Qo/Qu = 4. ReF a 24.300. and at z = 2 cm. 141 6.3. Indirect Conclusions Suggested by the Results of this Research Mechanism 604.1he Foamation 05 Multipfie Axial Celia Figures 4.6. 5.1. and 5.22 show that the mass flux through the conical boundary layer is independent of the split ratio. At a very high split ratio (or Rossby number) the vortex flow in the underflow chokes because of the relatively large back pressure com~ pared with the fluid pressure in the apex zone. The result of this is an ejection of mass from the viscous boundary with a strong jet- like flow appearing at high split ratios (see Figure 3.4b and the central core region of Figure 4.1). On the other hand. as the split ratio decreases more of the boundary layer fluid can discharge through the underflow. If an upward flow on the axis occurs for 00/0u = 0.25. it is not strong enough to be detected. Hhat is observed is an axi- symmetric downward core flow. which forms in the vortex finder and covers the entire length of the hydrocyclone. As the split ratio increases. an upward jet-like flow forms on the axis in the apex as previously mentioned (see Figure 3.2). The annular reverse flow caused by the contraction in the vortex finder at 00/0u = 4.0 is unable to penetrate the axial jet and the net result is a four-celled vortex structure. What is the physical origin of the jet-like behavior? Based on the experiments developed in this research. it is conjectured that the axial upward jet flow results when the viscous boundary layer begins to interact with the core flow. This happens deep within the apex region. Above this region and in the primary flow. 142 the e-component of the Reynolds equation is balanced by the inertial terms. The coriolis force of the mean field and the radial convec- tion of e-momentum balance the axial and radial transport of fluc- tuating e-momentum. The Reynolds equation in cylindrical coordinates is given by Boysan et al. [1982]. The above physical situation implies the following simplification: 3 0 r 0 3 I I 3 I I r 9 ur 3r r 32 uzue 3r r urue r (6.1) In this equation. the viscous transport terms have been neglected as well as . (U) 9 - ~ 32 a 0 . (6.2) Equation (6.2) follows from the experimental data obtained in this research. Equation (6.1) cannot hold deep in the apex where the viscous boundary layer begins to interact with the core flow. Furthermore. our previous analysis of suggests that = 0 at some r > 0 and we conjecture that this will also occur for z < 8.cm. Now. if begins to decay slightly in the apex. then Eq. (6.2) is no longer true. Indeed. for z‘+ 0, we anticipate that 3 32 > O 143 and that the viscous transport of momentum dominates the turbulent transport. Whence. the e-component of the Reynolds equation reduces to a balance between the axial convection of the average B-momentum and its viscous transport. i.e.. 3 6 3 3 32 = V'EF E%'§F (r)]. (5-3) > decays slightly. then (6.3) is balanced by a positive value of . Thus. an upward jet-like (if the right-hand-side za . is large and-——5;—-is small) flow is expected. Mechaniam 50n.the Fonmation ofiMultipLe Mantiea Figure 4.11 shows that for low back pressures several reverse flows may occur in the upper portion of the primary flow domain. The possibility of multiple mantles has been discussed before (see Bradley and Pulling. 1959). The idea is illustrated by the follow-1 ing simple schematic. 144 D) X _I m Pumping action Ejection of Ekman layer /c/, V I ‘1’. Entrainment '/ \ Conical boundary I layer \_l/. Multiple mantles with Taylor—like vortices If the angular momentum r decreases with increasing r in the outer region of the primary flow. Rayleigh's criterion for centrif- ugal instability is satisfied (see p. 272 in Greenspan. 1968). In physical terms. if a fluid particle at r with tangential velocity (r) moves to r + Ar. then it attains a swirling velocity equal to ( r W) ("I . provided angular momentum is conserved. For this situation. the centrifugal force which acts on the particle at r + Ar is 2 r 2 m (U6) (1‘) . 145 This force opposes the inwardly directed pressure force with magnitude 2(r + Ar) 0‘ + Afi Obviously. this force cannot balance the outward centrifugal force if the angular momentum decreases with r. Thus. based on an inviscid analysis. the outer primary flow is unstable and Taylor-like roll cells would occur. However. viscosity tends to stabilize the flow and obviously plays an important implicit role in maintaining the two-dimensional character of the outer primary flow. In the upper portion of the hydrocyclone additional effects must be considered. Here the angular momentum distribution is more complicated because of the vortex finder and the presence of an Ekman layer. which can eject mass in the vicinity of the maximum swirl velocity (see Section 1.2.1). Loss of mass from the boundary layer is compensated by entrainment of fluid from the primary flow so a pumping action on the roof occurs. In this way. a local region of circulation develops which supports the basic Rayleigh instability mechanism. The net result is the appearance of multiple mantles. Suppaesaionofigjunbuzenz:Thanspont in .the Conelzegggn.bnyo£ymen.AdstZueb In the vortex core. the mean tangential velocity is approxi- mately independent of z and linear in r, i.e.. = Kr . (6-4) 146 where K is a constant. Donaldson and Snedeker [1962] have shown that for the special case when the tangential velocity depends only on r. the r.6-component of the total stress (viscous and turbulent) can be determined from a measurement of the mean velocity profiles and . The r. e-component of the viscous stress is zero for given by (6.4). The result. which also assumes that u'u' z 9 is independent of z. is ' r .I.._1_ .3.- - - r2 I; r 3r (r) dr . (6.5) With given by Eq. (6.4) and using the continuity equation, the above result implies that I , _____ .5 32 I r=0 4 r (6'6) near the axis of the forced vortex. This equation can be used to determine the effect of a polymer additive on the turbulent flux. Figure 4.13 shows that the addition of Separan AP-30 (100wppm) does not affect the parameter K. However. an analysis of a/az between 2 = 8 cm and 20 cm shows a significant effect. From Fig- ure 4.1. it follows that with no polymer additive 8 32 150 ~— 3 r=D 1 147 With the addition of polymer. the experiments shown in Figure 4.12 yield the following estimate ' a I 3'59 32 r=0 12 ' Thus. according to Eq. (6.6). the Reynolds stress near the core axis is reduced by about a factor of 3 when polymer is added. Unfor- tunately, these two experiments were run at different temperatures and the viscosity of the solvent (water) at 12°C is about 24% higher than the viscosity at 20°C. Our tentative conclusion. however, is that this reduction in the Reynolds' stress is due to the visco- elastic nature of the polymer solution rather than to differences in viscosity (see the discussion in Section 1.2.4 on the effect of Separan AP-273 in vortex flows). This interesting result obviously deserves further study. CHAPTER 7 RECOMMENDATION FOR FURTHER RESEARCH Experimental findings of this research have raised many new questions regarding the mechanism of solid-liquid separations within hydrocyclones. Additional experiments outlined in Section 7.1 are required to answer completely some of thequestions in this study. In Section 7.2. new experiments are suggested which could provide additional fundamental understanding of the flow phenomena within hydrocyclones. Hopefully. the extension of this research will even- tually lead to some new applications as outlined in Section 7.3. 7.1. Experiments Needed to Complete This Study Three important problems remain for further study. These include the following. Radial Velocity In this investigation. it was necessary to make theoretical corrections for the refraction of the laser beam (see Section A.3).‘ Because of the lens effect. direct measurements of the radial com- ponent of the velocity were difficult and. unfortunately. were not made. It is strongly reconnIended that the experimental test section be designed to eliminate the refraction phenomenon. This can easily be done by matching the refractive index of the test fluid and the glass walls of the hydrocyclone. For this situation. a direct 148 149 measurement of the radial velocity is possible without any diffi- culties. The radial velocity component plays an important role in the performance of hydrocyclones. Tunbuzent Intenaétiea Unfortunately. because of difficulties in obtaining proper computer facilities. turbulent measurements were not made. Use of a computer to record velocity fluctuations can provide additional information related to the velocity field inside the hydrocyclone. Any theoretical modeling of the flow field within the hydrocyclone would require this information. FeediPaeaauae All of the experiments in this research were conducted for relatively low inlet flow rates and pressures. It is desirable to insure that the basic results obtained from this study are also applicable at other operating conditions. especially for the experi- ments with polymer. The effect of 100 wpmeSeparan AP-30 may have very important consequences in the core region (e.g., on flow rever- sal). The effect of polymer concentration (for optimum condition of polymer solution (see Appendix B) for relatively high inlet flow rates) will provide some fundamental information related to the effect of polymer in the core region. Results of polymer experi- ments compared with the Newtonian solution with high viscosity will illustrate the viscoelastic property of the polymer. 150 7.2. Problems Which Would Provide New Fundamental Understanding Among the numerous problems which could provide significant new information only the following are mentioned. The 56525: 55 2W2 Loadgzg Fine glass beads with the same index of refraction of water can be used to study the effect of solids loading on the velocity field. Simultaneous velocity and grade efficiency measurements should provide valuable information related to the performance of the hydrocyclone under different operating conditions. This infor- mation should provide the means to improve either specific applica- tions or design. Flow'ViauaLization Flow visualization for these studies was conducted under cir- cumstances where the hypodermic needle was inside the hydrocyclone. In order to trace the dye path precisely. it is recommended to con- duct a full set of experiments where an organic dye is excited by a high energy laser beam. A high speed camera is necessary to record the stream of dye for further analysis. Shoat Ciaeuiz Phenomena Interesting flow features above the vortex finder are expected due to short circuit and multiple mantles. Information related to the interaction between the boundary layer on the top roof of the 151 hydrocyclone and the main fluid body of the hydrocyclone should provide some understanding of the short circuit phenomenon. These short circuits and multiple mantles are related to residence time and primary separation of particles. which influence separation efficiency. The Efleet (1 Au Cone LDA measurements of the velocity field using the back scatter- ing mode in a hydrocyclone operating with an air core are recommended. Comparison of velocity profiles obtained with and without an air core may provide some additional understanding of flow reversals as well as the ejection phenomenon in the apex and the vortex finder. Inxzaaetion 06 Vontex Findea and the Hgdeoggczone During the course of these experiments it was noticed that flow reversals begin inside the vortex finder. It is highly recom_ mended that the flow characteristics of this part of the hydro- cyclone be probed quantitatively using LDA. The flow fields inside the vortex finder are apparently coupled and. in order to clarify this phenomenon. it is suggested that a series of experiments for a hydrocyclone without a vortex finder be conducted and the flow field compared with those measured with the vortex finder. 7.3. Problems Which Could Lead to New Applications Two novel applications of hydrocyclones which exploit the underlying secondary flows are suggested. 152 0E£.Waxzn.3epanazion The stable and coherent structure of the core region. obtained for a hydrocyclone with a 2:1 contraction and confirmed by flow visualization. might be used as a means to separate oil contaminate from water (e.g. crude oil). Figure 7.1 illustrates the possibility. which should be compared with the strategy being pursued by Thew and his colleagues at Southampton (see. esp. Colman and Thew [1980]). The application of hydrocyclones to this separation problem may be possible provided the following conditions are met: (i) Stable and coherent flow structure is formed in the core region for split ratios of 0.0 and 4. (ii) The flow pattern in the core region is not affected adversely by the presence of a suction tube inside the hydrocyclone. The Hgdnoggezone a5 a.Chemica£.Reacxnn. Trajectory calculations can be used to estimate the residence time of particles inside a hydrocyclone. A hydrocyclone may be uniquely suited for suspension or for crystallization processes. In reactions where residence time may be crucial to product quality or selectivity. hydrocyclones may be used to perform separation and reaction simultaneously (see. esp. Beenackers [1978]). 153 .covuuacucou a gap: ego—oaoocc»: a.:_ ocoo _.o mo :o—umseom .~.~ meam_e .xu.__amum ocou nape; :ovuuacueou cove.» xmaco> on mac _mmcm>oc zap; ppc .mcou mg» :_ mcopaaauuspe «copaaczu mm~_epepe coax—ca agape: capaum—oe gap; "so—w mepIumn a :. mmomopuoo ten mmumcape _wo cougmpa Tull swam: + pwo ~o \a-D Loans. 0'. APPENDIX A APPLICATION OF LASER DOPPLER ANEMOMETRY T0 VELOCITY MEASUREMENTS IN A HYDROCYCLONE APPENDIX A APPLICATION OF LASER DOPPLER ANEMOMETRY T0 VELOCITY MEASUREMENTS IN A HYDROCYCLONE A.1. Basic Principles A laser anemometer uses a laser beam to measure the doppler frequency of light scattered by small seed particles moving with the local fluid velocity. The particle diameter may be as small as the wavelength of the laser light so the particles have approxi- mately the same local velocity as the flow medium, even in turbulent streams. The velocity component measured is the one normal to the bisector of two beams with an intersection angle 61 and parallel to the beam plane. The magnitude of the velocity is determined by the following expression (see Durst et al.. 1976; and‘Durani, 1977) u = AfD/Z sin 61 . (A.1) Here A is the wavelength of light in the test medium and fD is the doppler frequency. Yeh and Cummins [1964] were the first researchers to use LDA. They developed the basic theory for a reference beam laser doppler velocimeter and obtained excellent data for laminar flow in a circular tube. George and Berman [1973] showed that unsteady measurements of fluid velocities with LDA are difficult because of doppler ~154 155 ambiguity and broadening. Doppler ambiguity arises from the random arrival and departure of the scattering particles in the sampling volume. Turbulence. velocity gradients. refractive index fluctua- tions. and noise all contribute to broadening. George and Lumley [1973] have discussed the effect of this phenomenon on the doppler signal and have presented an analysis of its effect on the beat frequency and signal demodulation. They illustrated how two-point velocity correlations could be made using two independent anemometers to eliminate ambiguity. * 0 The components of the laser anemometers may be arranged in three different modes of operation (see Durst et al.. 1976): the dual beam mode. the reference beam mode. and the dual scatter modes. The selection of a suitable mode depends on the specific application and the signal-to-noise ratio at the photodetector (see Figure 2.11). Some of the variables which affect the selection include (1) the number of the velocity components measured; (2) the phase (liquid or gas) of the test fluid; (3) the type of seeding; (4) the acces- sibility of the measuring point; and (5) the type of data required. . The dual-beam mode. applied in this research, uses two inci- dent beams of equal intensity. This configuration is often referred to as a "fringe" mode because interference fringes are formed when the two beams intersect. In this case. the scattered light can be picked up over a wide angle. Since the differential doppler fre- quency is independent of direction of detection. 156 There are two types of operation in dual-beam mode: one uses forward scattering (see Figure A.1) and the other uses backward scattering. In the backward scattering mode. the flow is viewed from only one Side. Since the dual beam mode is independent of the direction of the scattered light. all derivations are valid for both forward and backward scattering. To obtain a better intuitive understanding of the LDA tech- nique. a brief derivation of Eq. (A.1) using the real fringe concept will be given. The distribution of scattered light on the photo- detector surface is used to discuss the fringe spacing. probe volume. and the relationship between the velocity and the doppler frequency. The doppler shift was named after Christian Doppler in 1842. Basically. an observer moving away from a fixed source of light. sees the light at a lower frequency than a stationary observer. Thus. light scattered from a moving particle appears to have a lower or higher frequency. depending on its relative motion. The frequency relationship between a scattered light wave and a stationary light source can be deduced from the vector diagram shown in Figure A.2. The frequency of the stationary light source shown in Figure .A.2 is f = 11 (A.2) >40 where c is the speed of light and x its wavelength. The frequency seen by a moving observer (i.e.. seed particle) is simply 157 .muoe Emma Fuse 0 cowaomewu zopu. .H.< mc=m_e Lemme 158 c. speed of light £41 incident unit vector .951 scatter unit vector Light - Particle , source : |\\\\‘ Stationary | : ;u observer F I I c HI I : I l 1 I I I c - -e. l %_ 2- 11 (vi; Figure A.2. Vector diagram of light propagation. 159 which can also be written as u-e. a ——11 fp f11[1 - c ] . (A.3) Now the frequency of the light scattered by the moving particle is "'9 where f$1 is the frequency detected by astationary observer. Equating Eqs. (A.3) and (A.4) shows that f$1 and fi1 are related by [c-uefll A similar expression can be written for the second beam: [C ' 2.9.12] f$2 - fi2‘[E_:_§?§;;]" (A.6) The two scattered light waves Will interfere with each other and yield a beat signal which has a frequency given by — fsz . (A.7) Because both beams are from the same light source f11 = fiZ' More- over. because the light is collected in the same direction for both beams. the unit vectorsgs frequency fD is given by 1 and £52 are equal. Therefore. the beat 160 $912 ' £11) f = f , (A.8) Becauseg-g$2 << c. the speed of light. Eq. (A.8) can be approxi- mated by f g 1 no e _ e ) (A 9) D A-— i2 —i1 ‘ ° Thus. the component of g_which is measured by recording the beat (or doppler) signal is the one parallel to_e_,12 - sil (see Fig- ure A.1). and thus normal to the bisector of the beam intersection. If 61 represents the half-angle of the beam intersection. then 93.9.12 " IEI COS. (32" " 9}) 3.1.211 a 'IEI COS ("g ' 91)- Therefore. with cos (%-- 81) = Sin 61, Eq. (A.9) becomes l 2|g| sin e1I ‘ f0 = , A which is Eq. (A.1). Thus. lg_|=d1,-fD where the “fringe" spacing is given by I/(2sin 81). Optical interference between two light beams and the formation of a fringe pattern is fundamental in LDA. To illustrate the fringe pattern and intensity distribution in the probe volume. it is helpful to derive the interference formula for two coherent laser beams having the same intensity and polarization direction. 161 Consider the crossing of two light beams from the same source as illustrated by Figure A.3. The complex electric field vectors for the incident beams can be expressed as (see p. 28 in Durst et al., 1976) £1 = V1631 exp[i(wot - 2T(x cos 9 + 2 sin e))] (A.10) _E_2= “$32 exp[i(w0t - ‘ZXEI'X cos 8+ 2 sin 6))] . (A.11) Here Iois the intensity of the incident beams. 8 is the half-angle of the intersection. A is the wavelength of light. and “O is the frequency (i.e.. c/A). £1 and £2 indicate the direction of polari- zation of the beams. The electric vector within the probe volume is just the superposition of;1 and E2. viz., _§_=_E_.1 +.§2 . (A.12) The intensity of light in the probe volume is given by 1 = $1.“? (A.13) where gr is the complex conjugate of g. From Eqs. (A.10) and (A.11). it follows that (see p. 33 in Durst et al. [1976] for details) I = 210 [1 + 31-32 cos (“lam 6)] . (A.14) This equation can be used to estimate the fringe spacing by recog- nizing that I is a maximum at the zeros of cos (21m). Thus. for 162 Figure A.3. Cross section of two beams from the same source. 163 X" 35-3412??- , ll = 0, 1, 2, ... “1.15) the intensity I is a maximum. The difference between these uniformly spaced zeros gives the fringe spacing. i.e.. ._L. df 2 sin 9 ° (“'16) The mainIum fringe visibility is obtained when 31°32 = 1. This condition is satisfied when the polarization direction of both: incident beams are normal to the plane containing the beams. In the other extreme.£_1-£_2 . O. and the interference pattern vanishes (cross polarized light waves). To analyze the probe volume and its dimensions. let us examine the light waves in three dimensions. The flow is along the x axis. with the y axis normal to the plane of x and 2; (x1. y]. 21) and (x2. y2. 22) are coordinates referring to the direction of the illuminating beams. The coordinate system is shown in Figure A.4. Assume that the photocurrent is generated only by light scattered from a particle in the probe volume and moving with a uniform flow velocity in the x direction. At any time. the location of the par- ticle in the coordinate system is xp = u t + xp0 where xp0 is the location of the particle when it entered the probe volume. 164 Beam 1 Surface of Beam 2 photomultiplier tube Figure A.4. The coordinate system used for a fringe system. 165 The photocurrent on the surface of the detector is given by (see p. 129 in Durrani. 1976) 2 2 2 . 2 , + + ‘F(t) a Zn I%'M CSC)2€02 exp(- x cos 0 y 2’ 2 Sin 0 ) x 0 9.1 [1 + cos ( A (ut + xpo)sin 0] , (A.17) where M is the magnification factor of the receiving optics. csc is called the scattering cross section of the particle. n is the sensitivity of the photodetector (i.e.. the ratio of the photocurrent to the incident power of the radiation). and $02 is the intensity of the beams at the point P(O. O. 0). Inspection of this result shows that the maximum current occurs when the particle passes the center of the coordinate system. The photocurrent drops by a factor e'1 when the particle moves to the edge of the probe volume defined by x2 cos2 0 +y2 + 22 sin2 e = r02 . Thus. the shape of the probe volume is an ellipsoid with dimensions 2r 8 O Ax s a co Ay = 2r0 2ro A2: sin (65' Here 2ro is the diffraction-limited beam diameter. Some researchers 166 define the probe volume as the region over which the photocurrent 2, rather than e'l. decays by e' The probe volume and finge pattern are shown in Figure A.5. Ax is the diameter of the cross section of the ellipsoidwhere the particles pass. and df is the fringe spac- ing. The ratio of Ax and df is defined as the total number of fringes within the probe volume: 4ro tan 6 ' —7\— (A.18) .912 Once a scattering particle enters the probe volume (t = 0). the maximum amplitude does not occur until the particle crosses the center. This occurs at tmax E |-xo/u| . Now the output of the photodetector includes a mean component (low frequency) and a periodic component (doppler frequency). The mean value of the Signal has a Gaussian distribution in time and is often referred to as the pedestal or dc-component. Figure A.6 shows a typical signal (also see Figure 2.11). An interpretation of Figure A.6 follows from Eq. (A.17). The intensity fluctuates because a particle of velocity u (see the cosine term in Eq. A.17)) moves across the fringe pattern and scatters light. The frequency of this scattered light has been shifted Slightly from the incident light. The distance between the maximum peaks in Figure A.6 is just the time required for the particle to 167 Beam 1 y 28I Az Beam 2 Figure A.5. Beam intersection region in the real fringe system. The inner ellipse represents Eq. (A.17) and the outer ellipse is the locus of points at which the signal falls to e"2 of its maximum. 168 Signal processor checks for consistant reading , ,.___ At = df/u Particle enters 1f probe volume ”pm...“~ Particle leaves M w . probe volume t Figure A.6. Typical signal from the photodetector. 169 move between fringes. i.e.. t = df/u. Basically. the LDA technique measures the time between peaks and. if a consistent set of At's can be detected in one doppler burst. then u can be calculated as described. A.2. Frequency Shifting. In measuring the velocity by LDA a difficulty often encountered is the inability to determine the direction of flow. Eq. (A.1) only gives the magnitude. The most convenient way of overcoming this ambiguity is to frequency shift one of the beams. This also shifts the observed doppler frequency. so the direction of movement can be determined by an increase or decrease in the doppler fre- quency. In the dual beam mode. shifting the frequency of one of the beams can be thought of as moving the fringe system (see Fig- ure A.3). If the velocity is in the direction of the fringe move- ment then the frequency will decrease. otherwise there will be an increase in the frequency of the doppler signal. For a stationary particle in the probe volume the output signal is the same as the frequency shift. This can be used to detect the presence of the hydrocyclone wall (see Figure 2.13). Frequency shifting permits measurements in reverse flows. in secondary circulations. and in periodic flows. The frequency shifting is also a useful tool to bring the doppler frequency into a range which will be easy to process (i.e.. the doppler signal can be increased relative to the pedestal signal). For low velocity convective flows and for measuring turbulent components perpendicular 170 to the flow. frequency shifting has been a useful tool. Measuring flow reversals is probably the most common application of frequency shifting. To measure negative flows. the direction of frequency shifting should be against the mean flow. To measure flow reversals successfully requires careful consideration of the number of fringes that the particle passes when it traverses the probe volume. Adrian_and Fingerson [1981] have developed an expression for the number of effective fringes in the probe volume when frequency . shifting is employed. The final result is ' f5 "5 g NFI(1 i'?E)I where Ns and NF represent. respectively. the number of doppler cycles ,generated by a particle with and without frequency shifting. fs is the amount of frequency shifting induced and fD is the doppler frequency without shifting. The “+" Sign is used if frequency shift- ing is counter to the flow. whereas the "-" Sign. is for shifting in the direction of flow. A.3. Corrections Due to Refraction Phenomena If the optical bisector of two intersecting laser beams is perpendicular to the z-coordinate direction. then the magnitude of at the point of intersection is given by || = sways) . (A18) 171 Frequency shifting unambiguously determines the sign of the velocity. A similar expression holds for . Eq. (A.18), although simple in appearance. may be difficult to apply if the laser beams pass through several different optical media with curved surfaces. The actual measuring point x_and the fringe spacing df may both be affected. df depends on the half- angle of the refracted beams and the wavelength of the laser light in the test medium. The relationship is A 3 de‘) g 2 sin 61(5) ' (A.19) If 81 and 83 represent. respectively. the index of refraction in air and in the test fluid (water). then (see p. 843 in Sears and Zemansky. 1964) A3 = 81A1/B3 (A.20). where 11 is the wavelength of the laser light in air (= 632.8 nm for the 35 mW He-Ne laser used in this research). The half-angle 61(5) must be determined theoretically by using principles of geo- metric optics (Snell's law) and trigonometry. Expressions for the actual measuring point x_and 61(5) are developed in what follows by assuming that the half-angle of the two incident laser beams is small enough to justify replacing sin 0 with 8. This is an excellent approximation in practice. Bicen [1982] and Durst et al. [1976] have also developed equations for these corrections in 172 cylindrical geometries but the development for the conical portion of the hydrocyclone appears new with this study. A.3.1. The cylindrical section Axial Veloelty Figure A.7 illustrates the path of the two laser beams through the hydrocyclone when the axial velocity is measured in the cylin- drical section. The angles in the figure have all been greatly exaggerated for clarity. The axis of symmetry lies in the optical plane and is perpendicular to the optical bisector. Because the outward pointing normal vectors at the two interfaces (points "A" and "8") are also in the incident optical plane. the refracted beams remain in the plane containing the optical bisector and the axis of symmetry of the test rig. However. differences in the index of refraction between the phases cause the two refracted beams to intersect at point "I" rather than the focal point "F". The region between R1 and R2 contains glycerin. the glass wall of the hydrocyclone. and the wall of a pyrex box (see Fig- ure 2.4). This is considered as one homogeneous optical medium with an index of refraction 82. 81 and 82 represent. respectively. the indices of refraction for air and the test fluid (water). For this situation. the refraction phenomenon is symmetric about the optical axis; however. this will not occur for the conical section. From Figure A.7 it follows that 51 =‘xA +'ATg&3 (A.21) 173 Optical bisector Side view O -l —.—O——.—. -O-O-O—.+ 6A “7‘ ' 0 ..._._.- _._._... _._._. _._._ ...._L._‘.~.._'. _.L O I I Angle presentation Figure A.7. Refraction of laser beams during axial velocity measure- ment in the cylindrical section. 174 where 5A = -R2£_x + R2 tan eA-Qz (A.22) and 33 = cos GI 9x - sin 81 e . (A.23) "Z Symmetry about the x-axis implies thatxlfigz =10. so 5Afigz x =x - (-—,—)§ . (A.24) I —A £3.g2 3 For the small angle approximation. Eq. (A.24) gives the x-component of the beam intersection point in terms of R2. 61. and GA: 6A Repeated application of Snell's law for small angles gives (see Figure A.7) B2 B1 . (A.26) This is an important result because it shows that the fringe spacing formed for axial velocity measurements does not change with posi- tion. Thus. Eqs. (A.19). (A.20). and (A.26) give df = Al/(ZK) . (A.27) For 1.1 = 632.8 nm. and K = 5.71°. Eq. (A.27) shows thatdfs 3177 nm. 175 An equation for 6A follows easily from the relationships (see Figure A.7). 5A = -R1g_x + C332 + EK£2 (A.28) 5A = --R2 3x + D—A_e_z (A.29) 32 = cos 62B-9x - sin 928-92 (A.30) e ) , (A.31) tan K a I<==C§/(R1 +§F.—x The vector'xF locates the focal point "F" relative to the coordinate system at "O". Equating the x- and z-components of Eqs. (A.28) and (A.29) gives two equations for DA and DAL Applying the small angle approximation to these equations yields DA 3 R1 - R2 > (A.32) and 5 6A a: “”1 + xe - E:- (R1 - R2)] (A.33) where Snell's law has been used to replace 628 with file/82. Now. because tan 9A a 9A = DAYRZ. Eq. (A.25) can be rewritten as 33 B1 XIx ='§; [R1 + XFX -'§; (R1 - R2)] - R2 . (A.34) Note that xIx = xe for 81 = 82 = 83. 176 In making a traverse across the hydrocyclone. the test.rig moves toward the laser source by a known increment e (> 0). If.i{ andgF denote the initial positions of the real probe volume and the focal point of the unrefracted laser beams. then &=&+e& m”) and xIx ' x1x 5 AxIx ‘ B3 5’31 ' (“'35) This expression applies for traverses in the cylindrical section when the optical plane passes through the axis of symmetry and when the bisector angle K is small. Tangential Veloeltg Asymptotic expressions for the fringe spacing and the coordi- nates of the probe volume are developed here using a similar strategy as before. Figure A.8 shows the path of the two laser beams through the cylindrical section of the hydrocyclone when the tangential component of the velocity is being measured. Once again the angles have been exaggerated for clarity. "1" represents the point of intersection of the refracted beams. and "F" is the focal point of the unrefracted beams. Here the optical plane containing the two incident laser beams is perpendicular to the axis of symmetry and its optical bisector intersects the axis of symmetry. Unlike the orientation for measuring , the outward pointing normal 177 y 2 i ._2 . N ! B .." I \‘~ A 33 . 1 . 2 ~ “~\ I ________ .c-p — -— — —— —— ‘;h—— -+ Symmetric about 0. I F x x-axis ! 3 I I 4 R1 2 Top view Angle presentation Figure A.8. Refraction of laser beams during tangential velocity measurements in the cylindrical section. 178 vectors at "A" and "B” are not oriented in the same direction. How- ever. they are in the same plane. From Figure A.8 it follows that 51 8 5A + figs . (A.37) Because of sannetry about the y-axis. 11°35: 8 0and 'A'I'.8 -xA-gy/_g3-_ey. The vectors‘xA and_23 can be written as 5A 8 -R2 cos 8A.g* + R2 sin eA-Sy (A.38) 33 8 cos 8I (A.39) 3x - sin BI 3y . Combining Eqs. (A.38) and (A.39) and assuming that 8A and BI are small angles. we obtain 3A ' XIX '5 R2(-e—I - 1) (A.40) which is similar to Eq. (A.25). Here 8A and 81 will have a different dependence on K and the indices of refraction. It follows from Figure A.8 that 8I 8 8A - 83A. Application of Snell 'slaw at "A" gives for small angles 8 .. _2 Once again. an analysis of Figure A.8 shows that ezAgn-BZB-(1T'6A)86A-BZB, 179 so with another application of Snell's law at point_"B". which gives 928 8 BlK/BZ, an expression for BI in terms of 6A follows. viz., 32 B1 91: (1 - B—3)9A 4' .E3- K . (A.42) With 82 8 83. this gives the same result as Eq. (A.26). The angle eA can be determined from the observation that 5A can also be written as (cf. Eq. (A.29). xA8-ngx+C§gy+§A§2 (A.43) where 2.. 2y - Equating the x- and y-components of Eqs. (A.38) and (A.43) yields for the small angle approximation BA 8 R R2 (A.45) 1 - and B 8 R28A + (R1 - R2)82B , (A.46) where 928 8 Elk/82 and tan K = K 8 Cfi/(Rl + ijgx). Collecting these results shows that B R_ x 1 1 (—-1- - 1) + -F—" (A.47) 2 R2 and. from Eqs. (A.42) and (A.40). 180 31 x _ ir_________,, A.48 2 B2 2 .2 Note that for 81 8 82 8 B3. ‘xIx. reduces to xe. Also. with ilx 8 0 for e 8 0. it follows from Eq. (A.48) that 0 Bl XFX 3 - (1 - BE) R1 (A.49) and (cf. Eq. (A.37)) 2 52 6 BB 3 13 A"1x 5 -5 5 (“'50) 1 +-—§—§—-§ (l +-§Z~£L) -2 12 Eq. (A.50) reduces to Eq. (A.36) when 83 8 82 and. when all the phases are optically equivalent. AxIx 8 6. Because 6I changes with position for tangential velocity mea- surements. the fringe spacing varies with c. Combining the defini- tion (A.19) with Eqs. (A.20) (A.42). (A.47). and (A.49) shows that for the small angle approximation d 8 ‘ (A.51) where l a: = 5211/253ch (A-SZ) 181 Fore 80 and 81 = 82. Eq. (A.51) gives the same result as Eq. (A.27). However. for c > O and-B2 < 83. the fringe spacing for the tangential velocity measurements is less than df for the axial velocity measurements. On the other hand. Eq. (A.51) also shows that for 82 > 83 the opposite occurs. Thus. the curvature of the cylindrical section together with the differences in the indices of refraction gives a significant lens effect which can be corrected by applying the foregoing results. Figure A.9 sum- marizes the main conclusions of this section for the specific experi- mental system studied. A.3.2. The conical section Axial Veloeltg Figure A.10 shows the path of the laser light through the conical section of the hydrocyclone when the axial velocity is being measured. As before. the optical plane defined by the two incident beams passes through the axis of symmetry; however. the problem here is. not symmetric about the optical bisector. The intersection point "I“ does not necessarily fall on the x-axis as defined in Figure A.10. so in the analysis which follows both beams must be considered separately. The notation used here employs: a tilda (*) to denote geometric parameters related to the lower beam--for instance. compare the physical meaning of A and A. The three optical phases have the same physical meaning as before; the basic difference here is that the outward pointing unit normal vectors at "A" and "A" are rotated by an amount 6H relative to their counterparts at 1.0 182 0 8 Tangential Axial 0.6 . AxIx R2 0.4 . Intersection point 0.2 . 0 0:2 074 0:5 0.82/5 1.2 . d tangential f ‘6 1. df axial Fringe spacing 0.8 §~ , , . r O 0.2 0.4 0.6 0.8 e/R2 Figure A.9. Refraction corrections for the cylindrical section. 183 1 8 Optical bisector E I ' R 1 Axis of symmetry for IT \ I hydrocycl one ‘_ I R1 ‘ Side view €1p-_J. _‘_ K O ZAI - - {—0 .0-0 -o-o-o—o -. -. ’ I 6 a”’ 8H . Angle presentation I Figure A.10. Refraction of laser beams during axial velocity measure- ments in the conical section. 184 "B" and "B". 9H is the half-angle of the cone. which is about 11.30 for the apparatus used in this research. If 33 and;3 represent unit vectors in the direction of the upper and lower refracted beams in phase 3 (see Figure A.10). then the angle of intersection 6I can be determined by 33:13 8 cos (251) . (A.53) This requires. however. a representation ofg.3 and 23' Consider 33 first. Figure A.10 shows thatgs'gx 8 cos (eH - 63A) and that -&3792 8 sin (eH - 93A). Here 93A is defined as the angle between 79A andlg3 (i.e.. 7&375A 8 cos 63A). These considerations imply that g3 . cos (eH - 53,95:x + Sin (eH - 53A);z .. (A.54) A similar analysis for the lower beam leads to ~ where cos 53A 8 1§3fifiA. The negative sign appears becausefiA is the outward pointing normal vector at A. Eq. (A.53) relates 6I to 6“. 93A’ and 53A' For small angles. the trigonometric functions expanded through second order give (201)2 cos (291) .1- T + sin (eH - 63A) sin (eH - 83A) 8 (1.' 7?)(1 - 7?) + ab 185 where a E 9H"63A and b a 6H - 53A. Equating these two expressions and simplifying the algebra yields (201)2 = (03A - 63A)2 . (A.56) which does not depend on the relative magnitudes of 93A and 53A. Equation (A.56) is important because the angles can be related to K by several applications of Snell's law as follows (see Fig- ure A.10). For the upper beam. 63 sin 63A 8 82 sin 62A 52A 8 5H + 028 82 sin 628 8 81 sin K . For the lower beam. 83 sin 53A 8 82 sin 52A 9R ' 625 82 sin 523 8 81 sin K . For the small angle approximation. the above set of equations implies that B B 2 1 3A 83 H 32 and ~ 3 B O 2 1 3A=E;(9H--B-2-K) . 186 Therefore. Eq. (A.55) becomes B1 61 g E; K 9 (A057) which is the same result derived for 81 in the cylindrical section (see Eq. (A.26)). Obviously. Eq. (A. 27) gives the fringe spacing and shows that this does not vary across the traverse (i.e., -R(z) 5 x _<_ +R(z)). Determining the coordinates of "I" is a little more difficult. From Figure A.10 it follows that x4 8 5A + AI g3 (A.58) and 51 =‘iA + AI;3 . (A.59) Equating (A.58) and (A.59) gives two equations for AI and AI. viz., _ ~ _ AI 13 — AI £3 = 5A - (A.60) :2“ Rather than relating 5A (and )2“) to an angle as before. it is more convenient here to recognize that 5A8-Re +fi3}. ' (A-51) and = -R e - CA 5. (Ii-52) 187 where g“ s - Sin 8“ 2% + cos 9H.£2 . (A-53) Equations for CA and CA follow by realizing that;A and‘gA are also equivalent to (see Figure A.10) #‘%*Mh _ W“) 5A 8-58 + 8A3;2 . f (A.65) The foregoing set of equations can now be organized to give “Ag-1&28Rgxi-xg (A.66) -d#-fig-R%ng. mu) Here R is the radius of the hydrocyclone at the axial position 2; It“ is defined by Eq. (A.63); and. according to Figure A.10. .gz 8 cos 62B 9x - sin 628-22 (A.68) and ~ 22 8 cos 528 9x + sin 528'31 . (A.69) We have previously shown how 628 and 523 are related to K by apply- ing Snell's law at B and 8. Thus. once we find CA and CA from Eqs. (A.66) and (A.67). respectively, (Eq. 60) can be solved for AI— and AI. The vectors 58 and i8 are given by (see Figure A.10) 188 58 8 -R1 2x f (R1 + xFfigx)tan K's: (A.70) ~ 53 -Rl‘§x - (R1 + xF-gx)tan K'gz (A.71) For the small angle approximation. the x- and z-components of Eq. (A.66). together with the other definitions mentioned. yield (0" 1)(fl).(R1-R > 1 82B BA K(R1 + xe) K(R1 + xe) - 628(R1 - R) (1 - 5" 528) - . (A-72) Thus. Likewise. the x- and z-components of Eq. (A.67) can be written as H (D W! ml K(R1 + xe) which gives —: K(R +x)-0(R-R) CA: 1 Fx 251 . (1 + 6H 828) (A'73) Finally. we return to Eq. (A.60). which is equivalent to '1 1 r1 -6“ . '6H m 189 Inserting Eqs. (A.72) and (A.73) into this result and solving for AI yields 2[K(R1 + xe) - 823(R1 - R)][1 + 2eH(eH - 53A) 'AI 8 (A.74) (93A ' 63A)(1 ' 6H 625) The components of the beam intersection can now be determined by combining Eqs. (A.58). (A.61). (A.72). and (A.74). The result for the small angle approximation is B1 B2 81 [R1 + ny ' E; (R1 ‘. R)][1 T 611(6“ " '3; (6H " F2- K)” B *'§§ T"* B 2 1 1 2 2 [1 - (—) K 9 I 52 H (A.75) 81 [R1 I xe "E’ (R1 ' RI] sz = K 81 [1 - -—-K 8 H] 32 53 52 31 82 51 [K ’ (‘8‘; ' BT’BHJr'Rl + xe ‘ 'B‘Z'(R1 'R)][1+6H(9H"‘B'3'(9H' EEK)” B12 [1-(—)- .2931 (A.76) Eqs. (A.75) and (A.76) are the main results of this section. It is easy to show that for 6H 8 0. Eq. (A.75) reduces to Eq. (A.69) 190 and that x12 8 0. Furthermore. if 81 8 82 8 83, Eqs. (A.75) and (A.76) give x 8 xe and xI 8 0. respectively. Ix z It is convenient to express the above results relative to some specific intersection point "I". With xe 8 in + e and x1 8 51 - 31. we finally have B B K 2 1+ehlvl(1+GLln 8 H 5 B Ax 835: 3 2 (A 77) 1" 31 23152 . . ° 1 - BHITTE) which is (A.7l) for 8H 8 0. Also. 2 “I 83 B 1 + 9H + 8283 K2 11sz B (1 - 8 ).:5H 2 chr (A.78) 1 - 6H (g) It is noteworthy that the first order effect of curvature oan-gx is negligible (i.e.. AxIx 8 835/81) yet 83 82 Asz 8-§I (I --§g)e 8H . (A.79) Eq. (A.79) again shows why phases 2 and 3 should be optically equivalent. Unfortunately. this is not a characteristic of the set-up studied in this research; but. for the parameters which do . define our experiments. the variation in Asz across the traverse is insignificant. None of the axial velocities reported in this dissertation were corrected for axial refraction--therefore. some 191 of our quantitative remarks regarding flow reversals may be only preliminary. For 818 1. 82 8 1.47. 83 8 1.33348 (water at 14°C). 6H 8 5.65°. and K 8 5.71°. Eq. (A.78) becomes Ax 8 -0.0136 8 . (A.79) Iz In the experiments. 2 8 O on the axis of the hydrocyclone and e 8 28.7 mm at the wall (i.e.. AxI 8 R 8 38.1 mm according to x Eq. (A.77). Therefore. Eq. (A.79) shows that |Asz| 50.4 lllll, which Should not modify any of the qualitative (or quantitative) conclu- sions regarding the mean axial velocity. Tangmgae Velocity IThe top view of the laser path illustrated in Figure A.8 for measuring the tangential velocity in the cylindrical section applies here with one fundamental difference not shown in this projection. The outward pointing normal vectors at ”A" and "B" are still rotated relative to each other but now they are no longer in the plane defined by_!l.2 and 98' Thus. although the refracted beams retain their symmetry about the x-axis so the y-coordinate of the inter- section point is still y 8 0. the probe volume does not remain in a plane of constant 2. . For the cylindrical geometry. (II-“cylinder ‘ ' °°5 eA'ix * 5'" 6A fly but for the conical geometry. (2A) = (9A)cylinder cos 6H 2y ’ 51" eHIE-z ’ 192 which reduces to BA for the cylinder when 6H 8 O. This difference in geometry expresses itself whenever Snell's law is applied across the conical boundary. For example. suppose the direction of some incident light beam has been specified relative to a coordinate system. That is. we know g2. Furthermore. if‘gA has been specified as well as the indices of refraction 82 and B3 of the two phases in contact. then Snell's law includes the following five mathematical statements: 1) 82 sin 62A 8 B3 sin 63A 2) (£2 A £3)'flA = 0 3) - fiz’flA 8 cos 92A 4) - gajnA 8 cos 63A 5) 33-33 8 1 . In our previous examples. some of the above requirements were used implicitly (especially "2") because of the simplicity of the beam unit vector (i.e., £3)and the two angles 92A and 93A must be determined by (i.e.. £3) and the two angles 62A and 63A must be determined by using the foregoing explicit statement of Snell's law. For the small angle approximation. the intersection point is not expected to change much in the z direction but the effect of 9H > O on 91 and xI could be important. The main contribution. x however. will still be the local curvature of the cone (i.e., 1/Rz 8-8 as 2 +40). Therefore. with R2 8 2 tan 8H 8 2 8H ’ . (A.80) 193 the tangential data were corrected by applying (A.50) and (A.51) using (A.80). We emphasize that additional corrections should be investigated carefully by following the theory outlined here. A.4. Additional Remarks on Experimental Apparatus and Techniques A.4.1. Light source The 35 mW He-Ne laser used in this study consists of a plasma tube. two reflectors mounted at an angle (adjustable). a support structure. and a magnet system designed to enhance the laser opera- tion. The plasma tube is mounted using four clamps inside a heat shield structure. The laser is mounted to a black-anodized aluminum base with adjustable feet. ' The He-Ne gas laser operates with a simple DC glow discharge in a plasma tube containing 90% helium and 10% neon. The total pressure is 2-3 torr (mm Hg). To achieve continuous wave laser oscillation. a mirror is placed at each end of the plasma tube to form a cavity resonator. This stores most of the optical radiation by reflection back and forth along the axis of the tube. Only a small transmission is permitted through one of the reflectors giving a spatially coherent and monochromatic output beam. The coherent length of the He-Ne laser. which has a wavelength of 632.8 nm. may exceed many kilometers. The laser light is linearly polarized by placing Brewster angle windows (0 8 35°. 31") at the ends of the plasma tube so that only one component of polarization is transmitted 194 with negligible loss. The other component suffers a loss greater than the amplification per pass. A.4.2. Optical components A polarization rotator (see Figure 2.7; component 2) is a half wave retarder that changes the direction of polarization of an outgoing light beam. This is a necessary function because the beam splitter divides the incident laser beam into two equal inten- sity beams. provided the polarization of the original beam is per— pendicular to the plane of the two exiting beams. Thus. the polari- zation rotator takes the light from the laser. which is linearly polarized in the vertical direction. and rotates it a desired amount_ so that the polarization of the two resulting beams leaving the beam splitter is consistent with the measuring plane in the test section. The polarization rotator is mounted to a threaded plate and can be attached either directly to the laser or to the back of the first optics module. The beam splitter (component 4. Fig- ure 2.7) is used to split the laser beam into two parallel beams 'of equal intensity 50 mm apart from each other. A back-plate. with a small hole in the center. is used to help prevent extraneous light from entering the beam splitter. The beam splitter can be rotated at 45° intervals in order to measure other components of the velocity. but special care should also be taken to rotate the polarization of the beam for optimum Signals. Frequency shifting was employed in this study to define the direction of flow and to reduce fringe bias. The system consists 195 of an optics module (component .5. Figure 2.7) and a down-mix circuit. The optics module contains an acousto-optic cell (Bragg cell) which Shifts the frequency of one of the laser beams from the beam splitter by 40 MHz. The cell is installed in a cylindrical housing that aligns it with the other modules. A compensating wedge is used to correct shifted beam deflection. A tilt adjustment allows Shift- ing of the light intensity to the deflected beam. The brightness of the shifted light beam is controlled by adjusting the voltage of an RF potentiometer. A frequency shift of 40 MHz is too high for the range of velocities encountered in this study. A down-mix circuit permits shifting the frequency in twelve unequal increments from 2 KHz to 10 MHz. If other down-mixing frequencies are desired. an external oscillator can easily be interfaced with the system. However. maximum frequency from the down-mix circuit to the signal processor is only 25 MHz. The optical lenses (component 7. Figure 2.7) have been designed to minimize spherical and chromatic aberrations. They are cemented . doublet achromats and have dielectric. multi-layer. anti-reflective coatings. These lenses have a focal distance of 250 mm. The trans- mitting lens. with standard 50 mm beam spacing. results in a mea- suring volume that has 64 fringes and is typically 1.9 mm long and 0.18 nm in diameter. The half angle for the transmitting lens is 5°. 43" for the forward scatter mode of operation (see Figure A.1). 196 The receiving assembly (component 8) focuses the scattered light collected by the receiving lens onto the pinhole of the photo- multiplier. The photodetector mount has positive X-Y adjustments for the photomultiplier aperture. The front lens of the module is adjustable by two thumb screws for optimum focus. This adjustable lens has anti-reflective coating. with 90% reflectance at a 45° incident angle for the He-Ne laser beam. The photomultiplier system consists of the photomultiplier tube with a removable aperture. integral preamplifier. and photo- multiplier power supply. The photomultiplier tube is made of a cathode with a quantum efficiency of 13% for a He-Ne laser beam (A 8 632.8 nm). The removable aperture has a diameter of 0.2 mm. The photomultiplier high quantum efficiency and low noise preampli- fier generates good signal amplitude for low light levels. For high levels of light. the built-in automatic control circuit decreases cathode voltage (fail safe system). The photomultiplier is capable of handling frequencies of 200 MHz. The high power supply operates with an input voltage of either 115 or 230 volts (1 10% volts) and has an adjustable maximum cathode potential level knob. Maximum anode current is indicated by a red light on the front panel. This light should be off during the course of the experiment. The termina- tion box connects the photomultiplier output into a capacitor-coupled 50 ohm load. which reduces the signal amplitude ten fold. This is useful when viewing the signal on an oscilloscope directly from the photomultiplier and signal processor. 197 A.4.3. Data processing ‘ The counter processor is designed specifically for processing Signals from the photonultiplier ofa laser doppler anemometer. The counter processor consists of three major components: (a) an input conditioner; (b) a timer; and (c) a data display module. The input conditioner converts the doppler signal from the photomultiplier to an envelope to be timed. This envelope represents either the time for N cycles or the time for the total burst mode. Input and output jacks. installed on the front panel. receive signals from the photomultiplier and transmit signals to the oscilloscope for monitoring. To remove low and high frequency noise and to make the doppler signal symmetric. two filter banks using low and high pass filters are employed. The ranges of the low and high pass filters are from 10 KHz to 100 MHz and from 1 KHz to 10 MHz. respec- tively. The amplitude limit sets the maximum input signal allowable to help eliminate large particles from the measurement. The gain knob adjusts the gain of the input signal, and a light above it indicates the condition of the Signal. A green light indicates proper signal amplitude. red indicates high amplitude. and off is low amplitude. The minimum cycles/burst sets the number of cycles 1 to be timed. It is important to remove the pedestal and asymmetry component of the signal. As mentioned above. this is accomplished by using a high pass filter. When frequency shifting is employed. the setting of this filter is not crucial. To reduce high frequency noise. the 198 range of a low pass filter is decreased to the threshold of the maximum expected doppler signal. The gain controls the amplitude of the signal from the photo- multiplier tube to the counter processor. The increase in gain results in a higher data rate but. unfortunately. this could cause the noise to exceed a threshold level of 5 mV. which would make the Schmitt-trigger inoperable. This introduces “false“ data to the counter processor. For best operation the amplitude of the doppler burst should exceed a threshold level of 5 mN while the "noise” is.kept below i 2 mV around the zero. Therefore. the gain level is adjusted on the input conditioner to a reasonable sample rate to eliminate "false“ data. If "false“ data are present in the signal. they can be detected by viewing the analog output from the signal processor on an oscilloscope screen. False data would create spots on Figure 2.13. The purpose of the amplitude limit is to minimize the number of doppler signals from unusually large particles which may not be following the flow. There is no clear criterion to set the ampli— tude limit. However. during the course of this study. the location of the amplitude limit did not seem to affect the results. so it was kept at 90%. The function of the timer is to measure the length of the envelope provided by the input conditioner. The timer is a high speed clock which can measure 108 cycles per second with an accuracy of 1% Hz. An analog voltage proportional to the time of the envelope from the input conditioner is available through a jack. Another 199 function of the timer is to compare the envelope time between N cycles and-g-cycles in order to reject false data. The number of fringes per burst is designated as "N". These fringes are timed by the counter processor. The selection of the number of fringes mainly depends on optics and flow condition. and should not exceed more than half of the total number of fringes which exist in the probe volume. A parameter “n" defines the maximum time envelope to be accepted by the counter processor. The time envelope larger than the selected exponent "n" will be rejected by a reset pulse. The value of the exponent time not only depends on the flow condition. but also the total nunber of fringes "N". In the auto-position for a given “N". the upper and lower limit of the exponent time "n" will be selected. The transfer function is the ratio between input frequency and output voltage. where "X" is the transfer function for a given combination of "N" cycles and exponent number "n" (see Table 11-3 on page 24a in TSI [1982b]). The maximum frequency is equal to the transfer function for a 10:1 range. For this range setting. the minimum frequency should not be lower than X/lO; otherwise. the results are less accurate. Attempts are made to read all data on the middle range of the transfer function for better accuracy. When the data reaches the limit of the transfer function. a new exponent value is selected and the calibration procedure repeated. The time constant and data rate resolution on the readout module 200 are set on the desired values. The manual/auto button is pushed into the manual position. and the exponent buttons are set corres- ponding to the selected transfer function. Manual option is most appropriate when analog output is used. The location of the low pass filters is rechecked by the readout module or oscilloscope information for noise clean signals. A convenient feature of the readout module is the flexibility to display either frequency (i.e.. fd) or velocity directly by setting a calibration factor. For an optic orientation. the fringe spacing remains constant as the hydrocyclone is moved relative to the laser source. Thus. for axial velocity measurements (see Section A.3) the readout module is calibrated to read directly in velocity units (see p. 54 in TSI. 1982b). However. for tangential velocity mea- surements the half angle of the refracted beams changes with position and this results in a variable fringe spacing. For this case the readout module is calibrated to display frequency rather than' velocity. The data display module gives a direct readout of the analog voltage on a digital voltmeter with a selectable time constant of 0.1. 1. and 10 seconds. The data rate readout is a frequency counter that counts the data ready pulses from the timer. The resolution of the data rate can be selected as 10. 1. and 0.1 seconds. The digital data lines are all brought into the 37 pin plug-in connector on the front panel and can provide time. exponent. cycles. and other information to a computer for further processing. Unfortunately. 201 this capability was unavailable during this study. Instead. during each experiment the mean frequency of the doppler signal at each location in a traverse was read directly off the display panel and recorded. The mean velocities were calculated using Eq. (A.19) at a later time taking into account proper corrections for the effect of refraction on the fringe spacing. A.5. Example Problem: Laminar Flow in a Circular Tube LDA measurements for fully developed laminar flow in a circular tube were made for two different flow rates. The test section was located about 34 diameters from the tube entrance as indicated in Figure A.11. The circular tube was connected to the flow loop shown in Figure 2.1. A plexiglass box was built around the test section and filled with glycerin to minimize curvature effects. A dual-beam. forward-scattering mode (see Section A.1) was used to measure the axial velocity. The test section was mounted on a milling table which permitted up to 5 cm movement in the hori- zontal plane. The volumetric flow rate was measured very accurately by weighing a known amount of fluid collected during a specified increment of time. The results are shown in Figure A.12 and agree nicely with the expected parabolic behavior. The parameters for the counter processor were chosen from an estimate of the maximum velocity using the theoretical parabolic profile. 202 .mmPPeoca opponmcma mo mucmeoeameme oc appam so» mpwwocq zuvuopm> cam: .NN.< mean?“ m m comwcmqeoo mo a oON A . . ooe A mama mama NH e“ c m m z coamEmcas coycsoo m\:u .N: m N m m e m. N N c ,oN mm a N m m e m. N H 0 AN- I . . . . I I . . NI .0. g V 0H1 I v .Hl ~zx ON I uewgm xucmzcmcm newsm aucmscmcm oz m I! . .8538 9. . o :I. I.I:I.i:.:I:.:1:I:I:II:I:I:I:IaIizIazIIII.o pmwcmm + .H._ . as L O . . . N fiwwfl 8: 2. Es mcouocaccm zo—e 204 A.6. ‘gxample Problem: Axial and Tangential Velocity Profiles in a HydrogyElone In this study, the axial and tangential components of the mean velocity were measured by recording traverse of the milling table and the mean velocity (or, frequency) on the data sheets. During the course of experiments, the flow conditions were recorded (flow rates and pressures) to define the experiment. The velocity profiles were all corrected for index of refraction effects (see Section A.3). Samples of the raw data for axial and tangential velocity profiles are presented in Tables A.1 and A.2 to illustrate the procedure for obtaining the corrected velocity profiles. The data presented in this section are all taken in a hydrocyclone with a configuration shown in Figure 2.5a. Table A.1 presents the mean axial velocity for the axial loca- tion of z I 20 cm and 00/0; 2 4. Column 1 of the table gives the increment advancement of the milling table. Column 2 shows the velocity data directly obtained from counter processor, which includes frequency shift. Column 3 is obtained from column 1, taking into consideration that increment advancement of the milling tables moves the test section 0.024 mm; therefore, this column represents the traverse of milling table (or test section). Real probe volume distance from axis of the hydrocyclone is presented by XI‘ Similar expressions as those of (A.77) have been used to correct for radial position due to refractive index problem (see Appendix A.3). Because the fringe spacing does not change with position for axial velocity no correction is required for axial velocity. For the experiment 205 Table A.1. Mean axial velocity data measured in the hydrocyclone which satisfies Rietema's optimal design criteria. 2 (20; 5.3; 4), ReF = 24,300 Increment advancement Velocity x of the (include shift) 8 1 Velocity milling table m/sec mm mm m/sec 0 3.680 0.00 0.00 -0.503 40 3.534 1.02 1.36 -0.357 40 3.278 2.03 2.7 -0.101 40 2.894 3.05 4.05 0.283 80 2.376 5.08 6.75 0.801 80 2.613 7.11 9.45 0.564 60 2.8 8.64 11.48 0.377 80 3.011 10.67 14.18 0.166 80 3.15 12.70 16.88 0.027 80 3.281 14.74 19.59 -0.104 80 3.365 16.77 22.27 -0.188 80 3.473 18.8 24.97 -0.296 100 3.287 21.34 28.34 -0.11 206 reported in Table A.1 frequency shift of 1 MHz is applied to define the flow reversals. Using Eq. (A.18) and fringe spacing of 3177 nm the velocity corresponding to this frequency shift is 3.177 m/sec. This velocity is subtracted from the velocity recorded in column 2 to obtain the velocity of the fluid (column 5). Table A.2 presents the mean tangential velocity for the same flow condition of axial velocity. 0ne MHz frequency shift is applied to define the flow direction. The first column of Table A.2 is similar to those described in Table A.1. Because fringe spacing varies with position for tangential velocity measurements, the fre- quency is recorded instead of velocity directly from the counter processor. Column 2 presents this frequency including frequency shift. The third column obtained from column 1 is the same as those described for axial velocity. The amount of frequency shift (1 MHz) is subtracted from the frequency obtained from the counter (column 2) to obtain the frequency of fluid particles (column 4). A similar expression to that of Eq. (A.50) is used to obtain the real position of the probe volume (column 5). In order to obtain tangential velocity of fluid, the fringe spacing is calculated at each radial position (see Section A.3). Then Eq. (A.18) is applied to calculate the tangential velocity. The result of this calculation is presented in column 6. Mean axial and tangential velocity shown in Tables A.1 and A.2 are graphically presented in Figures 5.7 and 5.8, respectively. 207 Table A.2. Mean tangential velocity data measured in the hydro- cyclone which satisfies Rietema's optimal design criteria. 9(20; 15.3; 4), ReF 3 24,300 Increment advancement . Frequency x of the (include Shlft) 6 Frequency I Velocity milling table .MHz - mm MHz mm m/sec 0 1.022 0 0.022 0 0.077 5 1.11 0.13 0.11 0.21 0.386 40 1.427 0.89 0.427 1.45 1.504 40 1.61 1.90 0.61 3.13 2.161 40 1.688 2.92 0.688 4.83 2.452 20 1.695 3.43 0.695 5.68 2.485 20 1.688 3.94 0.688 6.54 2.466 80 11.697 5.97 0.697 10.04 2.202 80 1.515 8.00 0.515 13.62 1.891 80 1.45 10.03 0.45 17.29 1.673 40 1.388 12.06 0.388 21.05 1.46 80 1.273 14.10 0.273 24.90 1.04 80 1.211 16.13 0.211 28.78. 0.809 APPENDIX B DRAG REDUCTION CHARACTERISTICS OF SEPARAN AP-30 APPENDIX B DRAG REDUCTION CHARACTERISTICS OF SEPARAN AP-30 A series of experiments were conducted to study some of the parameters which influence the ability of Separan AP-30 to reduce the drag in fully developed pipe flow. The information developed here helps to define the system studied in Section 4.4. Basically, we examine the effect of temperature and pH on the friction factor and apparent viscosity for a range of dilute polymer-water mixtures. B.1. Viscosity Measurements Viscosity measurements were made at various temperatures using a Cannon-Fenske pipette type viscometer. Polymer solutions were made from a stock solution of 2000 wppm. The polymer used in the experiments was Separan AP-30, which has a molecular weight of about 106; the chemical structure of the repeat unit is shown in Figure 8.1. The concentrated polymer solution was prepared in a 200 liter plastic container filled with distilled water. The polymer was sprinkled on the surface and gently stirred with a plastic rod. This stock solution was made prior to the experiment and was later diluted to a desired concentration in a 100-liter plastic container. The stock solution was made at least one day prior to any experiment and checked under the light for homogeneity. The pH of the distilled water was adjusted by adding NaOH or HCL; salt was used to prepare a 0.514NaCl 208 209 P — .CH — 0H2...CH2 CH —— l I c=0 c=0 I l NH ON 2 3 a I- & law-106 Figure 8.1. Chemical structure of Separan AP-30, a co-polymer of polyacryl amide and polyacrylic acid. 210 polymer solution. An automatic temperature control tank was used to maintain the viscometer at a constant temperature. Figure 8.2 shows the apparent viscosity of polyacrylamide solutions at a low shear rate for the temperature ranging from 10°C to 40°C. The solvent is distilled water with pH = 6. Figure 8.3 illustrates the effect of water composition on the viscosity. It is noteworthy that the viscosity of the polymer increases signifi- cantly with pH, up to a certain point. Thereafter, the viscosity declines. The lowest viscosities observed were those at low pH and salt composition. Polyacrylamide is an anionic polyelectrolyte which easily uncoils in solution because of the electrostatic repulsion of chain ionic groups. The decrease in viscosity at high pH levels may result from a shielding effect caused by a large number of counter-ions in the domain of the polymer molecules. This phenomenon seems to sug-. gest molecular expansion diminishes at higher pH levels after the maximum value is attained. After comparing the polymer-distilled water results with the polymer-salt water results, it is apparent that the added electrolyte has essentially neutralized the electro- static repulsions between the chains. This type of behavior is well known for polyelectrolytes. The expansion of the poly-ions eventually reaches an upper limit and the decrease in the apparent viscosity upon further dilution merely reflects the decreasing inter- ference between the expanded chains. 211 4.0 T = 10°C 0. o 0 3.0.. 20 C >, I f. 25°C 8 U ""3 A o > 30 C 2.0.. I . J ‘ 40°C 1.0 A Low shear rate experiments I ' 1 I l 0.0 50 100 150 200 250 Polymer concentration, wppm Figure 8.2. The effect of polymer concentration on the apparent viscosity of aqueous AP-30 mixtures at different temperatures. Viscosity. cp 212 o dist. water PH = 8.3 5 0. o dist. water PH = 13.3 ' A salt water (0.5 M NaCl) D dist. water PH = 3.1 4.0« T = 25°C 3.0'- 2.0 d o O O #_ 1 0 a 4““ Ar —é\lj- é 0' I l I t ' I , ' 0 50 100 150 200 250 300 350 400 Polymer concentration, wppm Figure 8.3. The effect of polymer concentration and PH on the apparent viscosity of aqueous AP-30 mixture. 213 The viscosity behavior of dilute solutions of partially hydrolyzed polyacrylamide has also been investigated by Sylvesterand Tyler [1970]. Three different solvents were used: distilled water, salt water (0.55 M NaCl), and simulated sea water. The intrinsic viscosity found for salt water and distilled water are comparable with the findings reported here. Figure 8.3 also shows the effect of low pH on the apparent viscosity. Apparently, the molecules possess a compact, impermeable structure in the acidic range. This phenomenon has also been observed by Katchalsky'and Eisenberg [1951] for other high molecular weight polyelectrolytes (polymethacrylic acid) and by Kim et al. [1973] for polyacrylamide. 8.2. (DragReduction Results° Drag reduction experiments were conducted to characterize the polymer used in the hydrocyclone experiments reported in Sec- tion 4.4. The data are plotted using Von Karman coordinates 9c(-AP)D fE—r— ZprL R g Dpr e u Ap is the observed pressure drop; Ub’ the bulk average velocity; 0, the inside capillary diameter; and L is the length over which Ap is measured. The density and viscosity of the fluid are p and p, respectively. 214 To minimize the mechanical degradation of the polymer, 6 once- through flow system was designed. The system is basically the one shown in Figure 2.2. A small tube (1.0. = 3.55 mm) made from stain- less steel was connected to the flow loop. Pressure taps were located 229 mm apart with the location of the first tap more than 250 diameters from the entrance. The pressure taps were connected to a mercury manometer, and the flow rate was measured by recording the time necessary to fill a precalibrated plastic container. The homogeneous polymer solution was poured into the pres- surized tank and sealed for leaks. Pressure was applied to the supply tank through a regulator, and the polymer flow rate was adjusted by a valve. After the pressure stabilized and the flow rate became steady, the flow rate, pressure drop, and temperature were recorded. The pH of the solution was also measured. Special care was taken to maintain stable flow rate and pressure drops. Viscosities were determined by requiring f = 16/Re for the low Reynolds number data. Figure 8.4 shows the efect of pH on the friction factor for a 100 wppm polymer solution. The % drag reduction increases sharply with increasing ionization. This enhancement can be interpreted as a result of molecular expansion due to electrostatic repulsion between like charges in the polymer chain. No significant effects due to polymer concentration were noticed from 50-300 wppm. ‘ 215 30- 0 water f . 16 “g a water (repeat RR? 0A . e polyur (PH - 3.0) A polynr (PH - 10.5) o ‘ . o polymer (PH - 10.5 ‘ repeat) Maxin- drag . reduction . A #- 19. log Ralf-32.4 204 . .0 O. a O f _ 0.079 5I - . Me 0.25 > O ' . 0 fl 1...“ e g ’ v " 10" A7; "" .3 . ’,. " prendtl-Kereen . ° ‘ I a" d ” ' d o v v v v ' 1 I I r v T I v t t 1 m m2 z 3 4 56 7691& 2 Figure 8.4. The effect of PH on drag reduction characteristic of Separan AP- 30 (100 wppm). 216 If the polymer solution is continuously sheared in a turbulent flow. the drag reduction effect is lost. This phenomenon is demon- strated very vividly in Figure 8.4 which shows the Z drag reduction versus time. The decreasing nature of this graph is an indication of polymer degradation. Two pH levels are examined. Both solutions reach the same asymptote after two hours of shearing through the flow loop; the last two points on the graph represent the same poly- mer solution after sitting eighteen hours. Note that the % drag reduction continues to decay slightly without shear. Thomas [1982] has recently shown that if 100 wppm Separan AP-30 solution sits for eighteen hours without shearing, then the Z drag reduction is comparable to the sheared results shown in Figure 8.4. This suggests that the effect of shear is to disentangle the polymer rather than degrading them by breaking molecular bonds. If this occurs then degradation should be reversible. All the solutions used in the hydrocyclone experiments recorded in Section 4.4 showed significant drag reduction properties after the velocity traverses were recorded. The % drag reduction was = 50% at Re - 104 for all the solutions. The important conclusion here is that these solutions, which passed through the flow loop only once, were viscoelastic and any significant differences between the results with and without polymer could very well be attributed to the elastic properties of Separan AP-30. Unfortunately, very few data are available for direct comparisons since the temperature of the experiments recorded in Section 4.4 are almost 8° below the ones in Section 4.1; however, for a comparison see Figure 5.18. 217 .om-a< :mcmaom sag: ooH com covuozumc once u as» no cmmsm mo uommmm one .m.m manor; ma:=.2 .meph ONH o: :3 cm cm 2 cm on ov on cm 3 o.o ocpcmmsm cosapoa oz - P h b p p P L F P o.o O O , 1 e o o 8 .0 O O 0 Au 0 O . .0. 0 cm I. nu .7 nv . 3 m. mom O on e .I .u com u .aEm u h .I Ng w E o e .8 v.o~ m :a .I L om loo uogqonpad 624p % APPENDIX C FLOW CHARACTERISTICS FOR THE 3"-HYDROCYCLONE APPENDIX C FLOH CHARACTERISTICS FOR THE 3"-HYDROCYCLONE Pressures and flow rates for different split ratios and vortex finder configurations were measured to characterize the hydrocyclone. A schematic of the flow loop used to carry out the experiments is shown in Figure 2.1. The hydrocyclone overflow and underflow were not connected to the atmosphere. The valves labeled 1, 2, and 3 in Figure 2.1 were used to control the flow through the loop and provide the desired flow rates and split ratios. The pressures at the inlet and outlets were measured by a mercury manometer and checked by.a standard pressure gauge. The underflow and overflow rates were measured by a precalibrated plastic container over a period of time. A series of experiments were conducted to characterize the 3"-hydrocyclone with a 2:1 contraction above the vortex finder (see Figure 2.3). Inlet and outlet flow rates at different inlet pres- sures were measured for Qo/Qu 8 0.25, 4, and Q5 The results are summarized in Figures C.1-C.2. The effect of feed pressure on the overflow and underflow rates are also indicated. Table 0.1 compares the exponent on Ap in the empirical correlation 0,: = MAP)” found in this study and with other results in the literature. 218 219 530 T 450 - QF = 369 (Ap)o'38 370 . Flow rate. cm7sec 280 . 210 A 130 .9. - AP Pressure, Psig T = 25°C Vortex finder: see Figure 2.9 Np, 0.0 4 I i 1 12 16 20 24 PF’ psig mi Figure C.1. Operating characteristics of a 3"-hydrocyclone for 00/0u = 0.25. 220 500 . 0F = 310 (AP)°°4 0F 400. Flpw rate, 00 cm7sec 3001 200. 100. 0.0 W I I 1 1 16.- PF Pu 12.. . P Pressure, . o psig 8.. T = 25°C 4" Vortex finder: see Figure 2.9 o ' T T T I *1 0.0 4. 8 12 16 20 24 PF. psig Figure 0.2. Operating characteristics of a 3"-hydrocyclone for QO/QU = 4- Fl rate, cmgysec Pressure, psig 221 560- PF = 339 (AP)°°44 480- QF = 00 400« 320- 240! 160 . e . . . . P P“ 12- F P o. 10- 3. AP I'll 6. T = 25°C 4. Vortex finder: see Figure 2.9 ? (T I I I r fl 0 4 8 12 16 20 24 PF’ 9519 Figure 0.3. Operating characteristic of a 3"?hydrocyclone for 00/00 8 ”- 222 Table C.1. Values of exponent 'm' for various size hydrocyclones. Source 0. mm 'm' Remarks .Kelsall [1953] 76 0.416 air core Bradle and Pulling {1959] 76 0.425 air core Present study 76 0.42 no air core 223 The variation in the split ratio with changes in inlet pres- sure is given in Figure C.4. Several researchers have shown that the split ratio is a weak function of the inlet flow rate (equivalent to PF)’ geometry, and viscosity. For a 13 mm hydrocyclone, Haas et al. [1957] have observed a weak dependence of split ratio on the flow rate. The important effect of viscosity on the split ratio is summarized by Bradley [1965, see p. 143]. Usually, an increase in viscosity causes the split ratio to decrease. This decrease is related to the suppression of the swirl velocity, which reduces the upward reverse flow. Changes in the inlet flow rate have no significant effect on Qo/Qu over the range investigated. However, at low flow rates with a high split ratio setting, Qo/Qu increases with decreasing inlet pressure; the opposite behavior was observed for low split ratio settings. The flow characteristics of the other vortex finder configura- tions shown in Figure 2.5 were also measured and analyzed, but are not reported here. 224 Q 0F 0 5.5 - T = 25°C Fixed valve setting 00/0“ = 4! PF g 15.6 4.7- Qu 00/0u 3.9. 3.14r‘ . 1 , . Ii 0 4 8 12 16 20 PF. psig Figure C.4. Effect of inlet pressure on the 00/0 of a 3"-hydrocyclone with a 2:1 contraction above the vorgex finder. APPENDIX 0 DATA FOR THE 3"-HYDROCYCLONE WHICH SATISFIES RIETEMA'S DESIGN CRITERIA APPENDIX 0 DATA FOR THE 3"-HYDROCYCLONE HHICH SATISFIES RIETEMA'S DESIGN CRITERIA Data for Rietima's optimum hydrocyclone design are presented in Table 0.1. All velocity traverses are obtained at a temperature of 12°C. The following convention has been adopted to present the mean axial and tangential velocity. For the mean axial profile, ("2) (F,Z) = 2(2; PF; 00/0“) - For the mean tangential profiles, (r.2) = 6(2; PF; Qo/Qu) . The characters z(cm), PF(psig), and Qo/Qu represent the axial loca- tion, inlet feed pressure, and split ratio, respectively. For each velocity traverse in Table 0.1, the actual displacements of the test section (e,mm) and either mean velocity or mean frequency are presented. For tangential velocity measurements, the fringe spacing varies withe:(see Eq. (A.51). Data for tangential velocity profiles are shown in the term of mean frequency rather than mean velocity. An asterisk * is used to denote which profiles appear in graphical form, which are presented in Chapters 4, 5, and 6. 225 226 Table 0.1. Mean velocity data measured in the hydrocyclone which satisfy Rietema's optimal design criteria. Z(8; 15.3; 0), ReF = 24,200 c,mm m/sec e,mm m/sec .e,mm m/sec 0.00 - 4.296 4.58 -0.387 11.2 -0.537 0.51 - 4.180 5.08 -0.227 11.70 -0.079 1.02 - 3.83 5.60 -0.17 12.21 -0.02 1.53 - 3.209 6.10 -0.186 2.03 - 2.468 6.62 -0.221 2.54 - 1.774 7.12 -0.266 3.05 - 1.281 8.14 -0.391 3.56 - 0.825 9.16 -0.485 4.07 - 0.546 10.17 -0.556‘ _ e,mm MHz e,mm MHz e,mm MHz 0.00 0.049 2.67 1.28 8.76 0.454 0.13 0.226 3.17 1.16 9.27 0.429 0.38 0.944 3.68 1.061 0.63 1.42 4.19 0.951 0.89 1.534 4.70 0.853 1.14 1.593 5.20 0.71 1.39 1.588 .6.22 0.646 1.65 1.535 7.24 0.563 2.10 1.428 8.25 0.485, 2*(20; 15.3; 0), Rep = 24,100 e,mm m/sec c,mm m/sec e,mm m/sec -21.36 4 0.042 ' - 3.05 -1.409 12.72 -0.066 -19.33 -- 0.282 - 1.53 -3.047 14.75 -0.181 -17.29 - 0.252 - 0.51 -3;538 16.78 -0.255 -15.26 - 0.163 0.51 -3.615 18.82 -0.315 -13.22 - 0.053 2.54 -1.635 19.84 -0.324 -11.19 0.063 4.58 -0.376 20.85 -0.273 - 9.15 0.187 6.61 -0.024 21.36 -0.179 - 7.12 0.115 8.65 0.135 - 5.09 - 0.183 10.68 0.083 227 Table 0.1 (cont'd.). 0*(20; 15.3; 0). REF . 24.100 g,mm MHz e,mm MHz g,mm MHZ 0.0 0.0 2.79 1.24 11.43 0.435 0.25 0.561 3.30 1.17 12.44 0.403 0.59 0.912 4.32 0.928 13.46 0.374 0.76 1.512 5.33 0.768 14.48 0.35 1.02 1.58 6.35 0.645 15.49 0.322 _1.27 1.59 7.36 0.587 1.52 1.684 8.38 0.54 1.78 1.523 9.40 0.498 2.28 1.392 10.41 6.468 2(32; 15.3; 0), Rep = 24,200 e,mm m/Sec e,mm ("2) m/Sec e,mm m/sec 0.00 -3.243 8.14 0.018 20.34 -0.071 0.51 -2.933 9.16 0.066 22.38 -0.113 1.53 -2.003 10.17 0.087 24.42 -0.069 2.54 -1.258 11.19 0.104 26.45 -0.019 3.56 —0.773 12.218 0.121 4.58 -0.423 13.22 0.11 5.59 -0.245 15.26 0.054 6.61 -0.124 16.28 0.025 7.63 -0.012 18.31 -0.024 6(32; 15.4; 0), Rep = 24,100 e,mm MHz e,mm MHz e,mm MHz 0.00 0.006 2.16 1.239 9.78 0.468 0.13 0.18 2.67 1.132 10.79 0.444 0.38 0.526 3.17 1.064 11.81 0.424 0.63 0.782 3.68 0.961 12.83 0.409 0.89 0.999 4.70 0.828 13.84 0.393 1.14 1.119 5.71 0.699 14.86 0.383 1.40 1.23 6.73 0.615 15.87 0.374 1.65 1.292 7.75 0.55 716.89 0.367 1.90 1.286 8.76 0.50 17.91 0.363 18.92 0.361 19.94 0.348 20.95 0.33 228 Table 0.1 (c0nt'd.). 2(20; 5.1; 0), ReF = 14.100 e,mm m/sec e,mm m/sec e,mm m/sec 0.00 -1.97 6.10 -0.13 15.26 -0.081 1.02 -1.83 7.12 -0.032 16.27 -0.113 2.03 -1.36 8.14 0.053 17.29 -0.133 3.05 -0.91 9.16 0.098 18.31 -0.153 4.07 -0.51 10.17 0.089 19.33 -0.164 5.09 -0.28 11.19 0.054 20.35 -0.167 12.21 0.016 21.36 -0.05 13.22 -0.02 14.24 -0.047 z*(2.2; 15.3; 0.25). REF = 24.200 .;,mm m/sec e,mm z m/sec e,mm m/sec -8.14 -0.623 0.00 -5.081 -7.63 -0.844 1.02 -4.037 -7.12 -1.046 2.03 -2.361 -6.10 -0.991 3.05 -1.281 -5.09 -0.932 4.07 -0.721 -4.07 -0.967 5.09 -0.733 -3.05 -1.639 6.10 -0.801 -2.03 -2.93 7.12 -0.733 -1.02 -4.161 7.63 -0.713 2(7; 15.3; 0.25), ReF = 24,200 e,mm m/sec m/sec e,mm m/sec 0.00 -4.109 4.57 -0.291 9.14 -0.516 0.51 -3.723 5.08 -0.193 9.65 -0.553 1.2 -3.133 5.59 -0.179 10.16 -0.574 1.52 -2.635 6.10 -0.202 10.67 -0.526 2.03 -2.157 6.60 -0.256 11.17 -0.024 2.54 -1.666 7.11 -0.322 3.05 -1.204 7.62 -0.383 3.56 -0.835 8.13 -0.435 4.06 -0.493 8.64 -0.465 229 Table 0.1 (cont'd). 2*(8; 15.3; 0.25), Rel: = 24,200 e,mm 'mysec c,mm lmysec e,mm m/sec 0.00 -3.872 4.58 -0.211 9.66 -0.437 0.51 -3.628 5.09 -0.107 10.17 -0.485 1.02 -3.205 5.60 -0.081 10.68 -0.525 1.52 -2.773 6.10 -0.113 11.19 -0.553 2.03 -1.998 6.61 -0.177 11.70 -0.532 2.54 -1.493 7.63 -0.223 12.21 -0.343 3.05 -1.103 8.14 -0.29‘ 12.72 -0.313 3.56 -0.723 8.65 -0.345 4.06 -0.413 9.16 -0.39 e*(8; 15.3; 0.25), ReF - 24,300 e,mm MHz e,mm MHz e,mm MHz 0.00 0.03 3.94 0.807 8.50 0.411. 0.13 0.105 4.44 0.758 9.02 0.251 0.38 0.283 4.95 0.7 . 0.89 0.624 5.46 0.647* 1.40 0.788 5.97 0.603 1.91 0.846 6.48 0.538 2.41 0.864 6.98 0.503 2.92 0.862 7.49 0.471 3.43 0.842 8.00 6.438 2(19; 15.3; 0.25), Re; = 24,300 e,mm m/sec esmni m/sec e,mm m/sec 0.00 -3.13 7.62 0.36 13.21 -0.09 0.51 -3.008 8.13 0.345 15.24 -0.202 1.02 -2.643 8.64 0.296 16.26 -0.238 2.03 -1.881 9.14 0.262 17.27 -0.271 3.05 -1.13 9.65 0.21 19.30 -0.311 4.06 -0.53 10.16 0.165 20.32 -0.33 5.08 -0.134 10.67 0.115 21.34 -0.233 6.10 0.185 11.18 0.073 21.84 -0.211 7.11 0.347 12.19 -0.012 = 230 Table 0.1 (cont'd). 2*(20; 15.3; 0.25), Re; = 24,300 g,mm m/sec e,mm m/sec e,mm m/sec 0.00 -3.038 7.63 0.35 16.28 -0.211 0.51 -2.968 8.14 0.337 17.29 -0.244 1.53 -2.108 9.16 0.261 18.31 -0.276 2.54 -1.392 10.17 0.169 19.33 -0.308 3.56 -0.778 11.19 0.077 20.35 -0.341 4.58 -0.3 12.21 -0.003 20.85 -0.039 5.60 0.026 13.22 -0.062 ' 6.61 0.264 14.24 -0.117 7.12 0.331 15.26 -0.174 6*(20; 15.3; 0.25). ReF = 24.300 e,mm MHz e,mm MHz c,mm MHz 0.00 0.022 4.318 0.72 11.43- - 0.416 0.25 0.184 4.826 0.69 12.44 0.392 0.76 0.453 5.334 0.64 12.46 0.355 1.27 0.642 5.842 0.61 14.48 0.333 1.78 0.733 6.350 0.59 2.29 0.77 7.366 0.54 2.79 0.792 8.382 0.51 3.30 0.782 9.398 0.47 3.81 0.754 10.414 0.45 Z(31; 15.34; 0.25). Rep = 24.400 e,mm m/sec e,mm m/sec e,mm m/sec -0.25 -1.948 7.37 0.315 17.02 0.011 0.25 -1.959 7.87 0.34 19.05 -0.033 0.76 -1.903 8.38 0.33 21.08 -0.094 1.27 -1.783 8.89 0.296 23.11 -0.144 2.29 -1.39 9.40 0.248 25.15 -0.176 3.31 -0.912 10.92 0.099 26.16 -0.193 4.32 -0.498 12.95 0.007 27.18 -0.088 5.33 -0.126 13.97 0.003 6.35 0.149 14.99 0.014 Table 0.1 (cont'd). 231 2*(32; 15.3; 0.25),.ReF = 24.400 e,mm m/sec e,mm m/sec c,mm m/sec 0.00 -1.803 7.12 0.317 16.79 -0.003 0.51 -1.728 7.63 0.288 18.82 -0.05 1.02 -1.552 8.65 0.198 20.86 -0.107 2.03 -1.123 9.66 0.099 22.89 -0.155 3.05 -0.625 10.68 0.051 24.42 -0.117 4.07 -0.248 11.70 0.033 25.43 -0.077 5.09 0.087 12.72 0.041 25.94 -0.019 6.10 0.274 13.73 0.049 6.61 0.311 14.75 0.038 0*(32; 15.3; 0.25). ReF = 24.300 e,mm MHz e,mm MHz e,mm MHz 0.00 0.012 4.57 0.713 13.21 0.386 0.51 0.173 5.08 0.694 14.22 0.373 1.02 0.325 6.10 0.631 15.24 0.364 1.52 0.478 7.11 0.564 16.26 0.363 2.03 0.586 8.13 0.517 17.27 0.354 2.54 0.644 9.14 0.475 18.29 0.346 3.05 0.696 10.17 0.445 19.30 0.343 3.56 0.713 11.18 0.424 20.32 0.34 4.06 0.722 12.19 0.407 21.34 0.331 22.35 0.328 0*(20; 18.6; 4), Rep = 26,600 e,mm .MHz €,mm MHz‘ €,mm -MH2 0.00 -0.014: 3.94 0.78 11.05 0.48 0.13 0.056 4.45 0.773 12.06 0.441 0.38 0.18 '4.95 0.756 13.08 0.407 0.89 0.385 5.97 0.698 14.10 0.39 1.91 0.612 6.99 0.643 15.11 0.35 2.41 0.686 8.00 0.60 16.13 0.076 2.92 0.732 9.02 0.553 3.43 0.77 10.03 0.516 232 Table 0.1.(cont'd.). 2*(1; 15.3; 4), REF a 24,300 2 ,mm z> m/sec c,mm m/sec e,mm m/sec -7.121 -0.295 -2.543 -0.321 2.035 -0.061 -6.612 -0.642 -2.035 -0.229 2.543 -0.126 -6.104 -0.84 -1.526 -0.108 3.052 -0.251 -5.596 -0.799 -1.018 -0.068 3.561 -0.257 -5.087 -0.696 -0.509 -0.025 4.069 -0.138 -4.578 -0.596 0.0 -0.009 4.578 -0.086 -4.069 -0.501 0.509 -0.066 -3.561 -0.442 1.018 -0.074 -3.052 -0.408 1.526 -0.054 2*(2.5; 15.3; 4). Rep . 24,300 e,mm z) m/sec e,mm m/sec e,mm m/sec -7.63 -0.383 -3.05 -0.077 1.53 -0.102 -7.12 -0.367 -2.54 -0.037 2.03 -0.094 -6.61 -0.278 -2.03 -0.005 2.54 -0.213 -6.10 -0.208 -1.53 -0.008 3.05 -0.341 -5.60 -0.074 -1.02 -0.03 3.56 -0.39 -5.09 0.071 -0.51 -0.111 4.07 -0.393 -4.58 0.125 0.00 0.175 4.58 -0.47 -4.07 0.062 0.51 -0.22 5.09 -0.502 -3.56 0,037 1.02 ‘ -0.164 5.60 -0.015 2(7; 15.4; 4), Rep = 24,300 e,mm m/sec e,mm m/sec eaflfll m/sec -2.79 0.177 1.27 -0.245 6.86 -0.043 -2.29 -0.016 1.78 -0.18 7.87 -0.17 -1.78 -0.152 2.28 -0.045 8.89 -0.274 -1.27 -0.252 2.79 0.143 9.91 -0.343 -0.76 -0.295 3.30 0.334 10.92 -0.038 -0.25 -0.319 3.81 0.461 _ _‘ 4.32 0.477 0.25 -0.319 4.83 0.397 0.76 -0.295 5.84 0.169 233 Table 0.1 (c0nt'd.). Z(8; 15.3; 4),.ReF = 24,300 e,mm °mlsec e,mm ("2) m/sec e,mm j" m/sec 0.00 -0.356 4.58 0.343 9.16 -0.351 0.51 -0.336 5.09 0.237 9.66 -0.404 1.02 -0.3 5.60 0.15 10.17 -0.44 1.53 -0.21 6.10 0.037 10.68 -0.443 2.03 -0.68 6.61 -0.058 11.19 -0.226 2.54 0.127 7.12 -0.137 11.70 -0.019 3.05 0.315 7.63 -0.184 3.56 0.433 8.14 -0.245 4.07 0.421 8.65 -0.308 0(8; 15.3; 4), Rep = 24,300 e,mm MHz €,mm MHz €,mm MHz 0.00 0.058 4.19 0.7 0.13 0.114 4.70 0.655 0.63 0.294 5.21 0.614 1.14 0.436 5.71 0.575 1.65 0.545 6.73 0.499 2.16 0.65 7.75 0.42 2.67 0.715 8.76 0.231 3.17 0.756 9.27 0.075 3.68 0.75 Z(19; 15.3; 4), ReF = 24,300 e,mm m/sec e,mm (“2) m/sec c,mm m/sec 0.00 -0.487 4.32 0.68 11.43 0.128 0.25 -0.472 4.83 0.788 13.46 -0.024 0.76 -0.418 5.33 0.815 15.41 -0.013 1.27 -0.326 5.84 0.784 17.52 -0.229 1.78 -0.212 6.35 10.71 19.05 -0.266 2.29 -0.048 6.86 0.633 20.07 -0.283 2.79 0.109 7.37 0.564 21.08- -0.215 3.30 0.313 8.38 0.487 21.59 -0.106 3.81 0.512 9.40 0.338 234 Table 0.1 (c0nt'd.). 2*(20; 15.3; 4). Re; = 24.400 e,mm m/sec e,mm m/sec e,mm mysec -20.09 -0.287 - 4.43 0.681 2.29 -0.051 -19.58 -0.35 - 3.81 0.39 3.37 0.336 -18.06 -0.267 - 2.80 0.033 4.32 0.675 -16.02 -0.196 - 1.78 -0.282 5.34 0.785 -13.99 -0.087 - 1.02 -0.434 6.87 0.597 -11.95 0.041 - 0.51 -0.478 8.90 0.327 - 9.92 0.2 0.00 -0.495 10.94 0.126 - 7.88 0.43 0.51 -0.46 12.97 -0.021 - 5.85 0.694 1.27 -0.346 15.00 -0.19 ' 17.04 -0.195 19.07 -0.326 20.09 -0.359 21.11 -0.006 0*(20; 15.3; 4). Re; a 24,300 e,mm MHz e,mm MHz e,mm MHz 0.00 0.022 3.94 0.688 13.08 0.346 0.13 0.11 4.95 0.651 14.10 0.273 0.38 0.297 5.97 0.607 15.11 0.25 0.89 0.427 6.98 0.559 16.13 0.211 1.40 0.531 8.00 0.515 1.90 0.61 9.02 0.482 2.41 0.654 10.03 0.45 2.92 0.688 11.05 0.413 3.43 0.695 12.06 0.388 2(31; 15.3; 4), Rep - 24,200 e,mm m/sec c,mm m/sec c,mm m/sec 0.00 -0.515 7.11 0.693 16.26 0.027 0.51 -0.497 8.13 0.522 18.29 -0.008 1.02 -0.437 9.14 0.377 20.32 -0.073 1.52 -0.352 10.16 0.213 22.35 -0.131 2.54 -0.206 11.18 0.126 24.38 -0.19 3.56 0.144 12.19 0.064 25.91 -0.224 4.57 0.492 13.21 0.027 26.92 -0.125 5.08 0.776 14.22 0.017 27.43 -0.051 6.10 0.807 15.24 0.03 Table 0.1 (c0nt'd.). 235 Z(32; 15.4; 4). Re; . 24.200 e,mm m/sec e,mm m/sec e,mm m/sec 0.00 -0.515 5.60 0.791 18.82 -0.041 0.51 -0.476 7.12 0.521 20.85 -0.098 1 02 -0.404 9.16 0.199 22.89 -0.171 1.53 -0.263 10.17 0.096 24.42 -0.191 2.03 -0.103 12.21 0.004 25.43 -0.211 2.54 0.055 14.24 0.013 26.45 -0.12 3.05 0.232 15.26 0.023 26.96 -0.012 3 56 0.435 16.28 0.013 4.58 0.736 17.29 -0.008 0(32; 15.3; 4), ReF * 24,400 e,mm MHz e,mm MHz c.mm MHz 0.00 0.0 4.19 0.787 12.32 0.404 0.13 0.054 4.70 0.702 13.33 0.39 0.63 0.238 5.21 0.683 14.35 0.378 1.14 0.401 6.22 0:635 15.37 0.365 1.65 0.507 7.24 0.584 16.38 0.358 2.16 0.589 8.25 0.537 17.4 ‘ 0.35 2.67‘ 0.64 '9.27 0.495 18.41 0.346 3.17 0.675 10.29 0.457 19.43 0.334 3.68 0.701 11.30 0.428 20.45 0.329 21.46 0.087 Z(34.2; 15.3; 4), Re; = 24,200 c.mm m/sec e,mm m/sec e,mm m/sec 0.00 0.599 4.32 0.88 12.21 0.109 0.25 -0.573 4.58 0.922 14.24 0.103 0.76 -0.487 4.83 0.956 16.28 0.075 1.27 -0.35 5.09 0.953 18.31 0.022 1.78 -0.127 5.60 0.907 20.35 -0.052 2.29 0.085 6.10 0.799 22.38 -0.139 2.80 0.292 7.12 0.563 23.91 -0.204 3.31 0.523 8.14 0.367 24.92 -0.025 3.81 0.74 10.17 0.153 25.94 -0.328 26.96 -0.021 236 Table 0.1 (c0nt'd.). z*(20; 10.2; 4). Re; = 20.000 e,nln m/sec €,lllll m/sec €,lllll m/sec -21.105 -0.024 - 6.61 0.476 4.58 0.592 ~20.85 -0.021 - 4.58 0.625 6.61 0.384 -19.84 -0.273 - 3.05 0.258 8.65 0.187 -17.80 -0.213 - 2.03 -0.031 10.68 0.044 -16.79 -0.17 - 1.02 -0.274 12.72 -0.057 -14.75 -0.099 0.00 -0.431 14.75 -0.151 -12.72 -0.027 0.51 -0.427 16.79 -0.216 -10.68 0.099 1.02 -0.381 18.82 -0.285 - 8.65 0.245 2.54 0.187 19.84 -0.1 20.34 -0.039 0*(20; 10.3; 4). Re; . 20.100 e,mm MHz e,mm MHz e,mm MHz 0.00 0.003 4.06 0.551 11.68 0.332 0.51 0.145 5.57 0.554 12.70 0.327 1.02 0.304 5.59 0.509 13.72 0.312 1.52 0.395 6.60 0.471 14.73 0.24 2.03 0.474 7.62 0.441 15.75 0.181 2.54 0.518 8.64 0.402 3.05 0.542 9.65 0.379 3.56 0.548 10.67 0.352 2*(20; 5.4; 4). Re; = 14,500 e,mm m/sec e,mm m/sec e,mm m/sec -20.85 -0.011 - 4.07 0.401 6.61 0.352 -19.84 -0.193 - 3.05 0.274 8.65 0.196 -18.31 -0.174 - 2.03 -0.054 10.68 0.086 -16.28 -0.131 - 1.02 -0.163 12.72 -0.001 -14.24 -0.084 0.00 -0.29 14.75 -0.058 -12.21 -0.015 1.02 -0.173 16.79 -0.125 -10.17 -0.067 2.03 0.016 18.31 -0.181 - 8.14 0.179 3.05 0.228 19.33 -0.207 - 6.10 0.323 4.58 0.454 20.35 -0.072 236-3 Table 0.1 (c0nt'd.). 0*(20; 5.1; 4). Re; = 14.300 c,mm MHz e,mm MHz e,nnl MHz 0.00 -0.009 6.22 0.316 14.35 0.153 0.13 0.021 7.24 0.291 15.36 0.116 0.63 0.116 8.25 0.275 1.14 0.226 9.27 0.262 2.16 0.313 10.29 0.242 3.17 0.354 11.30 0.227 4.19 0.353 12.32 0.213 5.21 0.336 13.33 0.16 2*(20; 2.2; 4), Re; = 9,200 e,mm m/sec e,mm m/sec e,mm m/sec -20.346 -0.065 - 5.087 0.283 7.63 0.14 -19.838 -0.109 - 3.052 0.194 9.665 0.06 -18.820 -0.133 - 2.034 0.055 11.699 0.004 -17.295 -0.101 - 1.018 -0.056 13.734 -0.041 -15.260 -0.053 0 -0.13 15.768 -0.073 -13.225 -0.013 1.018 -0.075 17.803 -0.101 -11.19 0.026 2.035 0.036 18.82 -0.109 - 9.156 0.093 3.561 0.292 19.838 -0.131 - 7.121 0.178 . 5.596 0.23 20.855 -0.066 2*(20; 15.4; 6.8), Re; = 22.400 e,mm m/sec c,mm m/sec e,mm m/sec - 8.14 ' 0.346 1.02 0.001 11.19 0.075 - 7.12 0.469 2.03 0.156 13.22 -0.047 - 6.10 0.592 3.05 0.447 15.26 -0.125 - 5.09 0.707 4.07 0.678 16.28 -0.166 - 4.07 0.631 5.09 0.775 17.29 -0.229 - 3.05 0.395 6.10 0.666 18.31 -0.28 - 2.03 0.178 7.12 0.46 19.33 -0.319 - 1.02 -0.021 8.14 0.345 20.35 -0.128 0.00 -0.077 9.156 0.251 237 Table 0.1 (c0nt'd.). 2(7; 15.3; on). Re; a 24.100 e,mm m/sec e,mm m/sec e,mm m/sec - 0.25 0.445 4.32 0.556 8.89 -0.319 0.25 0.445 4.83 0.414 9.40 -0.359 0.76 0.515 5.33 0.186 9.91 -0.404 1.27 0.604 5.84 0.071 10. 41 0.435 1.78 0.737 6.35 -0.004 10. 92 -0.212 2.286 0.847 6.86 -0.116 11. 43 -0.083 2.79 0.88 7.37 -0.161 3.30 0.832 7.87 -0.235 3.81 0.697 8.38 -0.283 2*(8; 16.3; a»). Re; a 24.100 e,mn m/sec c.1110 m/sec e,nm m/sec -11.44 -0.072 -2.29 0.814 6.36 0.029 -10.43 -0.276 '1-27. 0.564 7.37 -0.126 - 9.41 -0.198 -0.25 0.417 8.39 -0. 25 - 8.39 -0.063 0.25 0.434 9. 41 -0. 363 - 7.37 0.08 1.27 0.617 10. 43 -0. 403 - 6.87 0.263 2.29 0.832 11.44 -0.184 - 5.34 0.546 3.30 0.802 - 4.32 0.864 4.32 0.511 - 3.30 0.99 5.34 0.217 8*(8; 15.3; no), ReF -- 24.100 emu MHz e,mn MHz e,mn MHz 0.00 0.008 4.32 0.672 9.40 0.305 0.25 0.101 4.83 0.64 0.76 0.284 5.33 0.605 1.27 0. 385 5.84 0.573 1.78 0. 495 6.35 0.538 2.29 0.591 6.86 0.5 2.79 0.655 7.37 0.456 3.30 0.684 7.87 0.438 3.81 0. 69 8.89 0.38 238 Table 0.1 (c0nt'd.). Z(19; 15.4; 99), ReF = 24,100 e,mm m/sec e,mm m/sec c,mm m/sec - 0.25 0.333 6. 86 0.59 16.00 -0.163 0.25 0.334 7. 87 0.449 17.02 -0.205 0.76 0.383 8. 89 0.34 18.03 -0.239 1.78 0.514 9. 91 0.227 19.05 -0.279 2.79 0.718 10. 92 0.158 20.07 -0.294 3.81 0.881 11.94 0.037 20. 57 -0.303 4.32 0.92 12.95 -0.023 21. 08 -0.306 4.83 0.906 13. 97 -0.094 21. 59 -0.313 5.84 0.757 14. 99 -0.143 22.10 -0.316 22.61 -0.293 23.11 -0.287 23.62 -0.032 2*(20; 15.4; a»). Re; = 24.100 e,mm m/sec ‘ e,mm m/sec e,mm m/sec -20.09 -0.01 - 5.34 0.847 8.90 0.277 -19.07 -0.293 - 4.32 0.931 10.94 0.094 -18.06 -0.28 - 2.80 0.734 12.97 -0.041 -16.53 -0.198 - 0.76 0.365 15. 00 -0.153 -14.50 -0.08 0.25 0.329 17.04 -0.229 -12.46 0.05 1.78 0.589 19. 07 -0.292 -10.43 0.216 3.81 0.871 20. 09 -0.308 - 8.39 0.445 4.83 0.831 21.11 -0.203 - 6.36 0.763 6.87 0.525 21.62 -0.111 e*(20; 16.3; on). ReF = 24,100 e,mm MHz e,mm MHz e,mm MHz 0.00 -0.001 4.19 0.648 11.81 0.39 0.13 0.046 4.70 0.661 12.83 0.368 0.63 0.242 5.21 0.649 13.84 0. 338 1.14 0. 39 5.71 ' 0.629 14. 86 0. 315 1.65 0.473 6.73 0.584 15. 87 0.27 2.16 0. 517 _ 7.75 0.547 2.67 0.556 8.76 0.482 3.17 0.596 9. 78 0.457 3. 68 0.632 10. 79. 0.426 Table 0.1 (c0nt'd.). 239 Z(31; 15.4; 0°), ReF = 24,100 e,mm m/sec e,mm m/sec e,mm m/sec 0.0 0.236 7.62 0.51 16.76 0.021 1.02 0.281 8.64 0.366 17.78 -0.001 2.03 0.493 9.75 0.237 18.80 -0.032 3.05 0.745 10.67 0.127 19.81 -0.063 3.56 0.957 11.68 0.075 20.83 -0.095 4.06 1.011 12.70 0.03 21.84 -0.124 4.57 0.979 13.72 0.019 22.86 -0.154 5.59 0.915 14.73 0.026 23.88 -0.178 6.60 0.709 15.70 0.028 24.89 -0.195 25.91 -0.198 26.92 -0.176 _ 27.94 -0.11 2*(32; 15.4; 9°), Re; = 24,100 e,mm m/sec c,mm m/sec e,mm m/sec -9.66 0.217 5.09 0.912 22.89 -0.133 -7.63 0.448 7.12 0.514 23.91 -0.148 -5.60 0.847 9.16 0.222 24.92 -0.151 -3.56 1.032 11.19 0.07 25.94 -0.027 -1.53 0.554 13.22 0.027 26.45 -0.011 0.00 0.230 15.26 0.032 1.02 0.287 17.29 -0.004. 2.54 0.656 19.33 -0.05 4.07 0.959 21.36 -0.099 e*(32; 15.3; no), Re; .. 24.100 c,mm MHz c,mm MHz e,mm MHz 0.00 0.021 3.94 0.681 12.57 0.402 0.13 0.076 4.44 0.678 13.59 0.387 0.38 0.157 5.46 0.651 14.60 0.377 0.63 0.242 6.48 0.608 15.62 0.367 0.89 0.296 7.49 0.572 16.64 0.357 1.40 0.387 8.51 0.52 17.65 0.354 1.90 0.448 9.52 0.48 18.67 0.348 2.41 0.503 10.54 0.442 19.68 0.338 2.92 0.568 11.56 0.481 20.70 0.25 240 Table 0.1 (c0nt'd.). 2*(34.2; 16.3; co), Re; -- 24.100 m/sec e,mm m/sec e,mm m/sec 0.968 9.16 0.306 25.94 -0.279 0.829 11.19 0.157 26.96 -0.08 0.129 13.22 0.124 27.47 -0.02 0.032 15.26 0.092 0.372 17.29 0.047 0.924 19.33 -0.025 1.203 21.36 -0.103 1.151 23.40 -0.183 0.642 24.92 -0.248 Z(34.6; 15.3; a), Re; = 24,100 m/sec e,mm m/sec c,mm m/sec 0.856 3.56 1.138 8.14~ 0.353 0.432 4.07 1.23 8.65 0.25 0.124 4.58 1.225. 9.16 0.176 0.04 5.09 1.11 9.66 0.137 0.021 5.60 0.975 10.17 0.125 0.129 6.10 0.827 10.68 0.124 0.396 6.61 0.704 11.19 0.129 0.761 7.12 0.584 12.21 0.129 0.979 7.63 0.459 13.22 0.13 - 10.2; 99), Re; = 19,700 MHz c,mm MHz e,mm MHz 0.00 -0.001 4.32 0.514 12.95 0.28 0.25 0.074 4.83 0.5 13.97 0.254 0.76 0.205 5.84 0.466 14.98 0.242 1.27 0.3 6.86 0.429 16.00 0.058 1.78 0.38 7.87 0.396 2.28~ 0.433 8.89 0.371 2.79 0.473 9.91 0.344 3.30 0.513 10.92 0.321 3.81 0.52 11.94 0.299 241 Table 0.1 (c0nt'd.). 2(20; 5.2; m), Re; = 14,000 e,mm m/sec e,mm m/sec e,mm m/sec -21.87 -0.084 - 3.56 0.478 14.75 -0.096 -19.84 -0.169 - 1.53 0.276 16.79 -0.158 -17.80 -0.137 0.51 0.279 18.82 -0.191 -15.77 -0.094 2.54 0.473 20.85 -0.104 -13.73 -0.026 4.58 0.534 21.36 -0.019 -11.70 0.059 6.61 0.32 - 9.66 0.185 8.65 0.176 - 7.63 0.337 10.68 -0.063 - 5.59 0.535 12.72 -0.024 0(20; 5.17; 9°), Re; = 14,000 e,nln MHz c.0311 MHz 6,010 MHz 0.00 0.012 4.19 0.332 12.83 0.198 0.13 0.024 4.70 0.328 13.84 ' 0.184 0.63 0.139 5.71 0.314 14.86 0.17 1.14 0.196 6.73 0.295 15.87 0.141 1.65 0.247 7.75 0.278 2.16 0.29 8.76 0.259 2.67 0.307 9.78 0.244 3.17 0.32 10.79 0.227 3.68 0.327 11.81 . 0.209 LIST OF REFERENCES LIST OF REFERENCES Adrian, R.J., and Fingerson, L.M., 1981, TSI Short Course. 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