‘39.... ._. v , 1 “— vwn-W Lt"*a 312?th 3-435'J'L-& I {$9 _9 . .-:.,. _TT_‘3 P‘ ,7"? n. ,"H*‘“"‘-u—- div.-- 3...! ‘4‘ 1"" tr ”5"- (1 '3 A "’ 3h. ;&i—-* This is to certify that the thesis entitled INDEX OF REFRACTION CORRECTIONS FOR LASER DOPPLER ANEMOMETER MEASUREMENTS IN CYLINDRICAL AND CONICAL GEOMETRIES presented by Robert Alan Pincus has been accepted towards fulfillment of the requirements for M.S. degfimin Chemical Engineering ataxia PW Major professor Charles A. Petty Date JULY 19, 1985 0—7 639 \ MS U is an Affirmative Action/Equal Opportunity Institution MSU LIBRARIES m \— RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. INDEX OF REFRACTION CORRECTIONS FOR LASER DOPPLER ANEMOMETER MEASUREMENTS IN CYLINDRICAL AND CONICAL GEOMETRIES 3? Robert Alan Pincus A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemical Engineering 1985 ABSTRACT INDEX OF REFRACTION CORRECTIONS FOR LASER DOPPLER ANEMOMETER MEASUREMENTS IN CYLINDRICAL AND CONICAL GEOMETRIES By Robert A. Pincus A phenomenon that must be accounted for when using laser doppler anemometry to make local velocity measurements in fluid flow systems is the refraction of the light beams. Corrections for the refraction phenomenon are deve10ped that provide accurate position and velocity information. A parametric study of the variables that affect the refraction corrections is performed to provide insight into the design of good LDA experiments and test sections. Results indicate that refraction- related measurement corrections are affected by the refractive indices of all media through which the light beams pass, the geometry of the test section, and the position of the beams. These corrections can be used to choose test section materials, test fluids, and test section geometry so as to minimize measurement errors. TABLE OF CONTENTS LIST OF TABLES ......OOOOOOOOOOOOO......OOOOOOOOOOOOOOOOOO LIST OF FIGURES 0.00............OOOOOOCOOOOOOOO0.0.0.0.... NOMENCLATURE CHAPTER 1. 2. 3. INTRODUCTION O.......OOOOOOOOOOOOOOOI...0.0.000... 1.1 1.2 1.3 1.4 1.5 1.6 Motivation for this Study ................... A Review of Other Stategies in the Literature O.......O.........OOCOOOOOO......O Objectives of this Research ................. Basic Principles ............................ 1.4.1 Laser Doppler Anemometry ............. 1.4.2 Geometric Optics ..................... Illustration of the Refraction Phenomenon ... MethOdOIOEY O.......OOOOOOOOOCOOOOOOOO...O... DEVELOPMENT OF THE REFRACTION CORRECTIONS ........ 2.1 2.2 2.3 2.4 Coordinate System ........................... Normal Vectors .............................. Analysis to Obtain BI ....................... Analysis to Obtain 51 ....................... 2.4.1 Axial Measurements ................... 2.4.2 Tangential Measurements .............. DISCUSSION 0.0.0.0.0.............OOOOOOOOOOOCOOOOO ii PAGE iv v vii 10 11 17 17 17 21 25 25 3O 33 3.1 Corrections for Axial Measurements .......... 33 3.2 Corrections for Tangential Measurements ..... 33 3.3 Procedure to Obtain Corrected Velocity Measurements ......CCOOOOCOOCCOCCOOO0.0.0.... 33 3.4 Effects of the Refractive Indices ........... 37 3.4.1 Effect on XIx ........................ 37 3.4.2 Effect on XIZ ........................ 37 3.5 Use of Small-Angle Approximations ........... 40 3.6 Conclusions ................................. 43 3.6.1 Design of an Experiment .............. 43 3.6.2 Finding the Axis of the HYdrocyCIOne .........COOOOOOOOOOOOOCO 44 3.7 Recommendations ............................. 44 APPENDIX PAGE A. COMPUTER PROGRAMS ................................ 46 B. SAMPLE COMPUTER OUTPUT ........................... 56 C. ILLUSTRATION OF THE CORRECTIONS .................. 61 REFERENCES 0.00.00.00.000.00............OOOOOOOOOOOOCOI... 69 iii Table Table Table Table Table Table Table Table Table Table Table Table Table Table 3.1. 3.2. 3.3. 3.4. A.1. A02. A.3. A.4. A.5. 3.1. B02. C01. 0.2. 0.3. LIST OF TABLES PAGE The working equations for the axial correCti-ons 0.00............OCCOOCCOOOOOOOOOOO 34 The working equations for the tangential corrections 00......O.......OCOOOOOOCOOCCOOOOO 35 The corrections for the axial measurements using small-angle approximations ............. 41 The corrections for the tangential measurements using small-angle approximations ............. 42 Relevant parameters for the computer programs .........OOOCOOO......IOIOCOOOOOOOOOO 48 Flow chart of computer program used to develop corrections for axial measurements ........... 49 Computer program used to generate corrections for 8x181 measurements ......OCCCCCCCOOOOOOOCC 50 Flow chart of computer program used to develop corrections for tangential measurements ...... 52 Computer program used to generate corrections for tangential measurements .................. 53 Sample computer output of corrections for axial measurements ........................... 56 Sample computer output of corrections for tangential measurements ...................... 57 Relevant parameters of the experimental system required by the computer programs in Appendix A to generate the corrections in the measurements ............................. 63 Raw data and corresponding axial velocity measurements in a hydrocyclone ............... 65 Raw data and corresponding tangential velocity measurements in a hydrocyclone ............... 67 iv Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11. 2.1. 2.2. 2.3. 2.4. 2.5. LIST OF FIGURES Hydrocyclone Cross-sections of the test section .......... Components of a dual beam differential doppler LDA system ..............................I... Orientation of the beams relative to the velocity vector of interest ................. Fringe pattern due to the intersection of two coherent laser beams .................... Reflection and refraction of light at an interface ...................O............OOO Illustration of the refraction of the beams when axial measurements are made ............ Illustration of the refraction of the beams when tangential measurements are made ....... The position vector for axial measurements .............................C.. The position vector for tangential measurements Illustration of the vector for each of 51:- the measurements The coordinate system Mobility of the test section The normal vectors .......................... Orientation of the beams with respect to the test section Refraction across the first interface PAGE 3 5 8 12 13 13 15 15 16 18 19 20 22 24 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 2.6. 2.7. 2.8. 2.9. 2.10. 3.1. 3.2. 3.1. 3.2. 8.3. 0.1. C.2. C.3. Relation of the beams with respect to the second interface The half-angle eI The position vector £1 The position vector for afiol measurements ........ The position vector measurements £8 for tangential g5 Sensitivity of the corrections for axial measurements with respect to the refractive indices Sensitivity of the corrections for tangential measurements with respect to the refractive indices ............................... Corrections in the radial position for tangential measurements ............... Corrections in the axial position for tangential measurements Corrections in the half-angle for tangential measurements Dimensions of the hydrocyclone used in the study Axial velocity measurements Tangential velocity measurements vi 24 26 31 38 39 58 59 60 62 67 NOMENCLATURE Angles k characteristic half-angle of the focusing lens 6H half-angle of the hydrocyclone 9 28 . 92A angles which describe the paths of the beams relative 62A to the normal vectors of the various interfaces 33A 3A , . 6A angle with respect to the optical bisector that describes where the beams intersect the hydrocyclone wall Vectors describing the path of the light 14 } vectors describing the path of the light through the 1. first medium E; } vectors describing the path of the light through the 1_ second medium 2 is vectors describing the path of the light through the ;3 } third medium Position Vectors g3 locates the origin with respect to the characteristic focal point of the lens ;A } locates where the beams strike the hydrocyclone wall EA relative to the origin £1 vector describes position of the point of intersection of the beams relative to the origin e ....x ‘gy } unit base vectors describing coordinate system e ....z 3A. normal vector at hydrocyclone wall QB normal vector at wall of box Other df(§) fringe spacing fd doppler frequency Ai wavelength of light in medium i Bi refractive index of medium i (“i> time-averaged velocity in i direction a distance test section is moved W half-width of box vii CHAPTER 1 INTRODUCTION 1.1 Motivation for this Study Laser doppler anemometry (LDA) is a nonintrusive method of making local velocity measurements. A problem inherent in this technique is that refraction of the light beams occurs due to differences in the indices of refraction of the media through which the light beams traverse. The refraction phenomenon affects the focal length and angle of intersection of the light beams. The measuring point and velocity measurement are dependent upon these parameters. LDA techniques take advantage of the frequency shift of the light to obtain velocity information about the flow field as follows (Durst et al., 1976): (ui>(§_) - dF(§_) <£d(§_)> (1.1) where (§) is the time-averaged velocity component of interest; dF(§) and fd(§) are the fringe spacing and doppler frequency, respectively. When the flow is confined, the light beams refract due to the optical inhomogeneity of the various media. The refraction changes the focal length and half-angle of intersection of the light beams which are characteristic of the focusing lens. Thus, the actual measuring point, 5, and the fringe spacing, dF(5)’ change. dF depends on the angle of the refracted beams and the wavelength, A, of the light in the test medium. The relationship is (Durst, 1982) dF(§) = X . (1.2) 2 Sin 61(E) The position vector §_and the half-angle 61(5) of intersection of the light beams are dependent on the geometry of the test section, the Optical properties of the media involved, and the position of the optics relative to the test section. The test section under investigation is a hydrocyclone (see Figure 1.1). Although the hydrocyclone is a centrifugal separator commonly used, the flow field within the device is not well understood. Previously, the flow patterns in the hydrocyclone were investigated by means of flow visualization techniques (Dabir, 1983; Knowles, 1971; Bradley and Pulling, 1959; Kelsall, 1952). The advent of laser d0ppler anemometry provides a much more precise method and allows for a much more detailed study of flow fields. Unfortunately, the geometry of the hydrocyclone does not lend itself to a direct application of the method. Refraction of the light beams occurs due to the curvature of the hydrocyclone and the optical inhomogeneity of the various media. 1.2 A Review of Other Strategies in the Literature The corrections presented in the literature are developed with the incident beams either in a plane perpendicular or coaxial with the axis of symmetry of the hydrocyclone (depending on whether tangential or axial velocity measurements are made). This is done to assure that the two beams will intersect. If the beams are oriented differently, the curvature of the vessel wall makes it difficult to position the beams at the desired point of measurement such that they intersect and allow the proper velocity vector to be measured. Hydro cyclone- Figllre 1.1 o Boadway and Karahan (1981) developed corrections for axial and tangential velocity measurements in cylindrical geometries (with the beams oriented in the aforementioned manner). Corrections for the refraction at both surfaces of the vessel wall were made. Small-angle linearizations were used in the development of their corrections. The error incurred by doing this is unknown. Durst et a1. (1981) suggest that the test fluid be selected to match the index of refraction of the vessel confining the flow. This eliminates the refraction at the inner wall of the vessel. This may not always be done however if particular fluid properties are desired. T81 (1980) suggests building a box around the test vessel with a material of the same index of refraction of the vessel. The space between the vessel and the box is then filled with a fluid that matches the refractive index of the chamber walls (box and vessel). This method eliminates the need to correct for the refraction due to the curvature of the outer vessel wall while allowing a free selection of test fluid. Cross sections of the test section are shown in Figure 1.2. Durst (1976) and T81 (1980) present methods of correcting for the refraction of a light beam passing through three plane parallel layers of different refractive indices using geometric optics. These methods are also valid for correcting axial velocity measurements in test sections which include cylindrical geometries if the plane containing the incident beams also contains the vertical axis of symmetry of the cylinder. These corrections were developed using a small-angle approximation (i.e., 9 << 15°). Corrections for velocity measurements in conical test sections were not deve10ped. 59/ // / HYDROCYCLONE % SIDE VIEW é % TEST FLUID % % __.__———-FLUID /m / / //////BOX 4/ ; é/ TEST FLUID // / TOP VIEW /' /' é \HYDROCYCLONE /i/// ///////\erv Figure 1.2 . Cross-sections of the test section. Dabir (1983), also developed corrections for the test section shown in FigureIJ2. Correction were developed for axial and tangential velocity measurements in both the conical and cylindrical sections. For test sections which include these geometries, however, the corrections are again dependent upon the plane of the incident beams with respect to the axis of symmetry of the test section. Small-angle approximations were used in the deve10pment of the corrections. The range of system parameters over which the small—angle approximations are valid remains unexplored. 1.3 Objectives of this Research The primary objective of this work is to develop the corrections required due to the refraction phenomena when laser doppler anemometry is used to make velocity measurements in test sections of cylindrical and conical geometries (see Figure 1.2). The corrections are necessary if accurate quantitative information is to be obtained about the flow using laser doppler techniques. Many researchers report uncorrected laser d0ppler velocity measurements and one of the goals of this study is to determine the measurement error due to the neglect of this phenomenon. This is particularly important if the radial velocity profiles are to be calculated using the measured axial velocity profiles (Dabir, 1983). Because the movement of the beams does not directly correspond to the movement of the laser or test section (depending upon how the profiles are made) the sensitivity with which the measurements may be made with respect to the design of the test section and choice of building materials and fluids will be examined. This will provide insight into the design of a test section which allows measurements to be made on a finer scale. 1.4 Basic Principles 1.4.1 Laser Doppler Anemometry The ability to use LDA as a method of obtaining velocities stems from light scattering theory (Born and Wolf, 1959; Mie, 1908). The generation of laser doppler signals in a dual-beam differential doppler system results from particles scattering light while moving through a region in space common to both incident light beams. This region is referred to as the measuring or probe volume of the LDA system. Figure 1.3 shows the components of a typical LDA dual-beam system. As shown in the figure, the light collection arrangement is in the forward scattering mode. The performance of the forward and backscattering light collection arrangements is compared by Cheung and Roseff (1982). The paths of the beams relative to the velocity component of interest are presented in Figure 1.4. The velocity vector lies in the optical plane of the incident beams and is perpendicular to the optical bisector. When two coherent light beams intersect, a fringe pattern results due to the interference of wave fronts (see Figure 1.5). The fringes may be detected using square-law detectors for electromagnetic waves which can resolve the fringes spatially and which have a much larger response time than the period of the light waves (Durst, 1970). Two interpretations have been used to explain the generation of the doppler signal. Rudd (1969) showed that the signal received by the photodetector can be understood to result from particles crossing the interference fringes and scattering light (this interpretation is termed the "fringe model" in the literature). As the particle crosses the fringes it blocks off and scatters varying amounts of light. This Test Section Aperture Lens Photodetector Laser L @3333}: Signal Transmission processing optics system Figure 1.3 . Cdmponents of a dual beam.differential doppler LDA system. optical plane incident beam Optical \\ . . / velocity blsqugi;;7L—-—- -—---—- vector Figure 1.4 . Orientation.of the beams relative to the velocity vector of interest. incident bean .mamon Momma unouonoo 03» mo qoauoomnmpna on» on one nuouumm museum . m.H magmam monoumosoa ss 2, a, % ,.1§ \ ”an”??? _ 55$ soapsnfispmfia spamqopnm assume 3333 10 variation in the intensity of the signal is used to determine the frequency shift of the light (DISA, Publ. no. 8208E). Durst (1982) pointed out that for a particle to experience a fringe it must interact with it for a time longer than the inverse of the wave frequency, but this does not actually occur. Instead, the signal is due to the scattering action of the particle and the properties of the photodetector. However the analytical result which relates the velocity to the frequency shift is the same regardless of the physical interpretation. 1.4.2 Geometric Optics In geometric optics, the wave nature of light is ignored and the path of light is represented by rays. This may be done as long as the apertures or obstacles in the path of the light are much larger than the wavelength of the light. The law of rectilinear propagation and the laws of reflection and refraction form the foundation of geometric optics. The law of rectilinear propagation states that in a homogeneous medium light travels in straight-line paths. The laws of reflection and refraction are analytically expressed as follows: reflection Y = V" (1.3a) 1&1”) «gemn . 3= 0 (1.3s) refraction Blsin Y = 82 sin T’ (1.48) [11”) * £20m . 3= 0 . (1.41:) F, Y’, and T" are, respectively, the angles of the incident, refracted, and reflected beams relative to a unit normal vector 2 (see Figurelub). 34, £2, and_;__re are vectors describing the paths of the incident, 11 refracted, and reflected beams. The refractive index (B) of a medium is defined as the ratio of the speed of light in a vacuum to the speed in the medium. Equations (1.3b) and (1.4b) state that the reflected and refracted rays lie in the plane formed by the incident ray and the normal to the surface at the point of incidence. Equations (1.3a) and (1.4a) define the angles of the reflected and refracted rays with respect to the normal to the surface, respectively. Derivations of the laws of geometric optics from electromagnetic theory are presented in Rossi (1957). The change in wavelength as light travels from one medium to another may be determined using Bi‘i - Bj‘j (1.5) if the indices of refraction of the various media (81, 82, B3) and the wavelength of the incident light are known (Halliday and Resnick, 1974). 1.5 Illustration of the Refraction Phenomenon Figures 1.7 and 1.8 show the paths of the beams through the test section when axial and tangential velocity measurements are made. Point I is the actual place where the two beams intersect. Point F is where the beams would intersect if all the media were Optically homogeneous. Point 0 denotes the point of intersection of the optical bisector and the axis of symmetry of the hydrocyclone. 1.6 Methodology Because it is desired to make all measurements relative to the center of the cyclone, point 0 is chosen as the origin of the coordinate system from which all positions (g1) will be described. The vectorggI describes 13 -——_-. . Figure 1.6 . Reflection and refraction of light at an. interface. 13 Figure 1.7 . Illustration of the refraction of the beams when axial measurements are made. Figure 1.8 . Illustration of the refraction of the beams when tangential measurements are made. 14 where the light beams intersect relative to the origin (see Figures 1.9 and 1.10). The origin (point 0) is not stationary in space however. In obtaining velocity profiles, measurements are made across the radius of the cyclone. To do this, the test section is moved. Thus, some stationary point of reference is needed from which to locate the origin. Point F is chosen because it is fixed in space, being only dependent on the radius of curvature of the focusing lens. An alternate choice for the stationary point of reference could have been the focusing lens. The difference between the two possible points of reference is the characteristic focal length of the lens. The vector g3 locates the origin relative to the fictitious focal point "F". Figure 1.11 shows the physical significance of £3 for each of the velocity measurements. xe’ defined as g? . Ex, is XFx = XOFX + 6 . (1.6) X0Fx is the distance from the origin to the lens focal point (point F) when the beams intersect at the origin; and, e is the distance the test section is moved off center. The problem now becomes one of determining g1 and BI(§). An analytical solution to the problem is presented in Chapter 2. 15 o—Q—o-o—O' N O . L l I Figure 1.9 . The position vector for axial measurements. —.< Figure 1.10 . The position vector for tangential measurements. l6 Figure 1.11 . Illustration of XFX for each of the measurements. CHAPTER 2 DEVELOPMENT OF THE REFRACTION CORRECTIONS 2.1 Coordinate System Solution of the problem requires an analytical representation of the paths of the beams. A coordinate system (see Figure 2.1) is established such that the optical bisector (9;) is colinear with the x-axis and the axis of symmetry of the hydrocyclone (AS) is colinear with the z-axis. é§_- $2 (2.1a) 91.3. ' 2,; (2.11:) (_8 “.li) - 2y = 1 (2.1a) An orthogonal rectilinear coordinate system is used because it corresponds to the directional (X,Y,Z) mobility of the test section. The test section may be moved from side-to-side and up-and-down with the use of a milling table and hydraulic jack as shown in Figure 2.2. 2.2 Normal Vectors The normal vectors illustrated in Figure 2.3 can be written as (2.2) 2; = ~24: 3A = {-cos eAgx + sin GAngcosBH - sineagz (2.3) For axial velocity measurements, the plane of the incident laser beams is set up to include the axis of symmetry of the cyclone. 9A is therefore zero. The normal vectors are then 2;, = ~2x (2.4) 9A = -cos SEEK - sinGng . (2.5) 17 18 Box Hydrocyclone 2 I I I Optical ——»—— ————I~——-— -2x Bisector ' 9v (OH) I I I Axis of Symmetry, (AS) 5y Box Hydrocyclone Optical — __ _ Bisector (OB) Axis of l Symmetry I (AS) Figure 2.1 . The Coordinate System. (ex, ey, and ez are unit base vectors) .eoanoom pass on» do massage: . m.m magmas \ K \ Q 4 393. 333: T mu I_|.. II_ _II l9 _ xomh ofiasmuoa: u Ixom mnoaomoonvhm 20 .noapoom Hmoflmoo on» ma oncogendmmoe Hmapmomdme Avv .mOApomw Hmofluosaaho one ma munoamnsmmoe Hmflpmowmme on .nOHpomm Hmofimoo on» ma nauseousmmoa Hdfix< Any .moauoom Hmoanomwaho on» ma mudmaonsmmoe amax< Amv "mu0poo> Hmenoc one . m.m ossmam A3 A3 _ . .. <- a I _ \ a) n11 _ an Bil II I c Q'OIII 1 ll lwvll II C _ _ _ _ _ m< Mm II xom xom A3 oeoaomoouczm mcoaomoomomm m Bun I WY .1: ...flv c \ _ _ _ < M: xom .MN xom 21 For the cylindrical section, SE = O and the normal vectors reduce to 3A = EB ‘3X . (2.6) For tangential velocity measurements, the plane of the incident beams is set up to be perpendicular to the axis of symmetry of the hydrocyclone. 9A which defines where the beams intersect the wall of the cyclone with respect to the x-axis is nonzero for this case. Equations (2.2) and (2.3) represent the normal vectors for the conical section. For the cylindrical section, 6H = 0; therefore, Equation (2.3) simplifies to EA = ~cosGA_e_x + SinQAEy . (2.7) 2.3 Analysis to obtain BI The position of the incident beams with respect to the test section depends on whether axial or tangential velocity measurements are made (see Figure 2.4). For axial measurements, the equations describing the incident beams (11 and ii) are: 11 . 11 = 1 (2.8) _1_1 . ‘Y = o (2.9) 11 . _t_1_ = cos k (2.10) 11 -= g . i1 . (2.11) For tangential measurements, Equation (2.9) is replaced with ll o 9-2 a 0 o (2.98) The incident beans 14 and ii are related through a reflection matrix g_ which is defined as follows 1 o 0 _Q - o --1 0 . (2.12) o o -1 L. .. The equations that describe the path of a beam through the second medium (see Figure 2.5) are: 22 Box z 9 -% flflfl",,_m~—Hydrocyclone —k—-b-- B X 0 ___I“.__ /B 2. (a) Box Y 2. -h__ --___ B Hydrocyclone /\ . \V iii/3 (b) Figure 2.4.. Orientation of the beams with respect to the test section: (a) axial measurements; (b) tangential measurements. 23 81 sin k = 82 sin 623 (2.13) (1.1 “1.2) . £8 = 0 (2.14) .1. - 1.2 = 1 (2.15) —l2 . 25 = cosBZB . (2.16) Equation (2.13) defines the angle of the refracted beam relative to the normal of the first interface. Equations (2.14)-(2.16) define the three components of the vector 14. The paths of the beams through the second medium remain reflections of each other when considered with respect to the first interface using the angle 92B as defined above, the 12 may be found using ~ h=3.1 . can The refracted beams are then described relative to the second interface as follows: ~12 - 2A = coaGZA (2.18) fie . EA = cosGZA . (2.19) These equations provide the angles (see Figure 2.6) that describe the beamm relative to the normal of the second interface. The following equations describe the paths of the beams through the thud medium: stinBZA - B3sin 63A (2.20) -1. . 3A = cos 63A (2.21) 13 . 13 = 1 (2.22) (1% “‘13) . EA = O (2.23) and stinéu = B3sin 83A (2.24) -_l:3 . PTA = cos 63A (2.25) i; . is - 1 (2.25) (g-g>.g=o . (Ln) 24 Box Figure 2.5 . Refraction across the first interface. Box Hydrocyclone Box 9'33 1 B 19.19 N A on y Hydrocyclone (a) (b) Figure 2.6 . Relation of 2a to the interfaces: (a) axial measurements; (b) tangential measurements. 25 Because the vectors 12, 13, and 3A are all in the same plane, the vector 13 can be written 13 = 312 + bgA . (2.28) This equation may be used in place of one of the Equations (2.20) - (2.23). 13 may be determined similarly. Once is and 13 are known, the half-angle of intersection of the laser beams in the test section (see Figure 2.7) may be obtained from 2.4 Analysis to obtain xx The position vector (see Figure 2.8) may be written 51 - 5A + AI 13 . (2.30) Because of the way the optics and test section are aligned with respect to the coordinate system, there is no y-component in the vector describing the point of intersection of the light beams. £1 . gy = O (2.31) The other components of 51 are found from and The vector 5A and the angle 6A (which is required in defining 3A for the tangential measurements) are as yet undetermined. These unknowns are obtained from a geometric analysis of the system. 2.4.1 Axial Measurements When the beams are positioned for axial measurements, 5A (see Figure 2.9) is defined by 26 Box Hydro cyclone (a) /Hydro cyclone \\ E3 23 \m (b) Figure 2.7 . The half-angle 6; : (a) axial measurements; C b) tangential measurements. 3A = -R ix + DA 111 (2.34) and Q = 343 + BA _1_2 (2.35) where 1H = -sin GEEK + cos eng (2.36) 5-13 = "WE-X + a _e_z (2.37) and CE = (W + XFX) tan k . (2.38) Equating the two expressions for EA results in the following component equations: 3x: -R 452 sin 6H 8 —W + EA cos 623 (2.39) gz: ‘DA cos 6H - (W + xe)tan k - EA sin 623 (2.40) Manipulating these equations results in the following equation for DA. 'DA ' (W+XEx)tan k - (W-R) tan 623 (2.41) cos 6H - sin 6H tan 923 Equation (2.34) may then be written as follows: EA a ”32x + [(W+X£x)tan k.- (V'R)t8“94nj[-sinegsx + coseagz] . cos eH - sin 9H tan 623 (2.42) The position vector is also defined by ~ ~ 51 = Q + AI :23 (2.43) where EA may be determined in the same manner as g,. Similarly, XI may be found by equating the two equations for g1, Equations (2.30) and (2.43). The equation for AI is Xi _ Mlcos 6H + sin enptan(9H-~93e) sin(e3A - 9H) 1 +tan(eH - QBAI. (2.44) tan(63A — 9H)] where M .{ (W+XEx)tan k - (W—R)tan 02],+ (W+XFK)tan k - (W—R) tan 6213 ] cos 6H - sin 6H tan 62B cos 6H + sin 6H tan 923 . 28 Hydrocyclone Hydrocyclone (b) Figure 2.8 . The position vector.z_ : (a) axial measurements; (b) tangential measurements. 29 S ”<1 (a) i Hydrocyclone \ (b) Figure 2.9 . Refraction of laser beams during axial (velocity measurements : (a) vectors; (b) angles. 30 Equations (2.32) and (2.33) may then be used to obtain XIx and XIZ' Using Equation (1.6) for XFX’ and noting that when 6 equals zero, XFX = XOFX' Therefore, XOFX can be calculated from Equation (2.32) with the result that R(cos2 9 -sin2 9 tan2 9 ) + [(W-R)tan 9 ] H H 28 23 cos 6H + sin 6H tan(9g-635) tan(e3A-ea) + tan(BH-63A) 2 cos eH-(cos on + sin OH tan 923) sin OH x° =-w+ . (2.45) cos 9H + sin 9H.tan(eB-e3fl) 2 cos 6B tank _ (cos 93 + sin 93 tan 923)sin entank 2.4.2 Tangential Measurements For tangential measurements (see Figure 2.10), xA is defined by the following equations: gA = -R cos 9A 3X + R sin 9A§¥ (2.46) EA ' -ng + 6323 + EZiq (2.47) ()nce again, using Equation (1.6) for XFX' and noting that 9A = 623, when a = 0 yields the following equation for 9A in terms of e: sineA - coseA .tanezn = gaggnk (2.48) From Equation (2.31), AI can be described in terms of the other quantities as follows: T1 = — :AX - 21 (2.49) 13 ° 32 31 Box .11. I: B .\ 9‘9 dro \ By C clone \ g a“ A y \ a \ \. \ __ ... _. 75____ __ __._. __ vv Figure 2.10 . Refraction of laser beam during tangential velocity measurements. 32 The preceeding development encompasses all the basic equations that are required in the solution of the problem. The working equations that result from the analysis and which are used in the development of the computer solution to the problem are presented in the following chapter. CHAPTER 3 DISCUSSION 3.1 Corrections for Axial Measurements The working equations used to deve10p the corrections for axial measurements‘are presented in Table 3.1. For axial measurements, there is a linear relationship between the movement of the test section and the movement of the beams. This can be seen from Equations (A) and (C) in Table 3.1. Also, the half-angle of intersection BI (and therefore the fringe spacing) is independent of the movement of the test section and the position where the measurement is made. This result was presented by Dabir (1983) for cylindrical geometries. From Equation (D) it can be seen that it is also true for conical geometries. 3.2 Corrections for Tangential Measurements The corrections for tangential measurements are presented in Table 3.2. The position vector and the angle 61 are nonlinear functions of k, (h, the refractive indices, and 8. Because a greater correction is required at the outer wall of the hydrocyclone (greatestki), observations in the corrections at the outer wall are used to determine trends in the results. 3.3 Procedure to Obtain Corrected Velocity Measurements A procedure to pr0perly obtain velocity measurements is outlined below. 33 34 c; 2: A: 3 l3 CV lav Adi. as. c.. N. . s an. ~. . n . a n AN. + :0 I has already been obtained, the corrections are made using steps 4-9. An example of how to apply the correction is presented in Appendix C. Computer programs that may be used to generate the corrections in the position of measurement and the half- angle GI are presented in Appendix A. 3.4 Effect of the Refractive Indices 3.4.1 Effect on XIX For the axial measurements, the correction for XIX is linear in e. The axial correction is symmetric about the test section with respect to the 1? movement of the test section. Also, maximum sensitivity in the XIX measurement occurs when 82 > 83. The sensitivity also increases with increasing 82:83 ratio (see Figure 3.1). For the tangential measurements, the correction for XIX is nonlinear in a. When 82 > 83, maximum}!Ix sensitivity is obtained when the test section is moved away from the laser (move negativeta), obtaining profiles in the half of the test section closest to the laser. When 82 < 83, maximum XIX sensitivity is obtained when moving the test section in the positive 6 direction (toward the laser). For this case (82 < 83), there is greater sensitivity for all s over the 82 > 83 case (see Figure 3.2). 3.4.2 Effect on x12 As the 82/83 ratio moves away from 1, the refraction in the z-direction increases. When 82 < 83, the refraction is in the positive dle d5 2.0- 1.8 -~ 38 1.0 .04 -- .02 . 4!- 1.1 1.2 0 «III- 0 1mm -.02 d b“ __04 .. Figure 3.1 . measurements with respect to the refractive indices- (Ba/Ba: 111047, k: 5.710 , e“: 5.70 ) Sensitivity of the corrections for axial 39 2.04 1 1.8 -r- L6—~ dxn< d5 L4-— 1.2 «- db 1.0 I 4. I I .8 .9 1 1.1 1.2 .04T dex dE -402-r -.o4i Figure 3.2 . Sensitivity of the corrections for tangential measurements with respect to the refractive indices. (3./,9,= 1/1.t.7. k== 5.71" . e": s.7° ) 4O z-direction (above the optical bisector). When 82 > 83, refraction is in the negative z-direction (below the Optical bisector). 3.5 Use Of Small-Angle Approximations Tables 3.3 and 3.4 present equations for the corrections for axial and tangential measurements that result from simplifying the equations in Table 3.1 and 3.2 using the small-angle linearizations for the angles (sin 9~6, cos 6-1). The equations for XIx and GI in Table 3.3 are the same equations as those derived by Dabir (1983). While his derivation for XIZ is correct, an error was made in the presentation Of the final equation. The correct equation for XIZ is presented in Table 3.3. The equations for the tangential corrections presented in Table 3.4 result from a rigorous derivation that is then simplified using the small- angle approximations. These equations are different from those presented by Dabir (1983). Dabir derives the correction for tangential measurements in a cylindrical geometry initially making the small-angle assumption for the angles (and then carrying out the derivation). Again, while his derivation is correct, the final presentation Of the equation is incorrect. Dabir’s equation should be EL _ B XIX - 1 (3.1) was 1+5... (82H 1R) For corrections to tangential measurements in cylindrical geometries, either the above equation or the equation presented for XIX in Table 3.4 may be used. Dabir did not rigorously derive an equation for the corrections to tangential measurements in conical geometries. He assumed that the effect Of (h on the refraction in the z-direction would be negligible (XIZ'O) due 41 2; Amy :3 .I.. l. N a l N . .. z .. . = 51.14.21... .... N.. . -. ..N . m x u Iw... _. Ix. ......NNx ... N.-N. ... N N . a a .-l N .NI. N smut"... N N N am ._ h; ...: . a N N N . . . ..co«u-s«xoumnq oumcsuuanso new-a eucalouanqol ”can. Os» mom occuuoouuoo ssh .n.n MAH3 3. This provides maximum sensitivity of the measurements. For tangential measurements, the test section should be designed with ‘2 < (5 for maximum sensitivity of the measurement. Also, for the tangential measurements, the measurements should be made on the side of the test section closest to the laser for maximum sensitivity in the measurements (see Figure 8.1). For cases where both types Of measurements must be made, the refractive indices Of mediums 2 and 3 should be chosen to be as similar as possible. While a trade-off in the sensitivities of the two types of measurements exists, this will minimize the refraction in the z-direction. 44 Making the measurements on the side of the test section closest to the laser will also minimize the refraction in the z-direction (see Appendix B). 3.6.2 Finding the Axis of the Hydrocyclone In the past, profiles were made by starting from the wall of the hydrocyclone and moving across the hydrocyclone, or by finding where the maximum or minimum occurs in the velocity profile and calling that point the center (or axis). .Velocity measurements across a radius were made by assuming symmetry. Unfortunately, there is no conclusive evidence to indicate that the velocity profiles within the hydrocyclone should be symmetric about the axis of symmetry (although various flow visualization studies appear to indicate this); thus, there may be error in assuming that the point of inflection in the velocity profiles (either axial or tangential) always occurs at the center of the hydrocyclone. The following alternative procedure should be used to position the intersection of the beams at the axis of the hydrocyclone. Dabir (1983) discusses how to verify that the beams actually intersect at the wall. He also discusses how to align the beams and verify that they are in the proper Optical plane. Equations (A) or (J) may then be used to locate the axis of the hydrocyclone by determining the e that corresponds to the distance the test section must be moved to be at the center Of the hydrocyclone. The test section is then moved the specified amount, positioning the beams at the axis of the hydrocyclone. 3.7 Recommendations The present study allows one to determine the position of the measurement with respect to the movement of the test section. Thus, one 45 moves the test section and then determines where the measuring point is. It is desired to be able to choose the location of measurement (both radial position and axial position) and then adjust the test section so that the position is obtained. The set-up examined allows measurements to be Obtained across a radius by moving the test section with the use of a milling table (see Figure 2.2). While the results Of this study permit measurements at particular points across a radius, the measurements may not be located in the same z-plane. Although results indicate that the z-refraction in most cases is small, it is still preferred to be able to make measurements at the exact location desired. To do this, adjustment Of the axial position (z-position) of the beams is necessary. The milling table provides such an adjustment to be made. The analytical changes to the corrections developed occur in the XFx vector that describes the position Of the origin with respect to the fictitious focal point F. TO allow for 2-dimensional movement Of the test section, the vector 5? should be written as follows: a. = 2°. + 8x9): + 6222 (3.3) where ex accounts for the radial movement of the test section and 62 accounts for the axial movement of the test section. The corrections should then be deve10ped with this modification. The end result will allow one to pick a particular radial and axial position and determine the necessary movement in 8X and £2 to attain that position. This will allow profiles to be obtained in the same axial plane (z-plane). APPENDIX A COMPUTER PROGRAMS 81 82 83 RT R8 HL A1 A2 A4 A5 A6 A7 A8 EL F0 XIX XIZ LIST OF COMPUTER NOTATION refractive index Of medium 1 refractive index Of medium 2 refractive index Of medium 3 radius at top of hydrocyclone radius at apex of hydrocyclone length of conical section radius at Optical bisector Z-position of optical bisector k 911 number of increments across a radius a lower bound of e O xrx xrx XIX x12 46 47 VX = 113x W = "BY VZ = '32 VXT = 13x V” 3 E33: var = 1:32 PI = 3.141529 Sl,SZ,S3,S4,SS,S6,A,B = storage variables for intermediate calculations 48 Table A.l. Relevant parameters for the computer programs RT RB HL Table A.2 to develop corrections Flow chart of computer program used 49 for axial measurements. We)“ when I1. I2. I}. e, 11. n. II. n. I an“ n (n-peeitien at optical lieecter) oi yee '9 n 2 ll. cone cylinder Determine II, 0. Calculate nil neglee I menus. 1;, I Determine lower bend (or C I Determine I (increment nine) I Ste; screen cyclone I S 1. no 1 I ES 4'9 ”-114“: I 1“: [hot I Cnlcalnte I" 8 X1; yen ‘xx’ ' INCH". "“18. 50 Table A.3. Computer program used to generate corrections for axial measurements. PROGRAM REF(INPUT,OUTPUT,TAPES=INPUT,TAPE6,=OUTPUT) c ENTER PARAMETERS DATA 81, 82, B3/1.0,1.47,1.33348/ DATA W/60/ DATA A1/5.71/ DATA RT,RB,HL/38.,6.08,320./ PI = 3.141529 WRITE(6,9) 9 FORMAT(2X,"AXIAL CORRECTIONS"/) C ENTER Z-POSITION AT OPTICAL BISECTOR IN MM READ *, 2 C CALCULATE R AT OPTICAL BISECTOR IF (2 .LT. ML) 00 T0 3 R - RT A2 = 0.0 WRITE(6,499) Z,R 499 FORMAT(2X,"CYLINDRICAL SECTION",10X,"Z - ",F7.3,10X, §"R = ",F7.3/) so To 5 3 20 - RB * HL / (RT - RB) R = RT * ( (z + zo) / (UL + zo) ) A2 = ATAN( RT / (HL + zo) ) * (180./PI) c OUTPUT SYSTEM PARAMETERS WRITE(6,599) z, R 599 FORMAT(2X,"CONICAL SECTION",10x,"z . ",F7.3,10X,"R - ",F7.3/) 5 WRITE(6,199) Bl,B2,B3,RT,RB,HL,A1,A2,W 199 FORMAT(32X,"81 - ",F6.3,6X,"B2 = ",F6.3,6X,"83 - ",P6.3/,32x, 8 "RT = ",F7.2,5X,"RB = ",F7.2,5X,"HL - ",F7.2I,32X,"THETAK =", s F5.2,3X,"THETAH = ",F5.2,3X,"W - ",F7.2/) WRITE(6,799) 799 FORMAT(20X,"E",10X,"THETAI",7X,"XIX",9X,"XIZ"/) WRITE(6,899) 899 FORMAT(91("*")/) c CALCULATE ANGLES A1 = A1 * (PI/180) A2 = A2 * (PI/180.) A3 = ASIN( 81/82 * SIN(A1) ) A4=A2+A3 A5 = ASIN( 82/83 * SIN(A4) ) A6 = A2 - A3 A7 = ASIN( 82/83 * SIN(A6) ) A8 = (A5 - A7) /~2. A8 = A8 * (180./PI) O 0000 CO 999 10 299 51 COMPUTER PROGRAM CALCULATE F0 31 = COS(A2) - SIN(A2) * TAN(A3) $2 = COS(A2) + SIN(A2) * TAN(A3) S3 = ( COS(A2) + SIN(A2) * TAN(A2-A7) ) / ( TAN(A5-A2) + $ TAN(A2-A7) ) 54 = (R * $1 * S2) + ( (W-R) * TAN(A3) ) * ( -sz * SIN(A2) + $32 * $3 + 83 * Sl ) 35 = TAN(A1) * ( 82 * S3 + 83 * $1 - $2 * SIN(A2) ) FO = 84/85 — w CALCULATE THE COMPONENTS OF THE POSITION VECTOR AS TRAVERSE THE HYDROCYCLONE ENTER THE NUMBER OF EQUALLY SPACED MEASUREMENTS ACROSS A RADIUS . N ' 10 DETERMINE LOWER BOUND FOR E EL - (W-R) * (TAN(A3)ITAN(A1)) - w - F0 DETERMINE SIZE OF INCREMENT DE = EL/N STEP ACROSS HYDROCYCLONE DO 10 I=1,3*N E - EL + (I-1)*DE F = FO + E CALCULATE THE COMPONENTS OF THE POSITION VECTOR $6 = ( (W+F) * TAN(A1) - (W-R) * TAN(A3) ) XIX - -R -S6/Sl * SIN(A2) + $6 * S3 * (l./Sl + 1./S2) x12 - S6/Sl * COS(A2) - $6 * s3 * (l./Sl + l./82) * TAN(A5-A2) IF ( ABS(XIX) .GT. R ) GO TO 10 OUTPUT RESULTS WRITE(6,999) E, A8, XIX, XIZ FORMAT(18X,F7.3,SX,F6.3,5X,F7.3,5X,F7.3) CONTINUE WRITE(6,299) FORMAT(/,4X,"ALL POSITIONS IN MM AND ALL ANGLES IN DEGREES"/) STOP END 52 Table A.4 . Flow chart of computer program used to develop corrections for tangential measurements. Dater eyeten parameters 81. 82. B}. k, RT, R8, 81. I Enter a (z-posiuon at optical usector) yes no a 2 HI. I Determine H. OH cone .8 .I 03 N cylinder Determine 623 I Determine lower bound for g. I Set increment nice I Step across cyclone I Calculate 8‘ I Calculate 8 I Calculate O I Obtain coaponente of 4!. vector 2A Output reeulte Table A.5. 499 599 199 799 899 53 Computer program used to generate corrections for tangential measurements. PROGRAM REF(INPUT,OUTPUT,TAPE5=INPUT,TAPE6=OUTPUT) ENTER PARAMETERS DATA 81,82,83/1.0,1.47,1.33348/ DATA RT,RB,HL/38.,6.08,320./ DATA A1/5.71/ DATA EPS / 1.0E-5 / PI - 3.141529 WRITE(6,9) FORMAT(2X,"TANGENTIAL C0RRECTIONS"/) ENTER z-POSITION AT OPTICAL BISECTOR IN MM READ *, z CALCULATE R AT OPTICAL BISECTOR IF ( 2 .LT. HL ) GO TO 3 R - RT A2 - 0.0 WRITE(6,499) z, R FORMAT (2X,"CYLINDRICAL SECTION",10x,"z = ",F7.3,10X, $ "R - ",F7.3/) GO To 5 20 = R8 * HL / (RT - R8) R - RT * ( (z + zo) / (HL + zo) ) A2 - ATAN( RT/(HL + zo) ) * (180./PI) OUTPUT SYSTEM PARAMETERS WRITE(6,599) z,R FORMAT(2X,"CONICAL SECTION",10X,"Z - ",F7.3,10X,"R - ",F.3/) WRITE(6,199) 81,82,83,RT,R8,HL,A1,A2 FORMAT(32X),"81 - ",F6.3,6X,"82 - ",F6.3,6X,"B3 - ",F6.3/,3ZX, 8 "RT - ",F7.2,5X,"RB - ",F7.2,SX,"HL - ",F7.2/32X,"THETAK -", $ F5.2,3X,"THETAH - ",F5.2/) WRITE(6,799) FORMAT(18X,"E",1OX,"THETAI",7X,"XIX",9X,"XIZ"/) WRITE(6,899) FORMAT(85("*")/) A1 - A1 * (PI/180.) A2 - A2 * (PI/180.) DETERMINE THETA28 - (A3) A3 - ASIN( 81/82 * SIN(A1) ) DETERMINE LOWER BOUND FOR E EL - -R * TAN(A3)/TAN(A1) ENTER THE NUMBER OF EQUALLY SPACED MEASUREMENTS ACROSS A RADIUS 20 40 42 45 75 54 N = 10 DETERMINE SIZE OF INCREMENT DE = -EL/N PROCEED TO STEP ACROSS HYDROCYCLONE Do 10 I=l,5*N E = EL + (I—l) * DE DETERMINE THETAA - (A9) GUESS = A3 A9 = ABS( ASIN( COS(GUESS)*TAN(A3) + E*TAN(A1)/R) ) IF ( ABS(GUESS-A9) .LT. EPS ) GO TO 40 GUESS = A9 GO TO 20 IF ( E .GT. (R/TAN(A1)) ) A9 = PI -A9 DETERMINE THETAZA - (A4) 81 - cos(A2) * ( cos(A3) * cos(A9) + SIN(A3) * SIN(A9) ) A4 = ABS( ACOS(Sl) ) CHECX FOR REFLECTION IF (83 .GT. 82) GO TO 42 SS - ASIN( 82/83 ) IF (A4 .68. SS) GO TO 80 DETERMINE THETA3A - (A5) A5 = ASIN(82/83 * SIN(A4) ) CALCULATE THE COMPONENTS OF THE VECTORS DESCRIBING THE PATHS OF THE BEAMS THROUGH THE HYDROCYCLONE IF ( A4 .EQ. O. ) THEN A a 0. GO TO 45 END IF A - ABs(SIN(A5) I SIN(A4) ) 8 = A * COS(A4) - cos(AS) vx = A * cos(A3) - B * COS(A2) * cos(A9) VY = -A * SIN(A3) + B * cos(A2) * SIN(A9) vz = —B * SIN(Az) VXT = VX VYT . -VY VZT = vz DETERMINE THETAI - (A8) 52 = VX * VXT + VY * VYT + vz * VZT A8 = ACOS(82) / 2. A8 = A8 * (180./PI) CALCULATE THE COMPONENTS OF THE POSITION VECTOR XIX = ( - R * cos(A9) ) - ( R * SIN(A9)/VT ) * VX XIz = -Vz * ( R * SIN(A9)/VY ) IF (XIX .GT. R ) GO TO 100 999 15 299 80 99 100 55 OUTPUT RESULTS WRITE(6,999) E, A8, XIX, XIz FORMAT(15X,F7.3,5X,F6.3,5X,F7.3,5X,F7.3) CONTINUE WRITE(6,299) FORMAT(/,4X,"ALL POSITIONS IN MM AND ALL ANGLES IN DEGREEs"/) GO TO 100 WRITE(6,99) FORMAT(/,4X,"TOTAL REFLECTION OCCURs"/) STOP END APPENDIX B SAMPLE COMPUTER OUTPUT TABLE 8.1. 56 Sample Computer Output of Corrections for Axial Measurements Axial Corrections R =26.030 Conical Section Z = 200.000 81 = 1.000 82 = 1.470 83 = 1.333 RT = 38.00 R8 = 6.08 HL 8 320.00 THETAK = 5.71 THETAH = 5.70 W = 60.00 E THETAI XIX XIZ -19.521 4.284 -26.030 .000 -17.569 ’ 4.284 -23.427 -.027 -15.617 4.284 -20.824 -.054 -l3.665 4.282 -18.221 -.081 -11.713 4.284 -15.618 -.108 -9.761 4.284 -l3.015 -.135 -7.808 4.284 -10.412 -.l62 -5.856 4.284 -7.809 -.189 -3.904 4.284 -5.206 -.216 -1.952 4.284 -2.603 -.243 .000 4.284 .000 -.270 1.952 4.284 2.603 -.297 3.904 4.284 5.206 -.324 5.856 4.284 7.809 -.351 7.808 4.284 10.412 -.378 9.761 4.284 13.015 -.405 11.713 4.284 15.618 -.432 13.665 4.284 18.221 -.460 15.617 4.284 20.824 -.487 17.569 4.284 23.427 -.514 19.521 4.284 26.030 -.541 All positions in mm and all angles in degrees. 57 TABLE 8.2. Sample Computer Output Of Corrections for Tangential Measurements Tangential Corrections Conical Section 2 = 200.000 R = 26.030 81 = 1.000 82 = 1.470 83 = 1.333 RT = 38.00 R8 = 6.08 HL = 320.00 THETAK = 5.71 THETAH = 5.70 E THETAI XIX XIZ -17.660 4.279 -26.030 -.000 -15.894 4.239 -23.648 -.024 -14.128 4.199 -21.220 -.049 ~12.362 4.159 -18.745 -.075 -10.596 4.119 -16.222 -.100 -8.830 4.079 ~13.650 -.127 -7.064 4.040 -11.028 -.154 -5.298 4.000 -8.353 -.181 -3.532 3.960 -5.624 -.209 -1.766 3.920 -2.841 -.237 0.000 3.881 0.000 -.266 1.766 3.841 2.899 -.296 3.532 3.801 5.859 -.326 5.298 3.761 8.881 -.357 7.064 3.722 11.968 -.388 8.830 3.682 15.122 -.420 10.596 3.642 18.344 -.453 12.362 3.602 21.638 -.487 14.128 3.562 25.005 -.521 All positions in mm and all angles in degrees. $8 Figure 3.1 . Corrections in the radial position for tangential measurements. X -1; x 101+ RH T40 "30 "2O .LIO / ‘ . ‘ - 1 : £ : .0. .07 .03 .05 .0; ..3 ..2 “.1 0 .‘ .2 .3 .‘ 05 .6 .7 .8 6'—/R Figure 8.2 . Corrections in the axial position for tangential measurements. 60 W10 .1 .1 d e7 .06 -.5 .e4 '.3 ‘.2 “.1 O .1 .2 .3 e Figure 3.3 . Corrections in the half-angle for tangential measurements. APPENDIX C ILLUSTRATION OF THE CORRECTIONS 61 APPENDIX C ILLUSTRATION OF THE CORRECTIONS Illustration of the Corrections An illustration of the position and velocity corrections is presented in this appendix. The system examined is the one used by Dabir (1983) in his hydrocyclone studies. This system was chosen because it is considered a typical experimental system in that the laser sits in air, the hydrocyclone is made of glass, and the test fluid is water. The box is constructed of plexiglass with the same index of refraction of the glass (3: 1.47). The space between the box and the hydrocyclone is filled with glycerine, also Of refractive index of the glass. The hydrocyclone is constructed tO Rietema’s specifications (Rietema, 1961). The dimensions Of the hydrocyclone are presented in Figure 0.1. The laser system used is a TSI He-Ne laser (1 = 632.8 nm) with a lens that focuses the beams with a 5.710 half-angle Of intersection. The relevant parameters for the computer programs (presented in Appendix A) are the refractive indices (81, 82, 83), the half-angle of the incident beams (k), the radial dimensions Of the hydrocyclone at the top and apex (RT and RB), and the length of the conical section (HL). The dimensions of these parameters are presented in Table C.l. The output to the computer programs is presented in Appendix B. The corrections are presented in a nondimensionalized form in Figures 8.1 - 8.3. 62 U C::i——-—‘ II I93 0 I H2 1 I __Ji. I Rietema I Dimension This Stud . [19611 I I D 761mn 761nn I 01 0.28 0 0.28 0 : 0 0 34 0 O 34 0 9\ ' H1 2 ° ' >1\ I 03 0.16 0 -- i H 5 0 5 0 I 1 1:0 --——- '1 Hz 0.67 0 -- 03—”,— 1” H3 0.4 0 0.4 0 u e Il.3° -- _‘I .1. ___II_ Figure Col e in the study Dimensions of the hydrocyclone used (nominal lengths given in mm). 63 Table C.1. Relevant parameters of the experimental system required by the computer programs in Appendix A to generate the corrections in the measurements. 81 = 1.00 82 = 1.47 8 = 1.333 E= 5.71° Z = position = 200 mm RT = 38 mm R8 = 6.08 mm HL 8 320 mm 64 The corrections influence both the location of the measurement and the velocity measurement itself. From Equation (D) in Table 3.1 it can be seen that 61 is independent Of position for axial measurements. It follows from Equation (1.2) that dF(§) is independent of position. dF is dependent upon the wavelength Of the light in the test fluid. The wavelength of the light is dependent upon the refractive indices of the media through which the light beams traverse (see Equation 1.5). Thus from Equation 1.1, the magnitude of the velocity is implicitly dependent on the refractive indices of the media through which the light beams pass. Also, from Equations (A) and (C) in Table 3.1 it can be seen that the refraction has changed the position of measurement. A sample of Dabir’s raw data along with the corrected results for axial velocity measurements are presented in Table 0.2. The results are plotted in Figure 0.2. For tangential measurements, it can be seen from Equations (M) - (P) that GI and therefore df is dependent on the position of the measurement. From Equations (J) and (L) in Table 3.2 it can be seen that the measuring point is also dependent on 6. Therefore for tangential measurements, the refraction phenomenon Changes both the magnitude of the velocity and the position of the measurement. A sample Of Dabir’s raw data for tangential velocity measurements along with the corrected results are presented in Table C.2. The results are plotted in Figure C.3. 65 Table C.2. Raw data and corresponding axial velocity measurements in a hydrocyclone. raw data * velocity measurement E (m) fd(-X-) (MHZ) ‘ XIX (mm) (U2) (111/8) 0.00 -.1559 0.0 -.4952 0.51 -.1448 .68 -.4602 1.27 -.1089 1.694 -.346 2.29 -.1606 3.054 -.051 3.37 .1058 4.494 .3362 4.32 .2125 5.761 .6753 5.34 .2472 7.121 .7853 6.87 .1880 9.161 .5973 8.90 .1030 11.868 .3271 10.94 .0397 14.588 .1261 12.97 -.0066 17.295 -.021 15.00 -.0598 20.003 -.1901 17.04 -.0614 22.723 -.1951 * Dabir’s data for the following conditions: 2 (20; 15.3; 4), ReF = 24,400. 66 Table C.3. Raw data and corresponding tangential velocity measurements in a hydrocyclone. raw data * corrected measurements 0.0 .003 0.0 3.8828 3.505 .0105 0.51 .145 .781 3.8714 3.516 .5098 1.02 .304 1.718 3.8588 3.527 1.072 1.52 .395 2.525 3.8485 3.536 1.397 2.03 .474 3.306 3.8354 3.548 1.682 2.54 .518 4.100 3.8228 3.560 1.844 3.05 .542 4.842 3.8114 3.571 1.935 3.56 .548 5.831 3.803 3.579 1.961 4.06 .551 6.638 3.7914 3.590 1.978 5.59 .509 9.423 3.7446 3.634 1.850 6.60 .471 11.271 3.7343 3.644 1.716 7.62 .441 13.015 3.7144 3.663 1.616 8.64 .402 14.863 3.6858 3.692 1.484 9.65 .379 16.659 3.6601 3.718 1.409 10.67 .352 18.481 3.6401 3.739 1.316 11.68 .332 20.303 3.6144 3.765 1.250 12.70 .327 22.126 3.5916 3.789 1.239 13.72 .312 24.104 3.5705 3.811 1.189 14.73 .24 26.03 3.5459 3.838 .921 * Dabir’s data for the following conditions: a (20; 10.3; 4), Ref -= 20,100. 60.- 40‘. 20.. 67 uz(cm/s) 0 5 -20.. -40.. -60... -30.. Figure C.2 . Axial velocity measurements. Mm) 68 200 1- ‘ 150“ “QCCm/s) 4. 100*- 50' r(mm) Figure C.3 . Tangential velocity measurements. REFERENCES Adrian, R. J., and Fingerson, L. M., 1981, T81 Short Course. Atkins, P. W., 1978, Physical Chemistry, W. B. Freeman & Co. Boadway, J. D., and Karahan, E., Feb. 1981, "Corrections of Laser Dappler Anemometer Readings for Refraction at Cylindrical Interfaces", DI§A Information: Measurement and Analysis, No. 26, p. 4. Boadway, J. D., Feb. 1983, "Corrections of Laser Doppler Anemometer Readings for Refraction at Cylindrical Interfaces", DISA Information: Measurement and Analysis, No. 28, p. 31. Born, M., and WOlf, E., 1959, Principles of Optics, Pergamon Press Inc. Bradley, D. and Pulling, D. J., 1959, "Flow Patterns in the Hydraulic Cyclone and Their Interpretation in Terms of Performance", Trans. Instn. Chem. Engrs., 81, p. 34. Cheung, T. K., and Koseff, J. P., Feb. 1983, "Simultaneous Backward-scatter Forward-scatter Laser Doppler Anemometer Measurements in an Open Channel Flow", DISA Information: Measurement andgAnalysis, No. 28, p. 3. Crosignani, 8., DiPorto, P., and Bertolotti, M., 1975, Statistical Properties of Scattered Light, Academic Press, Inc. Dabir, 8., 1983, "Mean Velocity Measurements In A 3"- Hydrocyclone Using Laser Doppler Anemometry", Ph.D. Thesis, Michigan State University. DISA Electronics, August 1972, "DISA Type 55L Laser Doppler Anemometer", Leaflet No. 2005/8. DISA Electronics, 1978, "Laser Anemometer Equipment Catalog", Publ. No. 8208E. Durrani, T. S., and Created, C. A., 1977, Laser Systems in Flow Measurement, Plenum Press. Durst, P., Melling, A., and Whitelav, J. H., 1976, Principles and Practice of Laser-Doppler Anemometry, Academic Press. 69 70 Durst, F., Keck, T., and Kleine, R., 1981, "Turbulence Quantities and Reynolds Stress In Pipe Flow of Polymer Solutions Measured", Turbulence In Liquids, University of Missouri-Rolls. Durst, F., September 1982, "REVIEW-Combined Measurements of Particle Velocities, Size Distributions, and Concentrations", Tansactions of the_ASME - Journal of Fluids Engineering, 104, p. 284. Halliday, D., and Resnick, R., 1974, Fundamentals of Physics, John Wiley & Sons, p. 669. Kelsall, D. F., 1952, "A Study of the Motion of Solid Particles in a Hydraulic Cyclone", Trans. Instn. Chem. Engrs., 89, p. 87. Knowles, N. S. R., 1971, "Photographic Fluid Velocity Measurement in a Hydrocyclone", M. Eng. Thesis, Dept. of Chem. Eng., McMaster University, Hamilton, Ontario. Mie, G., 1908, Ann. d. Physig,‘g§_(4), p. 377. Rietema, R., l96la-d, "Performance and Design of Hydrocyclones -- I, II, III, IV", Chem. Egg. Sci., lé, p. 298, 303, 310, 320. Rossi, 8., 1957, Optics, Addison-Wesley Publishing Co., Inc., p. 366. Rudd, M. J., 1969, "A New Theoretical Model for the Laser Dapplermeter", Journal of Scientific Instruments (Journal of Physics E), Series 2, g, p. 55. Till I U15 YMd ”"4 HMS E 7 mummy: "1:113qu