MSU LIBRARIES -_ RETURNING MATERIALS: PIace in book drab—to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. V‘- SIMILARITY STRUCTURE OF AN AXISYMMETRIC VISCOUS VORTEX WITH VARIABLE CIRCULATION By Hsin-Chih Chen A DISSERTATION Summitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILISOPHY Department of Chemical Engineering 1987 ABSTRACT SIMILARITY STRUCTURE OF AN AXISYMMETRIC VISCOUS VORTEX WITH VARIABLE CIRCULATION BY Hsin-Chih Chen Vortex flows occur in many engineering operations. Recent interest in oil-water separation by using hydrocyclones has increased the need for a better understanding of the flow structure of a viscous swirling core flow. Experimental observations in confined vortex chambers show that a locus of zero axial velocity divides an inner region from an outer region of the flow field and that the swirl component of the velocity field in the outer region can be represented by the empirical expression “0 - KrN, -l S N < 0 . The total viscous dissipation of the vortex is unbounded for —l s N s ~2/3 and bounded for -2/3 < N < O. The core region of the vortex is studied theoretically by constructing a class of similarity solutions which satisfy the boundary layer equations and the free—vortex-like swirl condition in the outer region. The flow model is used to analyze the behavior of a dispersed phase in the core by using the equilibrium orbit theory for particle - fluid separation. The model predicts three types of flow behavior in the vortex core reverse flow, undulated flow, and jet-like flow. The flow regime is uniquely determined by specifying the empirical index N, and a local spin parameter related to the distribution of vorticity on the axis. Most significantly, the theory predicts the existence and location of a 'mantle' (surface of zero radial velocity) within the core region of the vortex. The theory is qualitatively and quatitatively consistent with recent experimental studies of the viscous core using laser doppler anemometry. The particle - fluid separation study explains the difficulty of capturing very fine particles near the axis of a viscous vortex . To my family ii ACKNOWLEDGMENTS The author would like to express his deepest gratitude to Professor Charles A. Petty for his guidance and encouragement throughout this research and in preparation of this manuscript. Thanks are also due Professors John Foss, K. Jayaraman, and Chang; Yi Wang for their useful comments and criticism. Special thanks are given to my friend Spring Wu for her help and moral support. Special thanks are also given to my wife, Emmy Chen” for her many encouragements, moral support, and help during this study. The author gratefully acknowledges the Department of Chemical Engineering for finacial support. Partial support of this work by the national Science Foundation (Grant NO. CPE-80-15537) and the Exxon Foundation is appreciated. iii TABLE OF CONTENTS LIST OF TABLES ................................................ LIST OF FIGURES ............................................... NOTATION ...................................................... CHAPTER 1. INTRODUCTION ........................................... 1.1 Motivation For This Study ......................... 1.2 Objectives ........................................ 1.3 Methodology ....................................... 2. LITERATURE REVIEW ...................................... 3 2.1 Experimental Studies .............................. 2.2 Theoretical Studies ............................... 3. MATHEMATICAL FORMULATION ............................... 3.1 Vortex Models ..................................... 3.2 Similarity Conditions ............................. 3.3 Asymptotic Behavior ............................... 3.4 Boundary Value Problem Studied .................... 4. SOLUTION BEHAVIOR NEAR THE AXIS ........................ 4.1 General Solution .................................. 4.2 Different Flow Regimes ............................ 4.3 Mechanical Energy Balance On The Axis ............. 5. SOLUTION METHODOLOGY ................................... 5.1 Transformation To A System Of First Order Equations 5.2 Elements Of The Shooting Method ................... iv vii ix xii 11 ll 16 22 22 27 35 37 42 42 45 49 55 55 61 10. MODEL I : VORTEX FLOW WITH CONSTANT CIRCULATION ........ 6.1 General Behavior Of The Solution .................. 6. 6. 6 6. 2 3 .4 5 I Behavior Of Velocity Field ........................ Comparison With Long’s Vortex ..................... Macroscopic Properties ............................ Discussion ........................................ MODEL II : VORTEX FLOW WITH VARIABLE CIRCULATION ....... 7.1 Comparison Between Models I and II For N - -1 ..... 7.2 Different Flow Regimes ............................ 7.3 General Behavior Of The Solution .................. 7.4 Behavior Of The Velocity Field .................... 7.5 Macroscopic Properties ............................ AN ANALYSIS OF EXPERIMENTAL RESULTS IN THE LITERATURE .. 8. 8. 8. 8. l 2 5 6 Experimental Results .............................. Flow Regimes ...................................... Similarity Scaling Of The Centerline Velocity ..... Similarity Scaling Of The Angular Momentum Profiles Viscosity Estimates ............................... Entrainment Rates ................................. PARTICLE EQUILIBRIUM ORBITS WITHIN THE CORE REGION OF VORTEX FLOWS ........................................... 9. 9 9. 9 9. 9. 1 .2 3 .4 5 6 Background ........................................ Theory ............................................ Stability Of The Equilibrium Surface .............. Results ........................................... Conclusions And Recommendations ........................ 10.1 Summary Discussion ............................... 73 73 79 83 85 94 99 99 100 103 109 113 124 124 129 135 136 138 143 147 147 149 154 158 159 169 172 172 vi 10.2 Conclusions ...................................... 10.3 Recommendations .................................. APPENDICES A. SOLUTIONS (a,b,c,5) FOR MODEL I AND MODEL II ........... B. COMPUTER PROGRAMS ...................................... C. TABULAR EXPERIMENTAL DATA USED ......................... D. COMPONENTS OF THE STRESS TENSOR FOR AXISYMMETRIC, INCOMPRESSIBLE FLOWS OF A NEWTONIAN FLUID .............. LIST OF REFERENCES 175 177 181 185 207 226 227 TABLE LIST OF TABLES Mathematical models ................................... Physical properties calculated for the two vortex models ................................................ Criteria for transition between flow regimes .......... Representation of the vortex model as a system of first order equations ....................................... Boundary conditions on the vortex models .............. Representation of the rescaled vortex models .......... Flow conditions of the experimental studies ........... Theoretical evaluation of ¢(N,b) ...................... Predicted values of b and flow behavior ............... Viscosity estimates based on the similarity theory .... Parameters used to calculate the equilibrium surfaces . Solution (a,b,c,na) for Model I ....................... Solution (a,b,c,q*) for Model II (N - -l.0) ........... Solution (a,b,c,n*) for Model II (N - -0.75) .......... Solution (a,b,c,n*) for Model II (N - -0.5) ........... Experimental data set la .............................. Experimental data set 1b .............................. Experimental data set 1c .............................. Experimental data set 1d .............................. Experimental data set 1e .............................. vii 40 41 51 56 58 62 126 131 134 142 160 181 182 183 184 207 208 209 210 212 .10 .11 .12 .13 viii Experimental data set lf .............................. Experimental data set lg .............................. Experimental data set 2a .............................. Experimental data set 2b .............................. Experimental data set 2c ............................. Experimental data set 3 .............................. Experimental data set 4 .............................. Experimental data set 5 .............................. Components of the stress tensor for axisymmetric, incompressible flows of a Newtonian fluid (cylindrical coordinate) ............................... 214 216 218 220 221 222 223 225 226 FIGURE 1.1 3.1 LIST OF FIGURES Viscous vortex region of a hydrocyclone ............... Coordinate system and physical boundary condition for the vortex Model I .................................... Coordinate system and physical boundary condition for the vortex Model II ................................... Surfaces of similarity (constant values of n - r/6(z)). Qualitative behavior of the dependent variables on the similarity surfaces ................................... Geometric meaning of the expansion coefficients ....... Classes of vortex flows allowed by the similarity model Classification of vortex flows based on the mechanical energy balance ........................................ Solution strategy based on a shooting method .......... Domain of convergence for the three dimensional search with b - 0.3 (Model 11, N - -l) ....................... The effect of the accuracy parameter c on the behavior of the solution for b - 0.3 (Model II, N - -1) ........ ZSCNT search for a one dimensional problem ............ The behavior of the solution for Model I .............. The effect of b on the behavior of M for Model I ...... The effect of b on the behavior of F for Model I ...... The effect of b on the behavior of h for Model I ...... The effect of b on the tangential velocity profile for Model I ............................................... The effect of b on the axial velocity profile for Model I ............................................... 26 28 32 34 44 50 54 64 67 70 72 74 76 77 78 81 82 The effect of b on the radial velocity profile for Model I ............................................... 84 Comparison between Model I and Long's vortex for the stream function ....................................... 86 Comparison between Model I and Long’s vortex for the radial velocity profile ............................... 87 The effect of b on the macroscopic axial force for Model I .............................................. 89 The effect of b on the macroscopic axial torque for Model I .............................................. 91 The effect of b on the macroscopic volumetric flow rate for Model I ..................................... 92 The effect of b on the macroscopic pressure drop for Model I .............................................. 93 The effect of b on the energy of a fluid particle on the axis ............................................. 97 Comparison between Model I and Model II for axial velocity profile ...................................... 101 Regimes of different flow behavior for Model 11 ....... 102 The behavior of the solution for Model II ............. 104 The effect of b on the behavior of M for Model II ..... 106 The effect of b on the behavior of F for Model II ..... 107 The effect of b on the behavior of h for Model II ..... 108 The effect of b on the tangential velocity profile for Model II .............................................. 110 The effect of b on the axial velocity profile for Model 11 .............................................. 112 The effect of b on the radial velocity profile for Model II .............................................. 114 The effect of b on the macroscopic axial thrust for Model II ............................................. 115 The effect of b on the macroscopic axial torque for Model II ............................................. 116 The effect of b on the macroscopic volumetric flow rate for Model II .................................... 117 The effect of b on the macroscopic pressure drop for xi Model II ............................................. 120 .14 The effect of b on the macroscopic dissipation for Model II ............................................. 122 .1 Coordinate system for various vortex chambers ......... 125 .2 Deviation from ideal behavior in the outer region of several experimental vortex flows ..................... 128 .3 Theoretical evaluation for ¢(b,N) ..................... 130 .4 Comparison between theoretical and experimental flow behavior .............................................. 133 .5 Similarity scaling of the centerline velocity ......... 137 .6 Similarity scaling of the angular momentum (data from Escudier, et a1., 1982) ............................... 139 .7 Similarity scaling of the angular momentum (data from Dabir, 1983) .......................................... 140 .8 Estimates of entrainment rates ........................ 145 .9 Comparison between theoretical and experimental values of ho?) .............................................. 146 .1 Definition of a stable and unstable equilibrium surface 155 .2 Stability of equilibrium surfaces for forward and reverse flow vortices ................................. 156 .3 Equilibrium surfaces for jet-like flow ................ 160 .4 Equilibrium surfaces for undulated flow ............... 161 .5 Equilibrium surfaces for reverse flow ................. 162 .6 Equilibrium surfaces for jet-like flow ................ 166 > NOTATION uz(0,z) N , dimensionless centerline axial velocity K6 Initial guess for a duo 2( 3r ) r — 0 N-l , local spin parameter K6 M(0), dimensionless excess pressure on the axis Initial guess for c Total dissipation Particle diameter 2 a puc2( 2— - c), excess mechanical energy of the fluid on the axis E 2 , dimensionless excess mechanical energy on the axis pu c rug 6n ) c Dimensionless tangential velocity profile ( - Asymptotic behavior of F Centrifugal force acting on the particles Viscous drag on the particles xii xiii I 23 I (puz2 - P)rdr, axial component of the average force 0 acting on «:2 Dimensionless inertia force Dimensionless pressure force Dimensionless viscous force Fz/(2n62pucz), dimensionless macroscopic axial force 11 z Dimensionless axial velocity profile ( - ) c Asymptotic behavior of G ¢(r.2) Dimensionless stream function ( - -—————§ ) u (2)6 c Asymptotic behavior of h Circulation constant 2 Dimensionless pressure drop profile (P/puc ) Asymptotic solution of M Exponent of free-vortex-like flow p° - pgz - p(r.2) P(~.z) - p(r.w) (p(5a,z) - p(0,z))/puc2 , dimensionless pressure drop > 90 ur(r,z) uz(r,z) xiv 3 2x I uzrdr, volumetric flow rate across the area «62 0 Q/(2162uc) , dimensionless volumetric flow rate Inlet flow rate Radial distance Radial position of equilibrium orbit Inlet Reynolds number Intrinsic Stoke's number 3 2x I [pruo]uzrdr, axial torque over the cross section «62 0 Tz/(26n62puc2), dimensionless axial torque Radial velocity profile Axial velocity profile KSN , velocity scale Inlet velocity uz(0,z), centerline axial velocity Tangential velocity profile The radial component of the particle velocity M(n) XV h(n) C(n) G'(n) F(n) F'(n) Asymptotic solution of X Small purturbation from the asymptotic solution of X Axial distance from singular point Axial position on the axis where the pressure is zero Singularity point Reference coordinate for experimental data Axial position of equilibrium orbit Axial position of equilibrium orbit on the axis YS/(5N+1) Y6/<5N> Greek 6(2) 6*> 0 u -------s - g2 (1.10) m + a] HI t lo .4. filifi 73 2 r where u is the kinematic viscosity Of the fluid (or a constant 'eddy' viscosity if the vortex is turbulent) and K is a constant defined by lim uo - K/r, K > 0. (1.11) r46 The solutions developed.by Long are consistent with experimental observations (a), (b), (c), and (h); however, as indicated by Eq. (1.10), the axial velocity surrounding the vortex is always positive and decreases to zero very slowly (uz a l/r). For confined vortex chambers, a reverse annular flow (i.e. , observation (d)) surrounding the vortex generally occurs. Long noted that this also occurs for unbounded flows such as tornadoes and that the asymptotic result given by Eq. (1.10), although an.gx§g£ solution to the Navier - Stokes equations, was probably too restrictive. Although Long's vortex shows the qualitative properties expressed by observations (e), (f), and (g), it requires N - -1 (see Eq. (1.11)), B - -l, and.AI-'l. Unfortunately, these exponents are not observed experimentally. However, Long's vortex has the interesting feature that the circulation is constant on similarity surfaces (see Eq. (1.9)) Similarity scaling also implies that the viscous dissipation per unit volume on surfaces of constant n will depend on the axial coordinate as follows u c - r : Vu o: (— )2 on (zB’A)2. (1.12) v - 6 Thus, Long's vortex has the property that V or 2-4. However, the data of Dabir [1983] suggest that B - A > -2, so experimental flows do not appear as dissipative as Long's vortex. If @v is integrated.Iover the entire flow domain, then the total viscous dissipation is infinite for Long's vortex, but is bounded for similarity flows having the property that -1/2 < B < O. This intriguing theoretical possibility partly motivates this study. In the past decade, Thew and his colleagues at Southampton University (U.K.) have made some progress on removing Oil from water more efficiently by using a cyclone separator. They (see Colman et al., 1980) were able to obtain 99% separation for a drop size of 55 microns. However, the efficiency apparently drops off sharply when particle sizes become less than 40 microns. What causes the efficiency to drop at smaller particle sizes? Is this phenomenon a result of the specific design of the cyclone or is it an intrinsic property of all vortex flows? This research intends to examine this question by studying the behavior of a dispersed phase in a viscous vortex. Hopefully, this theory will provide new insights on how to improve the separation efficiency for hydrocyclone separators. 1,2 ijggtivgg The primary objective of this research is to apply tine techniques of boundary layer theory to investigate the similarity structure of a special class of swirling flows related to the experimental observations (a) - (h). Two models will be studied. Model I, which has the feature of constant circulation on similarity surfaces, is used to understand the flow structure induced by a free vortex defined by Eq. (1.10). This model was previously studied by Long [1961] and by Burggraf and Foster [1977]. Model II, which has the feature of variable circulation>0 (2.1) where -1 s N < 0. For a free vortex, N - -l. N can be determined from the swirl velocity profile in the outer region. Fortein and Dijksman [1953], Escudier et a1. [1980,1982], and Dabir [1983] have all shown that N has the value near -0.75 and appears to be insensitive to the flow rate and the contraction ratio of the vortex chamber. However, Knowles et a1. [1973] reported a much flatter tangential velocity in the outer region (N - -0.3). Near the axis, the tangential velocity profile has the same form as a forced vortex, whereas in the outer region, the tangential velocity always has a free-vortex-like behavior (i.e., Eq.(2.l)). Many experimental studies about swirling vortex flows have been conducted in the last 30 years. One of the most important findings is the existence of flow reversal phenomenon. In an earlier study of 11 12 swirling flow in a circular pipe, Nuttal [1953] found that three types of flow patterns occur for Reynolds number in the range 10,000 to 30,000. In his study, the Reynolds number is defined as ”F“? ReF - —v where DF is the inlet diameter of the pipe, uF is the inlet velocity, and v is the molecular kinematic viscosity. As the swirl component of the velocity increases, the flow changes from a one-celled to a two- celled and, finally, to a three-celled vortex structure. Reverse flows are characteristic of these flows. Although this finding is important, Nuttal gave neither a physical nor a theoretical explanation for this phenomenon. In a study of flow through a conical nozzle, Binnie and Tear [1956] showed that reverse flows also exist in a cone, but they only observed a two-celled vortex structure. Binnie [1957] also investigated the different flow patterns in the low Reynolds number range by injecting swirling water into a circular pipe, he found that the transition from a single-celled vortex structure to a three-celled structure depends not only on the Reynolds number but on the ratio of swirl to axial flow. One very important conclusion, which Binnie did not emphasize, is that the reverse flow was induced by increasing the swirl component for a fixed Reynolds number. In his measurements of velocity profiles using an optical technique based on flow visualization, Kelsall [1952] observed a surface of zero axial velocity which increases with axial position. Bradley and Pulling 13 [1959] also observed this phenomenon. Moreover, they reported that this surface was insensitive to operating variables and had a diameter of 0.43 times the cyclone diameter. For a flooded hydrocyclone (or more generally, a confined vortex chamber), Dabir [1983] , Knowles [1973], Escudier et a1. [1980,1982] , have noted that the surface of zero axial velocity increases in diameter in the direction of decreasing magnitude of the centerline axial velocity. The growth of the vortex core is a nonlinear function of the axial coordinate. Dabir, also observed that this surface (zero axial velocity) divides the inner region Of upward flow from an outer region of downward flow, even when the underflow rate (see Figure 1.1) is zero. The magnitude of the radial velocity in the core flow was also measured by Kelsall (with an air core) and Knowles (without an air core). It is not surprising to find that this number is very small compared with the axial and tangential components of the velocity. In the study of a hydrocyclone with an air core, Bradley and Pulling (see p.15, Bradley, 1965) found that a locus of zero radial velocity, or ”mantle", exist in the cylindrical section and terminates at a level in the conical section of the cyclone. For the study of a flooded hydrocyclone, an outward radial flow in the inner region and inward radial flow in the outer region was noted by Knowles (see Figure 8, Knowles et al., 1973). This phenomenon was also observed by Ohasi and Maeda (see p.38, Svarvosky, 1984) and by Dabir (see Figure 5.11, Dabir, 1983). It is interesting that the outward radial flow in the inner region has only been observed in flooded hydrocyclones. The most recent velocity measurements have been done by using Laser Doppler Anemometry (LDA) (see Durst et al., 1976; Escudier et al., 1980, 1982; Dabir, 1983; and Thew et al., 1980). More recently, Escudier et 14 a1. [1980,1982] , in their study of confined turbulent vortex flows by using a laser doppler anemometry and a flow visualization technique, showed that different flow patterns occur depending on the contraction ratio of the vortex chamber (Do/D) and the Reynolds number. Transition from jet-like core behavior to wake-like behavior (including undulated and flow reversal) occurs either as the Reynolds number or the ratio Do/D increases. Another interesting result is that the maximum swirl velocity (“0 max) increases significantly as Do/D decreases. These two results imply that jet-like flow behavior corresponds to larger values of u than wake-like flow. 0,max In the measurement of velocity profiles in a 3" flooded hydrocyclone by using laser doppler anemometry, Dabir and Petty [1984a,l984b] have revealed that multiple reverse flows occur in the vortex core if using a 2:1 contraction vortex finder. The occurance of jet-like and reverse flows strongly depends on the size and shape of the geometry. The radial velocity calculated from their data showed that an outward flow (positive radial velocity) occurs near the axis for flooded hydrocyclones. A large number of relative experimental observations were also found. These include the approximate axisymmetric structure, a reversal annular flow surrounding the surface of zero axial velocity, a non-linear growth of the viscous core (r - 6*(2)) in the direction of decreasing centerline axial velocity, and a free-vortex-like swirl velocity in the outer region. Long [1956], in his earlier experiments on withdrawing water through a hole at the centre of a bottom plate, observed that : (1) An intense narrow vortex formed when water was extracted from a sink of a 15 slowly rotating cylinder; (2) When the steady draining vortex is achieved from an initial slow rotation, the circulation was almost independent of radius outside the core; (3) The core seemed to spread almost linearly with increasing axial distance. These observations, which imply that N - -1, motivated Long's theoretical study. The features of vortex flows, in many aspects, are similar to those reported in a swirling jet. Chigier and Chervinsky [1967] indicated that the distribution of the axial and swirl velocities were similar at least for moderate swirl numbers. The swirl number, as represented by S, is usually defined as T2 3...— Fz an and leads to a stagnation point at the origin without physical justification. Sullivan [1959] , Donaldson [1956] , Donaldson and Sullivan [1960] , and Donaldson and Snedeker [1962] improved Burgers’ vortex by considering the velocity field as the form of : ur - ur(r), uo - u9(r), uz - zf(r) (2.3) Their solutions represents a rather large class of three - dimensional, viscous vortex motions. They found ,in addition to Burgers’ one-celled vortex, three - dimensional one-celled and two-celled vortices. However, in their work, the interaction between the vortex and the ground boundary layer flow is neglected. The Sullivan vortex is, however, of considerably interest to meterologists because two-celled structures have been found in tornadoes. An extensive review of confined vortex flows was reported by Lewellen [1971] . Long [1958,1961] , motivated by his early experimental observations (see Section 2.1), studied an intense vortex in an unbounded viscous fluid. Unlike Burgers' and Sullivan's vortices, Long's vortex has the velocity field at the outer region as : K +g+ 9 J? r I33 I a “I t HI 7: e g , r >> O (2.4) '1: Z All three components of the velocity are independent of z in the outer region. Furthermore, the tangential velocity is a free vortex. Long's vortex is one of the first models proposed to describe tornadoes. It 18 shows that the various flow structures can be determined by a single parameter. Long's vortex also shows the characteristics of constant circulation on similarity surfaces (i.e., constant :7). An interesting feature is that the macroscopic axial force in Long's vortex does not uniquely determine the flow behavior. It is noteworthy that the quasi - cylindrical approximation, in which variations in the axial direction are taken to be small compared with variations in the radial direction, was adopted by Long as well as by Bloor and Ingham [1975, 1984] , and Qing [1983]. Long's theoretical work initialed many studies about geophysical vortices. A review these concentrated vortex flows was reported by Hall [1966]. Another type of sink vortex was studied by Pedley [1969] . In his study of a so called bath-tub vortex, Pedley considered a point sink with strength 4a'Q situated on the axis and a uniform circulation at large radius. The flow is irrotational (i.e. , vorticity equals zero) at large radius and must be rotational at some region around the sink, so that steady vortices will ultimately be set-up. Although the theoretical study of geophysical vortices is far from satisfactory due to few experimental observations, some progress has been achieved in the last 25 years. An excellent review of geophysical vortices was reported by Morton [1966,1969]. Geophysical vortices, such as tornadoes, waterspouts and dust devils, which have narrow vortex cores, terminate at a lower boundary, and are maintained dynamically aloft by some convective system which prevents the lower pressure core from filling with upper air. More theoretical treatments have dealt with the lower part of tornadoes, which have been regarded as steady vortices (see Morton, 1966). Long’s vortex, as he recognized, cannot be applied to tornado as close to the ground. Therefore, to improve Long's vortex, 19 many researchers have tried to treat the effect of the ground boundary. Serrin [1972], adopted the same equations as Long, but assumed the no-slip condition at the surface and allowed a singularity at the origin. He proved that there can exist only three types of motion. In the first type, the radial velocity is directed inward along the boundary and upward along the axis. In the second type, the radial velocity moves inward along the boundary and downward on the axis, with a compensating outward flow at an intermediate angle. In the third type of motion, the radial velocity is outward near the plane and downward on the central axis. In the study of conical vortices, Yih et a1. [1982] found a class of exact solutions of the Navier - Stokes equations. Their solutions include different angles of cones which extend the right rectangular angle of Serrin. Wu [1986], in his recent study, has considered a conical turbulent swirling vortex with variable eddy viscosity. He argued that if the eddy viscosity varies only in the boundary layer near the core or surface, the solution outside the layer will approach to one of the laminar solutions of Yih et a1. or that of Serrin. He also found that for a class of deliberately chosen "eddy" viscosity functions, a steady turbulent vortex can satisfy both the regularity condition at the core and the adherence condition at the surface, except at the singular point. In a further study of vortex breakdown of Long's vortex, Burggraf and Foster [1977] showed that breakdown (reverse flow) occured when the global axial force, F2, was less than a critical number. They also found that Long’s FZ is about 3 % lower than their calculations. Foster and Duck [1982], in the study of stability analysis of Long's vortex, pointed out the most dangerous mode in Long's vortex are those with 20 positive azimuthal wave number n, and that the growth rates increase with n at least for the values of 0 s n s 6 computed. In a cyclone chamber, where vortex core flow are characteristized by a reverse annular flow , Bloor and Ingham [1975,1984] have made a systematic analysis. To investigate the core flow structure, they first solved for the basic flow in the outer region by using an inviscid, rotational flow (i.e., vorticity is not equal to zero). From this information, they were able to solve for the velocity profile in the core flow. In order to have solutions which satisfy the momentum equation, Bloor and Ingham assumed that the "eddy" viscosity could be expressed as p a z- Qing [1983], in his study of velocity and turbulence distributions within a cyclone, concluded that different radial distributions of ur and we have only a slight influence on the radial distribution of no , but strongly affect the shear - stress distribution (Tr0)' However, in his model uo is only a function of r, which he recognized as too restrictive. It is noteworthy that Qing adopted, for the first time, a continuous "eddy“ viscosity profile which consists of different models of we for different regions of the vortex. Another interesting and important theory about the outer region of a confined chamber has been developed. by Petty [1985] . The flow in the outer region could be Beltrami for which the velocity and vorticity vectors are colinear. Mathematically, Beltrami flow can be expressed as IE R If: 21 where ggrepresents the vorticity vector and 9 represents the velocity vector. For axisymmetric and inviscid flows, if the velocities in the outer region are also Beltrami, then the tangential velocity profile is approximately u a r-3/4. 0 The result is very interesting and important because the value of N (- - 0.75) is consistent with most of the experimental data (see Chapter 8). CHAPTER 3 MATHEMATICAL FORMULATION zil_ysr£sx_ugdels The balance of linear momentum and the continuity equation for a constant property Newtonian fluid govern the behaviour of the vortex flows studied in this research. For steady, axisymmetric, swirling flows, the velocity and pressure fields depend only on the cylindrical coordinates r and z : a - ur(r.2)er + u0(r.2)so + uz(r.z)s.z (3 1) P - p(r.2). (3.2) If the axial transport of momentum by viscous forces is neglected, then the boundary layer equations governing g and p over the spatial domain 0 s r 5 3(2), 0 < z < on are (see Long, 1961) ‘2 u 0 6p P T ‘ "a? ‘3'” a a a l a pur - (rug) + puz - (rue) - #r - ( -- --(ru9)) (3.4) Br 82 ar r 6r auz auz 6p [1 a auz put—+puz—--—+pgz+--——(r-—) (3.5) Sr 82 62 r 8r 8r 1 a 8 ‘E'a—r (rur) +5-—zuz - 0. (3.6) 22 23 Eq. (3.3) assumes that the radial component of the velocity is small and that the large centrifugal force per unit volume, puoz/r, causes the radial pressure drop. The axial pressure gradient in Eq. (3.5) is an important and unique aspect of vortex models examined. It is also noteworthy that Eq. (3.4), which is the tangential component of the equation of motion, can also be interpreted as a transport equation for the axial component of angular momentum. Eqs. (3.3) - (3.6) govern the behaviour of the three components Of g and the pressure. Because of the viscous nature of the fluid and the axisymmetric assumption, the boundary conditions on the axis (i.e. , r - 0 and z > 0) are ur(0,z) - O (3.7) u0(0,z) - O (3.8) auz 5;_ _ o (3.9) r - 0 Far from the axis, the tangential component of the velocity equals the experimental expression discussed in Chapter 2 : u - KrN, 3(2) 5 r s m. (3.10) 0 The surface defined by (r,z) - (3(2),z) is not known a_priori, but must be calculated for each vortex model studied. If Eqs. (3.4) - (3.6) are each integrated over the cross sectional area «62(2), then the following macroscopic balance equations result for the axial component of the angular momentum, the axial thrust, and the volumetric flow rate 24 de ~2 d3 r-6 sz d3 d6 Ez" - -2«3’ (Ur - Efz' uz)puz - rrz + d—z p (3.12) r-6 dQ a? a; - -2«3‘ ur - E uz (3.13) r-6 In the above equations, P a po - pgz - p(r,2), (3.14) 3' Tz a 21rI [pru0]uzrdr , (3.15) 0 3 F2 . 2x] [,mz2 - P]rdr , (3.16) 0 3' Q s 211} uzrdr . (3.17) 0 T2 represents the axial component of the macroscopic torque on the cross section «3'2 induced by the swirling flow; Fz is the axial component of the average force acting on «6'2; and, Q is the volumetric flow rate across this area. The above equations show what properties of the flow acting on the surface (F(z),z) determine the variation of T2, F2, and Q in the axial direction. 25 Two models of a viscous vortex are studied. Model I, previously analyzed by Long [1961] and Foster (see esp., Burggraf and Foster [1977]). is defined by Eqs. (3.3) - (3.6), Eqs. (3.7) - (3.9), Eq. (3.10) with N - -1, and the following two auxiliary conditions lim F < a, finite macroscopic force; (3.18) 3 4 o 2 lim P - 0, hydrostatics at infinity. (3.19) r-vao A sufficient condition for Eq. (3.18) is puzz - P, for 6a(z) s r s a, z 2 O. (3.20) Figure 3.1 shows the definition of the coordinate system and the boundary conditions for vortex Model I. As previously discussed in Chapter 1 and 2, many vortex flows are characterized by a reverse (or downward) annular flow far from the axis. In Chapter 7, a vortex is studied which approximates this behavior by assuming that a surface of zero axial velocity surrounds the axis. Mgle 11 accomodates this physical possibility by imposing specific boundary conditions on a finite surface of similarity rather than the asymptotic conditions defined by Eqs. (3.18) - (3.20). Therefore, on,a surface defined by (r,z) - (5*(2),z), which is not known a_priori, Model I; assumes that 1. the axial component of the velocity is zero; and 2. the tangential velocity is given by Eq. (3.10) for -1 s N < 0 and 6*(2) s r s w. 26 It as 2 F - 2x I (pU -P)rdr < m 2 z / / r - 0 / / 6(2) U - O / a r / / U-O / 9 / / 6U /ar - O / lim P - O, hydrostatic Z / r-om / z 2 0 / / / / 6 (z) < r < m , 2 2 0 / a / / U - K/r / 0 / z / / r Figure 3.1 Coordinate system and physical boundary condition for the vortex Model I 27 Although the velocity and pressure fields for r > 6*(2) are not specified explicitly, we still assume continuity of viscous stresses, * velocity, and pressure across the surface (6 Ufl,z). Ihus,the auxiliary conditions which define Model ll are no - KrN, -1 s N < O, 5*(2) s r s a (3.21) uz(6*.z) - o (3.22) rr9<6* - pK1N'1 (3.23) 102(6*(2),z) - O (3.24) The other components of the viscous stress are related to the radial and axial components of the velocity (see Appendix D). The behaviour of * outer flow (i.e. , r > 6 (2)) must be consistent with the predicted behaviour of the core flow (0 s r s 6*(2), z > 0). Figure 3.2 shows definition of the coordinate system and the boundary conditions vortex M2dsl.ll. Wm Because the three-dimensional flow studied is axisymmetric (see (3.1)), the radial and axial components are closely coupled through continuity equation. A convenient way to satisfy Eq. (3 automatically is to use a stream function to represent ur and uz, l a¢ ur _ _ f 52 (3.25) 1 6w “2 _ + f 52 (3.26) the the for Eq. the .6) r - 0, z > 0 an /8r - 0 Z 28 -1 s N < 0 * U (6 (z),z) - 0 2 * * N-l r (6 (2).2) - #K(N-l)[5 (2)] r9 * r (6 (z),2) - O 62 Figure 3.2 Coordinate system and physical boundary condition for the vortex Model II 29 Boundary conditions on p(r,z) stem from the previous conditions on ur and uz and will be explicitly noted momentarily. Firstcfimerve, however, that the boundary surface at (r,z) - (r,0) and at (r,z) - (m,z) have similar conditions on u9(r,z). This observation suggests the following representations for the three dependent variables 2 P(r.2) - puc (2)M(n) (3.27) ru9(r.2) - 6(2)uc(2)F(n) (3.28) 2 ¢(r.2) - 6 (Z)uc(2)h(n) (3 29) where n - r/6(z) (3.30) In the above expressions, 6(2) and uc(z) represent, resp. , intrinsic length and velocity scales. These are determined as part of the solution to a specific vortex model. Inserting Eqs. (3.27) - (3.29) together with Eqs. (3.26) and (3.27) into Eqs. (3.3) - (3.5) yields F .. -,,3Mv (3.31) ath' + azh’F - nF" - F' (3.32) I I 2 I I! l -a1hG + a3h G - 2a3nM - can M + nG + G (3.33) where uz(r,z) 1 G - - -— 3.34 uc(z> n dn ( ) and 3O 35..) a4udz The coefficients a1, a2, a3, and a4 must be constants. a's are not independent inasmuch as and Therefore, a necessary condition for similarity is that 2 d uc6 d; ( y ) - constant, and uc6 d6 T 32- - constant. It follows directly from (3.35) and (3.36) that 1 uc a 6N and 6 a (z)N+2 However, all the (3.35) (3.36) The physical constants K and u can be used to scale uc(z) and 6(2). Thus, 31 uc - K6 , (3.37) and V2 l/(N+2) 6 - ( K ) . (3.38) With these results, the a - coeffients become a1 - 1 N+l “2"N'T-2 N “3'311—2 1 “a'NTz' Eqs. (3.31) - (3.33) can now be written as 2 F M' - - 3 (3.39) q N+1 97F -F+Fh-fi:§Fh (3.40) d 1 2 2(N+1) 2 E[flG-mflM+hG]-—m—[G-M]n (3.41) h'- nG . (3.42) When N - -1, the set of equations given above reduce to a model previously studied by Long [1961] and reexamined in Chapter 6. The boundary conditions for the sixth order system will be presented in Section (3.4) after the asymptotic behaviour of the solution has been developed. Meanwhile, Figure 3.3 illustrates the shape of the surfaces of constant r; in the r,z - plane. The dependent variables M(n), F07), and h(n) are constants on these surfaces. Note that for the special 32 Z A N - -1 zocr n < n 1 2 r N - -0.5 3/2 zocr n < n 1 2 >r 2+N in general : 2 a r Figure 3.3 Surfaces of Similarity ( constant value of n - r/6(2) ) 33 case N - -1, the similarity surfaces form a family of cones. For -1 < N < 0, the surfaces have a parabolic-like shape. It follows from Eqs. (3.37) and (3.38) that for a fixed value of n, the dependent variables (see Eqs. (3.27) - (3.29)), depend on the axial coordinate as follows P(r,z) a (2)2N/(N+2) ruo(r Z) a (z)(N+l)/(N+2) ¢(r.2) « 2 As illustrated in Figure 3.4, the stream function is proportional to 2 for -l S N < 0. This result implies that the entrainment rate, defined. by Eq. (3.13), is constant on surfaces of constant n inasmuch as 6 a. Q a 2x I uzrdr 0 68. 1 6¢ - 2x I E 5? rdr 0 2«¢(6a(2).2) 22u2h(n) This results by letting ¢(0,2) - 0 which implies that h(0) - 0. Figure 3.4 also shows that the axial component of the angular momentum, ru is constant on surfaces of similarity for N - -1 only. 0’ However, for -l S N < 0, the pressure field is unbounded for 2 -' O on surfaces of constant a. By inserting Eq. (3.29) into Eqs.(3.25) and (3.27), it follows that 34 N - -l, n - constant P ru W 0 4} A constant 0 2 0 Negative pressure N - -0.75, n - constant P ru W 0 0 1/5 0 a 2 0 2 0 z 0 Figure 3.4 Qualitative behavior of the dependent variables on the similarity surfaces 35 N d6 h ur - K6 E [h - (N+2);7-] (3.44) h! u - K6N — (3.45) z n Thus, for constant values of n, Eqs. (3.44) and (3.45) show that u « z'l/(N+2) and r z a zN/(N+2) As 2 -> 0 for constant 97, both the radial and axial components of the velocity become unbounded; however, as 2 e on, these components decay to zero. Both vortex models studied in Chapters 6 and 7 are subject to the same conditions on the axis. The physical boundary conditions given by Eqs. (3.7) - (3.9) imply that the similarity functions defined by Eqs. (3.27) - (3.29) must satisfy the following conditions ' h lim [ h' - (N+2) - ] - o (3.46) «4 O ’7 n F lim [ - ] - 0 (3.47) n -' 0 " hi! hi lim [— - — ] - o (3.48) 2 n -* 0 '7 0 These results, and h(0) - 0, will restrict the behaviour of the swirling flow near the centerline of the vortex and will be explored further in Chapter 4. 3 3 t c e av our 36 The asymptotic condition on the tangential velocity expressed by Eq. (3.10) implies that F(n), defined by Eq. (3.28), has the following behaviour for large values of n 6a(2) N+1 n < n < m . (3.49) “Fa"? '37)" a" - Thus, for N - -1, Fa - 1 and the axial component of the angular momentum is bounded; however, for -1 < N s 0, Fa becomes unbounded for n 4 m. Because P(w,2) - 0, M(m) - 0 also (see Eqs. (3.51) and (3.27)). Therefore, by integrating Eq. (3.31) with F - Fa, it follows that 2N V M 4 Ma - (TEN) , "a S n S m. (3.50) If Eq. (3.20) holds, then 2 Ga - Ma , "a S n S w . (3.51) Thus, h ' N a n G -—- ’ S S 3.52 a n j'Tifi na 0 m ( ) Eq. (3.52) can be integrated to give flN+2 h + h - + 1 , n s n S m . (3.53) a (N+2) J -2N a With Ga2 - Ma’ Eq. (3.41) reduces to 37 ' 1 2 2 "Ga - N+2 0 Ga + h C - constant. (3.54) aa Inserting Eqs. (3.52) and (3.53) into Eq. (3.54) yields N n (1 + N ) -— - constant . (3.55) ./ ~2N This equation is satisfied by setting the arbitrary constant of integration to zero and 1 - -N. Thus, Eq. (3.53) becomes N+2 H h a - ‘ N , n S r, S m 0 (3°56) (N+2) J -2N a For N - -1, Eq. (3.40) is balanced. This is not suprising because the free vortex (i.e., u - K/r) is an exact solution to Eq. (3.4). 0 However, because the empirical swirl velocity employed in this study (see Eq. (3.10)) is not a solution to Eq. (3.4), the asymptotic functions given by Eqs. (3.49), (3.50), and (3.56) only satisfy the radial and axial components of the boundary layer equations and the continuity equation. If F3 and ha are introduced into Eq. (3.40), then N . N+l F h - "Fa + Fa - Fa ha] - (N+l) r; (3.57) [17:2 aa This result shows that the tangential balance of linear momentum is satisfied for N - -1 only. Thus, similarity solutions on the unbounded M213. which simultaneously satisfy all three component equations of the momentum balance and Eq. (3.10) with -l < N < 0, do not exist. 3 4 ou d V lue P o em tudied ‘ '1 38 The physical boundary conditions given by Eqs. (3.7) - (3.9) require the following restrictions on the similarity functions at n - 0 (also see Eqs. (3.46) - (3.48)) h'(0) - 0 (3.58) F'(0) - 0 (3.59) h"'(0) - 0 (3.60) Moreover, Eqs. (3.28) and (3.43) also require F(0) - 0 (3.61) h(0) - 0 (3.62) These conditions are satisfied by both vortex Models studied. Eqs. ((3.49), (3.50), and (3.56) with N - -1 define the asymptotic behaviour of a vortex with bounded circulation (Model I). Eqs. (3.39) - (3u42) with.N - -l govern the behaviour of Model I for 0 s n s "a' The constant "a is calculated as part of the solution. Model II also satisfies Eqs. (3.58) - (3.62) and the differential equations (3.39) - (3.42) for 0 S r) s 17*: However, as previously mentioned, similarity solutions on the unbounded domain for -1 < N < 0 do not exist (see Eq. (3.57)). Eqs. (3.21) - (3.24), which completes the mathematical definition of Model II, require the following restrictions on the similarity functions at n - 0* - 6*(2)/6(z) N+1 11' * F fl 39 Eq. (3.65) stems from either Eqs. (3.23) or (3.24). This can be seen by first writing the stress components in terms of the similarity function F(n) as follows u a 9 N-l 1 dF fro - pr 5; ( E_ ) - ”K6 -;§ [0 a; - 2F] 3 N-l d6 1 dF 702 - fl 5; (U0) - - 4K5 3; —;— [n a; - (N+1)F] :1- The above two expressions apply for 0 .<_ r .<. 6 (2). Eq. (3.65) follows directly by setting I: - 6*(2) and using the continuity conditions given by Eqs. (3.23) and (3.24). Table 3.1 summarizes the basic elements of the two vortex models studied and Table 3.2 lists the properties calculated for each model. The two models are qualitatively different inasmuch as the macroscopic axial thrust induced by the flow for Model I is constant (i.e. , 6uc - K for N - -1) whereas this property varies with axial position for Model II (see Chapter 6 and 7). 40 Table 3.1 Mathematical models Differential Domain Conditions Edge Equations at n - 0 Conditions Model I Eqs. n s n s w Eqs. a Vortex Flows ( 3.40 ) Infinite ( 3.59 ) F - l with I O S n S m I Constant ( 3.43 ) M - l/(2n) ( 3.63 ) Circulation with N - —l h - n/J—E + l * Model 11 Eqs. 0 - n Eqs. Vortex Flows ( 3.40 ) Finite * N+1 (3.59) F-(n) with I 0 S n S n I Variable ( 3.43 ) h' - 0 ( 3.63 ) Circulation -1 S N < 0 * N+1 F'-(N+1)(fl ) 41 Table 3.2 Physical properties calculated for the two vortex models Property Equations Comment Similarity scales: 1 v2 -—- N - -1 Length 6 - ( T< )N+2 Model I N -1 S N < 0, Velocity Uc - K6 Model II Angular on - 6U F c momentum Radial 2 n ' r/6(z) pressure P - pUc M difference Stream 2 function W - 6 Uch O S n S "a’ Tangential U _ U P Model I velocity 0 c n h! Axial * velocity U2 - Uc n 0 S n S n ’ Model-II d6 h Radial , _ velocity Ur - Ucdz (h ' (N+2)n ) Fz n 2 ~ Axial thrust - I (G - M)ndn n - n , 2«p(6U )2 a c 0 Model I T2 0 N+2 ~ ~N+1 Axial torque 2 - I FGndn - 2NI3 [h(n) + (N-l)]n 22p6(6U ) c 0 Q ’7 ,_ ,_ ,. Flow rate 2 - I Gndn - h(n) n - n , 226 U Model II c O a 5 2 p(6,z) - p(o,2) F ~ Pressure 2 - I -§ndn - M(O) - M(n) drop [)0c 0 n CHAPTER 4 SOLUTION BEHAVIOR NEAR THE AXIS 4 e a utio The solution to Eqs. (3.39) - (3.42) for small values of r) can be developed using Taylor series representations for M, F, and h. In general, 1 2 M(n) - M(O) + M'(0)n + 2? M"(O)n + ---- (4.1) 1 2 Mn) - 1121 + may: + g F"(0)n + (4.2) 1 2 hm - m + 11121» + 2—, h"(0)n + (4.3) The coefficients underlined in Eqs. (4.2) and (4.3) are zero because Cd? boundary conditions (3.58), (3.59), (3.61), and (3.62). All of due coefficients multiplying odd powers of the independent variable are zero because of local symmetry about n - 0. Because Eqs. (3.39) - (3.42) are invariant if n -* -n. M(n) - M(-n); F07) - F(-n); and, Mn) - h(-n). Thus, the power series representations through fourth order can be expressed as 2 4 MO - c + cln + czn + ---- (4.4) b 2 4 F0 - 2 n + bln + ---- (4.5) 42 43 a 2 4 ho - 2 n + aln + ---- (4.6) The coefficients a1, b1, c1, c2, --- can be related to a, b, and c by inserting Eqs. (4.4) - (4.6) into Eqs. (3.39) - (3.42) and equating like forms of the independent variable. This gives 2 N a a1 - E?N:§) ( 5— - c ) (4.7) ab b1 " ' m (“-8) b2 °1 ' ' T (“'9) ab2 c2 - 64(Nli) (4.10) The general solution near the axis depends on the empirical index N and the coefficient a, b, and c. The asymptotic behavior for 2) >> 0 (see Model 1, Chapter 3) will determine three of the four coefficients (a, b, c, na); the remaining coefficient can be used to parameterize the behaviour of the vortex (see Chapter 6, 7). Figure 4.1 portrays the mathematical meaning of these coefficients and provides the interesting a_priori observation that as 'b' increases the angular momentum changes from zero to its asymptotic value over smaller spatial domains. Thus, the vortex becomes more 'concentrated' about the axis. In Chapters 6 - 7, the limiting behavior of the solution for small values of n will be connected to the asymptotic behavior for the two vortex models defined previously (see Table 3.1). 44 b2fl2 M"f M-c-._8__ c \ \\ 1 \ ‘\ \ M - ___ \ \\ a 2 2 \ \ / ’7 l \ l i 0 V ’i n J 8c /b 1.0” /| 2 fl MID / / ./ a > o///:/ ' l . o-<\ 7 angular momentum / l h - o n 1'0 """""" 1] WKFa - 1 stream function Figure 4.1 Geometric meaning Of the expansion coefficients 45 4 ow e imes The local behavior of the axial and radial velocity near the 'axis' (i.e., small values of n) is determined by the coefficient 'a'. Because (see Eq. (4.6)) N a2 2 4 Ito-“2'17 +W(—2—-C)n +---- (4.11) it follows that (see Table (3.2)) N 2 V2 §:§ N a 2 uz(r,2) - K ( fi— ) [a + §?§;§)( 2_ - C)n + "" ] (4.12) and l 2 V2 - N12 N (2-N)N a 3 ur(r’Z)-V(K_) [-m)an+—-—2-(2—-c)n+---] 8(N+2) (4.13) Eq. (4.12) shows that the transition from forward flow (i.e., uz(0,z) > 0) to backward, or reverse, flow (i.e., uz(0,2) < 0) within the vortex is determined by the sign of the coefficient 'a'. Therefore, transition from forward to reverse flow on the axis of a vortex corresponds to a - 0. Likewise, Eq. (4.13) shows that for -l S N < 0 forward flow around the axis is accompanied by an outward radial flow (i.e., ur > 0); and 46 reverse flow around the axis is accompanied by an inward radial flow (i.e., ur < 0) Whereas - ¢>< a < + m, the coefficient b on the other hand is always positive. This follows from Eq. (4.5) and Eq. (3.28) which shows that N v2 -—— b u,(r.z> - K < K— >N+2 [ 5 n + ---1 (4.14) Positive values of 'b’ correspond to counter clockwise rotation about the axis; negative values of 'b' correspond to clockwise rotations. .Because Eqs. (3.3) and (3.4) are invariant to a sign change in u only 0. counter clockwise rotations are considered. The empirical parameter K in Eq. (1.1) is positive. As noted earlier, large values of 'b' will give a 'concentrated' vortex motion about the axis. Because of the physical interpretation of 'b' as a measure of the frequency of the forced vortex motion about the axis and because of its clear qualitative effect on the flow (see Figure 4.1), the global solution will be parameterized by this dimensionless group. Note that due 2( 6r ) r-0 b ' uz (N-1)/(N+2) ' (4'15) K(K) 47 which is a relatively easy property to measure experimentally (see Chapter 9). This alone justifies its use in the theoretical development as an independent parameter. The parameter 'c', which represents the dimensionless 'excess' pressure on the axis (see Table 3.2), is always positive for Model I because (see Eq. (3.39)) c - M(O) - M(fi) + an > o (4.16) OV—xél é | '11 u N For Model II, computer experiment for the range of b studied in this research (see Chapter 7) shows c is positive for any value of N. For small values of n, 2N 2 ”2 N12 b2 2 abz 4 P(r,z) - PK ( K. ) [ C ' g" n + ng§137 n + ---' 1 (4.17) Note that 32? < 0 , ar2 r - 0 so for a fixed cross section (constant 2), the pressure on the axis always corresponds to a local minimum (maximum for P). Because c > 0, P(0,z) is positive and always decreases with increasing 2. Therefore, N-2 a (~2N) VZ N—+2’ 5; (p + pgz) - CKp _N:§_ ( K_ ) > 0 . (4.18) r - 0 48 Eq. (4.18) shows that forward flow on the axis is always against an O O o adverse pressure gradient. There will always be a region, 0 < 2 < 2 , near the singularity point for which the pressure is less than zero. 20 can be calculated from Eq. (3.27) by replacing uc with Eq. (3.37) and letting p(0,2°) - 0. Thus, negative values of p(0,2) occur on the axis 0 for 2 < z , where (4.19) Because -1 s N < 0, it follows from Eq. (4.19) that 2° increases as c increases inasmuch as p0 >> pgzo. For N - -l, 20 a co'5 whereas for N - -0.75, z0 a c5/6. For N - -1, Eq. (4.19) reduces to P 0.5 K 0.5 z° - K ( o_'_6) ; (c) (4.20) P ' P82 For jet-like, forward flow with c z 0.1, z° is about 0.08 m for a gas vortex with a backpressure (i.e., po - pgzo z p0) of 14.7 psi (1.01x106 g—cmz/secz), K z 3050 cmZ/sec(306 Boysan at 81..1982 and Table 8.5), p z 0.0013 g/cm3, and K/u z 231. However, for a liquid vortex with N - -0.75 (see Dabir, 1983 and Table 8.5), c - 0.1, p° = 14.7 psi, K z 270 -(1+N) cm1-N/s (see Table 8.2), p - 1.0 g/cm3, and K/u 342 cm the R onset of negative pressures occurs at 2° z 0.056 m. 49 The behaviour of the axial velocity near the axis depends on the relative values of 'a’ and 'c'. Figure 4.2 illustrates four flow patterns consistent with Eq. (4.12). With the exception of Flow IV, all of these features have been observed experimentally (see Chapter 2) and calculated theoretically (see Chapter 6, 7). Table 4.1 gives the conditions for transition between these flow regimes. The qualitative behaviour for the radial flow follows directly from Eq. (4.13) and is also noted for each transition. The sign of the 'excess' mechanical energy, E, of the fluid on the axis relates directly to the flow structure illustrated in Figure 4.2. The value of E at r - 0 can be calculated as follows 1 E - 5 ptuz(0.z)12 + [p(0.2) - 1 (4.21) By using the results listed in Table 3.2, the above expression can be written as 2 2 a E-puc(2—--c). (4.22) Thus, forward, jet-like flows (Type I) as well as reverse, undulating flows (Type IV) correspond to m value of the 'excess' mechanical energy on the axis; negatlve values of E give a wake-like behaviour which includes both undulated (Type II) and reverse, jet-like flows (Type III). Transition between Type I and II flows correspond to E - 0. 4 a nce 0 e is A mechanical energy balance consistent with the approximate boundary layer equations, defined by Eqs. (3.3) - (3.6), has the Notes huwumHHEHm 0:» an poaoaam macaw xouuo> mo mommmao N.¢ enamHm 50 3OHm HNHUNM . o v N N\ o v o O o N\ o A o u o cumacH N N N N N o I u on o V D o o v 46\ so o A om\ an o v Nm\ am N N u N N N 3osm : D : D o I H on o I u on o I u on omuo>om >H HHH son Hawtmm o A m N\ m A o u o N\ a V o w o UNN3uso N A . N m N N N OIHUNOAD o o A 66\ pm o v 46\ an o A um\ Dm N N N N N N 3o~m f: 3: o I u um o I u on o I u on pumsuom HH H HON>m£om couaaspca HOH>N£om oxNH-NOH 51 Table 4.1 Criteria for transition between flow regimes Transition 1 Criteria Comment Forward, 2 a I ta II a > 0, c - —§- Outward Flow II - III a - 0, c 2 O Inward Flow Backward, 2 a III t» IV a < 0, c - -§- Inward Flow Impossible for IV +» I a - 0, c - 0 Nonzero angular momentum ; see Eq. (4.16). 52 interesting feature that the viscous dissipation is zero on the axis. Thus, the change in E (see Eq. (4.21)) along the axis is governed by the following equation DE 6E(0,2) 1 6 DE - uz(0,z) -——-— - lim - -— r(rr6u6 + r 62 r 6r u ) (4'23) r-O r g 0 r2 2 The term on the right-handmside of Eq. (4n23) represents the rateIof work done on the fluid per unit volume by viscous force. By using the definitions of Tre and Trz (see Appendix D) as well as the boundary conditions on the axis, the above expression simplifies to DE Dt - A ( 2 u ) (4.24) r-O Eq. (4.24) and the results listed in Table 3.2 yield 3 —2N put a —za<2—‘C> (“'25) DE E r-O where uc is defined by Eq. (3.37). For N - -1, Eq. (4.25) reduces to an earlier result reported by Long [1961] 6 DE pK a2 IT: ‘234a(2—'°) r-0 [’2 Eqs. (4.22) and (4.25) can be combined with the conclusion that on the axis 53 DE -2N uc E In - N—+2- ) ‘Z—' aE(O,z) . (4-26) r-O Figure 4.3 illustrates the relationship between the direction of flow on the axis (a < 0, backward; a > 0, forward), the mechanical energy of the fluid on the axis, and the rate of change of mechanical energy due to convection. Thus, the four types of structures shown in Figure 4.2 can also be classified by using energy considerations. Figure 4.3 shows that for Type I and Type III vortices the fluid on the axis moves faster than the surrounding fluid. In this case, energy is transfered.to the slower moving fluid by viscous work. On the other hand, Type II and Type IV vortices have the feature that the fluid on the axis moves slower than the sourrounding fluid so, hatflds case, energy is transfered to the axis by viscous work. Transition between the various flow regimes is characterized by DE FE - O . (4.26) r-O +. ,/____:___ 1} + X v, l a /2 - c I : a /2 - c ' I ‘ l I l l 0 DE/Dt I l I l I + l l I I ' > Type IV Type I U A A A j) :2; T? Figure 4.3 Classification of vortex flows based on the mechanical energy balance CHAPTER 5 SOLUTION METHODOLOGY 5 a mat o o e 0 First rder E uations Eqs. (3.39) - (3.42) can be rewritten as a system of first order ordinary differential equations by introducing the following dependent variables : Y1 - M; Y2 - h; Y - G; Y - F'. The '- G'; Y - F; and, Y 3 4 5 6 resulting equations are listed in Table 5.1 along with the power series representations of Yi(") for small values of n (Eqs. (5.1) - (5.12)). The equations can be expressed more concisely using vector notations: Y' - £(Xn7). (5.13) Solutions to Eq. (5.13) depend on the three parameters a,1>,and c discussed in Chapter 4. These coefficients determine the trajectory of 1(a) near n - O inasmuch as 1(0) - cgl + ag3 , and (5.14a) d! N a2 "limo d_'l - m (2— - c)_e_4 + bg6 (5.14b) gi denotes the igh unit base vector in a 6 - dimensional Euclidean space. Eq. (5.14a) follows directly from Eqs. (4mQ)-(4u6) and the 55 56 e : ANN.mV ...+m:Nn¢+enno> Ao.mv own» mwm + - II I .o» e» .a N+2 ost m» STE ...++:Hn+nemnmw 3.3 m.» I .m» m» m ewwwm ma s is.mc ANs - Nmsv mnm N + - N+2 quw e : AoN.mv ...+eNmmqu um: mwm - m» - N» mwm + In - u .q» s» .o m N N N s N w w .z Ao.mv ...+N=Nme+mnmw Am.mv e» n .mw my o uNwm m¢ Am.mv ...++er+N=anN> AN.mv m»: I .N» N» : aowwwmm : Au.mv ...++:No+NsNo+onN> As.mv IMI - I.N> H» z .musmmmum m» N mwxm can uaonm m50wumado manmwum> manmuum> scamcmaxo mmwuom Hmauaoummmaa 3oz zuuumawsam anewumsvo Hocuo umufiw mo amumhm a ma coauoa xouuo> on» no coaumuammmuamm H.n canoe 57 definition of C(n) (see Eq. (3.34)). The parameter 'a' and 'c' must be calculated as part of the solution; however, 'b’ can be specified and, as indicated by Eq. (5.14b), determines the initial trajectory of Y6(n). The asymptotic behavior of XM) for large values of r; follows directly from the analysis developed in Section 3.3 and the definition of 1(a). These specific results are listed in Table 5.2 and apply to the vortex on an unbounded domain (Model I). The bounded vortex, * defined by Table 3.2, satisfies the conditions at n shown in Table 5.2 (Model II). The set of first order equations were used to calculate the components of 1(a) and the integral properties for each model over their * respective domains once 'a', 'c', and "a (n for Model II) were found (see Section 5.2). Eq. (5.13) was integrated numerically using the computer program listed in Appendix B. For Model I, the stream function far from the axis is given by 7! h»h -—-+1, 97 a JF2 a IA :3 IA 8 The numerical solution approaches this theoretical result, but eventually runs "parallel" to it depending on the precision in determining the parameters 'a' and ’c'. This offset error, which occurs for n z "a’ is more apparent with h than the other similarity variables. This can be understood from a perturbation analysis. If Xd represents a small perturbation from the asymptotic solution, then 58 Table 5.2 Boundary conditions on the vortex models Variable Condition Model I Model II 1 Y -M c --- 1 2"2 1; Y2 - h 0 + l --- 2 1 Y - G a O 3 2 n 1 Y - G' O - --- 4 2 n2 *N+l YS-F 0 l (n) * N Y - F' O 0 (N+l)(n) 59 X - X + Y . (5.15) Inserting this decomposition into Eq. (5.13) and neglecting nonlinear terms yields a linear equation for id : g ‘2 ) , Y (5.16) dn - ( 6 The six first - order, linear differential equations represented by Eq. (5.16) are led 2 an“ ' ' ‘3st (5.17a> n dY2d dn ' ”st (5.17b) dY3d an ' Y4d (5.17c) dY4d 1 2 J‘E 1 2 ———--(—+-)Y -—Y + Y +2Y -—-Y d 4d 3d 3 2d d 0 2 n n j’? n l "2 5d (5.17d) dYSd ‘53“ ' Y6d (5.17e) dst 1 —d—'l—- - EY6d (5.17f) The above six equations are not fully coupled. For instance, it follows directly from Eq. (5.17f) that 60 -n//5 Y6d °‘ 9 ' so Y6 approaches itsasymptotbcbehavior exponentially. This is also the case for Y5d and Yld' However, Eqs. (5.1flx>-(5.l7d) have a different behavior. Because Y1d and Y5d decay exponentially, the general solution for Y2d - Y4d can be expressed as q Y2c1""" q-2 Y3d « qn , and -3 Yhd « Q(Q'2)nq - Substituing Y2 - Y d. into Eq. (5.17d) and neglecting the term which 4d decays exponentially, implies that q - l. Therefore,timeperturbation variables Y2d - Y4d approach their asymptotic behavior algebraically : Y2c1°"7 3d 4d QIH alt-a no 61 and Y on the other remains proportional to n. Y 4d’ As n v w, Y2 3d d hand, decay to zero. Because Y2d a n, the offset error between Y2 and Y2a for large values of n is determined by the accuracy of the solution defined by Ei < e, i - 3,5,6 (see Eq. (5.29)). For instance, if e - 0.0001, numerical calculations show that Y2d is less than O‘OOSYZa; however, if e - 0.001, Y is about 0.04 Y The 2d 2a' difference between these results illustrates that the control of e is very important. In order to have an accuracy of Y2 within O.99Y2a for n 2 "a’ 6 must be 0.0001. e ts Th ot n et od Although the two vortex models studied have different conditions surrounding their cores, the mathematical structure of each problem and the solution strategy have a common methodology. Both models are represented.as a.system of differential equations on a finite domian E. * The actual values of 'a', 'c', andna (n for Model II) for a given value of ’b' and N were found by rescaling the two - point boundary value problem and using a shooting method. The rescaled problem is d; A gg-£(§.€;N.5).0<€Sl where 62 {a ,ModelI n- a a: n , Model II m I ml. Table 5.3 defines the component equations and boundary conditions. Only three of the six conditions listed under Model I for 5 - l are independent. The conditions at 6 - 0 are the same for both vortex models. The solution of both vortex models involves a three dimensional search for vectors with components (a,c,'fi). The above two - point boundary value problem was solved by using a "shooting" method. Figure 5.1 illustrates the four elements of the algorithm and a brief discussion of each component follows. The Fortran program which implements the algorithm is listed in Appendix B. A. Initial Guess The program designed here has been used successfully for a wide range of initial guesses : a0, co, and 3°. These parameters satisfy -m ImN e N N+2 e . m z m . N z N N N . 0 Im ... =N =me e m In N N NNN me N N +qwz-mI n+Nwz-NIa N N+zNIV\ N N w . ANN.mV N w + IMI N+M + eNNN mN zNI N --- Am:m\V\N - m wwm - NN mwm. m - Iw - I sN ...+memNIqN e»: I «N N z zN N e . I N a o N emxv\N NoN.mv «NINN ...+szmNme+aImN mwnmN m a . m N ... N N N N N III t ~u I a.“ ...-I r I N \NN+m\\ v NmN my Nw N +¢NNI +Nwo NI\ s N --- A meNv\N ANN.mv Imw =-INN ...+ w N.NNJSINN quNN N m zNI . s NI N N NN Nunez N Nope: NN Noooz.I: I m ”N Nmooz.¢s I w o I w oNpmNum> H I w um mcowuwpcoo %prc:on HVon mcoHumsuo HoauaoummmNa usonu cowmcmaxm vonomom mampos xmuuo> Umamommu osu Mo GofiuMDCGmoummm n.n mHan 64 Specify -1, Model I N, b N - -1 s N < 0, Model II . ~ ~ "a’ Model I A. Guess a, c, n n - * n , Model II New B. Calculate 2(0) 2 _______. dz a.C.n and(ag)o J. B. Integrate d; A ~ D. Three 5? - £(Z.€;N,fl) dimensional search 0 < f s 1 No C. Check boundary conditions at 5 - l. Satisfied ? 1 Yes Figure 5.1 Solution strategy based on a shooting method 65 a0, b, and c0 for N - -1 follows by assuming Mo( JZ/b) z Ma(J2/b). The result is c° - b/2 - ao/16. (5.24) Eq. (5.24) is consistent with the general behavior of (a,b,c) for large values of b. Numerical studies indicate that c is roughly proportional to b. Therefore, with c° - b/3, (5.25) Eq. (5.24) gives a° - 8b/3. (5.26) Unfortunately, 5° z JZ/b is not a good initial guess for 5 and is replaced by the empirical result 5° - [8./2/b1'N (5.27) Eqs. (5.25) - (5.27) provide good initial guesses for Models I and II at large values of b (say b > 0.10) for any N. Figure 5.2 illustrates how the initial guess of (a,c,3) affects the convergence of the algorithm. The initial guess, denoted by point 1 in Figure 5.2, is obtained from Eqs. (5.25) - (5.27). The solid line shows 66 how the algorithm searches for the solution. After six iterations, the value of (a,c,n*) is very close to the solution. It took another 6 iterations to satisfy the criteria (6 5 0.0001) and reach the solution (point 12). The initial points denoted by (*) will not converge to the solution. These points give a rough idea of the size of the domain of convergence for b - 0.3. Although the projected values of points 4 and 5 on Figure 5.2 appear close to a nonconvergent point, the corresponding value of c for the two points, as indicated, are different. §:-?¥§129§9§2-£9§-;£§2 For small values of 6, the Taylor series expansion defines the solution in the vicinity of f - 0. The differential equations are integrated numerically from 6 - 0.001 to l by using the DGEAR subroutine (IMSL,1984). The required input conditions for the algorithm are listed in Appendix B. The DGEAR subroutine solves the set of differential equations by the ADAMS predictor - corrector method. (see Gear, 1971; Lapidus, 1971). The integration only reguires values of 210.001) and i(0.001) to start. The algirithm continues until the end point 5 - 1 is reached. DGEAR subroutine is also designed to solve "stiff" differential equations. In this purpose, it is more powerful than other algorithm. The effect of step size M on the behavior of M, F, and h is important. It was found that these three functions do not change much (three significant figures) between using 1000 steps (A5 - 0.001) and 100 steps (A5 - 0.01) over the domain 0 .<. 5 s 1 for (a,b,cfli) - 7a. 23.00 21.00 WM 0 II 9 F 19.00 17.00 LIIL;LLJIIIliLLlLllLlllJJl 67 C a «088‘ 7-l2 (.aopsl‘f {caafiq‘n 2 * c 30.076 3 Cg°.. came?“ A, Solution (a,c,q) =- (0.893,0.0881,21.3) 46, No convergence 0, Initial guess (a°,c°,'7°) - (0.8.0.1207) C=o.1 15000 T T T T FT I VT T T TTTT T I l I I T TTT ITT T T T] 0.7 0.8 0.9 1.0 Figure 5.2 Domain of convergence for the three dimensional search with b = 0.3 ( Model ll, N =- —1.0) 68 (0.893,0.3,0.0881,21.3). A6 - 0.001 gave satisfactory results for all values of the parameters studied. 9.-Qsésesée-§9§-999Y9§59999 Let ZA(£) represent an approximate numerical solution to the boundary value problem. The accuracy of the solution is determined by how close the boundary conditions at 6 - l are satisfied. If 3, denotes a three dimensional vector whose components represent the difference between the desired boundary conditions at 6 - 1 and the approximate results based on a given value of (a,c,;7'), then a solution to the two point boundary value problem occurs when 151(3) < e for i - 1,2,3 (5.28) where e is an accuracy parameter. The components of 5 are defined by E |23A(1> - z3<1>| 5 N [ a; ] - IzSA(1) - 25(1)| (5.29) Ea |26A(1) - 26(1)| 3 depends on a three dimensional vector x_, where (5.30) Ix l x can I atom 69 In this research, the three boundary conditions 23(1), 25(1) and 26(1) are used for both Models I and II. The results showed that once these three boundary conditions are satisfied for Model I, the other asymptotic conditions are automatically satisfied. For Model I, three boundary conditions will uniquely determine all six dependent variables (see Table 5.3). But these three boundary conditions cannot be arbitrarily chosen from the six asymptotic functions. Three groups are separated directly by Eqs. (5.18) - (5.23). The first group contains only 21 and 25 (Eq. (5.18)). The second group contains 22, 23, and 24 only (Eqs. (5.19) - (5.21)) since 21 and 25 in Eq. (5.21) will automatically drop out at 5 - 1 and N - -1. The last group includes 26, 25, Z3, and 22 (Eqs. (5.22) and (5.23)). Therefore, three different boundary conditions must be chosen from these three groups. One is chosen from either 21(1) or 25(1), the other is chosen among 22(1), 23(1), and 24(1), and the last one is 26(1) since 26 only appears in the last group. In this research, 23(1), 25(1), and 26(1) are chosen for Model I in correspondence with Model II. For Model I, 25 and 26 become asymptotic much sooner (i.e., small values of n) than 23. Figure 5.3 illustrates how the accuracy parameter, c, affects the soluthnn (a,c,m*), for Model II. For large values of e (e z 0.1 - 0.002), La,c,n*) changes significantly. When 6 is small enough ( 5 0.001), the values of (a,c,n*) become stable. In order to have three significant figure accuracy for a, c, and "a’ 6 must be less than 0.0001. 1.. O J LJLALI LII Ll 1.00 L L L L L L L]. 70 initial guess : (0° ,c° A") = (0.8.0.1207) Eq. (5.19) Solution : (a,c,‘7) = (0.893,0.0881,20.7) 0.90—4 a n a J. C no 0.80 T TTTTITT T TfiT—TII T j VTTII 10.0 100.0 1000.0 1.0E+04 1/6 Figure 5.3 The effect of accuracy parameter 6 on the behavior of the solution for b = 0.3 (Model ll, N = —‘l.0) 71 P;-¥9Ebeé-§9§-9992E§95-Se.9.§2 If the criteria in step C are not satisfied, the program will call ZSCNT subroutine in IMSL (see IMSL, 1984). This subroutine uses a secant method to solve a system of non-linear equations (Phillip, 1959), E(3) - Q. When an initial guess is specified, the ZSCNT subroutine uses a point nearby to search for a better solution. Figure 5.4 shows how the next solution "vector" is obtained for a one dimensional problem. If the initial guess of xk.1 is given, the subroutine calculates E(xk'l) and uses an arbitrary point nearby xk to find E(xk) . A new guess for x (i.e., xk+1) is determined from (see Figure 5.4) xkE(xk- l) _ xk- 1E(xk) 2(xk‘1) - sock) k+l x The process is repeated until E(xn) < e. For a system of non-linear algebraic equations, the algorithm is similar but more complex (see Phillip, 19.59). The search for (a,c,;7') continues until the desired accuracy (see Eq. (5.28)) is obtained. 72 Initial guess k- - (x 1.1~: "a’ is independent of b, the "pressure" drop across the vortex (i.e., M(O) - M(na)) increases with b. Thus, coherent vortex.structures require relatively large radial pressure drops. Note also that M z M3 for n z 10, which is significantly smaller than "a' This result was anticipated by the analysis of Section 3.3. Figure 6.3 shows that F obtains its asymptotic value for n z 15, provided b 2 0.03. For large values of b, F z Fa for n << "a also. Thus, if M and F alone were used to control the numerical search (see Section 5.2), then a significant error in "a would occur (see Section 6.3). For the same value of n within the core region, larger values of b yield larger values of F. In other words, the circulation for fixed '1 increases as the rotation around the axis increases (strong vortex) I‘ In §9 g.§_9rd0 - 21rKF(I7;b) 76 0.10 ITITITTTI 1T I I I l I I I I T r I I j l -I I «i l l " l "'l I -I l -l l ‘ —l 0‘08; I N = —1.0 l .l ' q l 3 = ill/K i -I t ... l -I I .1 .3 l ' 0.06" l --1 I l 1 Ma =- —- .. -I I1 213 -I l N l 0.04—l i, ._ ..l ‘l j 1 c1 \ -I .1 ‘ .. I .. \ 0.02-1 \ .. J - '1 b =- .03 . 0000 I T I r T— r r r r T r I T I r r T T r l r r r r 0.0 5.0 10.0 15.0 20.0 25.0 r/d Figure 6.2 The effect of b on the behaviour of M for Model l 77 1.20 I I j I I T I—TTT r T I l I I I l . . I _J. ...--J ,- N = —-1.0 —l 8=Ilz/K I 1 J J J 0'00“ T r T r r r I— r T T I I 1— I T— T I T r T4 0.0 5.0 10.0 15.0 20.0 25.0 r/IS Figure 6.3 The effect of b on the behaviour of F for Model I 78 2000 I I T I r ITI I r I I I I I I I If I I T r I r . j 3l 3 , 1 : / 3 15 0-7 3% ' j N = -1 0 _; .. I j 8=Jz/K 3 J l I ‘1 I 3 -l 10.0-: '1 .3 h =: —+\ / .1 ° :5 ,v .1 -l / 7' : / 3 4' // l .l / :1 ..I / .1 5.0" // 3 u / J / 3 / 3 1 b = .03 . l .l 1’ 3 0.04 - " ----------------------- =1 . l d 'l l 1 T's-O r T r r r T I I r ITT I TT r I T I T r I T I rj 0.0 5.0 10.0 15.0 20.0 25.0 r/d Figure 6.4 The effect of b on the behaviour of h for Model l 79 The behavior of the dimensionless stream function h, which relates directly to the axial and radial components of the velocity, deserves special attention. Figure 6.4 shows that for reverse flow (b - 0.03), h is negative for small values of n. This was expected based on the analysis of Section 4.1 (see Figure 4.1). For b - 0.03, note that h - 0 when I1 z 4. Because 11(7)) also represents the ma; entrainment rate (see Table 3.2), h(4) - 0 means that the local volumetric flow rate across a surface of fixed 2 and 0 S. r s 46(2) is zero. This also implies that uz(r,z) must be zero for some values of r between 0 and 46(2), if b - 0.03. For the other two values of b (b - 0.1, 0.3), h is always positive. For very large values of b (see Figure 6.8), h can overshoot ha for n < "a' For n > "a’ h parallels ha with a very small, and controllable offset. As discussed in Chapter 5, the offset always exists because the allowable error in the numerical search for a, c, and "a is always larger than zero (see Eq. (5.28)). This constant offset error does not affect the velocity components (see Section 5.2). Another result is that h increases as b increases for fixed 7). Thus, a "strong" (i.e., large b) vortex has a larger M volumetric flow rate (fixed a) than a relatively "weak" vortex. However, because a "weak" vortex has a larger value of "a (see Figure 6.1), the global or macroscopic flow rate through a "weak" (i.e., small b) vortex is large compared to the flow rate through a "strong" vortex. 6 Behavior Vel cit Field 80 The tangential velocity profiles for different values of b, shown in Figure 6.5, have a Rankine-type structure with a forced vortex behavior near the axis and a free vortex behavior in the outer region of the core. For r; > "a’ uo- K/r. For large values of b, the region of solid body rotation (i.e. , forced vortex) shrinks. The peak tangential velocity shifts to smaller values of 17 and becomes larger as b increases. If the size of a vortex is defined as the radial position where u is a maximum (see, esp., Donaldson and Sullivan, 1960), then as 0 b increases from 0.03 to 0.3, the size of the vortex decreases by a factor of 2.5, whereas "a only decreases by a factor of 1.4 over this range of b. An interesting observation in Figure 6.5 for b - 0.03 is that the zero Log], entrainment rate (h(4) - 0 in Figure 6.4) occurs within the forced vortex region (see Figure 6.5). Figure 6.6 shows three different axial flow profiles for different values of b. This result is consistent with many experimental observations (Escudier, 1980, 1982; Dabir, 1983). When b increases, the flow changes from a reverse flow on the axis to an undulated flow and, finally, to a jet-like flow. In the flow reversal region, it is observed that uz - 0 at n z 2.5 for b - 0.03. Thus, the fluid moves downward toward the singular point for n < 2.5 and moves upward for 17 > 2.5. This observations also follows directly from Figure 6.4 inasmuch as h'(2.5) - 0. Note also that for b - 0.3, the magnitude of uz near the axis is larger but falls below the axial velocity of the weaker vortices in the outer region. However, all the axial profiles approach K/(f2 r) for n > "a' It follows from Figures 6.5 and 6.6 that the local 81 0.25 I I r I l I I I I l I I I I T I f I I l I I I I J 4 4 CI " 1 -l> .. a.2o—l N = —1.0 _. 'l -1 J 5 = Jz/ K g " Ua =- Ka/Jz ~ 4 .3 J .l, O D .. I. i :3 .1 ...+ '1 '1 4 ~l 0’00 [ I I T T T 1 I I—r III I fifT I I I i I I T r 0.0 5.0 10.0 15.0 20.0 25.0 r/6 Figure 6.5 The effect of b on the tangential velocity profile for Model I Uz/Uc 82 r} r—flri T?IT r) I I I U I fiTTj r1 1....__1._J l l 1% Irrrl’W—TiirrITTITTT—TITTT'T 5.0 10.0 15.0 20.0 r/6 Figure 6.6 The effect of b on the axial velocity profile for Model I 83 peak of uz for b - 0.1 and 0.03 occurs at slightly smaller values of 77 than the peak of uo. The radial velocity profiles in Figure 6.7 exhibit many important features. As predicted in Chapter 4, an outward radial flow (ur > 0) occurs around the axis for forward flow and an inward radial flow (ur < 0) occurs around the axis for reverse flow. For all values of b, a zero radial component of the velocity always exists for a nonzero value of n. As previously discussed in Chapter 2, a surface characterized by ur - 0 is often called a "mantle". Because the flow at infinity must move radially inward toward the core (ur - - v/r for n > ”3), two internal mantles exist in the reverse flow regime due to the inward radial flow near the axis. For the forward flow regime (either undulated or jet- like), however, there is only one mantle. The inner mantle in the reverse flow case appears at very small values of 11 (n - 2.0 for b - 0.03) , and causes the flow within r; < 2.0 to move downward toward the singular point. In forward flow, the size of the mantle decreases as b increases, which implies that the stronger rotation around the axis will force the fluid to move into a much smaller core area. This phenomena is very important for the application developed in Chapter 9. W ' V t As discussed in Chapter 5, the computer algorithm employed in this study used a three dimensional search for a, c, and "a until the asymptotic conditions were satisfied. This was accomplished by controlling the allowable error on 23(1), 25(1) and 26(1) (see Section Ur/(UcdS/dz) 84 1000 I I I I i I I I I i I —T I I T I— I I I I I I I I -l .. -ll .. 0.80-4 N = -1.0 _4 5 = Jz/K ‘ .4 Uc = Kz/Jz .4 0.60-l _. " .4 ‘ 1 0.40—1 ... " 'l 0.20% _l J b = .03 ‘ .4 .1 _4 ,4 .3 1 0.00-l --- —————————————————— _____., d _a 1 . 4 .. “'0.20 r f I I r I I I I F I T— I I I I I I I T I I "T 0.0 5.0 10.0 15.0 20.0 25.0 r/6 Figure 6.7 The effect of b on the radial velocity profile for Model I 85 5.2). In contrast, the solution methodology used by Long [1961] determined the values of a and c by only controlling the asymptotic error of Y1 and Y5 (equivalently, 21(1) and 25(1)). The value of "a was determined by the asymptotic behavior of h, but, as illustrated in Figures 6.2 - 6.4, Y1 and Y5 (M and F) obtain their asymptotic behavior for smaller values of 7) than h. Thus, Long's procedure effectively determines "a inaccurately. Basically, his solution does not satisfy the asymptotic condition on h. Using the tabular values of a, b, and c found by Long (two decimal accuracy), the differential equation for 2(a), defined by Table 5.1, was integrated using the computer algorithm listed in Appendix B. For b - 3.82, Figure 6.8 compares the stream functions calculated using Long's solution and the solution developed in this study. The results are the same for small values of I), but differ significantly for large 7). Although the difference between (a,c) is small (only 1% difference), the behavior of h at large values of n is much different. Because h is very sensitive to the values of a and c, controlling the numerical error of ha is very important. A.similar comparison between the radial velocity profiles calculated using Long's parameter and the set developed here is given in Figure 6.9. The radial velocity for large values of n is very sensitive to h. For small values of n (q < 3.0), bath solutions are the same, but differ significantly when n become larger. However, both solutions show the existence of an internal mantle near the axis. 6 4 as o ' o ties 86 30.0 "I‘fi fTTTrIfirrI'YI’fI J N =- "1.0 b = 3.82 . 5 =Qz/K Long (1961) This Merck l4.5 4 LOns l4. 6 l. o 0.99 5 no. ler4 This research 0.0 - 0.0 5.0 10.0 r/6 15.0 Figure 6.8 Comparison between Model I and Long's vortex for the stream function YYYYYYY 4....L_._L_.ui\ 1 -J._._J.._J....J._.-J~—_J———.L_—L__1_—.L. J 20.0 Ur/(UcdS/dz) 87 1.60 . . r e. . , T i i - . . J 5 = — This 120.. n N 1.0 Long Research q '1 b = 3.82 G [4.6 “155 .4 -ll . 8 =QZ/K (1 Lo 0.995 . 0.80: UC =' Kz/JZ 'ia — l8.4- .. 0.40- ‘ l 4 Long [1961] 0.00“" ------------------------------ ..l l 1 —O,40-:l This research . ( see Table 3.2 ): -0.80-4 _ d '1 -1.20- .. ‘ -l l l _1.60 Tffi1.r.,j,TT,rTIf'Ij 0.0 2.0 4.0 6.0 8.0 10.0 r/6 Figure 6.9 Comparison between Model I and Long's vortex for the radial velocity profile 88 The macroscopic properties discussed in Chapter 3, including axial thrust, axial torque, flow rate, and pressure drop, can also be used to quantify vortex flows. The values for the macroscopic axial thrust computed in this research (see Figure 6.10) agree with the study by Burggraf and Foster [1977] and are a little larger than reported by Long A (see Figure 6.10). A minimum value of F2 occurs at b = 0.04; Fz increases significantly for very large and small values of b. The large values of the axial thrust at small and large values of b are due to different mechanisms. For small values of b, the vortex core becomes large and the axial momentum decreases. Thus, the large values A of F2 mainly come from the area integration (see Eq. (3.16)). On the other hand, for large values of b, the vortex core becomes small and the axial momentum increases significantly (jet-like flow). In this case, the large value of F2 stems directly from the large convective transport of axial momentum (see Eq. (3.16)). The macroscopic axial thrust does not uniquely determine the flow behavior (see Figure 6.10) . For N - -1.0, reverse flows exist for b< 0.043 and jet-like flows exist for b> 0.136. Between these two values of b, the axial velocity has an undulated flow behavior. For Fz larger than 0.75, either flow reversal or jet-like flow will occur (see Figure 6.10). For Fz less than 0.75 but larger than 0.60, either reverse or undulated flow occurs. If F2 is between 0.6 and 0.58, only undulated flows exist. Below 0.58, however, no steady state similarity solutions exist. 89 1.5 iIfoTIFjT?FrTVTIFTTTTTITVT[1*] i J N U .1 a A N : 1.1-1 8 l E “- J u w”- 1 Flow Reversal —..—-———_—-.——_—_~— N = -1.0 Model I (see Table 3.1) Long [1961] Burggraf and Foster [1977] Undulated C. 0 ('0' ...”..lJL... -L_..-.._L.__- --L ---- J___.w..1__._.i~._.....L- .. .1“.-- h- . J .. L... ---- - 33. (b .1. 0.00 Figure 6.10 The effect of b on the macroscopic TTrTTTjfirTTTTTTTTTF 0.10 b axial force for Model 1 90 The macroscopic axial torque and volumetric flow rate decrease monotonically as b increases in the range of 0 < b s 0.30 (see Figures 6.11 and 6.12). For b - 10.0, the axial torque is about 10 and the volumetric flow rate (dimensionless) is about 12. Unlike the axial thrust, the large velocity associated with jet—like flows does not make the axial torque or volumetric flow rate unbounded for large values of b. Instead, it appears (numerically) that these two properties may approach a nonzero lower bound. However, for small values of b, the large cross sectional area associated with the flow reversal makes both T2 and Q, like Fz, unbounded (see Figures 6.11 and 6.12). For the study range of 0 < b S 10, if the axial torque has a value larger than 10, then only one value of b and, therefore, only one type of flow is possible; however, if the axial torque is less than 10, a steady state similarity solution may not exist (see Figure 6.11). Similarly, if the dimensionless macroscopic volumetric flow rate is larger than 12, then this parameter uniquely determines the flow behavior (see Figure 6.12). Transition from flow reversal to undulated flow corresponds to T2 - 19.0 and Q - 21.0. Transition between A undulated and jet-like behavior occurs for T2 - 15.5 and Q - 17.5. Figure 6.13 shows that the macroscopic pressure drop Ap uniquely A determines the flow structure. Note that Ap increases monotonically as A b increases with an almost constant slope. Moreover, for Ap less than A 0.017, reverse flow exist and for Ap between 0.017 and 0.045, undulated flow behavior occurs. Finally, for Ap > 0.045, the axial velocity on 91 3000fiTIIITITIIIfiII1IITIjTIIjTTITWT I I J .I l l 25.04 .1 J N = -1.0 I 1 .. I 1 ~30 : .I‘ 0‘ 20.0-l I .. N” 1 "l g .l l l q “n? l I V 1 l l “ \ _‘ 1 l L‘ l ll 15.04 | l i l l “i" I l l .4 l I j l l i l ‘ ' ID {or b - I0 I, 10.04 ‘ ' -----------, '1 i : Numerical loWor bonndJ I I I _i I l . -l l l 1 -l l I J. sooqReverse: Undulated : Jet-like .I .l l I .11 Flow | l J J I I I . l l .‘ l l l 4 I l -; J . 000 T T ITTlT I T T T T T T r T T T I T I TTI' T T T T T I T I 11 0.00 0.05 0.10 0.15 0.20 0.25 0.30 b Figure 6.11 The effect of b on the macroscopic axial torque for Model I 92 30.0 I I I fr I I I I I I F I i l I I i I i I I Ifr I I IIIj ‘ i ‘ 42 25.0 «3 l 1 20.0-I : ...; ’3 I J. 3 l ; a: l 1 N l I .1 < . . . a O l l 1 15.0-I l l '1 ll 1 l l J <0 1 g g I ‘ I I I -_E_If’."_bf.'_°.-.1 1 i : Nunerical 10w" bound 1? 10.0-4 ' l _1l I I 1 i l I "E I I J 4 I l . 4 l I ~; 1 I l i 5.0JReverse: Undulated : Jet—like _j I I 9 lHow l I j 4 l I 'i J l 1 4| 1 I ’ “l l I '1 l l 0.0 TIII IIT IIII IIII rTTT—Trir] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 b Figure 6.12 The effect of b on the volumetric flow rate for Model I 93 0.10 fifitffirfifffifcj‘ruIiTir.w‘ffvifij « .I I ODE-II I “30 'I I m ' > I 1 fr? 9.; ODS-ll I O. .I I ' I 1? «I l 'l . I 5 J I q 3 Undulated Jet-like ' u 0.041 ; ~f l . <0 . I l I I ‘3 I 2 I l I 0.02--I ; 7 I I ' I I I l 'I I i I , I O‘OOITTT FTTI Tr IVY—Fl}— TTFT I'TT'TTfI I [I 0.00 0.05 0.10 0.1 5 0.20 0.25 0.3 b Figure 6.13 The effect of b on the macroscOpic pressure drop for Model I 94 the centerline has a jet-like behavior. Thus, once y, K, and N (--l) have been specified, the vortex flow with bounded circulation is uniquely determined by either assigning a numerical value to the local A spin parameter 'b' or to the global dimensionless pressure drop Ap (O < Ap < co) . This conclusion was not developed by Long [1961] or by Burggraf and Foster [1977]. They only noted the "unexpected" result A that Fz did not uniquely determine the flow, as it does for other A nonrotating flow problem. Here an explanation for the behavior of F2 has been given and more significantly, a macroscopic parameter has been identified which controls the flow behavior. W22 In Chapter 4, the "excess" mechanical energy of a fluid particle on the axis was examined qualitatively. This property (see Eq. (4.21)) and its substantial time derivative are related to the dimensionless parameters 'a' and 'c' as follows A E a2 P C and A 2 DE 6 /V DE A —7r - — — - -2aE Dt puc2 Dt A A A Figure 4.3 illustrates the relationship between E and DE/Dt for four different flow regimes. Although an undulated, reverse flow regime was 95 identified theoretically in Chapter 4 (Type IV pattern, see Figure 4.3), this flow behavior was not observed numerically, for 0.002 s b s 10. Additional physical understanding of the three flow structures for a vortex with bounded circulation (N - -1) results by examining the axial component of the force balance. The force acting on a fluid element situated on the axis includmscontributions from inertial, pressure, and viscous effects. Thus, by setting r - 0 in Eq. (3.5), it follows that FI + FP + Fv - 0 where - [puz(8uz/62)] A r-0 2 PI - 2 - a puc 6(d6/dz) 6? A a—z r-O F - - ~2c P puc26(d6/dz) p a [E 5;;(1‘8112/31') ] A r-0 2 F - - -a + 2c puc26(d6/dz) A The dimensionless "excess" mechanical energy, E, can also be expressed in terms of these dimensionless forces as 96 A A It is noteworthy that PI and PP do not change sign for 0 < b < co inasmuch as a2 z 0 and c is always positive (see Figure 6.1) . Thus, a fluid particle on the axis moving in the positive 2 direction (forward A flow) must overcome an adverse pressure gradient (F < 0). P Figure 6.14 shows how the individual terms in the force balance and the energy balance evaluted on the axis change with b. These results show that the three flow structures (jet-like, undulated, flow reversal) have the following features T299-I-£199:lé¥9.-§9§¥€§9-f19¥2 A DE FI>O,FPO,E925 05 co a do “cote on... :6 0.59... a cud m _..o N —.o 00.0 +0.0 00.0 bIPIPFbpr—hl-PbePhI—IPIPIPTLLWLLhIpIPLLhPFkbhbblpII—OFDI s 1 _ fl [iii/I _ W _ /I/ mu . n u I/ W n W “ lei/w, " I _ /, _ mod... fl _ /I _ . fl _ 1+./ H u w _ ./ . / / v — — I r / _ _ Il- . .I IIIIIIIIIIII _ .................. _ I“. 23 +1. _ _ \n. .. _ _ ... Y n \\ m O—tmm \\\4\ . Ma. \>\ . I _ n. _ no.0 m u . < n W _ \m. _ I _ \ ( _ r K . n _ 25 __ one. " ___ 25 ; n h p u . p I?» p rm? . r— p b n - PbLIFIplLII-ILILILILI—BILILLILIPI-l OFoo 98 acting on the fluid element is small and is balanced by a positive inertial and viscous forces. Transition from Type III to Type II behavior occurs when FI goes to zero and the viscous and pressure forces balance. At this point, (a,b,c,na) - (0,0.042,0.0l74,28.6) with E < 0 A A and DE/Dt - O. A E is less than zero for a Type II flow pattern, but as the fluid A moves away from the singular point B increases due to energy transfer A A from the surrounding fluid to the axis by viscous work (DE/Dt > O). In this regime, F is balanced by positive viscous and inertial forces (F P V >»0,1FI > 0). 'Transition from Type II to Type I behavior occurs where Fv goes to zero and the pressure and inertial forces balance. At this A A point, (a,b,c,na) - (0.301,0.136,0.0449,24.2) with E - 0 and DE/Dt - 0. When b > 0.136, a jet-like flow behavior occurs (Type I). The A kinetic energy term now dominates the "excess" potential energy (i.e., E > 0); however, as the fluid particle moves toward the hydrostatic A A condition at z - w, its "excess" energy decreases (DE/Dt < 0) because the high speed fluid on the axis transfers energy to the surrounding fluid by viscous work. The viscous force acting on the particle in this regime is negative so the large inertial force is countered by the adverse pressure and the viscous drag. The relatively large adverse pressure force acting on the axis for large values of b is due directly to the centrifugal force and the hydrostatic boundary condition for z e w. CHAPTER 7 MODEL II : VORTEX FLOW WITH VARIABLE CIRCULATION In Chapter 6, quantitative results for swirling flows with constant circulation (Model I) were presented. In this chapter, Model II (see Section 3.1), which includes the effect of N, will be discussed. An interesting aspect of these flows is that the dissipation integral (see Section 7.5) is bounded for -2/3 < N < 0. A comparison between Model I and Model II for N - -l is given in Section 7.1 and, in Section 7.2, the effect of N on the three different flow regimes is developed. The behavior of the velocity and pressure fields for -l < N < 0 is presented in Section 6.3 and 6.4; and, the effect of N on the macroscopic properties will be discussed in Section 7.5. These theoretical results will be compared with experimental data from several different laboratories in Chapter 8. 7 om e e de An or N - - Model II for N - -1 is similar to Model I. Model II, however, includes the effect of N. Because N is generally not equal to -1 (see Figure 8.2), Model II was developed in order to understand certain experimental results for confined vortex flows. As discussed in Chapter 3, the major difference between Model I and Model II is that the axial velocity for Model II is zero for (r,z) - (6*(2), 2). Also, for N > 99 100 -2/3, the growth of the viscous core near the singular point keeps the dissipation integral bounded. Figure 7.1 shows that a jet-like flow around the axis occurs for both models when N - -l and b - 0.30. The small difference in the axial velocity at n - 0 is due to differences in the values of 'a’ determined for each model (a - 0.886 for Model I and 0.893 for Model II). The computed values of 5 for the two models differ * only slightly : "a - 21.7 whereas r] - 21.3. It is also apparent that the axial velocity has an asymptotic behavior (uz a l/r) at n - "a for Model I, but is zero at n - "*for Model II. The differences in the macroscopic properties between the two models are important and will be discussed in Section 7.5. The criteria for transition between the different flow regimes (see Figure 4.2) for the two models (N - -l) are very similar. For instance, reverse flow passes to an undulated flow for (a,b,cfi) - (0,0.042,0.0149,28.6) for Model I; and (0,0.038,0.0151, 29.0) for Model II. The transition between undulated and jet-like flow occurs for (a,b,cji) - (0.301,0.136, 0.0449, 24.2) and, (0.321,0.14,0.0447,24.1) for Models I and II, respectively. 7 f e t F ow e e Three different flow regimes have also been found for Model II. As b increases, the axial velocity changes from flow reversal to undulated and,finally,obtains a jet-like structure. As N increases, the region of flow reversal and undulated behavior broadens (see Figure 7.2). For example, flow reversal occurs for b < 0.038 and N - -l.0. However, for N - -0.75, flow reversal occurs for 0 < b < 0.055. Similarly, undulated 101 0.80— . N = —1.0 ‘ b = 0.3 0.60— 6=JZ/K 4 Uc = Kz/Jz Uz/Uc .0 4:. o l Modell -..L._ .-_L..._m L - Model” (loo-r -------------------------- ---~* 4 -O-ZO . T T T T r T T T T ITT T T T I l T T T I T T I 0.0 5.0 10.0 15.0 20.0 r/6 Figure 7.1 Comparison between Model l and Model II for axial velocity profile 102 = .302 tow 52553 30: «cocoa—tn to moE_mom NH 059.“. a 00 ....o no «.0 to 0.0 Ll F b p L L F b lP I» h h h L L! lP _ lb r P L 00.0 r r r load 1 T r Amd. .m¢ro.o.ov a at.» .u.cv a I . rote .. 3mm .m and ..medv \ 69651 1 ,. 86325 26: T h f road # T r r ,. 2.9 .39... .31.. T .. 9:. u E. 55.3%...2x3 rows r T I r AZ." ~33... _«m ..ov A I ofid __mZd 0v _ r _ . _ . _ r _ / f 103 axial velocities occur between 0.038 < b < 0.14 for N - -l.0, but between 0.055 < b < 0.23 for N - -0.75. As N increases from -1 to -.75, the lower limit on b for jet-like behavior increases from 0.14 to 0.23. 'Thus, in order to have a jet-like axial flow, a strong rotation around the axis is necessary for larger values of N. This kind of effect becomes more apparent as N increases. 7 vo eouton Tables A.2 - A.4 in Appendix A list the values of (a,c,n*) for different values of N and b. Figure 7.3 shows the effect of N on the * parameter set (a,b,c,n ). The general behavior is the same as Model I * where 'a' and 'c' increase, but :7 decreases, as 'b' increases. 'c' is * less sensitive to 'b' than 'a'. Because 17 represents the size of the viscous core, it follows from Figure 7.3 that larger values of 'b' give a more coherent vortex. As N increases, c increases and 0* decreases. The size of the core is reduced almost 50 % when N changes from -1 to -0.75. This means that a more concentrated (or coherent) core will occur at larger'walues of N for fixed values of b. Because this effect is so large, it is important for Model II to be fully developed and analyzed. Once again, for fixed * N, the parameters 'a' , 'c' , and 97 change monotonically with b. Therefore, the flow structure is uniquely determined by any one of the * four dimensionless number (a,b,c,n ) together with K, v, and N. Figures 7.4 - 7.6 show the general behavior of M, F, and h. The *- similarity functions are defined on the finite domain 0 s n S n , so the 104 1.00— ___" N = ‘1-0 ~ ——-—— N = ~o.75 “0.20 FTIITIF—TTIIIFITIIITITITIITFIFI 010 0.1 0.2 b Figure 7.3 The behaviour of the solution for Model ll 0.3 105 curves terminate at n*. The derivatives of M, F, and h are all zero for n - 0 because of the boundary conditions (see Eqs. (3.58) - (3.59)). Figure 7.4 shows how b affects M for N - -0.75. Comparing Figure 7.4 and 6.2 (N - -l.0, Model I), M decreases as b increases but, for * Model II, goes to negative values at n - n for any value of b. M in Model I becomes asymptotic to l/(2n2) at r) - "a' A negative value of M in Model II implies that the pressure at n - n* is higher than hydrostatic. The parameter 'c', which represents M(0), is always positive (see Figure 7.3). The effect of b on M is significant (see Figure 7.4). For constant 17, when b increases, M becomes larger at small values of 0, but becomes * smaller at large values of 0. Because M at n - n becomes smaller as b * increases, the pressure drop across the vortex (i.e. , M(O) - M(n )) increases with b. Thus, coherent (concentrated) vortex structures require relatively large radial pressure drops. The behavior of F, shown in Figure 7.5, monotonically increases to a maximum value at n - 71*. For constant a, F increases as b increases. For fixed b and q, comparing Figure 7.5 and 6.3, F increases as N increases. Therefore, larger values of N give larger values for the circulation, or the angular momentum, at constant b and n. The stream function h for Model II (see Figure 7.6) in the outer region is different from h for Model I (see Figure 6.4) due to the boundary conditions imposed at 97*. However, the behavior of h near the axis is similar for both models. For forward flow, h increases monotonically. For reverse flow, h is negative for small values of r; 0.12 0.04 0.00 -0.04 106 4 IT I I 1 T I Tl r i l r r r I l I r T j 1 j 'l J l .l 4 ‘l 4 N =- —o.75 3 j 1 '1 .l N+2 “ 4 6 = (JZ/K) . a -l 4 w ...a _‘l -l .. -l -1 j b =- .o:s : d -l .—li- ———————————— —--—— —--: ——————— -1 ..l J ' I151 : _, 3 g 11.1 ' - ..l J . l 1 } q*= 7.43 j I I I T T r I T I T T T T 1 T Tr T I 1 0.0 50 10.0 150 20. r/6 Figure 7.4 The effect of b on the behaviour of M for Model ll 2.5 107 b=-.03 5.0 Figure 7.5 The effect of b on the behaviour of F for Model ll TTl’TTT VT] 10.0 r/J T 108 10 0 l I 1 r 1 IT I r I r F I r I j l I F I -l . --4 I a ' .1 15.1 7 l 1.1 7 8.04 I ..u 4 -l « -l 4 .4 6.0% ...-1 ..l -l N = -O.75 4.0— —1 .. N+2 " 8 == (02/ K) ‘ . .3 .1 b = .03 _ 1 -1 2.0~ - .1 ... ‘ 'l 0.0-A ---------------------------- Al 4 -1l 4 1 _‘2-0 F r I I r T I T I r T I T T 1' F T T T 0.0 5.0 10.0 15.0 20.0 r/J Figure 7.6 The effect of b on the behaviour of h for Model ll 109 and positive for larger values of n. The point where h - 0 occurs at n z 5 for N - -0.75 and b - 0.03. Because h(n) represents the local dimensionless entrainment rate (see Eq. (3.43)), h(5) - 0 means that the local volumetric flow rate across a surface of fixed 2 and 0 s r 5 55(2) is zero. Therefore, the axial velocity must be zero at some value of r between 0 and 56(2). It can be seen from Figure 7.6 that zero axial velocity occurs at n z 3.5 (also see Figure 7.8) where the slope of h is zero. One important difference between Model I and Model II is the stream function at the outer edge of the vortex care. In Model I, h is asymptotic to ha’ defined by Eq. (3.53). However, for Model II, the * ~1- slope of h at q - n equals zero because uz(6 ,z) - 0. For fixed N and 17 < 11*, the effect of b on h is to increase the local volumetric flow rate as b increases. However, the global entrainment rate, defined by Eq. (3.43), increases as b decreases. Thus, as the vortex core diameter increases, the volumetric flow rate becomes larger. For example, it follows directly from Figure 7.6 that * h(n ) z 9.5 for b - 0.03 and h(n*) z 6.0 for b - 0.3. So as b increases by an order of magnitude, the core size decreases by a factor of two and the volumetric flow rate through the vortex decreases by about 30%. 4 v’ 0 e V t e1 Figure 7.7 shows that the common features of the tangential velocity profiles for different values of b and N - -0.75 include : a Rankine-type structure with a forced vortex near the axis and a free- vortex-like flow in the outer region; the maximum tangential velocity Uo/Uc llO 0.35 I T—r T TTT I I T I—I T I IT] I I I I TT I ITTI I I_I T I I -l -1 : N = -0.75 : 1 _ 0.30-11 N+2 q 1‘ 6 = (JZ/K) . 1 -”_.s ., = N+2 q 0.25—1 .ft = 7,43 U° K9 Z/K) .. .4 | d .. l .. .l ‘ .1 .l .3 .4 0.20-11 - I 11.1 I -1 | -l « .1 ~ 0.154 ' 154- .. l - .4 . .1 I . ., - 0.10—l b a '03 -1 4 -1 1 d -l ..1 0.05-1 -+ 4 -1 - l 0'00 [I I VII I ITI I TI I I—r I ITITITTT I rI'I I ITI I ITT 0.0 5.0 r/G 1 0.0 Figure 7.7 The effect of b on the tangential velocity profile for Model II 15.0 111 shifts to smaller values of 17 and becomes larger as b increases; and, the core size decreases as the rotation frequency around the axis increases (i.e., b increases). It is also noteworthy that as b increases, the fraction of the cross sectional area of the vortex in solid body rotation decreases. The axial velocity behavior for Model II for general N has some features which are similar to Model I (see Figure 7.8). Note that reverse flow on the axis appears at smaller values of b and jet-like flow on the axis appears at larger values of b. At intermediate values of b, undulated flow exists (see Figure 7.2 for other values of N). In the flow reversal region (Figure 7.8), zero axial velocity occurs at n z 3.5 for b - 0.03 which corresponds to the zero slope of the stream function (see Figure 7.6). The flow moves downward within 7; < 3.5 and moves upward for 3.5 < n < 15.1. For forward flow on the axis, the flow moves upward everywhere (i.e. , 0 5 n < 11*). Note that strong rotation around the axis (large b) gives a higher axial velocity near the axis and lower axial velocity near the outer boundary. The O * * O axial veloc1ty goes to zero at n - n because uz(6 ,z) - 0. Again, the comparison between Figure 7.7 and 7.8 shows that the values of n at maximum axial velocity is smaller than the values of n at maximum tangential velocity for any value of b. Thus, the position of maximum axial velocity occurs within the solid body region. Characterization of different flow regimes using the mechanical energy balance on the axis is the same for Models I and II and has already been discussed in Section 6.4. The radial velocity behavior in Model II is very important to the understanding of light particle separation (see Chapter 9). The common 112 IT'I'T'TITIIIITTTIFTIF—IIVIIITTTF 'l d I _, 0304 N = -0.75 T "l 1 .. N+2 . 8 = (92/10 — -l N -l N+2 0.40-l 3 ”C = KHZ/K) __ '1 .0 :‘3 \ -l . 5‘ . ‘ l 0.20- ._ J '4 b = .03 A d .. .1 q ' l l 0.00-fl" ————————————————————————— ‘ .1 J 0* = 7.43 11.1 15.1. -l ~11 1 1‘ "0.20 TTTTTITIVI[VITIITTTTrTrTITTTTI—Tl—TTI 0.0 5.0 10.0 15.0 r/G Figure 7.8 The effect of b on the axial velocity profile for Model ll 113 features of radial velocity in Model II (see Figure 7.9) are the same as in Model I. (a). (b). (c). (d). (e). (f). (g). 7 Figu propertie Figure 7. In brief, these are : Positive radial velocity near the axis for forward flow, but negative radial velocity near the axis for reverse flow. Only one mantle (ur - 0) exists in undulated or jet-like flow region, but two mantles exist in reverse flow region. The inner mantle for reverse flow is about n z 3.0 for b - .03 and N - -0.75. Therefore, all the flow within this mantle will move downward toward the singular point. The peak of maximum radial velocity shifts to smaller values of n as b increases. The effect of b on the radial velocity is to reduce the mantle size in the forward flow region; therefore, a strong rotation about the axis causes a small mantle to form with a very strong jet-like flow behavior over a very small vortex care. For fixed values of n, the radial velocity near the axis becomes larger when b increases. However, the radial velocity near the edge of the vortex becomes smaller when b increases. A surface of zero radial velocity (i.e., mantle) always * exists within the vortex core (n < n ). O t res 7.10 - 7.13 show the effect of b on the macroscopic s for the vortex (see Table 3.2). An important feature of 10 is that the axial thrust does not determine the flow' behavior uniquely. The minimum value of F2 is about 0.5 for N - -l.0; and, 0.45 for N - -0.75. For Model I, the minimum thrust is 0.58 (see Ur/(Uc dS/dz) 1.20 —1.20 114 T ITfFi—TI'T 1‘le TTITF_I‘ T T F'Tfi‘ i‘T FF N —-0.75 - TTTTTTTII | s = 02/101"? 3 __J N ; Uc = l<(\)z/l<)“+2 7 d ~— .- .03 1.-.--1 \l 0*: 7.43 l -.. 11.1 15.1 0.0 IITTTITIITTIIfTYITITTIIITTTTTTTTJ 4.0 8.0 12.0 r/6 Figure 7.9 The effect of b on the radial velocity profile for Model ll 16.0 115 2.0 I I I I I I I r I I F I I I I I I I I [TI I ITT I I I I ‘ l l 4 4 l l I l N = -0.75 1 1 51-1 ‘l 1 l “u .l .4 D l 0- 3 “1,0 l J. ff, 3: 1 1 “- 1.0-1 -l 11 (Li J l J N = —-1.0 4, J J l . , 4 0 5-l \ J "J. l . l l 1 0.0 ITT TT rTT I T I I I TT rTTT T T TTTTTTT'i 0.00 0.05 0.10 0.15 0.20 0.25 0.30 b Figure 7.10 The effect of b on the macroscopic axial thrust for Model ll 116 20.0 ITITITFTTTWITIIIITVITIFIITTITTIII'ITTI T2 = Tz/(21l539uc 2) 5 0 J l l ’ ‘1 l ITI1TITI'II1TTTII[ITIITTIIII'WITTIw O 0.05 0.10 0.15 0.20 0.25 0.30 b u—lh—vcc- -—L—-—-—.—J .-- -. Figure 7.11 The effect of b on the macroscopic axial torque for Model ll ll7 20.0—[IVITrIIITjTITTITII—IIIIIIITIIIIIT‘ 1 i l l 4e "I l l .l l l I l l l ‘ N=-—O.75 3 5.0~ .1 l 1 I ‘ I J .5 l l 0'0 TTTIITTITIIFFTrTrITrTIIrrrrrr€ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 b Figure 7.12 The effect of b on the volumetric flow rate for Model ll 118 Figure 6.10) . For a fixed value of the axial thrust above the minimum value, two values of b are possible. For example, with N - -l and F2 > 0.5, three different cases are possible : (1) For 0.50 5 F25 0.52 (which corresponds to the transition point between flow reversal and undulated flow), two different undulated profiles are possible; (2) For 0.52 5 F2 5 0.70 (which corresponds to the transition point between undulated and jet-like flows) , either a flow reversal or an undulated flow behavior on the axis occurs; and, (3) When the axial thrust is larger than 0.70, either flow reversal or jet-like behavior obtains. For N - -0.75, the corresponding values for Fz are 0.45, 0.47, and 1.4, respectively . The axial torque and the volumetric flow rate decrease monotonically as b increases (see Figure 7.11 and 7.12). The slope of these curves are steeper in the small b region. For large values of b, small changes in the axial torque and flow rate will cause large changes in b. Although the qualitative behavior of Tz and Q for Models I and II are similar, the specific velues of these parameters are much smaller for Model II. For example, at b - 0.3 and N - -1, T2 is 14.5 for Model I but only 9.6 for Model II. If N increases to -0.75, then Tz - 5.8. The macroscopic flow rate Q decreases from 16.5 (Model I) to 11.6 (Model II, N - -l). For N - -0.75, Q - 5.9. 119 A A The axial torque T2 and the volumetric flow rate Q are both at :1- proportional to h(n ) (see Table 3.2). As b decreases, 0 increases A A * (see Figure 7.3). Because h(n*) increases with n , T2 and Q become very A large for b 4 0. On the other hand, both T2 and Q appear to approach a positive lower bound for very large values of b. For example, with b - 10 and N - -1.0, T2 - 5 and Q - 7. If N - -0.75 and b - 10, then Tz - 2.8 and Q - 4. An important parameter for both Models I and II is the macroscopic A A pressure drop, Ap. Figure 7.13 shows that for N - -l and -0.75, Ap increases monotonically with an almost constant slope. For fixed b, Ap increases as N increases. It is noteworthy that for b > 0.05, the macroscopic pressure drop for N - -0.75 is almost 1.5 time the macroscopic pressure drop for N - -l.0. Unlike the other global A properties, a one-to-one correspondence between Ap and b exist. Thus, as previously discussed in Chapter 6, the flow behavior is uniquely A determined by specifying K, v, N and either Ap or one of the local * properties : a, b, c, n . The dissipation of kinetic energy over the cross section of the vortex can be calculated as follows 3(2) <1, - ZWJ (1;: Vu)rdr (7.1) where 2 c (p(Sa-Z) -- p(O.Z))/?u A Ap== 120 0.15741 Ifi'I ITITTI 1 ITITITI r? rW—TTTT FITTT rT FFFF 1 '1' i l .‘ .l .1 0.10-1 .1 l ; l ‘ 1 « 43 1 I J l 4 J. 0051 4 I .l l I . .1 l 1 . 1 0'00 I TI T I I T FT I T T I T I I I r r ITT I I T T I J} 0.00 0.05 0.10 0.15 0.20 0.25 0.30 b Figure 7.13 The effect of b on the macroscopic pressure drop for Model H 121 6 us 2 auz 2 L=V9-#{[IE(T’] +[5r_]} (7'2) represents the irreversible conversion of kinetic energy into internal energy per unit volume by viscous dissipation (see p. 82 of Bird et al., 1960). Eq. (7.1) can be written in terms of the similarity functions defined in Chapter 3. Thus, by substituting Eqs. (3.28), (3.34), (3.37), and (3.38) into Eqs. (7.1) and (7.2), it follows that . 0 " F' 2F 2 2 4’ - '1 (— - —) + (G') fldfl- (7.3) Zapuc The dimensionless dissipation Q depends only on b and N. Figure 7.14 shows how b affects Q for N - -l.0 and N - -0.75. The behavior of Q is similar to the behavior of the macroscopic axial thrust. The values of A A 0 become large for small and large values of b. The minimum value of 0 is about 0.062 for N - -l.0 and 0.102 for N - -0.75. Although the :macroscopic.dissipation does not uniquely determine the flow structure, it nevertheless provides a useful characteristic of the flow. For a fixed value of N and b, it follows directly from Eq. (7.3) that 0 - Zapuc2¢(b,N) (7.4) where N uz -—- N N+2 uc - K6 - K( K) . Thus, § #20113) 0.4 122 lLLLleL .0 t4 ILLLLLL1L4LLALLLIILALL1 -_ 4.--.1.-J_e_1,- _ i "l J l l , ‘l -1' J -; i 0.2 4: N = -0.75 l l l l l 0.1 .-, d N=—1.0 J l «l ‘ l " 1 0'0 TIITrTTITTIIIIrTIIIITIITrTTTT 0.00 0.05 0.10 0.15 0.20 0.25 b Figure 7.14 The effect of b on the macroscopic dissipation for Model ll 0.30 123 2N a « zN+2. (7.5) Eq. (7.5) implies that 0 e m as 2 e 0 for -1 s N < 0; and, 0 e 0 for z 4 w. The total dissipation follows by integrating Eq. (7.4) from zD‘> 0 to z - L : 3N+2 N+2 N+2 L ,. D 2 D - I edz - 21rpLuc (L)(b.N)§N—+§ [1 - (f) 2 ] D For zD - 0, D is unbounded for -l S N 5 -2/3; however, if ~2/3 < N < 0, then 0 < D < m. For 2 D > 0 and -l s N < 0, D is positive and bounded. CHAPTER 8 AN ANALYSIS OF EXPERIMENTAL RESULTS IN THE LITERATURE e u t The major assumptions of this study are constant density and viscosity, axisymmetric flow, and similarity structure. Although the similarity theory necessarily neglects the effect of wall boundary layers in confined vortex flows, the results of several experimental (and computational) studies are used to explore the possibility that the analysis can be used to quantify the core region of flooded hydrocyclones or, more generally, of confined vortex chambers containing a single fluid phase (either gas or liquid, but not both). Several major studies of confined vortex chambers are used for the purpose of comparing theoretical and experimental (including computational) results. Figure 8.1 shows the various coordinate systems and geometries of these studies. Table 8.1 summarizes the important operating conditions for the experiments. The location of the reference coordinate 5, shown in figure 8.1, defines the orientation of the viscous core and will be discussed further in Section 8.3. Among the six papers listed in Table 8.1, Escudier et a1. [1980, 1982] and Dabir [1983] are the most important because they have very complete information. Moreover, the flows in these studies were axisymmetric, although under some conditions nonaxisymmetric behavior was observed. The tangential and axial components of the velocity were 124 125 u - (Volumetric flow rate)/(area) F l ——-> 1——t l l l l l. 00 .1 _ 2' ll ll it] I ——»—- --------- ir-----H--*Q f! Tl llj 0/2 I: L I; (l) Escudier et al., 1980 Q o T | —v I s————- D | o u -l_ _ F : “ ""l"" ‘ l l l l ‘— position I of l velocity 2' profiles I _ —v| 1 +-——- D u Q u (2) Dabir, 1983 (3) Kimber and Thew, 1974 Figure 8.1 Coordinate system for various vortex chambers Table 8.1 Flow conditions of the experimental studies 126 Flow Conditions Flow Reference Index u Q D D L F o o Regime 3 cm/s cm /s mm mm cm 1a 24 400 55 40 42.5 Reverse Escudier et a1. 1980 lb 24 400 55 25 42.5 Undulated (experimental) 1c 24 400 55 10 42.5 Jet-like 1d 6 100 55 18 42.5 Jet-like Escudier et a1. 1e 12 200 55 18 42.5 Undulated 1982 (experimental) 1f 24.6 411 55 18 42.5 Undulated lg 51 850 55 18 42.5 Reverse 2a 140 500 76 25.8 35.0 Undulated Dabir 1983 2b 140 500 76 12.9 35.0 Jet-like (experimental) 2c 140 500 76 12.2 35.0 Jet-like Kimber and Thew 1974 3 40 51 50 12.7 300 Undulated (experimental) Boysan et a1. 1982 4 1370 42000 203 64 33.0 Jet-like (computational) Pericleous et a1. 1984 5 56.7 1250 200 80 45.2 ------ (computational) 127 obtained by using laser doppler anemometry. The velocity data are tabulated in Appendix C. Both computational papers assume axisymmetric behavior and a model for the Reynolds stress. Boysan et al. [1982] used an algebraic Reynolds stress model and Pericleous, et a1. [1984] used a modified Prandtl mixing length model. These results, which are sensitive to the inlet and boundary conditions, will also be examined for similarity in this chapter. Because the similarity theory assumes that the viscous core is driven by a tangential velocity having the form u -I(r, r26*(z). the experimental (and computational) data should also show this behavior. The important idea is that K and N are independent of the axial coordinate. Specific values of N and K are developed for each data set. Figure 8.2 shows the behavior of the tangential velocity in the outer region for the data sets in Appendix C. The results of Dabir (Indices 2a,b,c) and of Escudier (Indices la,b,c) show that N is approximately -0.75. This is interesting because a Beltrami vortex (see Section 2.2) also shows this type of behavior. However, the vortex device studied by Kimber and Thew (Index 3) shows that N z -O.6; and, N is about -1.0 for the two computational studies (Indices 4 and 5). It is noteworthy that N and K do not change significantly with axial position. For example, la represents data at E - 19.35 cm and 19.85 cm,whereas 2a represents three axial positions: 2 - 8, 20, and 32 128 Index N may..." Ia,lb,lc -0.75 63 1054-04- 3 Id 4'75 ' ' , Ic -MS 36 j I 4 l-F -0.75 ‘7 4‘ I I 3 -ms 9 5 1 I 2a,2b,3c ”35 27° 3 -o.€, 64 4 -t.o 3050 .5 -l.o 1050 ‘1’“ 100.0 r(cm) ~"igure 8.2 Deviation from ideal behaviour in the 0 outer region of several experimental vortex flows 129 cm. For the data evaluated, N appears to be independent of the flow rate and insensitive to the contraction ratio of the vortex chamber (i.e., Do/D). The coefficient K, however, is a strong function of Q0, but not the contraction ratio Do/D (see experiments 1a - 1c). 8 ow Re e Following Eqs. (4.12) and (4.15), the values of 'a' and 'b' are defined by qul luol IaI-———-— u0(6,z) KSN and 6(2) can be eliminated between the above two expressions to give N- |a|‘N+2)/N(u/K>(E - 20) (8.3) A plot of |u0|(N+2)/N vs. 2 will provide N+2)/N slope - (IaIK)( (v/K) and E - 20 at Iuol(N+2)/N - 0 (extrapolated). (8.4) Figure 8.5 shows how the above expression can be used to estimate 20 from experimental values of no. The table in Figure 8.5 lists the values of zo determined in this manner. Because the data follow the predicted behavior defined by Eq. (8.3), Figure 8.5 provides additional evidence that the similarity theory can be used to describe the viscous core of confined vortex flows. 8 4 S m t ca in 0 The u ome tum Prof es The similarity theory requires the local circulathnn or axial component of angular momentum, to follow the scaling law ruo(r,z) r 6(z)uc(z) ' F(5(2)) 137 0.006— 4 Index z , cm i (see Table 8.1) o “I 0.005? M (2.5) 4 lb 2.5 N {.2 " 1c (2.5) ,1 ..l a? 0.0044 M 0 Q 44 4 1e 3.0 , J 1f 3.0 «v J 20 gfiz_15 4 1g 3.5 _1 O-OOIH N = “075 / 2.. _25 2b (-2.5) 2c (-2.5) 3 (0) 4 2.0 rrT?I TTTrIrTFTTrFIII’r] —10.0 0.0 10.0 20.0 30.0 40.0 z(cm) Figure 8.5 Similarity scaling of the centerline velocity 138 Because (see Eqs. (3.37) - (3.38)) 6uc a 6N+1 and 6 « sz/(N+2? a plot of (ru0)/(z(N+1)/(N+2)) vs. r/(zl/(N+2)) for different values of r and 2 should fall on the same curve. Figure 8.6 and 8.7 show that the data for Index 1d and Index 2a (see Table 8.1) fall approximately on the same curve, which indicates that a similarity structure obtains. All of the experimental velocity profiles used in this chapter fall outside the region about the singular point where the pressure is negative. With 2° defined by p(0,z°) - 0, Eq. (4.19) in Chapter 4 can be used to determine 20 for a specific data set. For example, for the data presented in Figure 8.6, the parameter 'b' was estimated to be about 0.57 (see Table 8.3, Index 1d). From Figure 8.2, N - -0.75 and K - 11 (cm/s)/cmN. A theoretical estimate for 'c' follows directly from Figure 7.3 : for b - 0.57, c - 0.13. Therefore, if p0 z 1 atm (1.01 bars) and if K/u - 340 cm'(N+1) (see Table 8.4), then 2° = 0.034 cm (see Eq. (4.19)). The smallest value of 2 used in Figure 8.6 is 10 cm >> 20. Similarly, z0 z 3.1 cm for the data analyzed in Figure 8.7, which is much smaller than 2 - 20 cm. 8 V Es ima e Eq. (4.15) relates b to the viscosity coefficient u. Thus, once b, K, N, 2°, and the angular velocity at the axis have been determined, the viscosity coefficient can be calculated from N+4 N+2 ] v *— 139 a ] . (cma/sec)/[(cm) ‘- ELL N lib/[(2) i“ o LLLLLLLLLLLLLJLLILJLLILILIIJllLLlllleLJ Index 1d (see Table C.1) u N--0.75,z-0,z-E O I 2, cm 10 19.85 20.55 29.45 31.65 F 0.1 0.2 0.3 0.4 l _i__ r/[(z)"" 1. cm/[“*’1 Figure 8.6 Similarity scaling of the angular momentum (data from Escudier et al., 1982) 140 N O .0 O 1.._41”.i_._1___l 1 E3 9 o 1 1_ N+l rue/[0)“ 1 . <<:rn‘/emc)/[(cm)'“'a 100.0-l % Index 2a (see Table C.8) __ -+ N - -0.75, z - -2.5 cm + o 2 _4 "’ ( z z - 20) ..l 50.04 Symbol 5, cm .1 O 20 A 32 TrrfYTTrTrrrr‘rrTI—rrI—TFrrTT—rTTl 0.0 0.1 0.2 0.3 __|__ __'._ r/[(2) "*2 1 . «cm/tom)“a 1 Figure 8.7 Similarity scaling of the angular momentum (data from Dabir, 1983) 141 N+2 mm mm] _ K 0 r 0 N-l v--——( ) ~ bK (z-zo) Table 8.4 gives the values of u for each experimental data set in Appendix C. For data sets 1a - c, the viscosity coeffient was found to be about 0.11 cmz/s. Obviously, :2 cannot be interpreted as a fluid parameter inasmuch as the molecular kinematic viscosity for liquid water is about an order of magnitude smaller. Thus, the similarity theory applied to laboratory scale vortex flows should be interpreted as a mathematical model for the mean field with a Boussinesq approximation for the shear components of the Reynolds stress (see Chapter 1, Hinze, 1975). The boundary layer approximations used in Chapter 3 only retained the r0 - and rz - components of the viscous stress. Thus, a consistent extension to the mean field equations for turbulent flows requires < ' ' a uo> < >-- — ur U0 yer 8r r ) (8.5) and 6 6 z r - - Ve( 81' + 62 ) (8.6) I where and no represent, resp., the mean and fluctuating components of the tangential velocity. The turbulent coefficient of viscosity (or "eddy" viscosity) appearing in Eqs. (8.5) and (8.6) depends on the flow 142 Table 8.4 Viscosity estimates based on the similarity theory Flow Conditions K, N see Table 8.2 b see Table 8.3 20 see Figure 8.5 Viscosity Index Coefficient K/v , Flow rate Contraction Flow Ratio Regime 2 - Qo , cm3/s Do / D V, cm /s cm (N+1) 1a 0.73 Reverse lb 400 0.45 Undulated 0.11 i 0.01 620 i 60 1a 0.18 Jet-like 1d 100 Jet-like 0.034 i 0.003 324 i 30 1e 200 Undulated 0.075 t 0.07 346 i 35 0.33 1f 411 Undulated 0.11 i 0.01 427 i 50 lg 850 Reverse 0.062 i 0.004 1500 i 100 2a 0.34 Undulated 2b 140 0.17 Jet-like 1.1 i 0.3 246 i 70 2c 0.16 Jet-like 3 40 0.25 Undulated 0.049 1300 4 1370 0.40 Jet-like 50 t 18 60 i 18 143 and may change significantly with either r or 2 (see, for example, Bloor and Ingham [1984]). Here, however, ye must be considered constant. Table 8.4 indicates that the contraction ratio Do/D, which determines the type of flow regime in the vortex care, does not affect the "eddy" viscosity ye (- v). However, as the flow rate decreases (see indices 1x1 - 1g). the estimated value of u changes from 0.062 cm2/s at 3 2 3 Q0 - 850 cm /s to 0.034 cm /s at Qo - 100 cm /s. The experiments of Dabir (Indices 2a - 2c) yield values of we about an order of magnitude larger than observed by Escudier at lower flow rates. It is noteworthy that both Dabir's and Escudier's data show that we may not be affected by the contraction ratio Do/D° The experiments of Kimber and Thew (Index 3) are also consistent with the above results and give ye - 0.049 cmZ/s for Q0 - 40 cm3/s and Do/D - 0.25. The computational results of Boysan (Index 4) suggest that "e z 50 (:18) cmZ/s for Q0 - 1370 cm3/s. 8.6 Entrainmen; Rates A dimensionless entrainment rate into the vortex can be calculated from Eq. (3.43), 1 dQ '27:; a; ' hm)- (8'7) For fixed values of N and b, h(n*) is uniquely determined by the similarity theory (see Chapter 7, Figure 7.6). The local volumetric 144 flow rate Q(z) at various values of z (or 2, see Figure 8.1) can be calculated from 6* Q(z) - 2x I uz(r,z)rdr O 31' where 6 (2) corresponds to u2 = 0. The actual graphs for uz(r,z) at different axial positions presented by Escudier et a1. [1980, 1982] were very small and difficult to read. Estimates for uz obtained from these figures, which have been tabulated in Appendix C, are subject to error. A rough judgment is that Q can only be determined to within 30 % using this information. Figure 8.8 illustrates how dQ/dz was determined from the experimental data, a process which introduces.significant errorn Nevertheless, a dimensionless entrainment rate was estimated by using the values of u listed in Table 8.4; the results are shown in Figure * 8.9. Figure 8.9 compares the theoretical values of h(n ) corresponding to (b,N) with the experimental estimate of h(n*) using 0(2) and u (see Eq. (8.7)). Although the correlation between these two methods for * determining h(n ) is less than desirable, Figure 8.6 provides a critical test of the similarity theory. 145 300.— ..l a N = --O.75 .1 250.4 1 4 200.4 fl '* 2 I d to ,J Index 20 A n ex dQ/dz, cm /s \ 4 M E 150.4 1a 7.2 ° .l 0': 4 lb 9.0 ..l lc 9.3 .ll 1d 2.0 100': 1e 3.9 .. 1f 8.7 l 1 g 5.8 .4 50._ 2a 7.7 " 2b 8.0 'i 2c 14.8 [Tr TrTrr—TrrIFTT—TTITITY ~10.0 0.0 10.0 20.0 30.0 40.0 2. cm Figure 8.8 Estimates of entrainment rates 146 20.0-— q .1 -l .1 l9 ’ 15.0-J .- ’g/ .l K. 'b // 5 rr / :g' i / g -l H: I, so / l" g 10.0-4 m / 9" 4 le /’/ LIJ ' .4 l" / -4 //'/ -l /,/ 5.0-l -l // / .4 // 3C. I' at: 20 l l" r 0.0 T “r r r I— r rT [7 T T T 1' 0.0 5.0 10.0 15.0 Theoretical Figure 8.9 Comparison betwaen theoretical and experimental entrainment rates CHAPTER 9 PARTICLE EQUILIBRIUM ORBITS WITHIN THE CORE REGION OF VORTEX FLOWS 9 a a An important application of swirling flows is the separation and classification of very fine particles. Oil - water separation and coal beneficiation are significant examples. Hydrocarbon contamination of the oceans by oil tankers and offshore platforms is obviously undesirable, and an efficient process for removing small amounts of crude oil from a continuous phase could lessen the enviromental impact of these energy sources. Equipment already exist for cleaning oily water but, unfortunately, these filter-coalescer systems require long residence time and are typically large in size and mass, a feature which is especially bothersome on offshore platforms where space is limited. An oily water clean-up system with a large throughput and a short mean residence time is clearly desirable and Colman et a1. [1980] have suggested that hydrocyclones be used for this application. Thew and his coelleagues (see, Colman and Thew, 1980; Colman et al., 1980; Smyth et al., 1980; Colman and Thew, 1983; Smyth et al., 1984; Thew et al. , 1980; Kimber and Thew, 1974) over the past decade have been studying the possibility of using hydrocyclones for oil/water separations. To stabilize the central vortex region within the hydrocyclone, they have developed the concept of a co-axial withdrawal of fluid through the vortex finder (Colman and Thew, 1980). With this 147 148 modification, they were able to increase the ratio of the overflow concentration of dispersed component (by volume) to the feed concentration from 20 : l to 160 : 1 in a 30 mm cylindrical hydrocyclone. The experiments were developed for Kuwait crude with density equal to 0.86 g/cm3; mean particle diameter, 41 microns; and, feed concentration, 1000 ppm by volume. Earlier, Kimber and Thew [1974] studied the separation of Roxstone oil from water with drop sizes in the range 40 - 50 microns. They were able to separate 90 % of the oil. For the hydrocyclone design shown in Figure 8.1 (design a) , Colman et a1. [1980] obtained 99% separation for a drop size of 55 microns, but only 75% separation for 23 microns. They used Forties crude with the following physical properties : pp - 0.84 g/cm; feed concentration - 1000 ppm. Similar effects were observed for Kuwait crude. Obviously, the particle size plays a critical role in the performance of hydrocyclones. More recent design modification have been developed at Southampton University, U.K. (see Colman et al., 1980) with some additional improvements in separation performance, but the ubiquitous drop in efficiency for particle sizes between 20-40 microns remains. Is this phenomenon a result of the specific design or an intrinsic property of the flow structure of a viscous vortex ? The above results of Thew partly motivated the investigation of this chapter, which explores the behavior of very fine particles in a vortex flow. The model developed in Chapter 7 is used to calculate the equilibrium orbits of spherical particles and to develop some understanding of how the complex flow patterns within a viscous vortex could possibly account for the apparent low efficiency of separation of very fine particles. 149 Within a cyclone separator, particles with densities larger than the continuous phase migrate to the outer region and are removed by a helical flow directed toward the underflow. Likewise, particles with densities less than the continuous phase migrate toward the core of the vortex and are removed by an upward helical flow. The viscous drag on very fine particles (<< 500 microns) resist the relative migration due to differences in density and may, in some situations, balance the net centrifugal force acting on the particles. Thus, if the particle residence time within the vortex is sufficiently long, then equilibrium orbits may occur. Many investigators (see p.44 Svarovsky, 1984) have used this idea to study the effect of hydrodynamic parameters on the separation performance of hea (i.e., p > p ) particles inasmuch as in W p f , the outer region of the vortex field an inwardly directed radial flow drags the particles away from the conical wall. An analogous phenomenon may also occur for light particles (i.e. , pp < pf) in the central core region of a hydrocyclone operating without an air core (see Dabir and Petty, 1984b; Listewnik, 1984; and, Chen and Petty, 1986). MM Criner and Driesser (see p.44 Svarovsky,l984) first proposed the concept of the equilibrium orbit. According to this concept, a particle in a hydrocyclone flow achieves an equilibrium orbit at a radial position where its terminal settling velocity equals the radial velocity of the continuous phase. If the equilibrium orbit lies inside the locus of zero axial velocity, the particle leaves the cyclone through the vortex finder (see Figure 8.1) due to the upward motion of the fluid. 150 Otherwise, it moves toward the underflow due to the downward motion of the fluid. In this study, several assumptions are made : 1. The axial and tangential components of the particle velocity are the same as the continuous phase; 2. The discrete particles are spherical and pp < pf; 3. Particle acceleration times are small, so the viscous drag on the particle balances the net centrifugal force on the particle everywhere in the flow field; and, 4. Stoke's law applies (see, p.59 Bird et al., 1960). Because pp < pf, the net centrifugal force acting on the particles is directed toward the axis and has a magnitude given by «d 3 u2 p 0 Fc - T (pf - pp)? 2 0. (9.1) In Eq. (9.1), dp represents the diameter of the particle with density p ; pf denotes the density of the continuous fluid phase. u is the p 9 tangential velocity of the fluid. The viscous drag on the particle is directed away from the axis if u > u , and toward the axis if u < u . The magnitude of this force, r pr r pr according to Stoke's law, is Fv - 31rpdp(ur - upr) (9.2) 151 where upr represents the radial component of the particle velocity and p is the mglecula; viscosity of the fluid. The difference between the fluid and particle velocities is often called the "terminal" velocity. Svarovsky [p7, 1984] argues that the time needed for a particle to achieve its terminal settling velocity is very small (2 milliseconds). Therefore, F -F (9.3) is a good practical approximation in the entire flow field. Eqs (9.1) - (9.3) imply that u -u -r-—— (9.4) where the characteristic time fp for the particle is given by (1 - —) (9.5) The sign of upr obviously depends on the relative magnitudes of ur and rpuoz/r. Obviously, if ur rpuOZ/r, then the particle moves away from the axis. The gggilibgigm_21b1§ is defined by upr(rE,zE) - 0. (9.6) 152 From Eq (9.4), this implies that [uo(rE.zE)]2 ur(rE,zE) - 7? r3 (9.7) Eq. (9.7) defines the locus of points (rE,zE) in the flow domain for which Eq. (9.6) holds. The radial and tangential components of the fluid velocity can be related to the previously developed similarity theory (see Table 3.2 and Chapter 7) d6 h . ur - ucaE(h - (N+2);) (9.8a) F u - KSN (9.8c> c l v z e fi:2 5 - ( T) (9.8d) The viscosity coefficient which affects the relaxation time fp is the W kinematic viscosity of the fluid. The viscosity coefficient vein the similarity parameter 6(z) should be interpreted as an "eddy” viscosity (see Section 8.5). By inserting Eqs. (9.8a) and (9.8b) into Eq. (9.7) and rearranging the result, it follows that 153 . h . fpuc ”(h' - (N+2); ) ——EE - F 2 (9.9) 8 _— dz 2-2 ( 3 ) n-n E ' ‘ E The left-hand-side can be interpreted as a intrinsic Stokes' number (cf. Svarovsky, p8). ILt depends explicitly on the properties of the particles (dP' pp/pf), the intrinsic properties of the continuous phase (p, pf?’ and the parameters used to characterize the vortex flow (N, K, we) . The right-hand-side depends explicitly on N and the dimensionless spin parameter b (see Eq. (4.15)). Thus, for a given similarity surface "8’ Eq. (9.9) gives an expression for zE. Because n - r/6(z), the corresponding value of r can easily be calculated as E 1 V2 r - 6(z ) - ( E E) N+2 (9 10) E"EE"EI< ' Thus, with the Stokes number for an arbitrary similarity surface defined by . h - "(h' (N+2); ) 2 v (9.11) Sk(n;b.N) - _ ) - . ’7 .’7’7E it follows from Eq. (9.9) that 154 r K(N+2)z [6(z )1N'2 - Sk(n 'b N) p E E E’ ’ which easily rearranges to Sk(nE;b,N) Zn: " < .puemm > (:7) (942) Thus, for a given similarity surface "E’ Eqs. (9.10) and (9.12) determine the position of the equilibrium orbit, defined by Eq. (9.6), for a given set of physical parameters : N, b, K/ue, rpye' The set of * points (rE,zE) calculated for 0 s "E < n (b,N) defines the equilibrium orbit surface for a specific flow structure. 9 3 ab t The u Su ac An equilibrium surface consists of all points (rE,zE) in the flow domain for which the terminal velocity of the dispersed phase equals the local velocity of the fluid phase. The equilibrium surface is logally stable if a particle in the neighborhood of the surface tends to move toward the surface. It is locally unstable if the particle moves away from the surface. For a given flow situation, the equilibrium surface can be very complex. It may have regions which are locally unstable and regions which are stable. Figure 9.1 illustrates the above stability definition and Figure 9.2 shows the type of behavior which occurs for forward flow and reverse flow vortices. Figure 9.1 shows a portion of an equilibrium surface. Above the surface, the radial velocity of the dispersed phase is outward; and, 155 Locus of points (rE,z E) 2 ur > guo /r Unstable \\ \ Equilibrium Surface A“ .f—n B Q‘D i / \.../ 2 \ l’ u - ru /r " Stable r p 0 2 ur < GPO /r 2 Figure 9.1 Definition of a stable and unstable equilibrium surface Equilibrium 2 surface 156 Equilibrium / 2 surface \ u -0 I stable / unstable 2 ur>$po /r unstable 21.: / -z . -- - '-z . E,min / unstable 7’ E,min stable Stable ur 0) and a reverse flow vortex (a < 0). It follows directly from Eq. (9.4) that on the axis upr -10, because both ur and uo are zero. Because ur < 0 near the axis for reverse flows (see Section 4.2), the set of points rE - 0, O <: z < m represents stable orbits provided a < 0 (reverse flow). E For forward flow, ur > 0 near the axis and eventually exceeds A rpuoz/r for large values of z. This value, defined as zE, follows directly from Eqs. (9.11) and (9.12) by setting "E - 0 : a Sk(0;b,N) - -2N-—-§ (9.13) b 2 N+2 N—2 A 'ZNB/b _ K — 2N 2N zE " ( er(N+2) ) (7:) ' (9'14) A Numerical calculations show that for z > 2E, ur > rpuoz/r near the axis; A 2 and, for z < 23, ur < fpua /r near the axis. Figure 9.2 shows multiple equilibrium orbits for a fixed axial position. Below 2E min’ the orbit is unique for both situations and occurs on the axis. For 2 > zE min and a < 0, three orbits exist. 158 Stable solutions occur on the axis and on the branch near the outer surface of zero radial velocity. An unstable solution occurs near A the inner surface of zero radial velocity. For a > 0 and z > zE, an unstable orbit occurs on the axis (as previously mentioned) and a stable solution exist near the 'mantle' , defined by ur - 0. For 2 < z < E,min E A 2E and a > 0, three solutions exist (see Figure 9.2a). For larger values of b, this region disappears and the branch leaving the axis at A 2E is stable. The surface zE(rE) in this case has a parabolic structure . W The W equilibrium surface (rE,zE) can be constructed for a specific set of parameters b, N, K/ve, and rpve. The W parameters b and N uniquely determine the similarity structure of the vortex: jet-like flow, undulated flow, or reverse flow (see Figure 7.2). Because the experimental studies of Dabir [1983] and Escudier et a1. [1980, 1982] correspond to N - -0.75 (see Figure 8.2), this value of N is used in the parameter calculations presented here. The effect of the flow structure on the quantitative behavior of the equilibrium surface is examined for b - 0.03 (flow reversal), b - 0.10 (undulated flow), and b - 0.30 (jet-like flow). Although the empirical dimensional parameter K/ue may depend implicitly on b (see Table 8.4), a nominal value of 500 cm-(N+1) is used for the equilibrium orbit calculations. 159 The numerical value of Tpue depends on several physical properties and, most importantly, the particle diameter dp' Eq. (9.5) implies that d 2 v p e rpve .- T8— (1 - pp/pf) T. (9.15) Figures 9.3 - 9.5 show the equilibrium surfaces calculated using Eqs. (9.10) and (9.12). Three different values of rpue are presented for each flow situation; Table 9.1 relates these values to a specific particle size, density ratio, and ue/u. The dimensionless Stokes number, defined by Eq. (9.11), parameterizes each equilibrium surface for a fixed value of ’pye' The magnitude of Sk(nE;b,N) at representative points are indicated on each curve. W Figure 9.3 shows an equilibrium surface for jet-like flow (b - 0.3, see Figures 7.7 - 7.9). As previously discussed, stable orbits only A exist on the axis for z < 2E and on the branch near the outer surface of the zero radial velocity (see Figure 9.2a). Therefore, a particle with diameter 32 microns (see Figure 9.3 for other parameters) will obtain its stable orbit through two possible ways. If the particle reaches the stable axis first, it will follow the axis and move upwards until it A reaches 2E. Above that point, the particle will move away from the axis; if the residence time is long enough, the particle will reach the outer equilibrium surface and continue to move upwards and away from the 2 (cm) 30.0 '1 L J. L J l L I L I 20.04 160 II I Llz = O / l l/ I dp = 3.2/u N = —O.75 b = 0.3 -(~+l) K/\)¢= 500, cm ”‘A’ = 10 0, 5% as noted I TrrrTTrTfIf'rT—I rTr] 0.6 0.8 1.0 1.2 1.4- r (cm) Figure 9.3 Equilibrium surfaces for jet—like flow 161 N = —0.75 b = 0.1 K/ge= 500, cm S1V9,t = 0.9 Ve/g = 10 o , SK as noted 'Trfrrr"l'r*rfirr1"-1 0.4 0.6 0.8 1.0 1.2 1.4 r (cm) Figure 9.4 Equilibrium surfaces for undulated fl ow 30.0- 2 (cm) 162 N = —0.75 b = 0.03 K/ge= 500. 9p/9f = 0.9 0, SK as noted Figure 9.5 Equilibrium surfaces for reverse flow 163 Table 9.1 Parameter used to calculate the equilibrium surfaces N - -0.75 , K/ue - 500 cm-1/4 pp/pf - 0.9 , Ve/V - 10 8 2 b r v x 10 , cm d , microns P e P (nominal) 2.21 6.3 (6) 0.3 jet-like flow 13.87 15.8 (16) Figure 9.3 55.56 31.6 (32) 2.21 6.3 (6) 0.1 undulated flow 55.56 31.6 (32) Figure 9.4 221.90 63.2 (63) 55.56 31.6 (32) 0.03 reverse flow 221 90 63.2 (63) Figure 9.5 889.01 126.5 (127) 164 axis. The particle may, however, move directly to the outer stable surface, depending on how it enters the vortex flow. Figure 9.3 shows that the equilibrium surfaces for large particles are positioned at large values of z and that all the equilibrium surfaces become asymptotic to the surface of zero radial velocity. Therefore, for constant zE, smaller particles have larger stable equilibrium orbits and are closer to the outer surface of zero radial A velocity. The value of 213 (see Figure 9.2a) increases as dp increases. A For 32p, 16p, and 6;: size particles the corresponding values of 2E are, resp., 17.2 cm, 5.4 cm, 1.2 cm (see Figure 9.3 and Eq. (9.14)). Likewise, the values of 21?. min decreases from 15.5 cm to 0.9 cm as dp changes from 32 p to 16 p. The intrinsic Stokes' number, defined by Eq. (9.11), can also be calculated by rearranging Eq. (9.12): 4 2N Bil—2' m 2 E (9.16) Sk - rpue(N+2)(K/ve) Thus, for a fixed equilibrium surface, the value of 2 (see Figure E,min 9.2) will produce the maximum value of Sk. This occurs because N < 0. For the case shown in Figure 9.3, Skmax equals 11.6 for <1p - 32p and 8.9ford -6. P 1‘ For undulated, forward flows, the behavior of the equilibrium surface (Figure 9.4) is very similar to the equilibrium surface for jet- like, forward flows. A positive radial velocity always occurs near the 165 axis (see Figure 7.9), which makes the existence of equilibrium orbits A possible. The values of 2E, compared with Figure 9.3, are only reduced slightly: 13.4 cm for 32 p particles (17.2 cm in Figure 9.3) and 1.0 cm for 6 p particles (1.2 cm in Figure 9.3). However, the minimum value of 2 decreases significantly from 15.5 cm to 3.2 cm for 32 p particles E when b decreases from 0.3 to 0.1. The equilibrium surfaces for reverse flow are topologically different from jet-like and undulated flow (Figure 9.5). Because a negative radial velocity always exists near the axis (see Figure 7.9), particles in the vicinity of the axis will always move inward toward the axis. Two mantles of zero radial velocity exist for reverse flow, and equilibrium surfaces can only exist between these two surfaces. Although at 2E > zE,min (see Figure 9.2) three equilibrium orbits exist, only two are stable: one on the axis and the other on the branch near the outer surface of zero radial velocity. Therefore, if the equilibrium orbit concept ultimately determines the location of a particle in the flow field, then it may move either upward following the outer surface or downward following the axis. The two possibilities depend on how the particle is introduced into the vortex flow. The position of z is about 8.0 cm for 127;: and becomes smaller as the E,min particle size decreases. For dp - 32p, is about 0.8 cm, a twenty zE,min fold decrease as the flow field changes from jet-like behavior (b - 0.3) to a reverse flow behavior (b - 0.03). The effect of N on the equilibrium surfaces is also significant. By comparing Figures 9.6 and 9.3, the local minimum orbit occurs at z - A 2E for N - -1.0, but is off the axis for N - -0.75. The size of the 166 i/ / / —.= O 30.0 i // uz 1 I ,/ I O / l u,r 5“ // i / \I / " / 0.0-IQ I\d gs 16)} / l P / l l / l / i / / a.” " 0: // 1 O 20.0 0.011 g / N 3 : ,1 , .. 0. l / . =3 00. : // K/Qe 5 0 ll , 6P / N I / ”/9 / I s new I II 0, SK Q I / ' / I 10.0 ‘ l/ l l 0.15 I / l / l I/ l l /’ ' / ' / l ' ’/ 1 2 1.4 I / 1.0 ' ‘ ’ 0.8 0 4 0.6 C”30.0 0'2 V (cm) like flow 167 mantle reduces by almost half when N decreases from -0.75 to -l.0; and, A the values of 2E for 6;: particles increase from 1.2 cm to 8.4 cm. Because rp a dpz, it follows from Eq. (9.14) that N+2 zad Thus, for large values of N, the flow field has a much sharper classification effect on the particles. For example, if N - ~1/2, then 2E 0: dp? which shows that the axial position of the equilibrium surfaces is relatively insensitive to particle size for small particles, but change significantly for large values of dp' As discussed in Chapter 7, b uniquely determines the similarity structure of the flow for a given value of N. The effect of b on the structure of the particle equilibrium surfaces is clearly seen by comparing Figures 9.3 - 9.5. For large values of b, the size of the outer mantle becomes smaller, and the particles move closer to the axis. At zE - 30 cm and dp - 32p, the equilibrium radius decreases from 1.2 cm to 0.4 cm as b increases from 0.03 to 0.3. Also note that for dp - 32p, the local minimum equilibrium orbit (ZE’rE)min changes in units of cm from (15.4, 0.15) to (0.8, 0.025) as b decreases from 0.3 to 0.03. The equilibrium orbit theory studied in this chapter provides useful insights provided the major assumptions are satisfied. There are, however, some limitations which should be discussed. For instance, the model assumes that Stoke's law is valid, which implies that the 168 particle Reynolds Number is less than unity (see p.68 Bradley, 1965). The particle Reynolds number, defined as dplur - upr' Re - , (9.17) p u can be roughly estimated on the equilibrium surfaces where the terminal velocity equals the radial velocity of the fluid. For d.p - 32;: (0.0032 cm), b - 0.3, N - -0.75, K/ve - 500 cm'o°25, pp/pf - 0.9, and ue/u - 10, the maximum value of ur, 2 cm/s, occurs on the similarity surface "E z 2 (see Figure 7.9). This corresponds to z - 15.6 cm on the equilibrium E surface (see Eq. 9.12 and Figure 9.3). Therefore, the maximum value of Rep is about 0.64 (v - 0.01 cmz/s), which shows that Stoke's law is a reasonable approximation in the neighborhood of the particle equilibrium surface. The model also assumes a balance between the centrifugal and viscous forces. Other forces, such as gravity, acting on the particle are neglected. For dp - 32p, the maximum tangential velocity occurs for "E - 3.5 and 2E - 20 cm (see Figures 7.7 and 9.3) and is about 1200 cm/s. Therefore, the centrifugal force acting on the particle at ZE - 20 cm, r - 0.3 cm is 0.0093 g.cm/sz. The mass of the 32p size particle E (pf - 0.9 g/cm3) is 1.7x10 -8g, so its centrifugal acceleration (Fe/mp) is 5.42x10S cm/s2 or about 550 times larger than the acceleration due to gravity. 169 9 6 onc u o The following conclusions result from the calculations presented in this chapter: A. For a fixed cyclone geometry, large particles are probably not limited by their equilibrium surfaces because they occur at relatively large values of 2. However, small particles are more likely to follow their equilibrium surfaces, which approach the outer surface of zero radial velocity, or 'mantle'. B. This model predicts the existence of equilibrium orbits for the entire range of b studied. C. Different flow behavior will have different types of equilibrium surfaces; the location of these surfaces change significantly with b and N. I). For forward.flow on the axis, the axis is stable only when 2 s 2E; however, it is always stable for reverse flow because ur < 0 near the axis. E. The effect of N on the classification of particles is very significant. From Eqs. (9.14) and (9.5), 2 a d for N - -1 E P A 5/3 2E « dp for N -3/4 2 a dp3 for N - -1/2 Thus , for larger values of N, the flow field has a much sharper Classification effect on the particles. The axial position of the 170 equilibrium surface for larger values of N is relatively insensitive to small particles but changes significantly for large particles. F. Because 'a' is almost proportional to 'b' at large values of 'b' (see Figure 7.3) and because N+2 A a 75'“ 2E or ( —2- ) , it follows that b N+2 2E (b) 2N A Thus as 'b' increases, 2 increases because -1 S N < 0. Therefore, a E A stronger rotation around the axis provides a large 2 provided b does El not affect K/ue. As previously discussed (see Section 9.1), the efficiency of a 3"- hydrocyclone drops sharply when the particle size approaches 20 - 40 microns. The model developed here may provide a theoretical explanation for this phenomenon because small particles are more likely to be limited by their equilibrium surfaces. Thus, in the apparatus of Colman et a1. [1980] , the jet-like behavior shows a very high efficiency for capturing the large particles because their trajectories in the vortex are not limited by equilibrium surfaces. However, as dp decreases and the geometry and flow parameters remain the same, the efficiency is expected to drop as the particle size decreases because the equilibrium surface will cause the particle to miss the vortex finder. 171 This theory provides new information which can be used for light particle separation in vortex flows. The equilibrium surfaces can be located by knowing the properties of the particles (dp’pp/‘OB’ the intrinsic properties of the continuous phase (p,pf), and the parameters used to characterize the vortex flow (b,N,K,ue). The stability of the equilibrium surface (see Section 9.3) yields important information about the ultimate location of the particles. The results show that the stable surfaces always approach a surface of zero radial velocity for large values of 2. Thus, once the above parameters have been estimated, the relationship between the separation efficiency and the hydrodynamics can be quantified. The efficiency can be improved by either adjusting the size of the outlet vortex finder or by adjusting the strength of rotation around the axis, which may be controlled by the inlet velocity. A relationship between the separation efficiency, the size of the vortex finder, and the inlet velocity could be developed experimentally and the results used to study certain aspects of this theory. It is clear from the quantitative calculations presented that a single design and set of operating conditions cannot give the same separation efficiency for all particle sizes. However, a more explicit connection between the design, the hydrodynamics, and the performance of hydrocyclones should be beneficial. CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS 0 a ’0 Some experimental observations, as discussed in Section 1.1, suggest that a similarity theory could be used to describe the flow structure in the core region of swirling flows. Among these observations are the algebraic decay of the centerline velocity, the free-vortex-like swirl velocity in the outer region, and the nonlinear growth of the viscous boundary layer. Model II, motivated by these experimental observations (see Section 4.1), has the feature of variable circulation on similarity surfaces by using a more general vertex in the outer region (see Eq. (3.21)). The boundary conditions require that a zero axial velocity and free-vortex- like swirl .velocity occur at the same care surface. Although these assumptions may not be true for some vortex flows, the theoretical results are consistent with many experiments (see Dabir, 1983; Escudier et al., 1982, 1984). The results have revealed the existence of various flow structures in the vortex core. Three types of flow behavior (reverse, undulated, and jet-like flow) were identified experimentally and theoretically (see Figure 7.8). A study of the solution behavior near the axis provides a_prigri criteria for different flow regimes (see Figure 4.2 and Table 4.1), which can be determined uniquely by one of the four parameters 172 173 local spin parameter ('b'), "excess" pressure on the axis ('a'), centerline axial velocity ('a'), and size of vortex core (n*) as well as K, v, and N. The analysis of the "excess" mechanical energy and the axial forces acting on a fluid particle on the axis has revealed that transition from reverse flow to undulated flow occurs when the inertia force (F1) is zero and the excess mechanical energy (E) is negative. However, the transition from undulated flow to jet-like flow occurs when the viscous force (Fv) and excess mechanical energy are both zero. The parameters K and ”e affect the behavior of the flow. As K/ue increases, the length scale (5) decreases (see Eq. (3.38)) and the velocity scale (uc) increases (see Eq. (3.37)). For K/ue - 500 cm-o'25, we - 0.1 cmz/sec, b - 0.3, N - -0.75, z z 15.6 cm, and n z 2 (see Section 9.5), the axial, tangential, and radial velocities are estimated as 200 cm/sec, 96 cm/sec, and 0.4 cm/sec, resp.. In general, the magnitude of the radial velocity is much smaller than the magnitude of the axial and tangential velocities. For the data evaluated (see Chapter 8), N appears to be independent of the flow rate and insensitive to the contraction ratio of the vortex chamber. The coefficient K, however, is a strong function of the flow rate (00), lnxt not the contraction ratio (Do/D). For N - -l, the macroscopic axial thrust (F2) is independent of the axial coordinate. However, for N i -1, the macroscopic axial thrust is zero at the singular point and increases as 2 increases. 174 Circulation is another feature which is quite different for N - -l and N i -l. A constant circulation on similarity surfaces always occurs for N - -1 (see Section 6.1). However, for N v‘ -1, the circulation on similarity surface (constant q) increases as 2 increases. The distribution of axial component of vorticity on the axis wz(0,z) in this study varies along the axis and is uniquely fixed by 'b'. For N - -l, wz(0,z) is proportional to 2-2. However, in the theoretical study of Ingham and Bloor, 1984, the certerline vorticity is independent of z. The tangential velocity profiles in this study always show a Rankine - type structure with a forced vortex near the axis and a free- vortex-like flow in the outer region. It is found that as b increases, the fraction of the cross sectional area of the vortex in solid body rotation decreases. the radial velocity, however, is quite different for various flow structures. This study revealed that positive radial velocities near the axis always occur for forward flow, whereas negative radial velocities result for reverse flows (see Figure 7.9). The forward flow on the axis in this study is always against an adverse pressure gradient (see Section 4.2). There will always be a region near the singularity point for which the pressure is less than zero. This region (2°), from previous specific calculations (see Section 4.2), is estimated to be about 8 cm for a gas vortex and 5.6 cm for a liquid vortex. In general, they are much smaller than the length of the vortex chambers for most of the experiments studied. Therefore, data for z > 2° are used to compare with theoretical calculations (see Chapter 8). 175 Similarity scaling (see Section 1.1) implies that the total dissipation for N - -l is always unbounded. However, for N vi -1, this study revealed that the total dissipation is unbounded for -l < N 5 -2/3 and is bounded for -2/3 < N < 0. As 'b' increases, the macroscopic flow rate decreases. This is also true for the macroscopic axial torque; however, the macroscopic pressure drop increases as 'b’ increases. Although the macroscopic axial thrust does not uniquely determine the flow behavior, the macroscopic pressure drop determines uniquely the flow structure . 12.2mm The major conclusions of this study can be summarized as follows : (1). This study predicted quantitatively the various flow structures for a viscous vortex (see Figure 7.1 and 7.2). The range of b for reverse and undulated flow behavior broadens as N increases. Transition values of b for three types of flow structures become larger when N is larger (see Figure 7.2). (2) . The viscous boundary layer increases linearly with the axial coordinate for Model I (6 at z) and is parabolic-like for Model II (6 or zl/(N+1) ). Model II reduces to Model I, with only some minor differences (see Figure 7.1), when N - -1. (3). The study of macroscopic properties revealed that the macroscopic axial force, Fz’ did not uniquely determine the flow structure (see Figure 7.10). The macroscopic pressure drop, however, uniquely determines the flow behavior because its one-to-one relationship with the local spin parameter, b (see Figure 7.13). For 176 fixed b, a larger value of N causes a larger macroscopic pressure drop, Ap (see Figure 7.13). (4). N appears to be insensitive to the flow rate and the contraction ratio of the vortex chamber (see Figure 8.2 and Table 8.1). An interesting result is that N has a value near -0.75 for most of the experimental data analyzed and is, thereby, consistent with a theoretical analysis for Beltrami flows in the outer region. (5) . Model II, which considered flow structures induced by a more general flow (-1 S N < 0), has the feature of variable circulation on similarity'surfaces. For the same similarity surface (constant 0), the circulation increases as 'b' increases (see Figure 9.5). For fixed 7) and b, large values of N cause an increase in the circulation (see Figure 9.5). (6). This study revealed that the macroscopic dissipation for Model I is unbounded. However, the total dissipation for Model II is unbounded for -l s N S -2/3, but bounded for -2/3 < N < 0. (7). The pressure on the axis always increases in the direction of increasing axial position for any type of flow behavior. A negative pressure, which exists near the singularity point, is estimated to be the same order of magnitude for gas and liquid vortices (see Section 4.2). However, it is an order of magnitude smaller than the length of many experimental vortex chambers (see Dabir, 1983; Boysan et al., 1982; and Escudier et al., 1980, 1982). (8). The effect of N on the size of the viscous core is significant. When N increases, the size of the core decreases (see 21* in Table A.2 to A.4) for fixed b. Therefore, a more coherent core will occur at larger values of N for fixed b. 177 (9) . When N increases, the lower limit on b for undulated and jet- like flows becomes much larger (see Figure 7.2). Therefore, in order to keep jet-like behavior, a stronger rotation on the axis (large b) is necessary for larger values of N. (10). This study, for the first time, provides a way to understand various flow structures through the excess mechanical energy and force balances on a fluid particle on the axis (see Section 6.5). The viscous force and the substantial time derivative of excess mechanical energy A (DE/Dt) have to change sign when the flow transfers from jet-like to undulated behavior. However, the flow transfers from undulated to A A reverse behavior only if DE/Dt changes sign. (11). The estimates of viscosity coefficient of various experiments based on this study have revealed that the viscosity coefficient cannot be interpreted as a molecular kinematic viscosity, instead it should be considered as a constant eddy viscosity (see Chapter 8). (12). This study, for the first time, provides a possible explanation why the separation efficiency of hydrocyclones drops sharply as the particle size of a light disperse phase decreases. Because the very fine particles are more easily controlled by the equilibrium surfaces, which move far away from axis for smaller particles, they are probably missing the vortex finder, if the vortex finder is too small. 0 c e Based on the results developed in this research, the following recommendations for additional study are suggested. (1) . Although Model II produces many useful results, it treats the problem with some restrictions (see Chapter 3). The model has assumed 178 that the similarity functions at the surface (i.e. , locus of zero axial velocity) require the free-vortex-like swirl velocity (ug - KrN, r z 6*) and continuity of viscous stress. The behavior of the outer flow (i.e. , r > 6*(z)) is still unknown and has to satisfy these boundary conditions at the surface. These restrictions are too fig;hgg and may not exist for some flaws. A study of an outer flow which can match the core behavior in Model II is recommended for development. Another recommendation is to remove the free-vortex-like swirl velocity condition at the surface of zero axial velocity. This may provide a way to match the core flow with an inviscid flow or Beltrami flow in the outer region. The relationship between the core flow and the outer flaw could provide a connection with the geometry and operating conditions. (2). This study has predicted some specific features which are important for the comparison between theoretical and experimental data. These features include : the decay of central axial.velocity, free- vortex-like swirl velocity at the outer region, and the decay of the axial component of angular momentum. Therefore, it is recommended to measure the following velocities very carefully and accurately. (a). Tangential velocity profile, which can be used to determine K and N; (b). Central axial velocity, which is used to determine the direction of the vortex care; (C). Angular velocity around the axis (duo/ar|r_o), which combined with (a) and (b) can be used to determine the value of b; (d). Axial velocity profiles, which provide an estimate of entrainment rate. 179 The above accurate measurements will provide a basis for the comparison between theoretical predictions and experimental results. (3) . In this study, radial velocity profiles play a very important role, esp., in the oil-water separation. Unfortunately, experimental data for radial velocity are very scarce. Because this study predicts a mantle (locus of zero radial velocity) always exists, it is very important to validate this finding. Therefore, it is strongly recommended that the experimental test, either direct measurement or visualization of radial velocity, should be carefully executed for a flooded hydrocyclone, esp., near the central core region. (4) . Although the equilibrium surfaces of light particles can be predicted from this study, the detail trajectory of particles are still unknown. Because the velocity field can be determined from this study directly, it provides a way to calculate the trajectory of dispersed particles. Based on the major assumptions in Chapter 9 and Eq. (9.4), the two differential equations for the spatial coordinates of individual spherical particles can be expressed as Because ur, uz and 7p can be calculated for any initial particle position, the particle trajectory at any time can be calculated by integrating these two non-linear differential equations. Once the 180 trajectory of specific particles have been determined, an estimate of how long it will take for particles to reach their stable equilibrium surfaces can be made. (5). In the study of oil-water separation, this model predicted large particles are probably not limited by the equilibrium surfaces while the small particles have more chance to be controlled by the equilibrium surfaces. Therefore, for collecting different particle sizes of oil droplets, a different design of the vortex finder is necessary. This theory also showed that the quantitative behavior of the ultimate particle location is determined by the local spin parameter, b, for fixed other parameters. Because b could be controlled by the inlet velocity for fixed geometry, it is recommended to study the relationship between inlet velocity (or flow rate) and b. The information could provide a way to improve the efficiency of oil-water separation for various particle sizes through the control of inlet velocity for fixed geometry. APPENDIX A SOLUTIONS (a,b,c,;) FOR MODEL I AND MODEL II 181 Table A.1 Solution (a,b,c,na) for Model I b a c "a .03 -0.0313 .0136 30.0 .04 -0.0540 .0168 28.9 .042 0.0 .0174 28.6 .05 0.0226 .0199 27.7 .06 0.0521 .0230 27.3 .08 0.114 .0289 26.6 .10 0.179 .0347 26.0 .136 0.301 .0449 24.2 .15 0.348 .0489 23.5 .20 0.524 .0627 22.5 .30 0.886 .0898 21.7 182 Table A.2 Solution (a,b,c,n*) for Model II (N - -1.0) b a c n* 0.03 -0.0251 0.0127 30.1 0.038 0.0 0.0151 29.0 0.04 0.00083 0.0158 28.7 0.05 0.0285 0.0190 28.4 0.06 0.0582 0.0219 27.5 0.08 0.120 0.0279 27.1 0.10 0.184 0.0337 26.8 0.14 0.321 0.0447 24.1 0.15 0.355 0.0475 23.4 0.20 0.531 0.0611 21.9 0.30 0.893 0.0881 21.3 183 Table A.3 Solution (a,b,e,n*) for Model II (N - -0.75) b a C 0 0.03 -0.0582 .0186 15.1 0.04 -0.0406 .0232 14.2 0.05 -0.0212 .0276 13.5 0.055 0.0 .0297 13.2 0.06 0.00061 .0318 12.9 0.08 0.0431 .0397 11.9 0.10 0.0889 .0471 11.1 0.20 0.335 .0786 8.79 0.23 0.412 .0860 8.33 0.25 0.464 .0910 8.03 0.30 0.596 .101 7.43 0.50 1.150 .121 5.86 Table A.4 Solution 184 at (a.b.C.n ) for Model 11 (N - -0.5) b a c n 0.03 -0.123 0.0385 13.7 0.04 -0.113 0.0475 12.9 0.05 -0.101 0.0558 12.3 0.06 -0.0873 0.0636 11.8 0.08 -0.0575 0.0780 11.0 0.10 -0.0254 0.0909 10.4 0.115 0.0 0.0963 10.2 0.20 0.152 0.138 8.28 0.25 0.248 0.151 7.55 0.30 0.348 0.157 6.94 0.43 0.634 0.203 5.96 0.50 0.784 0.222 5.31 APPENDIX B COMPUTER PROGRAMS 185 Table B.1 Objective of computer program and FORTRAN symbol list Program Objective ABCI.FOR This program determines (a,c,5) for Models 1 and II at given b and N YI.FOR This program gives Y1 to Y6 with respect to n for Model I and Model 11 DATA1.FOR This program sets up individual data files for non- dimensional momemtum transfer, stream function, angular momentum, pressure field, axial, tangential, and radial velocity profiles for Model I DATA2.FOR This program sets up individual data files for non- dimensional momemtum transfer, stream function, angular momentum, pressure field, axial, tangential, and radial velocity profiles for Model II ORBIT.FOR This program sets up individual data files for particle equilibrium orbits for various particle sizes 186 Subroutine Description ZSCNT This is a subroutine in IMSL library available at Michigan State University. It solves a system of nonlinear equations by secant methods. DGEAR This is a subroutine in IMSL library available at Michigan State University. It solves simultaneous first- order differential equations by implicit ADAMS methods. FORTRAN symbol list A a B b C c eta 5 D . Particle rsdius. F A vector of length N. F is given non-linear equations. FCN The name of a user-supplied subroutine which evaluates the system of equations to be solved. FCNl, FCN2 Name of subroutine for evaluating functions. FCNJ GAMMA H, H1 IER, IERl INDEX, INDEXl ITMAX KV METH, METHl MITER, MITERl N1 N2 NSIG PAR 187 Name of the subroutine for computing the N2 by N2 Jacobian matrix of partial derivatives. F(n,N,b) On input, H contains the next step size in X1. On output H contains the step size used last, whether successful or not. Error parameter Input and output parameter used to indicate the type of call to the subroutine. The maximum allowable number of iterations Defined by K/v Input basic method indicator. METH - 1, implies that the ADAMS method is used. METH - 2, implies that the stiff method of Gear is used. Input iteration method indicator. MITER - 1, implies that the chord method is used with an analytic Jacobian. The number of unknowns Power index where U0 - KrN. Input number of first-order differential equations. The number of digits of accuracy desired in the computed root Par contains a parameter set which is passed to the user supplied function FCN. Equilibrium orbit radius. SPE TOL X1, X2 XEND, XENDl 188 pp/pf Input relative error bound. A vector of length N. On input, X is the initial approx- imation to the root. On output, X is the best approxi- mation to the root found by ZSCNT. Independent variables. On input, X1 and X2 supply the initial value and are used only on the first call. On output, X1 nd X2 are replaced with the current value of the independent variable at which integration has been completed. Input value of X1 at which solution is desired next. 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ON ON 206 OZN szPNm NU.N<.m.—z ZOEZOO X.AZ.ZOOQ.AZO> ZOHmHUNmn. NJNDOO Z «NONPZH AOQ.>.X.Z:.ZON NZHPDOGNDN ozm zmnhmm x\Anmv> + A~O>9on> I on>9x9AmO>9a~+_zO\A..+.zOOuaoOm:~ma> Amv>namvmzuma> x\AA¢O> I A~99Amv>9x9_z + A~O>IA¢O>9A~+_2O I x\AmO>9an> I A_O>.x._z..~IO.A~+.zO\..Ouavvmzmma> Acv>uamvmzuma> Amv>9xuhuvm59ma> Am99xv\AmO>.Amv>Iua_Ouauaa> .u..<.m..z zozzou _u.—<.m._z.x.aszz~ma>.AzO> zowmuumma m4maoo z «momhzu Amzuma>.>.x.zvzum mzmhaoamam ozm aopm Ao.~_u~.xuvp<:¢Om Aw.o_m.. u a ..xo. \v.o_m.. u u ..xo. \v.o.u.. u m ..xo. \m.o_m.. u < ..xo. d d —ON APPENDIX C TABULAR EXPERIMENTAL DATA USED Table C.1 Experimental data set 1a 207 Reference : Experimental conditions QF - 400 cm3/sec, u Escudier, et al., [1980] (see Figure 8.1) - 24 cm/sec F Do 40 Q0 B—n-S-g’ q-Q E - 19.35 cm (from Figures 9 and 10 of Escudier) r,cm 0 .2 .4 1.0 1.2 1.4 1.6 1.8 2.0 2.6 N - -0.75 uo,cm/s o 32 64 67 58 so 44 39 37 33 x - 68 cml'N/s auo \1 ,cm/s -24 -12 0 29 - - 9 - 4 0 -- - 160 2 8r r-0 2 - 19.85 cm (from Figures 9 and 10 of Escudier) r,cm O .2 .4 1.0 1.2 1.4 1.6 1.8 2.0 N - -0.75 u0,cm/s o 32 64 67 58 so 44 39 37 K - 68 cml'N/s auo uz,cm/s -23 -11 O 25 — - 8 ‘ - 4 3?‘ - 160 r-O Table C.2 208 Experimental data set 1b Reference Experimental conditions QF - 400 cm3/sec, u : Escudier, et al., [1980] (see Figure 8.1) - 24 cm/sec F Do 25 Q0 D 55 ’ Qu E - 19.35 cm (from Figures 9 and 10 of Escudier) r,cm 0 .1 .2 1.0 1.2 1.4 1.6 1.8 2.0 2.6 N - -0.75 uo,cm/s o 36 72 67 53 46 39 36 3a 33 K - 68 cml'N/s duo u ,cm/s 3 45 35 28 25 20 15 10 2 -—— - 360 2 6r r-0 2 - 19.85 cm (from Figures 9 and 10 of Escudier) r,cm 0 .1 .2 1.0 1.2 1.4 1.6 1.8 2.0 N - -0.75 u0,6m/s o 36 72 67 53 46 39 36 34 K - 68 cml’N/s auo uz,cm/s 10 80 35 28 20 15 6 0 5;-' - 360 r-O Table c.3 Experimental data set 1c 209 Reference : Experimental conditions QF - 40 Do 10 D_'5'5 Escudier, 0 cm3/sec, uF et al., [1980] - 24 cm/sec (see Figure 8.1) 2 - 19.35 cm (from Figures 9 and 10 of Escudier) r,cm 0 .1 .2 1.0 1.2 1.4 1.6 1.8 2.0 2.6 N - -0.75 uo,cm/s o 80 160 67 55 48 42 38 35 33 x - 68 cml'N/s auo uz,cm/s 144 72 33 28 25 22 19 15 0 5;— - 800 r-0 2 - 19 85 cm (from Figures 9 and 10 of Escudier) r,cm 0 .1 .2 1.0 1.2 1.4 1.6 1.8 2.0 N - -0.75 uo,cm/s o 80 160 67 55 48 42 38 35 K - 68 cml'N/s duo uz,cm/s 450 72 4o 32 24 16 8 0 5E’ - 800 r-O 210 Table C.4 Experimental data set 1d Reference : Escudier, et al., [1982] Experimental conditions : (see Figure 8.1) QF - 100 cm3/sec, u - 6 cm/sec F Do 18 Q0 _-_, ——a) D 55 Qu 2 - 1 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 l-N u0,cm/s 0 19 15 10 8.5 6 9 K - 11 cm /s duo uz,cm/s - - - - - - 5?“ - - r-0 2 - 10 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 2.8 N - -0.75 l-N u0,cm/s 0 19 18 10 8.6 6.9 6.0 K - 11 cm /s aua uz,cm/s - 14 8 6 3 2 0 5E" - - r-O 211 E - 19.85 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 u0,cm/s o 29 17 10 7.9 6.9 K - 11 cml‘N/s duo u ,cm/s - 22 11 8 6 0 'EE- - r-0 2 - 20.55 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 u9,cm/s o 27 17 10 7.9 6.9 K - 11 cml’N/s aua uz,Cm/S 62 ' ' ' ' - 5r— - - r-0 2 - 29.45 cm (from Figures 5 of Escudier) r,cm O 0.1 0.2 0.25 0.50 1.0 1.5 1.6 2.0 N - -0.75 u9,cm/s o 10 20 21 14 10 7.9 7.7 6.9 K - 11 cml'N/s duo u ,cm/s 50 - - 40 36 20 2 0 5;- - 100 r-0 2 - 31.65 cm (from Figures 5 of Escudier) r,cm 0 0.1 0.2 0.25 0.50 1.0 1.5 1.6 2.0 N - -0.75 u9,cm/s o 10 20 14 18 12 9.6 9.4 8.0 K - 11 cml’N/s due u ,cm/s 48 - - 48 45 25 4 0 EE— - 100 r-O 212 Table 0.5 Experimental data set 1e Reference : Experimental conditions QF - 200 cm3/sec, u Escudier, et al., [1982] (see Figure 8.1) - 12 cm/sec F Do 18 Q0 ___, —-co D 55 Qu E - 1 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 1—N u9,cm/s 0 65 34 21 15 13 K - 26 cm /s auo uz,cm/s - - - - - - 5;_ - - r-0 2 - 10 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 2.8 N - -0.75 l-N u0,cm/s 0 69 34 21 15 13 12 K - 26 cm /s auo uz,cm/s - 25 10 6 4 1 0 5;_ - - r-O 213 E - 19.85 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 uo,cm/s o 75 62 22 19 14 K - 26 cml'N/s auo u ,cm/s — 40 3O 20 10 0 5;— - - r-0 2 - 20.55 cm (from Figures 5 of Escudier) r,cm O 0.25 0.50 1.0 1.5 2.0 N - -0.75 uo,cm/s 0 82 41 24 21 17 K - 26 cml-N/s auo u ,cm/s - - - - - - ‘8? - - r-0 2 - 29.45 cm (from Figures 5 of Escudier) r,cm 0 0.1 0.2 0.25 0.50 1.0 1.5 1.6 2.0 N - -0.75 u0,cm/s o 10 20 48 34 24 19 18 17 K - 26 cml'N/s auo u ,cm/s 23.8 30 50 6O 75 3O 10 O 5;— - 100 r-0 5 - 31.65 cm (from Figures 5 of Escudier) r,cm 0 0.1 0.2 0.25 0.50 1.0 1.5 1.6 2.0 N - -0.75 uo,cm/s o 12 24 17 27 22 21 21 19 K - 26 cml'N/s Bug uz,cm/s - - - 4O 55 15 0 EE— - 120 r-O 214 Table C.6 Experimental data set If Reference : Escudier, et al., [1982] Experimental conditions : (see Figure 8.1) QF - 411 cm3/sec, u - 24.6 cm/sec F Do 18 Q0 D 55 ’ Qu E - 1 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 l-N uo,cm/s 0 162 91 49 32 28 K - 47 cm /s aua uz,cm/s - - - - - - ‘a'-r— - - r-0 2 - 10 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 2.8 N - -0.75 l-N u9,cm/s 0 162 84 46 32 28 27 K - 47 cm /s 8u(9 uz,cm/s - 60 10 5 5 2 O ‘5;- - - r-O 215 E - 19.85 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 2.8 N - -0.75 uo,cm/s 0 162 77 46 35 28 26 K - 47 cml‘N/s auo u ,cm/s - 120 60 40 30 20 0 5;-' - r-0 2 - 20.55 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 u0,cm/s o 169 91 56 42 28 K - 42 cml'N/s auo u ,cm/s - - - - - - 5;. - - r-0 2 - 29.45 cm (from Figures 5 of Escudier) r,cm 0 0.1 0.2 0.25 0.50 1.0 1.5 2.0 N - -0.75 u9,cm/s 0 - - 84 70 42 35 28 K - 47 cm1-N/s auo u ,cm/s l7 - - 30 80 140 O -—- - 200 2 8r r-0 2 - 31.65 cm (from Figures 5 of Escudier) r,cm 0 0.1 0.2 0.25 0.50 1.0 1.5 2.0 N - -O.75 u0,cm/s 0 - - 84 71 42 35 28 K - 47 cml-N/s duo u ,cm/s 15 - - 25 80 150 20 0 5;. - 140 r-O vumr Table C.7 216 Experimental data set lg Reference : Experimental conditions QF - 850 cm3/sec, u Escudier, et al., [1982] (see Figure 8.1) - 51 cm/sec F Do 18 Q0 D 55 ’ Qu E - 1 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 uo,cm/s o 290 200 100 82 61 K - 95 cml'N/s auo uz,cm/s - - - - - - '5;- - - r-0 2 - 10 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 l-N u0,cm/s 0 320 175 87 58 44 K - 95 cm /s auo uz,cm/s - - - - - - 5;— - - r-O 217 E - 19.85 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 u0,cm/s o 335 175 93 64 58 K - 95 cml'N/s duo uz,cm/s - 175 80 4O 20 0 5;— - - r-0 2 - 20.55 cm (from Figures 5 of Escudier) r,cm 0 0.25 0.50 1.0 1.5 2.0 N - -0.75 u9,cm/s o 320 204 117 87 64 K - 95 cml‘N/s auo u ,cm/s -62 - - - - - 5;— - - r-0 2 - 29.45 cm (from Figures 5 of Escudier) r,cm 0 0.1 0.2 0.25 0.50 1.0 1.5 1.6 2.0 N - -0.75 uo,cm/s o 25 50 131 131 96 73 7o 58 K - 95 cml‘N/s aua uz,cm/s - - - 80 180 20 0 5;. - 250 r-0 2 - 31.65 cm (from Figures 5 of Escudier) r,cm 0 0.1 0.2 0.25 0.50 1.0 1.5 1.6 2.0 N - -0.75 u6,cm/s o 21 42 52 93 87 67 65 58 K - 95 cml'N/s aue u ,cm/s -52 -30 -5 O 100 190 40 0 5?- - 210 r-O Table C.8 218 Experimental data set 2a Reference : Dabir, [1983] Experimental conditions QF - 500 cm3/sec, u (see Figure 8.1) - 140 cm/sec F Do 25.8 Qo _- ’ —-