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Observations of the flow of a semi-dilute fibre suspension through a sudden expansion using positron… Heath, Stuart James 2006

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Observations of the Flow of a Semi-Dilute Fibre Suspension Through a Sudden Expansion Using Positron Emission Tomography by Stuart James H e a t h B . A . S c , T h e U n i v e r s i t y of B r i t i s h C o l u m b i a , 2003  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF M a s t e r of A p p l i e d Science in T H E FACULTY OF G R A D U A T E STUDIES (MECHANICAL ENGINEERING)  The University of British Columbia A u g u s t 2006 © Stuart James H e a t h , 2006  Abstract  T h e m o t i o n of  F radioactively labeled p a p e r m a k i n g fibres flowing t h r o u g h a n a x i s y m -  metric 1:5 sudden expansion has been studied using p o s i t r o n emission tomography ( P E T ) . Various length fractions of a mechanical p u l p were radioactively labeled a n d then introduced into a non-radioactive aqueous 0.4 wt. % (consistency) w o o d p u l p suspension. F u l l y three-dimensional images were reconstructed b o t h upstream a n d downstream of the expansion plane for cases i n w h i c h the upstream velocity U was set f r o m 0.5 to 0.9 m/s  (an  approximate R e y n o l d s number range of 7,000 to 13,000). T w o distinct flow regimes were clearly identified. W i t h U ~ 0.5 m/s we find that the fibre suspension was not  fluidized  and the tracer fibres passed t h r o u g h the expansion as a p l u g . N o m i x i n g was observed between the confined central jet a n d the static outer region. A t larger velocities, we observed that the suspension was fully fluidized,  fluidized.  O u r results i n this regime indicate that albeit  concentration inhomogeneities were evident. W e find that a particle depletion  zone was evident between the central jet a n d the recirculating zone resulting from the inlet concentration profile formed i n the upstream tube. P a r t i c l e a c c u m u l a t i o n was observed i n the vortices. N o significant differences were observed between the different length tracer fibres.  Contents  Abstract  "  Contents  iii  List of Tables  v  List of F i g u r e s  vi  Acknowledgements  xi  Dedication  xii  1  Introduction  1  2  Positron Emission Tomography  5  2.1  Detectors  6  2.2  Resolution  8  2.3  Attenuation  9  2.4  Scattered Events  10  2.5  Accidental Coincidences  11  2.6  Detector Deadtime  12  2.7  Physical Model  12  2.8  Filtered Backprojection  15  3  Materials and Methods  16  4  Results a n d D i s c u s s i o n  22  5  4.1  The Effect of Velocity  4.2  Effect of Particle Size  4.3  Far Downstream  Summary and Conclusion  References  22 . . .  29 30  35  37  Appendix A  Filtered Backprojection  42  Appendix B  Scan D a t a  46  List of Tables  3.1  A s u m m a r y of the scans conducted  21  B.l  A s u m m a r y of the p o s i t i o n a l d a t a from the scans conducted  46  List of Figures  2.1  Schematic of the detector arrangement i n a P E T t o m o g r a p h  2.2  A schematic of a detector from the m i c r o P E T R 4 system.  6 C o n s i s t i n g of  a n 8 x 8 L S O array, 10 c m fibre optic bundle, a n d a H a m a m a t s u R5900C 8 p o s i t i o n sensitive p h o t o - m u l t i p l i e r tube ( P S - P M T ) . W i t h the readout boards attached to the rear of the tube 2.3  7  S a m p l i n g p a t t e r n i n the t r a n s a x i a l plane for a P E T t o m o g r a p h . E a c h segment i n the detector r i n g represents one crystal. T h e solid lines show the parallel projections for the first angle, the dotted lines for the second angle, a n d the dashed lines for the t h i r d angle  2.4  8  D i a g r a m of a scattered event (left) a n d an accidental coincidence (right). P h o t o n s shown leaving the ring are scattered t h r o u g h a n oblique angle such that their paths do not intersect a detector  10  3.1  A schematic of the sudden expansion  16  3.2  A schematic of the experimental setup  17  3.3  A schematic of the portable flow loop  18  3.4  T h e fibre length d i s t r i b u t i o n s of the S B K p u l p a n d the three tracer fibres used (labeled R 1 4 , R 4 8 a n d R100). T h e nomenclature for the fibre fractions is retained f r o m the B a u e r M c n e t t device used to separate the fibre suspension. 19  4.1  E s t i m a t e s of the size of the recirculation zone for the cases i n w h i c h the suspension was fluidized after the expansion. h  s  4.2  is the step height  F o u r views of the a c t i v i t y profile for the case i n w h i c h the tracer  23 fibres  were w e l l - m i x e d , or fluidized, after the expansion. T h e image was acquired using the R 4 8 tracer fibres w i t h the upstream velocity set at 0.7 m/s  (see  Table 3.1). Image (a) represents the top view, (b) is a cross-sectional view at x/R  0  = 0.57, (c) is a side view, and (d) is a three-dimensional reconstruction  of the a c t i v i t y d i s t r i b u t i o n 4.3  26  F o u r views of the a c t i v i t y profile for the case i n w h i c h the tracer fibres were w e l l - m i x e d , or fluidized, after the expansion. T h e image was acquired using the R 4 8 tracer fibres w i t h the upstream velocity set at 0.7 m/s 3.1).  (see Table  Image (a) represents the concentration profile i n the r a d i a l direction  for a central h o r i z o n t a l slice at four different a x i a l positions.  Image  (b)  represents the concentration profile i n the r a d i a l d i r e c t i o n f r o m a central vertical slice at four different a x i a l positions. Image (c) is a n estimate of the growth of the depletion zone. Estimates are given of the inner a n d outer r a d i i of the depletion zone a n d their difference. Image (d) is a n estimate of the average concentration of the jet as a function of a x i a l p o s i t i o n  27  4.4  F o u r views of the a c t i v i t y profile for the case i n w h i c h the tracer fibres were not w e l l - m i x e d after the expansion. W e define this case as p l u g flow type behavior.  T h e image was acquired using the R 4 8 tracer fibres w i t h the  u p s t r e a m velocity set at 0.5 m/s (see Table 3.1). Image (a) represents the top view, (b) is a cross-sectional view at x/R  Q  — 1.00, (c) is a side view, a n d  (d) is a three-dimensional reconstruction of the a c t i v i t y d i s t r i b u t i o n 4.5  29  A view of the a c t i v i t y profile a n d jet expansion for the case i n w h i c h the tracer fibres were not w e l l - m i x e d after the expansion. T h e figures are from the image acquired using the R 4 8 tracer fibres w i t h the u p s t r e a m velocity set at 0.5 m/s (see Table 3.1). Image (a) represents the concentration profile i n the r a d i a l direction from a central horizontal slice at four different a x i a l positions. Image (b) represents the w i d t h of the a c t i v i t y profile as a function of a x i a l p o s i t i o n  4.6  30  T w o cases i n w h i c h the tracer fibres were w e l l - m i x e d after the expansion. I n images (a)-(c) we examine the case i n w h i c h the u p s t r e a m velocity was set at 0.9 m/s w i t h the R 1 0 0 tracer fibres. Images (d)-(f) represent the case w i t h the R 1 4 tracer fibres. T h e images i n the first c o l u m n ((a) a n d ((d)) represent the t o p view. T h e images i n the second c o l u m n represent the cross-sectional view at x/R  0  = 0.57. T h e r e m a i n i n g two images represent the side view. . .  31  4.7  F o u r views of the a c t i v i t y profile for the case i n w h i c h the tracer fibres were p a r t i a l l y - m i x e d after the expansion. T h e image was acquired using the R 1 4 tracer fibres w i t h the upstream velocity set at 0.5 m/s (see Table 3.1). Image (a) represents the top view, (b) is a cross-sectional view at x/R  0  = 0.57, (c)  is a side view, a n d (d) is a three-dimensional reconstruction of the a c t i v i t y distribution 4.8  32  T h e velocity profile of the suspension as measured using U D V . S i x experiments were conducted i n w h i c h the upstream velocity Vi set i n the range from 0.5 to 1.0 m/s.  T h e error bars represents the s t a n d a r d d e v i a t i o n de-  t e r m i n e d from the average of this d a t a set 4.9  33  T h e concentration profile downstream of the expansion, (a) Scan 1, (b) Scan 2, (c) Scan 5 a n d (d) Scan 6. Details of each experiment are given i n Table 3.1  34  A. l  Projection Geometry  44  B. l  Scan 1 2D Slices  47  B.2  Profiles for Scan 1. H o r i z o n t a l Profiles (a), Jet E x p a n s i o n (b), a n d D o w n stream C o n c e n t r a t i o n (c)  48  B.3  Scan 2 2 D Slices  49  B.4  Profiles for Scan 2. H o r i z o n t a l Profiles (a), V e r t i c a l Profiles (b), D e p l e t i o n Zone D a t a (c), Jet Densification (d), and D o w n s t r e a m C o n c e n t r a t i o n (e).  B.5  Scan 3 2D Slices  B.6  Profiles for Scan 3. H o r i z o n t a l Profiles (a), V e r t i c a l Profiles (b), D e p l e t i o n Zone D a t a (c), Jet Densification (d), and D o w n s t r e a m C o n c e n t r a t i o n (e).  .  50 51  .  52  B.7  Scan 4 2 D Slices  53  B.8  Profiles for Scan 4. D o w n s t r e a m C o n c e n t r a t i o n (a), a n d Jet Densification (b). 54  B.9  Scan 5 2 D Slices  55  B.10 Profiles for Scan 5. H o r i z o n t a l Profiles (a), V e r t i c a l Profiles (b), D e p l e t i o n Zone D a t a (c), Jet Densification (d), a n d D o w n s t r e a m C o n c e n t r a t i o n (e).  .  B . l l Scan 6 2 D Slices  56 57  B.12 Profiles for Scan 6. H o r i z o n t a l Profiles (a), V e r t i c a l Profiles (b), D e p l e t i o n Zone D a t a (c), Jet Densification (d), a n d D o w n s t r e a m C o n c e n t r a t i o n (e).  .  58  Acknowledgements  I would like to t h a n k K e n B u c k l e y , Suzy L a p i , a n d T h o m a s R u t h from the T R I U M F Life Sciences p r o g r a m for a l l their h a r d work, my supervisors M a r k M a r t i n e z a n d James O l s o n for their guidance a n d insight, a n d a l l my friends at the P u l p a n d P a p e r Centre. F i n a n c i a l support from the N a t u r a l Sciences a n d E n g i n e e r i n g Research C o u n c i l of C a n a d a a n d the T R I U M F Life Sciences p r o g r a m are gratefully acknowledged.  STUART  The University August 2006  of British  Columbia  JAMES  HEATH  To my family, R o b e r t , M a r i e - C l a u d e , A l e x a n d r e , and N a t h a l i e for a l l their support and encouragement.  Chapter 1  Introduction  T h e focus of the present work is an experimental study of the concentration d i s t r i b u t i o n of a semi-dilute fibre suspension-undergoing steady flow i n a n a b r u p t 1:5 sudden expansion. A l t h o u g h the flow of multiphase fluids t h r o u g h sudden expansions is found i n many i n d u s t r i a l a n d n a t u r a l settings, there are still many unanswered questions regarding the mechanism of particle dispersion (fluidization) or c l u m p i n g (flocculation). T h e m o t i v a t i o n for the present work stems from a n interest i n the p a p e r m a k i n g process. U n d e r n o r m a l processing conditions, p a p e r m a k i n g suspensions mechanically entangle to f o r m a network, which possesses a measurable y i e l d stress (Duffy & Titchener (1964a,b)).  (1975), T h a l e n & W a h r e n  D u r i n g processing the suspension is fluidized into i n d i v i d u a l fiocs or  fibres,  w i t h weakly correlated velocities, by turbulence created locally by a n a b r u p t expansion. T h i s aids i n evenly dispersing the suspension. I n a paper machine sudden expansions are found i n the cross-flow d i s t r i b u t o r , the turbulence generator, a n d the headbox nozzle. Suspensions w h i c h are evenly dispersed reduce the cloudiness or graininess of the sheet a n d  a i d i n the e l i m i n a t i o n of grammage (areal density) variations, a property that is i m p o r t a n t for a l l paper grades. W h i l e the objectives of the p a p e r m a k i n g process are clear, a n d the equipment on the paper machine well k n o w n , the exact m e c h a n i s m by w h i c h f l u i d i z a t i o n and m a t e r i a l r e d i s t r i b u t i o n occurs remains obscure. O u r objective is to provide insight into this phenomenon by v i s u a l i z i n g the flow of the p a p e r m a k i n g suspension t h r o u g h a sudden expansion using P o s i t r o n E m i s s i o n Tomography ( P E T ) . U n d e r s t a n d i n g the m o t i o n of a n aqueous fibre suspension flowing t h r o u g h a sudden expansion is difficult. Insight into this phenomena can be gained by first e x a m i n i n g the simpler case of the flow of single-phase fluid.  For N e w t o n i a n fluids, M a c a g n o & H u n g  (1967)  indicate that over a l l R e y n o l d s numbers, a vortex exists i m m e d i a t e l y downstream of the expansion.  T h e r e is general agreement that at low Re the size of the vortex increases  linearly w i t h R e y n o l d s number a n d then decreases w i t h Re > 635 ( L a t o r n e l l & P o l l a r d (1986)) . I n contrast w i t h this, N o n - N e w t o n i a n fluids exhibit vortex lengths that differ significantly from N e w t o n i a n fluids. W i t h y i e l d stress fluids, the r e c i r c u l a t i o n lengths were found to be smaller w h e n compared to N e w t o n i a n fluids at a comparable R e y n o l d s number ( H a m m a d et a l (1999), Jossic et a l (2002)). T h e case of expansion flows w i t h particle suspensions remains largely unexplored.  This  class of flow is s t r i k i n g l y different t h a n single-phase flows as particle concentration i n homogeneities are generated t h r o u g h particle collisions, shear i n d u c e d particle m i g r a t i o n (Leighton & A c r i v o s (1987), P h i l l i p s et a l (1992)), density differences between the particles and the carrier f l u i d , inertia; a n d , i n the case of p a p e r m a k i n g fibres, mechanical  floccula-  t i o n of the particles fibres (Kerekes et a l (1985)). To help i l l u s t r a t e this complexity, there is evidence that w i t h suspensions of neutrally buoyant monodisperse spheres, particle acc u m u l a t i o n or depletion is evident i n the vortex depending u p o n the r a t i o of the upstream  tube to particle diameters ( A l t o b e l l i et a l a recent study, M o r a c z e w s k i et a l  (1997a,b), K a r i n o & G o l d s m i t h  (1977)). I n  (2005) observe that a low concentration region exists  which divides the central jet a n d the recirculation region. T h e y a t t r i b u t e this to inhomogeneities i n the inlet concentration that were convected downstream.  T h e concentration  profiles i n this case were measured using N M R after the flow h a d been stopped. W i t h regards to p a p e r m a k i n g suspensions, there is evidence of seemingly two different behaviours. A r o l a et a l (1998), for example, imaged the a x i a l velocity profile of a 0.5% (wt) wood p u l p suspension flowing t h r o u g h a 1:1.7 sudden expansion using nuclear magnetic resonance i m a g i n g ( N M R ) . These authors report that the p u l p suspension exhibited behavior similar to that of a confined jet. I n recent work, S a l m e l a & K a t a j a  (2005) used an  optical technique to measure the floe size a n d fibre flow field of a semi-dilute suspension after the expansion. T h e y report that the recirculation eddy downstream of the expansion plane was found only to exist when the step height exceeds the mean fibre length. W h e n existing, the suspension was fluidized and behaved as a N e w t o n i a n  fluid.  O u r work is focused on c o m p l i m e n t i n g these previous studies by measuring the steadystate concentration profiles of p a p e r m a k i n g fibres as they pass t h r o u g h a sudden expansion. Here, the behaviour of F l u o r i n e - 1 8 ( F ) labeled p a p e r m a k i n g fibres flowing i n the midst of 1 8  non-radioactive fibres are studied using P E T . We measure the r a d i o a c t i v i t y d i s t r i b u t i o n , three-dimensionally, near the expansion plane a n d then far downstream of the expansion plane. T h e e x p e r i m e n t a l conditions were such that the b u l k concentration of the suspension was fixed while we varied the volumetric flowrate and size of the  1 8  F labeled fibres. T h e  key advantage of this measurement technique is that the concentration profile can be determined for each particle fraction directly w i t h o u t stopping the flow. I n a d d i t i o n , we measure the a x i a l velocity of the suspension far downstream of the expansion plane using  pulsed u l t r a s o u n d D o p p l e r anemometry ( U D V ) . In chapter 2 the technique b e h i n d P E T is quickly reviewed. C h a p t e r 3 describes the experimental apparatus a n d e x p e r i m e n t a l protocol used i n this work. T h e results from the six successful scans are discussed i n chapter 4. T h e results are presented i n three subsections. F i r s t the effect of velocity is examined, followed by the effect of p a r t i c l e size; a n d , finally, the observations made far downstream of the expansion. C h a p t e r 5 summarizes the major findings i n this work. A p p e n d i x A provides a more detailed look at the mathematics of P E T ; specifically, the filtered-backprojection a l g o r i t h m used to reconstruct a l l of the i m ages i n this study. A p p e n d i x B provides each of the 63 2D slices a n d associated figures for each of the cases e x a m i n e d i n this study.  Chapter 2  Positron Emission Tomography  P o s i t r o n emission tomography ( P E T ) is a non-invasive i m a g i n g technique developed i n the medical field for measuring the metabolic a c t i v i t y of cells in-vivo.  P E T is unique because  it produces images of basic biochemistry or function, rather t h a n other diagnostic imaging techniques such as x-rays, C T scans or magnetic resonance i m a g i n g ( M R I ) , w h i c h produce images of anatomy or structure (Ollinger & Fessler (1997)). P o s i t r o n e m i t t i n g isotopes of carbon ( C ) , nitrogen ( N ) , oxygen ( 0 ) 1 1  1 3  1 5  a n d fluorine ( F ) are p r o d u c e d i n an on-cite 1 8  cyclotron a n d are i n c o r p o r a t e d into compounds of biological interest. These isotopes have relatively short half-lives of 20.3 minutes, 9.97 minutes, 2.03 minutes a n d 109.8 minutes respectively. A P E T s t u d y begins w i t h the injection of a r a d i o p h a r m a c e u t i c a l . To allow for transport to, a n d uptake by, the organ of interest the scan is begun after a delay ranging from seconds to minutes. W h e n the radio-isotope decays it emits a p o s i t r o n , w h i c h travels a short distance before a n n i h i l a t i n g w i t h an electron. T h i s a n n i h i l a t i o n produces two highenergy (511 keV)  photons propagating i n nearly opposite directions. If two photons are  Front View  Side View  F i g u r e 2.1: Schematic of the detector arrangement i n a P E T t o m o g r a p h . detected i n a short (~10 ns) t i m i n g window, a n event is recorded along the line connecting the two detectors, sometime referred to as the line of response ( L O R ) . S u m m i n g many such events results i n quantities that approximate line integrals t h r o u g h the radio-isotope d i s t r i b u t i o n (Ollinger & Fessler  (1997)).  F o r 2 D i m a g i n g these line integrals form a  discrete a p p r o x i m a t i o n of the R a d o n transform (Deans  (1983)) of a cross-section of the  radio-isotope concentration, a n d can be inverted to form a n image of the radioisotope d i s t r i b u t i o n . A schematic d i a g r a m of a P E T tomograph is shown i n F i g u r e 2.1.  2.1  Detectors  T h e most c r i t i c a l components i n a P E T tomograph are the detectors ( D a h l b o m & Hoffman  (1988)). Detectors are arranged i n blocks as shown i n F i g u r e 2.2. Detector blocks  are formed by o p t i c a l l y c o u p l i n g a rectangular bundle of crystals to one or more photomultiplier tubes ( P M T s ) . W h e n a photon is incident o n the c r y s t a l , electrons are moved from the valence b a n d to the conduction b a n d . L i g h t is e m i t t e d as these electrons r e t u r n to  Readout Boards < ? J 3 - ' J 1 / / X P h o t o - m u l t i p l i e r Tube  F i g u r e 2.2: A schematic of a detector from the m i c r o P E T R 4 system. C o n s i s t i n g of a n 8 x 8 L S O array, 10 c m fibre optic bundle, a n d a H a m a m a t s u R 5 9 0 0 - C 8 p o s i t i o n sensitive photo-multiplier tube ( P S - P M T ) . W i t h the readout boards attached t o the rear of the tube. the valence b a n d at i m p u r i t i e s i n the crystal. Since i m p u r i t i e s i n the c r y s t a l usually have meta-stable excited states, the light o u t p u t decays exponentially at a rate characteristic to the c r y s t a l . T h e i d e a l c r y s t a l has: (1) h i g h density, so that a large fraction of incident photons scintillate, (2) h i g h light o u t p u t , for positioning accuracy, (3) fast rise t i m e , for accurate t i m i n g , a n d (4) a short decay time, so that h i g h c o u n t i n g rates can be handled (Ollinger h Fessler (1997)). T h e block is fabricated i n such a way that the amount of light collected b y each P M T varies uniquely depending o n the c r y s t a l i n w h i c h the s c i n t i l l a t i o n occurred ( D a h l b o m & H o f f m a n  (1988)).  Hence, integrals of the P M T outputs can be  decoded t o y i e l d the p o s i t i o n of each scintillation. T h e s u m of the integrated P M T outputs is p r o p o r t i o n a l to the energy deposited i n the crystal. T h e m i c r o P E T R 4 scanner used i n this study uses blocks of l u t e t i u m oxyorthosilicate ( L S O ) crystals, arranged i n a n 8 x 8 array,  fibre-optically  coupled t o a single p o s i t i o n sensitive p h o t o m u l t i p l i e r t u b e ( P S - P M T ) .  E a c h c r y s t a l is 2.1 mm wide i n the transverse plane, 2.1 mm wide i n the a x i a l dimension a n d 10 m m deep.  Figure 2.3: S a m p l i n g p a t t e r n i n the transaxial plane for a P E T t o m o g r a p h . E a c h segment i n the detector r i n g represents one crystal. T h e solid lines show the parallel projections for the first angle, the dotted lines for the second angle, a n d the dashed lines for the t h i r d angle.  2.2  Resolution  W h e n d a t a is acquired i n the 2 D slice-collimated mode, the L O R s connecting crystals can be binned into sets of parallel projections at evenly spaced angles, as shown i n F i g u r e 2.3. T w o characteristics are evident. F i r s t , samples are unevenly spaced, w i t h finer sampling at the edges of the field of view t h a n at the center. Second, the samples along the heavy line at angles one (#i) a n d three (63) are offset from the samples at angle two (#2) by one-half of the detector spacing (66N,N-I)-  Therefore, adjacent parallel projections can be combined  to y i e l d one-half the number of projection angles w i t h a s a m p l i n g distance of one-half the detector w i d t h (Ollinger & Fessler (1997)). T h e N y q u i s t c r i t e r i o n states that the s a m p l i n g distance be one-half the spatial resolution, expressed as the f u l l - w i d t h - a t - h a l f - m a x i m u m ( F W H M ) . T h e f u l l - w i d t h - a t - h a l f - m a x i m u m is defined as the distance between the half-value points of the impulse response.  T h i s is  the m i n i m u m separation required to resolve two distinct points. Hence, the m i c r o P E T R 4  detector block w o u l d support a spatial resolution of 2.1 m m . I n fact, a t o m o g r a p h w i t h this c r y s t a l size has a measured resolution that is somewhat worse; v a r y i n g from 1.8  mm  at the center of the field of view, to 2.5 mm at the edge. T h e best obtainable resolution is termed the intrinsic resolution. T h i s resolution is rarely achieved i n practice because unfiltered images are usually very noisy. T y p i c a l tomographs have intrinsic resolution of less t h a n 5 m m , the final resolution of the image usually being greater t h a n 8 m m .  T h i s is because the reconstruction algorithms trade off resolution  for reduced image variance. T h i s final resolution is called the reconstructed resolution. Therefore, the resolution of P E T images as they are t y p i c a l l y used is not determined by the detectors, b u t by the degree to w h i c h resolution must be degraded to achieve a n acceptable image variance. Since the variance is determined by the number of counts that can be collected d u r i n g the scan, the constraints that govern the resolution of P E T images are the amount of r a d i o a c t i v i t y used, the scan d u r a t i o n , the sensitivity of the tomograph, and the count-rate c a p a b i l i t y of the tomograph (Ollinger & Fessler (1997)).  2.3  Attenuation  For an incident 511 keV  p h o t o n there are two possible interactions; photoelectric ab-  sorption, a n d C o m p t o n scatter. I n materials w i t h low atomic numbers the incidence of photoelectric a b s o r p t i o n , for 511 keV photons, is negligible. I n a C o m p t o n interaction the photon interacts w i t h a n outer shell electron. I n doing so its p a t h is deflected, a n d it loses some of its energy. M o s t scattered photons are scattered out of the field of view a n d are never detected. T h e effect of these interactions is termed a t t e n u a t i o n . T h e p r o b a b i l i t y of a photon not i n t e r a c t i n g as it propagates along the line £, at transverse distance d, a n d  Scattered Event  Accidental Coincindence  F i g u r e 2.4: D i a g r a m of a scattered event (left) a n d a n accidental coincidence (right). Photons shown leaving the r i n g are scattered t h r o u g h an oblique angle such that their paths do not intersect a detector. angle 9 is termed the s u r v i v a l p r o b a b i l i t y ; a n d is given by (2.1) where pi is the linear a t t e n u a t i o n coefficient at position x. T h e significance of equation  (2.1)  is that the a t t e n u a t i o n experienced by a given pair of a n n i h i l a t i o n photons is independent of the p o s i t i o n of their a n n i h i l a t i o n along the L O R . T h i s makes possible a simple precorrection of the d a t a (Ollinger & Fessler (1997)).  2.4  Scattered Events  A n n i h i l a t i o n s i n w h i c h one or b o t h photons are scattered, b u t b o t h are s t i l l detected, are termed scattered events. T h i s is depicted on the left i n F i g u r e 2.4. These events are incorrectly positioned because the photons' paths are no longer collinear. T h e overall effect is to a d d an error signal to the d a t a at low spatial frequencies. Since photons lose some of their energy w h e n they undergo C o m p t o n interaction, they can be be d i s c r i m i n a t e d f r o m un-scattered photons by measuring the energy they deposit  i n the crystal. A l t h o u g h this measurement is o n l y accurate to w i t h i n + / -  10% o n most  tomographs, it can be used w i t h a simple threshold to reject a significant fraction of scattered events (Ollinger & Fessler (1997)).  2.5  Accidental Coincidences  W i t h so m a n y scattered photons a n d the relatively s m a l l solid angle presented by the detector r i n g , it is apparent that for many annihilations only one of the photons w i l l be detected. These events are termed singles. If two singles arising f r o m separate annihilations are detected w i t h i n the same coincidence t i m i n g window, they w i l l be recorded as shown on the right side of F i g u r e 2.4. These events are termed accidental coincidences, or randoms. T h e rate of accidental coincidences can be related to the singles rate by n o t i n g that for each single detected at detector i, on average TRJ singles occur at detector j d u r i n g the coincidence t i m i n g w i n d o w r ; where Rj is the singles rate at detector j. these TRJ singles results i n a coincidence, there are TR^RJ  Since each of  coincidences per unit time for  w h i c h the first detected p h o t o n is incident on detector i. T h e t o t a l n u m b e r of accidental coincidences is the s u m of those for w h i c h the first p h o t o n is detected at detector j a n d those for w h i c h the first p h o t o n is detected at detector i.  Hence, the rate of r a n d o m  coincidences along the L O R connecting detectors i a n d j is given by  Rr  = 2TRIRJ  (2.2)  E x a m i n a t i o n of equation (2.2) shows that reducing the coincidence t i m i n g w i n d o w reduces the counting rate of accidental coincidences. However, t i m i n g inaccuracies due to variations i n the rise-time of the c r y s t a l light output require a t i m i n g w i n d o w of 6-8 ns for L S O . Since the incident singles rates are p r o p o r t i o n a l to the amount of r a d i o a c t i v i t y , the accidental  coincidence rate increases as the square of the amount of r a d i o a c t i v i t y i n the field of view (for counting rates that do not saturate the detectors).  T h i s count-rate l i m i t a t i o n , along  w i t h detector deadtime, determines the upper l i m i t on the r a d i o a c t i v i t y used for many studies.  2.6  Detector Deadtime  T h e time required to process a single event limits the counting rate of a P E T scanner (Hoffman et a l  (1989)).  E v e n t processing begins w i t h the rising edge of the pulse for  the first detector involved. T h e pulse is integrated for some t i m e interval, then p o s i t i o n calculations a n d energy d i s c r i m i n a t i o n are performed. T h e detector is " d e a d " to new events d u r i n g this time. A t very low counting rates, randoms are negligible a n d the number of true events is linearly related to the amount of a c t i v i t y i n the field of view. T h e number of randoms increase as the square of the r a d i o a c t i v i t y i n the field of view u n t i l deadtime becomes significant. T h e n the number of true events begins to saturate. A s the counting rate increases further, the numbers of trues a n d randoms peak a n d t h e n decline because of detector s a t u r a t i o n . D e a d t i m e is the dominant effect that l i m i t s the amount r a d i o a c t i v i t y used.  2.7  Physical Model  If statistical effects are ignored, these factors can be incorporated into a m o d e l for the t o t a l number of recorded events to y i e l d YM = 7de[Vde d9M g p  + rf r  d  dg  12  de  + r} s \ s  dg  d6  (2.3)  where MdQ is the number of a n n i h i l a t i o n s w i t h photons e m i t t e d along a L O R (specified by (d, 0) i n F i g u r e 2.3), Pde is the s u r v i v a l probability, T&Q is the n u m b e r of accidental coincidences,  sae is the number of scattered events, rf  d6  is the p r o b a b i l i t y of detection  for true events, rf g is the p r o b a b i l i t y of detection for accidental coincidences, r) d  dd  is the  p r o b a b i l i t y of detection for scattered events, a n d 7^0 is the p r o b a b i l i t y of a n event not being lost due to deadtime. P r i o r to the emission scan, a t r a n s m i s s i o n scan is performed to characterize the effects of a t t e n u a t i o n from the subject. Here a point source c o n t a i n i n g 5 7  C o rotates a r o u n d the subject to provide a flux of photons along each line of response.  T h e measured d a t a y i e l d the number of t r a n s m i t t e d events, Tde, along each L O R . E v e r y m o r n i n g a b l a n k scan , i.e. a t r a n s m i s s i o n scan w i t h n o t h i n g i n the t o m o g r a p h , is performed to y i e l d a d a t a set, E>de- T h e s u r v i v a l p r o b a b i l i t y is a p p r o x i m a t e d by their r a t i o : P = T h i s estimate of s u r v i v a l probabilities w o u l d be exact if the d a t a were noiseless. However, they are not noiseless, so they contribute significantly to the overall image variance unless noise r e d u c t i o n a l g o r i t h m s are applied. These algorithms u t i l i z e s m o o t h i n g (Palmer et a l (1986)), segmentation a n d re-projection ( M e i k l e et a l (1993), X u et a l (1994)), or s t a t i s t i c a l image reconstruction a n d re-projection ( B o u m a n & Sauer  (1996), Fessler (1995), Fessler  et a l (1996), O l l i n g e r (1992)). A simple way to estimate the accidental coincidences is to note t h a t the a r r i v a l times of the photons due to r a n d o m s are u n i f o r m l y d i s t r i b u t e d i n t i m e while those of true coincidences fall w i t h i n the t i m i n g w i n d o w .  C o l l e c t i n g d a t a i n a second coincidence t i m i n g w i n d o w  that is offset i n t i m e , such t h a t it collects no true coincidences, yields d a t a w i t h nearly the same mean as t h a t of the accidental coincidences falling i n the trues t i m i n g window. T h e measured d a t a are given by the product Id6'ifd0 d6i so the detector efficiencies r  for  accidental coincidences, r] , do not have to estimated. Therefore, not only is the m e t h o d de  simple to implement, but it can be performed i n hardware before the d a t a are stored. T h e major drawback of this approach is that the variance of the estimate is of the same order of magnitude as the variance of the d a t a , i f a significant fraction of detected events are accidental coincidences.  I n this case, the subtraction can lead to a significant increase  i n the variance of the d a t a unless noise reduction methods are used (Casey &; Hoffman (1986)). T h i s variance increase can be avoided by counting the n u m b e r of singles at each detector a n d using the rate of r a n d o m coincidences, R .  Since there are m a n y more singles  r  t h a n true coincidences, the effect on variance is relatively m i n o r .  T h i s approach is not  widely used because of the a d d i t i o n a l requirements placed o n the a c q u i s i t i o n hardware, and because singles rates often vary over the course of a n acquisition. T h e detector efficiencies for true a n d scattered events are estimated from a scan of a calib r a t i o n source w i t h k n o w n characteristics (Hoffman et a l (1989)). D e a d t i m e is dependent on many factors related to the architecture and design of a specific machine, so its estim a t i o n is tailored to the t o m o g r a p h ( D a u b e - W i t h e r s p o o n & C a r s o n (1991)). It is usually assumed to be constant over the d u r a t i o n of the scan. These parameters can be used to estimate the number of e m i t t e d photons by using the expression M  de  = ^ - R  d  e ) - S  d  g  { 2 A )  1deV Pde de  where we assume that R g d  = j eri gE[r g], d  d  tion. T h e d a t a modeled i n equation  d  S g = Yd9V gE[ de], d  r  d  s  a n d E[-] denotes expecta-  (2.4) are often stored i n 2 D arrays w i t h the columns  indexed by d a n d the rows by 9. These d a t a arrays are often called sinograms. T h i s is because, for a point source, d varies sinusoidally w i t h 9.  2.8  Filtered Backprojection  One way to s i m p l i f y the p r o b l e m is to ignore the measurement noise altogether, a n d to assume that the measured d a t a approximate line integrals t h r o u g h the radioisotope d i s t r i b u t i o n . T h i s leads to the classical  filtered-backprojection  image reconstruction ( K a k & Slaney  (1988)).  ( F B P ) m e t h o d for tomographic  T h i s m e t h o d is r o u t i n e l y used for x-ray  C T , as well as for P E T a n d S P E C T (Single P h o t o n E m i s s i o n C o m p u t e d Tomography). Its p o p u l a r i t y stems f r o m its c o m p u t a t i o n a l simplicity, a n d not because of any advantage i n image quality. Appendix A .  A m a t h e m a t i c a l description of filtered b a c k p r o j e c t i o n is provided i n  Chapter 3  Materials and Methods  T h e portable closed-loop system used for these experiments consists of a 120 L tank, a centrifugal p u m p , a bypass loop, two magnetic flow meters, two pressure transducers, a test section a n d valves for control. T h e test section is made of clear polycarbonate pipe 70 mm i n diameter a n d 1.1 m i n length. T h e inlet pipe is 14 mm i n diameter, forming a 1:5 a x i s y m m e t r i c sudden expansion. F i g u r e 3.1 shows a cross-sectional view of the abrupt expansion. F l o w reaches a n d leaves the test section t h r o u g h 4.5 m of reinforced hose, w h i c h ensures  r  L mm  FLOW F i g u r e 3.1: A schematic of the sudden expansion.  mm  L  Full Bore Quick Disconnect  Lead Shielding  Test Section  Linear Stage  Full Bore Quick Disconnect  Lead Shielding  Linear Stage  F i g u r e 3.2: A schematic of the experimental setup. fully developed flow at the expansion step for a l l cases studied. B o t h ends of the test section are t e r m i n a t e d w i t h a p a i r of full-bore b a l l valves a n d a full-bore quick-disconnect coupling to facilitate placement a n d removal of the test section into the gantry of the tomograph. T h e test section is m o u n t e d to 760 mm linear stages on either side of the t o m o g r a p h so that the test section can be moved along its axis. To shield the detector blocks from r a d i a t i o n originating outside the t o m o g r a p h 19 mm thick lead shielding is positioned concentrically w i t h the test section, b u t t e d up against the camera. T h i s thickness of lead stops ~ 9 5 % of i n c o m i n g 511 keV g a m m a photons. F i g u r e 3.2 is a schematic of the e x p e r i m e n t a l setup and F i g u r e 3.3 represents the portable flow loop. T h e experiments were conducted by first radioactively l a b e l l i n g a selected B a u e r - M c N e t t fraction of T M P fibres w i t h  1 8  F a n d i n t r o d u c i n g t h e m into a non-radioactive p u l p suspen-  sion. T h e fibre length of each fraction of fibres a n d the whole suspension, as determined through use of a n o p t i c a l fibre analyzer, are provided i n F i g u r e 3.4. A s shown, the names of the fibre fractions are defined using the screen sizes by w h i c h they were retained i n the B a u e r - M c N e t t device. T h i s is the t r a d i t i o n a l m e t h o d of defining fibre fractions i n the p u l p  Bypass Loop  »- To Test Section "• Return Line  LT"  Transmitters  Pressure Transducers  Power Box Tank  F i g u r e 3.3: A schematic of the portable flow loop. and paper literature. T h e fibres were labeled by suspending t h e m i n a solution of acetic acid while  1 8  F — F2 was b u b b l e d t h r o u g h the suspension at 10 m l / m i n w i t h constant stir-  ring. A f t e r the a d d i t i o n of the fluorine the fibres were filtered a n d washed w i t h distilled water. A t this point the fibres were labeled w i t h radioactivity i n t r o d u c e d . tracers such as  1 5  0,  n  1 8  C , or  1 8  F w i t h a 10% y i e l d based u p o n the t o t a l  F has been chosen here i n preference to other p o s i t r o n e m i t t i n g 1  3  N because of its reasonably long half-life of 110 minutes a n d  its reactivity w i t h T M P p u l p fibres. T h e tomograph used i n this study is the Concorde M i c r o s y s t e m s m i c r o P E T R 4 .  The  m i c r o P E T R o d e n t 4-ring system (R4) has a 7.8 c m a x i a l extent, a 10 c m t r a n s a x i a l field of view ( F O V ) a n d a 12 cm gantry aperture. T h e system is composed of 96 detector modules, each w i t h a n 8 x 8 array of 2.1 x 2.1 x 10 m m l u t e t i u m oxyorthosilicate ( L S O ) crystals, arranged i n 32 c r y s t a l rings 14.8 c m i n diameter. E a c h of the detector crystals are coupled to a H a m a m a t s u R 5 9 0 0 - C 8 p o s i t i o n sensitive p h o t o m u l t i p l i e r tube ( P S - P M T ) v i a a 10 c m long o p t i c a l fibre bundle.  T h e detectors have a t i m i n g resolution of 3.2 n s , a n average  Fibre Length (mm) Figure 3.4: T h e fibre length distributions of the S B K p u l p a n d the three tracer fibres used (labeled R 1 4 , R 4 8 a n d R100). T h e nomenclature for the fibre fractions is retained from the Bauer M c n e t t device used to separate the fibre suspension. energy resolution of 18.45%, and a n average intrinsic s p a t i a l resolution of 1.75 m m .  The  system operates i n 3D mode w i t h o u t inter-plane septa, a c q u i r i n g d a t a i n list mode. U s i n g the 2D filtered back projection reconstruction a l g o r i t h m , the resolution i n the centre of the field of view ( F O V ) is 2.03 m m F W H M i n the tangential d i r e c t i o n (horizontal direction of the F O V ) , a n d 2.07 m m F W H M i n the r a d i a l direction (vertical d i r e c t i o n of the F O V ) . T h e tangential resolution slowly increases to 3.38 m m F W H M at the edge of the F O V . T h e r a d i a l resolution increases to 3.00 m m F W H M at 25 m m r a d i a l offset a n d t h e n deteriorates linearly to 3.68 m m F W H M at the edge of the F O V ( M o k et a l  (2003)). A l l images i n  this study were reconstructed using the 2D filtered back p r o j e c t i o n a l g o r i t h m . Before beginning a series of scans w i t h p u l p we started w i t h three test scans using water and 1 1  Cfmethyl-iodide]. T h e purpose of the p r e l i m i n a r y scans were two fold. F i r s t l y , we wanted  to characterize the effect of h a v i n g ~ 9 5 % of the a c t i v i t y located outside of the tomograph's  field of view; a n d secondly, to find out the count-rate c a p a b i l i t y of the t o m o g r a p h w i t h our particular apparatus. I n the first scan we set up the flow loop a n d filled the tank w i t h 40 L of water a n d added 3880 MBq  o f C [ m e t h y l - i o d i d e ] . T h e a c q u i s i t i o n was r u n for 2.5 hours 11  ( 7 half-lives). A f t e r reconstructing the d a t a we found that the a c t i v i t y h a d decayed m u c h faster t h a n expected. W e discovered that the b o i l i n g point of  11  C [ m e t h y l - i o d i d e ] was ~ 2 6  °C a n d that the a c t i v i t y was volatile. I n our second scan we set up a sealed sudden pipe expansion w i t h the same d i a m e t r i c a l dimensions as the test section, b u t shortened to 30 c m i n length. T h e new p h a n t o m was filled w i t h water a n d 330 MBq  of C [ m e t h y l - i o d i d e ] n  and scanned for 2.5 hours. T h e results indicated that the t o m o g r a p h h a d trouble w i t h very h i g h a n d very low count rates. A f t e r 3.3 half-lives the the t o m o g r a p h was able to measure the a c t i v i t y i n the p h a n t o m , a n d thereafter for a p e r i o d of 3.3 half-lives. F o r 40 L of p u l p suspension this translates into a m a x i m u m a c t i v i t y of 1110 MBq MBq.  W i t h this d a t a we repeated the first scan w i t h 1000 MBq  a n d a m i n i m u m of 110 of  11  C[methyl-iodide] i n  solution w i t h a s o d i u m bicarbonate buffer to prevent the a c t i v i t y from becoming volatile. T h e results were good a n d the a c t i v i t y levels confirmed for the subsequent series of p u l p scans. Table 3.1 provides the fraction labeled, the upstream b u l k flow rate, d u r a t i o n of scans, and a c t i v i t y added for each image successfully obtained i n this study. F o r each case an image was c a p t u r e d at the step a n d another 70 c m downstream of the expansion. E a c h scan begins w i t h the p r o d u c t i o n of the labeled fraction. M e a n w h i l e , the flow-loop is set up around the t o m o g r a p h . A f t e r the test section has been placed i n the gantry aperture of the tomograph a n d p l u m b e d i n , it is positioned i n the tomograph's field of view w i t h the a i d of a laser line. P r i o r to the emission scan a transmission scan is performed to characterize the effects of a t t e n u a t i o n of the photons, due to the test section a n d its contents.  Here  Scan  Tracer  Upstream Vel.  (fraction)  (m/s)  Activity  (MBq)  Image D u r .  Image D u r .  Phenom.  Step (s)  Downstream (s)  Behaviour  1  R48  0.5  585  3599  2736  plug  2  R48  0.7  480  2212  2922  fluidized  3 4  R48  0.8  1125  2708  2289  fluidized  R14  0.5  1015  2573  2197  partially-fluidized  5  R14  0.9  1045  1220  1134  fluidized  6  R100  0.9  475  2772  3072  fluidized  T a b l e 3.1: A s u m m a r y of the scans conducted. a r o t a t i n g point source containing  C o rotates around the object t o provide a flux of  photons along each line of response. Before the a c t i v i t y is i n t r o d u c e d , the p u m p is t u r n e d on a n d the system allowed to r u n for several minutes. Before the labeled fibres are added to the tank the flow rate is adjusted a n d the camera set to acquire d a t a . T h e radioactive fibres are then added to the tank a n d allowed to be p u m p e d t h o u g h the system. A t this point d a t a a c q u i s i t i o n has begun. E a c h scan is allowed to r u n u n t i l 100,000,000 events are detected or one hour has elapsed.  Chapter 4  Results and Discussion  T h e concentration profiles were obtained at various u p s t r e a m velocities for suspensions w i t h a b u l k concentration of 0.4 %( wt) w i t h three different tracer fibre sizes.  A s the  number of trials conducted was s m a l l , most figures have been i n c l u d e d . T h e discussion of the results w i l l be conducted i n three subsections. I n the first subsection we w i l l discuss the impact of velocity on the concentration d i s t r i b u t i o n . I n the second subsection we examine the effect of tracer particle size on the measured profile. F i n a l l y , we report the results far downstream of the expansion.  4.1  The Effect of Velocity  In this section we discuss the qualitative behavior of this suspension as a function of velocity.  A s shown i n Table 3.1, we recorded four cases i n w h i c h the tracer fibres were  well m i x e d after the expansion. We defined these cases as " f l u i d i z e d " , a n d the observed  <I o a  R14 R48 R100  8.5  CD  E  04  0.6  5  0.7  0.8  Upstream Velocity (m/s) F i g u r e 4.1:  E s t i m a t e s of the size of the recirculation zone for the cases i n w h i c h the  suspension was fluidized after the expansion. h  s  is the step height.  flow was q u a l i t a t i v e l y s i m i l a r to that reported by S a l m e l a & K a t a j a  (2005).  T h e size  of the r e c i r c u l a t i o n zone is shown i n F i g u r e 4.1 for these cases only. It should be noted that the length of the r e c i r c u l a t i o n zone L R has been scaled to the height of the step  h. s  It is difficult to compare our results quantitatively to either those reported by S a l m e l a & Kataja  (2005), as we have a different step size a n d suspension concentration, or to  those for a corresponding B i n g h a m fluid, as the rheological properties of the suspension are difficult to characterize properly. We begin the discussion of the effect of velocity by e x a m i n i n g Scan 2, see Table 3.1 a n d Figure 4.2.  A s w i t h a l l cases presented three views are p r o v i d e d ; a top view, a side  view, a n d a cross-sectional view.  T h e top a n d side views are slices f r o m the centre of  the test section i n planes orthogonal to the viewing direction. A s shown, these images are qualitatively similar to those reported by M o r a c z e w s k i et a l  (2005) i n that we see a  central jet surrounded by a r e c i r c u l a t i n g zone. There are three observations that can be  made i m m e d i a t e l y f r o m this result. F i r s t , the fibres are not d i s t r i b u t e d evenly t h r o u g h the imaged-volume. A n a s y m m e t r y is apparent i n F i g u r e 4.2(c) i n w h i c h there is a higher concentration of tracer fibres at the b o t t o m of the tube w h e n compared to the top. T h i s feature was found i n a l l fluidized cases examined. A s the images were integrated over some time the t e m p o r a l evolution of this concentration a s y m m e t r y is not k n o w n . R e b i n n i n g the datasets into shorter d u r a t i o n frames m a y provide insight into this particle accumulation. However, w i t h a reduced number of events i n each frame m u c h higher levels of noise are to be expected.  Second, there is a n annular region between the jet a n d the recirculation  zone w i t h a concentration that is lower t h a n the average concentration of the suspension. In other words, we observe a region w i t h particle depletion. F i n a l l y , i n the centre of the recirculation zones the concentration of tracer fibres is significantly larger t h a n the average concentration. T h e last two observations have been reported by M o r a c z e w s k i et a l (2005). We attempt to quantify these features by e x a m i n i n g the r a d i a l concentration profiles at different a x i a l positions, see F i g u r e 4.3 (a) a n d (b). I n these figures we have normalized the concentration to the b u l k concentration of the suspension, i.e. 0.4 w t % . R a d i a l a n d a x i a l distances have been n o r m a l i z e d by the radius of the larger pipe; a n d the a x i a l o r i g i n , x = 0, is set at the step. T h e horizontal a n d vertical axes are y a n d z, respectively, a n d their o r i g i n is placed along the central axis of the test section.  W e speculate that the  concentration-depletion zone results from the water layer formed i n the u p s t r e a m tube. It is commonly understood that a water layer forms o n the periphery of a pipe w i t h a flowing p u l p fibre suspension. W e have characterized b o t h the radius of the jet a n d thickness of the concentration-depletion layer i n F i g u r e 4.3(c) as a function of a x i a l p o s i t i o n . A s shown, we see that the radius of the jet remains essentially constant over the distance reported, while the size of the concentration-depletion layer decreases slightly. F i n a l l y , as shown i n F i g u r e  4.3(d) the average concentration of the jet increases w i t h a x i a l p o s i t i o n . I n this case we see a 50% increase i n the concentration i n the central p o r t i o n of the t u b e a n d advance the argument that this results f r o m the deceleration of the jet. A t this point we t u r n our attention to the second type of behavior observed, i.e.  plug  flow. We observed this i n one of the scans conducted; namely Scan 1, w h i c h was conducted at an u p s t r e a m velocity of 0.5 m/s.  A s shown i n F i g u r e 4.4, the tracer fibres were not  well d i s t r i b u t e d after the expansion plane a n d traveled t h r o u g h the region visualized as a plug.  C l e a r l y at this lower velocity the shear i m p a r t e d by the fluid is insufficient to  disrupt the fibre network. It must be noted that d u r i n g i m a g i n g , the central jet may be slowly meandering or folding as it travels d o w n the length of the tube. T h i s feature can not be captured as the image acquired is averaged over some t i m e . W e characterize these curves by showing the r a d i a l concentrations and the size of the central jet i n F i g u r e 4.5. A s shown, we see that the tracer fibres spread r a d i a l l y w i t h increasing a x i a l distance. T h e mechanism by w h i c h these fibres spread is difficult, i f not impossible, to ascertain from these figures alone as the jet may meander d u r i n g the i m a g i n g p e r i o d . I n other words, we can not ascertain i f the tracer fibre m i x i n g results from shear induced m i g r a t i o n or from the stability of the jet. O n e of the s t r i k i n g features i n this image is the relatively large localized concentration of a c t i v i t y near the top of the tube i n F i g u r e 4.5(c). W e are u n c e r t a i n of the o r i g i n of this but its presence confirms the fact that m i x i n g does not take place at this velocity.  T h e tracer fibres that accumulated at the top of the t u b e may have done  so gradually, or may have a r r i v e d as a floe. R e b i n n i n g the dataset into frames of shorter d u r a t i o n w o u l d allow the evolution of the a c c u m u l a t i o n to be e x a m i n e d . T h i s , however, would result i n images a n d d a t a w i t h higher noise content due to the reduced number of events i n each rebihned frame.  (b)  (c)  8  Figure 4.2: Four views of the a c t i v i t y profile for the case i n w h i c h the tracer fibres were well-mixed, or fluidized, after the expansion. T h e image was acquired using the R 4 8 tracer fibres w i t h the u p s t r e a m velocity set at 0.7 m/s (see Table 3.1). Image (a) represents the top view, (b) is a cross-sectional view at x/R  0  = 0.57, (c) is a side v i e w , a n d (d) is a  three-dimensional reconstruction of the activity d i s t r i b u t i o n .  (a) x/R =0.03 _ _ x/R°=0.20 x/R°=0.35 . . . . x/R°=0.68 Q  0.2  I  0  -0.2 -0.4 -0.6 -0.6  Normalized Concentration  (C) o 0 <  Outer Region Inner Region Depleted Regio  0 0 0 0 0 0  0.1  0.15  0.2  0.25  0.3  0.35  0.4  Distance Downstream of Expansion x/R  0.45  0  0.2  0.4  0.6  0.8  1  1.2  Distance Downstream of Expansion x/R  Figure 4.3: F o u r views of the a c t i v i t y profile for the case i n w h i c h t h e tracer fibres were well-mixed, or f l u i d i z e d , after the expansion. T h e image was acquired using the R 4 8 tracer fibres w i t h the u p s t r e a m velocity set at 0.7 m/s (see Table 3.1). Image (a) represents the concentration profile i n the r a d i a l direction for a central h o r i z o n t a l slice at four different a x i a l positions. Image (b) represents the concentration profile i n the r a d i a l direction from a central v e r t i c a l slice at four different a x i a l positions. Image (c) is a n estimate of the growth of the depletion zone.  E s t i m a t e s are given of the inner a n d outer r a d i i of the depletion  zone a n d their difference. Image (d) is a n estimate of the average concentration of the jet as a function of a x i a l p o s i t i o n .  (b)  •  (c) •  Figure 4.4: Four views of the a c t i v i t y profile for the case i n w h i c h the tracer fibres were not well-mixed after the expansion. We define this case as p l u g flow type behavior. T h e image was acquired using the R 4 8 tracer fibres w i t h the u p s t r e a m velocity set at 0.5 m/s Table 3.1). Image (a) represents the top view, (b) is a cross-sectional view at x/R  0  (see  — 1.00,  (c) is a side view, a n d (d) is a three-dimensional reconstruction of the a c t i v i t y d i s t r i b u t i o n .  (a) VV-. 08  x/R =0.03 _ _ x/R°=0.35  Cv-..  ...  x/R°=0.68  . . . . x/R°=1.00  0.6 0.4 02 o  a= ° c %--" (  -0.4 -0.6 -O.B -1  Normalized Concentration  Distance Downstream ot Expansion x/R  Figure 4.5: A view of the a c t i v i t y profile a n d jet expansion for the case i n w h i c h the tracer fibres were not w e l l - m i x e d after the expansion. T h e figures are f r o m the image acquired using the R 4 8 tracer fibres w i t h the upstream velocity set at 0.5 m/s (see Table 3.1). Image (a) represents the concentration profile i n the r a d i a l direction f r o m a central h o r i z o n t a l slice at four different a x i a l positions. Image (b) represents the w i d t h of the a c t i v i t y profile as a function of a x i a l p o s i t i o n .  4.2  Effect of Particle Size  A t this point we compare the effect of particle size by e x a m i n i n g two cases conducted at the same velocities using different length fractions. I n the first comparison, we examine scans 5 a n d 6 i n w h i c h we compare the d i s t r i b u t i o n of the R 1 0 0 a n d R 1 4 tracer fibres at 0.9 m/s (see F i g u r e 4.6). A s shown, the a c t i v i t y d i s t r i b u t i o n of these tracer fibres appear somewhat similar. I n b o t h cases the features reported i n the previous section are apparent; that is an a s y m m e t r y i n the v e r t i c a l direction, a depletion layer between the central jet a n d the recirculation zone, a n d particle a c c u m u l a t i o n i n the lower r e c i r c u l a t i n g zone. W e find that no significant differences i n the distributions are apparent between these two cases. In the second comparison, we examine the distributions of the R 4 8 a n d R 1 4 fibres at 0.5 m/s as given as scans 1 a n d 4 i n Table 3.1.  T h e results for scan 1 have been shown  earlier as F i g u r e 4.4. Here it is apparent that the tracer fibres are not w e l l - m i x e d after the expansion. W i t h the R 1 4 fibres however, we find that the tracer fibres are w e l l - m i x e d i n the upper p o r t i o n of the channel only (see F i g u r e 4.7 ). I n the upper p o r t i o n of the channel we observe the depletion zone between the central jet a n d the outer portions of the channel. It should be noted that d u r i n g this t r i a l it was v i s u a l l y observed that the lower p o r t i o n of the channel was static. W e do not interpret these results as q u a n t i t a t i v e evidence that their is a difference between the m o t i o n of these two classes of fibres. W e speculate that this result occurred due to the fact that the experimental p r o t o c o l was not the same as the other scans. T h e p u l p i n the test section was allowed to settle over several days prior to the scan a n d was not well m i x e d prior to the start of the scan.  F u r t h e r m o r e , when  the scan was conducted the flow rate began high a n d was lowered, as opposed to being r a m p e d up to the targeted flow rate as i n subsequent scans. T h i s resulted i n the region w i t h lower fibre concentration being fluidized first, a n d when the flow rate was dropped this region remained i n m o t i o n . W e have included this result as it is interesting to report the possibility of a stable, static region i n this type of device.  4.3  Far Downstream  Far downstream of the expansion (x/R  0  = 20) we were able to measure b o t h the concentra-  tion d i s t r i b u t i o n using P E T a n d the velocity profile using u l t r a s o u n d D o p p l e r velocimetry ( U D V ) , a c o m m e r c i a l device obtained from Signal Processing S A . F o r a l l flowrates tested, the velocity profiles at this point were similar a n d displayed p l u g like behaviour. A s shown i n F i g u r e 4.8,. a velocity b o u n d a r y layer exists near the walls of the pipe i n the region \y/R \ > 0.8. W e display the concentration profiles at this point for four different cases 0  (a)  1 (b)  1 (c)  (d)  1(e)  1(f)  F i g u r e 4.6: T w o cases i n w h i c h the tracer fibres were w e l l - m i x e d after the expansion. I n images (a)-(c) we examine the case i n w h i c h the upstream velocity was set at 0.9 m/s w i t h the R 1 0 0 tracer fibres. Images (d)-(f) represent the case w i t h the R 1 4 tracer fibres. T h e images i n the first c o l u m n ((a) a n d ((d)) represent the top view. T h e images i n the second c o l u m n represent the cross-sectional view at x/R  0  represent the side view.  = 0.57. T h e r e m a i n i n g two images  F i g u r e 4.7: Four views of the a c t i v i t y profile for the case i n w h i c h the tracer fibres were p a r t i a l l y - m i x e d after the expansion. T h e image was acquired using the R 1 4 tracer fibres w i t h the upstream velocity set at 0.5 m/s (see Table 3.1). view, (b) is a cross-sectional view at x/R  0  Image (a) represents the top  — 0.57, (c) is a side view, a n d (d) is a three-  dimensional reconstruction of the a c t i v i t y d i s t r i b u t i o n .  0.05  .0.01  1  -1  ' -0.8  ' -0.6  ' -0.4  ' -0.2  ' 0  y  / R  1  0.2  1  0.4  ' 0.6  ' 0.8  1  1  0  F i g u r e 4.8: T h e velocity profile of the suspension as measured using U D V . S i x experiments were conducted i n w h i c h the upstream velocity Vi set i n the range from 0.5 to 1.0  m/s.  T h e error bars represents the s t a n d a r d deviation determined from the average of this d a t a set. (see F i g u r e 4.9).  W h a t is apparent from these figures is that the concentration profile is  not necessarily similar to the velocity profile. We speculate t h a t the tracer fibres, t r a p p e d i n s m a l l floes at the inlet, are not sufficiently d i s r u p t e d by the shear at the step to redistribute themselves evenly downstream.  A s the flow rate is increased more m i x i n g is  induced a n d a more even r a d i a l d i s t r i b u t i o n is evident.  (a)  (b) 0.8 0.6 0.4  •  s  0.2 g°  0  -0.2  -0.2  -0.4  -0.4  -0.6  -0.6  -0.8  -0.8  Normalized Concentration  i  Normalized Concentration  F i g u r e 4.9: T h e concentration profile downstream of the expansion, (a) Scan 1, (b) Scan 2, (c) Scan 5 a n d (d) Scan 6. Details of each experiment are given i n T a b l e 3.1.  Chapter 5  Summary and Conclusion  P o s i t r o n E m i s s i o n T o m o g r a p h y ( P E T ) was used to investigate the d y n a m i c s of a 0.4% (wt) fibre suspension flowing t h r o u g h an a x i s y m m e t r i c 1:5 sudden expansion.  S i x scans  were conducted i n w h i c h b o t h the upstream velocity a n d the size of tracer fibres labelled were varied. Images were taken u p s t r e a m a n d downstream of the expansion plane w i t h the upstream velocity being varied from 0.5 to 0.9 m / s .  T h e expansion plane i m p a r t s shear  that disrupts the fibre network causing measurable changes i n the l o c a l fibre concentration. T w o distinct regions were clearly distinguished: p l u g like, i n w h i c h the tracer fibres were not m i x e d t h r o u g h the entire volume of the expansion; a n d fluidized, i n w h i c h the tracer fibres were well m i x e d .  O u r results for the fluidized case are w o r t h h i g h l i g h t i n g as we  found that concentration inhomogeneities exist.  These inhomogeneities are exemplified  by the fibre depleted annulus surrounding the i n c o m i n g jet, the densification of the jet immediately downstream of the expansion and the vertical a s y m m e t r y i n the recirculation region. R e b i n n i n g the datasets into shorter d u r a t i o n frames may provide insight into the  evolution of the concentration asymmetries, provided there are sufficient counts i n each frame to y i e l d d a t a w i t h acceptable noise levels. We consider these to be the most significant findings i n this work b u t are as yet unable to develop a mechanistic u n d e r s t a n d i n g of these phenomena.  Bibliography  ALTOBELLI,  S. A . F U K U S H I M A  E. & MONDY,  L . A . 1 9 9 7 N u c l e a r magnetic resonance  imaging of particle m i g r a t i o n i n suspensions undergoing e x t r u s i o n J. Rheol. 41(5), 1 1 0 5 1115  ALTOBELLI,  S. A .  measurements  GIVLER,  R.  C. &  F U K U S H I M A , E . 1 9 9 7 Velocity and concentration  of suspensions b y nuclear magnetic resonance i m a g i n g J. Rheol. 35(5),  721-734  AROLA,  D. P O W E L L , R. M C C A R T H Y ,  M . L I ,T.-Q. & ODBERG,  L . 1 9 9 8 N M R Imaging  of P u l p Suspension F l o w i n g t h r o u g h a n A b r u p t P i p e E x p a n s i o n AIChE  J. 44(12), 2 5 9 7 -  2606  B O U M A N , C . A . , & S A U E R , K . , 1 9 9 6 A Unified A p p r o a c h to S t a t i s t i c a l T o m o g r a p h y U s i n g Coordinate Descent O p t i m i z a t i o n IEEE  Trans. Imag. Proc. 5, 4 8 0 - 4 9 2  C A S E Y , M . E . , & H O F F M A N , E . J . , 1 9 8 6 Q u a n t i t a t i o n i n P o s i t r o n E m i s s i o n Tomography: 7. A Technique to Reduce Noise i n A c c i d e n t a l Coincidence Measurements a n d C o i n c i dence Efficiency C a l i b r a t i o n J. Computr.  Assist.  Tomogr. 10, 4 2 - 5 0  D A H L B O M , M . , & H O F F M A N , E . J . , 1988 A n E v a l u a t i o n of a T w o - D i m e n s i o n a l A r r a y Detector for H i g h R e s o l u t i o n P E T IEEE DAUBE-WITHERSPOON, M o d e l for P E T IEEE  Trans. Med. Imag. 7, 264-272  M.E., & CARSON,  R . E . , 1991 U n i f i e d D e a d t i m e C o r r e c t i o n  Trans. Med. Imag. 10, 267-275  D E A N S , S . R . , 1983 T h e R a d o n T r a n s f o r m and Some of its A p p l i c a t i o n s New York: DEFRISE, M . , TOWNSEND, D.W.,  Wiley  & C L A C K , R . , 1989 T h r e e - D i m e n s i o n a l Image R e c o n -  struction from C o m p l e t e Projections Phys. Med. Bio. 34,  573-587  D U F F Y , G . G . & T I T C H E N E R , A . L . 1975 T h e D i s r u p t i v e Shear Stress of P u l p Networks Svensk. Papperstidn.  13, 474-479  F E S S L E R , J . A . , 1995 H y b r i d P o i s s o n / p o l y n o m i a l Objective F u n c t i o n s for Tomographic Image R e c o n s t r u c t i o n f r o m T r a n s m i s s i o n Scans IEEE FESSLER,  J.A,  FICARO,  E.P.,  CLINTHORNE,  N.H.,  Trans. Imag. Proc. 4, &  LANGE,  K . ,  1996  1439-1450 Grouped-  coordinate Ascent A l g o r i t h m s for Penalized-likelihood T r a n s m i s s i o n Image Reconstruct i o n IEEE HAMMAD,  Trans. Med. Imag. 16, K.  J., OTUGEN,  M.  V.  166-175 VRADIS,  G.  &  ARIK,  E  1999  L a m i n a r F l o w of  Nonlinear V i s c o p l a s t i c F l u i d T h r o u g h a n A x i s y m m e t r i c S u d d e n E x p a n s i o n ASME 121,  a  J.  488-495  H E R M A N , G . T . , 1980 Image R e c o n s t r u c t i o n from Projections New HOFFMAN,  E.J., GUERRERO,  T.M.,  GERMANO,  G., DIGBY,  York Academic  W . M . , &  Press  DAHLBOM,  M.,  1989 P E T System C a l i b r a t i o n Corrections for Q u a n t i t a t i v e S p a t i a l l y A c c u r a t e Images IEEE  Trans. Nucl. Sci. 36,  1108-1112  J o s s i c , L . B R I G U E T , A . k, M A G N I N , A . 2002 Segregation under flow of objects suspended i n a yield stress fluid a n d N M R i m a g i n g visualization Chem. Eng. Sci. 5 7 , 409-418 K A K , A . C . , & S L A N E Y , M . , 1988 P r i n c i p l e s of C o m p u t e r i z e d T o m o g r a p h y New  York  K A R I N O , T . & G O L D S M I T H , H . L . 1977 F l o w behavior of blood-cells a n d r i g i d spheres i n an annular vortex  Phil.  Trans. R. Soc. London,  K E R E K E S , R . J . SOSZYSNKI,  fibres Fund. Res. Syrnp.,  Ser. B 2 7 9 , 413-445  R . M . & T A M D O O , P . M . 1985 T h e f l o c c u l a t i o n of p u l p  Oxford 1 , 265-310  K I N A H A N , P . E . , k. R O G E R S , J . G . , 1989 A n a l y t i c T h r e e - D i m e n s i o n a l Image Reconstruct i o n U s i n g a l l Detected Events IEEE  Trans. Nucl. Sci. 36,  1108-1112  L A T O R N E L L , D . J . , & P O L L A R D , A . 1986 Some Observations o n the E v o l u t i o n of Shear Layer Instabilities i n L a m i n a r F l o w T h r o u g h A x i s y m m e t r i c S u d d e n E x p a n s i o n s Phys. Fluids  2 9 , 2828-2835  L E I G H T O N , D . & A C R I V O S , A . 1987 T h e shear-induced m i g r a t i o n of particles i n concentrated suspensions J. Fluid Mech. 1 8 1 , 415-439 M A C A G N O , E . O . & H U N G , T . K 1967 C o m p u t a t i o n a l a n d E x p e r i m e n t a l S t u d y of a C a p t i v e A n n u l a r E d d y J. Fluid Mech. 2 8 , 43-64 MIEKLE,  S.R., DAHLBOM,  M . , $ CHERRY,  S . R , 1993  Attenuation  Correction Using  C o u n t - L i m i t e d T r a n s m i s s i o n D a t a i n P o s i t r o n E m i s s i o n T o m o g r a p h y J. Nucl. Med. 34, 143-144 M O K , S - P . , W A N G , C - H . , C H E N , J - C , k L i u , R - S . , 2003 Performance E v a l u a t i o n of the  H i g h R e s o l u t i o n S m a l l A n i m a l P E T Scanner Biomed.  Eng. Appl Basis  Comm.  15(4),  143-149 M O R A C Z E W S K I , T . , T A N G , H . , & S H A P L E Y , N . 2005 F l o w of a concentrated  suspension  through a n abrupt a x i s y m m e t r i c expansion measured by nuclear magnetic resonance imaging J. Rheol. 4 9 ( 6 ) , 1409-1428 N A T T E R E R , F . , 1986 T h e M a t h e m a t i c s of C o m p u t e r i z e d T o m o g r a p h y New York: OLLINGER,  Wiley  J . M . , 1992 T h e Use O f M a x i m u m A Posteriori A n d M a x i m u m L i k e l i h o o d  Transmission Images F o r A t t e n u a t i o n C o r r e c t i o n I n P E T Proc. 1992 IEEE  Med. Imag.  Conf. 1185-1187 OLLINGER, J . M . , & FESSLER, J.A.,  1997 P o s i t r o n - E m i s s i o n T o m o g r a p h y IEEE  Signal  Pro. Mag. 1 4 ( 1 ) , 43-55 P A L M E R , M . R . , R O G E R S , J . G . , B E R G S T R O M , M . , B E D D O E S , M . P . ,& P A T E , B . D . ,  Transmission Profile F i l t e r i n g for P o s i t r o n E m i s s i o n T o m o g r a p h y IEEE  1986  Trans. Nucl. Sci.  3 3 , 478-481 PHILLIPS, R. J . A R M S T R O N G R. C. B R O W N R. A . G R A H A M A . L . & A B B O T T J . R.  A constitutive equation for concentrated particle m i g r a t i o n Phys. Fluids  1992  suspensions that accounts for shear-induced  4 , 30-40  S A L M E L A , J . , & K A T A J A , M . , 2005 F l o e R u p t u r e a n d R e - F l o c c u l a t i o n i n T u r b u l e n t Shear F l o w Fund. Res. Symp.,  Cambridge  1 , 35-50  S H E P P , L . A . , & L O G A N , B . F . , 1974 T h e Fourier Reconstruction of a H e a d Section Trans. Nucl. Sci. 2 1 , 21-43  IEEE  T H A L E N , N . & W A H R E N , D . 1964 Shear M o d u l u s a n d U l t i m a t e Shear S t r e n g t h of Some P u l p F i b r e Networks Svensk. Papperstidn.  67(7), 259-264  T H A L E N , N . & W A H R E N , D . 1964 A n E x p e r i m e n t a l Investigation of the Shear M o d u l u s of M o d e l F i b r e Networks Svensk. Papperstidn. Xu,  M., LUK, W . K . ,CUTLER, P.D.,  67(11), 474-480  & DIGBY,  W.M.,  1994 L o c a l T h r e s h o l d for Seg-  mented A t t e n u a t i o n C o r r e c t i o n of P E T Imaging of the T h o r a x IEEE 41, 1532-1537  Trans. Nucl. Sci.  Appendix A  Filtered Backprojection F i l t e r e d b a c k p r o j e c t i o n was first applied to P E T by Shepp & L o g a n (1974). I n t r o d u c t o r y treatments of the a l g o r i t h m c a n be found i n K a k & Slaney (1988), a n d H e r m a n (1980). W i t h more comprehensive treatments i n Deans  (1983), a n d N a t t e r e r  (1986). T h e dis-  t r i b u t i o n of the radioisotope is modeled by the function X(x,y, z) G L . 2  F o r a given 2 D  slice, we assume that the mean of an i n d i v i d u a l measurement Y^g (equation (2.3)) is given by g {d) = / X(x,y,z)dxdydz Jl{d,9) e  where £(d,6)  (A.l)  is the line connecting the two detectors involved i n the coincidence.  In  practice, it is assumed that the mean gg(d) is equal to the corrected d a t a , M^g. I n the rotated coordinate system of figure A . l , d — xg, so the line integral c a n be expressed as oo  /  X(x ,yg)dyg,e e  -oo  e (0,n),xg,yg,e  (A.2)  where xg represents transverse distance i n the rotated coordinate system shown i n figure A . l . We w i l l refer to the f u n c t i o n (and the d a t a that it approximates) as a projection. T h e Fourier t r a n s f o r m of each p r o j e c t i o n is given by  =  k{ug,V ) 0  1^=0  (A.3)  T h i s result, k n o w n as the projection-slice theorem, has two i m p l i c a t i o n s . F i r s t , the Fourier transform of a p r o j e c t i o n yields samples of the 2D Fourier t r a n s f o r m of the image; a n d second, these samples lie along a line at the same angle, 6, i n the frequency d o m a i n as that of the p r o j e c t i o n i n the s p a t i a l d o m a i n . T h i s result can be w r i t t e n i n more s t a n d a r d n o t a t i o n as G (p) e  =  A(p,7) \ =e, m y  P  (A.4)  where the Fourier t r a n s f o r m of the image is now expressed i n p o l a r coordinates (p, 7). E q u a t i o n (A.3) can be used to reconstruct the image by c o n s t r u c t i n g the Fourier transform i n polar coordinates, i n t e r p o l a t i n g to rectangular coordinates, a n d t h e n t a k i n g the inverse transform. A more efficient m e t h o d can be derived as follows. T h e image \(x, y) is given by  (A.5)  x =rcos((j)-0) e  F i g u r e A . l : P r o j e c t i o n Geometry. v = psia.0, x — rcos(/>, a n d y — rs'mcj) yields r-27r  \(r,<j>) = / Jo  roo  pA{pcos9,psmB)e  / Jo  - d de  j27rprcos{,l> e)  P  (A.6)  R e w r i t i n g A(p cos 6,p sin 9) as A ( p , 0) a n d using the facts that cos(<^> — 6) = — cos(</> — 0 + ir) and Ge(p) = Gg (~p),  this c a n be rewritten as  +V  X( ,cp)= r  P7T poo / \ Jo J-oo '0 J-oo  p\A{p,6)e  tt- Updd  j27Tprc08  e  (A.7)  A p p l y i n g the projection-slice theorem leads to  0  * o  J-oo  |p |  G {p)e  g (rcos(<l>-0))d6 e  e  '- dpde  j2lxprcos{<i  e)  (A.8)  cretizing leads to the expression N -i e  Mr^) = ^X>(rcos(0-0,)) Nf  (A.9)  > 1=0  E q u a t i o n (A.9) shows that the value of the image at a point (r cos </>, r s i n <f>) i n figure A . l can be found by first filtering the projections w i t h a r a m p filter, t h e n s u m m i n g the filtered values at the coordinate xg = r cos(9 — <f>) over a l l projection angles 0 , . N o t e that the value x  l  at xg w i l l contribute to a l l pixels along the L O R s that c o n t r i b u t e d t o the measurement t  at this point.  T h e a l g o r i t h m c a n be efficiently implemented by filtering each estimated  projection, gg(d) = M^g, w i t h a r a m p filter to y i e l d gg(d) a n d t h e n a d d i n g each filtered value into a l l voxels along the corresponding L O R as shown by the dashed line i n figure A . l . T h e latter operation is called backprojection, so the a l g o r i t h m is u n s u r p r i s i n g l y called filtered-backprojection. T h i s a l g o r i t h m a n d its extension to three dimensions ( K i n a h a n & Rogers  (1989),Defrise et a l (1989)) is used almost exclusively for image reconstruction i n  P E T . It is i d e n t i c a l to the a l g o r i t h m used i n x-ray C T except for modifications to the filter necessitated by the noise properties of P E T data.  Appendix B  Scan Data  In this section each scan is presented i n c l u d i n g each of the 63 image slices, central horizontal and vertical concentration profiles, jet a n d depletion zone d a t a , jet densification d a t a , a n d downstream concentration profiles.  Scan  Tracer  Upstream Vel.  P o s i t i o n of Step  P o s i t i o n of Centre  Image Centre (Past Expansion)  (Fraction)  (m/s)  (Slice)  (X,Y)  R48 R48  0.5 0.7  50  (68,59)  2 cm  50  (68,59)  2 cm  R48  0.8  33  (60,61)  0 cm  0.5 0.9  33 50  (67,62)  5  R14 R14  (66,59)  0 cm 2 cm  6  R100  0.9  50  (66,59)  2 cm  1  2 3 4  9  1  F i g u r e B . l : Scan 1 2D Slices.  F i g u r e B . 2 : Profiles for Scan 1. H o r i z o n t a l Profiles (a), Jet E x p a n s i o n (b), and D o w n s t r e a m Concentration  (c).  F i g u r e B . 3 : Scan 2 2D Slices.  (a) - .  x/R°=0.20  ...  x/R°=0.35  . . . . x/R°=0.68  0.4 0.2 o  or  >s  0  j  -0.2 -0.4 -0.6  1  1.5  2  2.5  Normalized Concentration  (c)  o 0 <  0 0 0 0 0 0 H  » « « «  o 0.05  0.1  0.15  ° o o °  <  <  0.2  0.25  0  <  >  < > o  Outer Region Inner Region Depleted Region  0 o  o  <<.<, < <  0  <<<  <  0.3  0.35  0.4  Distance Downstream of Expansion x/R  o o o '  < 0.45  0.5  -0.2  0  0.2  0.4  0.6  0.8  1  Distance Downstream of Expansion x/R  Normalized Concentration  1.2  F i g u r e B . 5 : Scan 3 2D Slices.  Jl 0  , 0.05  , 0.1  , 0.15  , 0.2  , 0.25  , 0.3  1  , 0.35  „. I -0.8 7  0.4  , -0.6  Distance Downstream of Expansion x/R,  , -0.4 D i s t a n c e  , -0.2 D  o  w  n  , 0 s  t  r  (e) V  7  0.8  -  0.6  -  0.4  >  0.2  0  1  0.2  f  0.4  -0.6  -0.8  -1  —v—  1  Normalized Concentration  e  a  m  , 0.2 o f  E x p a n s i o n  , 0.4  , 0.6 m  1  0.8  F i g u r e B . 7 : Scan 4 2D Slices.  (a)  (b)  Normalized Concentration  Distance Downstream of Expansion x/R  o  Figure B . 8 : Profiles for Scan 4. D o w n s t r e a m C o n c e n t r a t i o n (a), a n d J e t Densification (b).  F i g u r e B.9: Scan 5 2D Slices.  0.5  1  Normalized Concentration  F i g u r e B.12: Profiles for Scan 6. H o r i z o n t a l Profiles (a), V e r t i c a l Profiles (b), D e p l e t i o n Zone D a t a (c), Jet Densification (d), and Downstream C o n c e n t r a t i o n 58  (e).  

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