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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 Heath B . A . S c , The University of B r i t i s h Co lumbia , 2003 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M a s t e r of A p p l i e d Science i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S ( M E C H A N I C A L E N G I N E E R I N G ) The University of British Columbia August 2006 © Stuart James Heath , 2006 Abstract The motion of F radioactively labeled papermaking fibres flowing through an ax isym-metric 1:5 sudden expansion has been studied using positron emission tomography ( P E T ) . Various length fractions of a mechanical pulp were radioactively labeled and then intro-duced into a non-radioactive aqueous 0.4 wt. % (consistency) wood pu lp suspension. Fu l ly three-dimensional images were reconstructed both upstream and downstream of the ex-pansion plane for cases i n which the upstream velocity U was set from 0.5 to 0.9 m/s (an approximate Reynolds 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 through the expansion as a plug. N o mix ing was observed be-tween the confined central jet and the static outer region. A t larger velocities, we observed that the suspension was fully fluidized. O u r results in this regime indicate that albeit fluidized, concentration inhomogeneities were evident. We find that a particle depletion zone was evident between the central jet and the recirculating zone resulting from the inlet concentration profile formed i n the upstream tube. Part ic le accumulation was observed in the vortices. N o significant differences were observed between the different length tracer fibres. Contents Abstract " Contents iii List of Tables v List of Figures vi Acknowledgements xi Dedication xii 1 Introduction 1 2 Positron Emiss ion 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 M e t h o d s 16 4 Results and Discussion 22 4.1 The Effect of Velocity 22 4.2 Effect of Particle Size . . . 29 4.3 Far Downstream 30 5 Summary and Conclusion 35 References 37 Appendix A Fi l tered Backprojection 42 Appendix B Scan D a t a 46 List of Tables 3.1 A summary of the scans conducted 21 B . l A summary of the posit ional data from the scans conducted 46 List of Figures 2.1 Schematic of the detector arrangement i n a P E T tomograph 6 2.2 A schematic of a detector from the m i c r o P E T R 4 system. Consist ing of an 8 x 8 L S O array, 10 cm fibre optic bundle, and a H a m a m a t s u R5900-C 8 posit ion sensitive photo-multipl ier tube ( P S - P M T ) . W i t h the readout boards attached to the rear of the tube 7 2.3 Sampl ing pattern i n the transaxial plane for a P E T tomograph. E a c h seg-ment i n the detector r ing represents one crystal . T h e solid lines show the paral lel projections for the first angle, the dotted lines for the second angle, and the dashed lines for the t h i r d angle 8 2.4 D iagram of a scattered event (left) and an accidental coincidence (right). Photons shown leaving the r ing are scattered through an 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 The fibre length distributions of the S B K pulp and the three tracer fibres used (labeled R14,R48 and R100). The nomenclature for the fibre fractions is retained from the Bauer Mcnett device used to separate the fibre suspension. 19 4.1 Estimates of the size of the recirculation zone for the cases i n which the suspension was f luidized after the expansion. hs is the step height 23 4.2 Four views of the act iv i ty profile for the case i n which the tracer fibres were wel l -mixed, or f luidized, after the expansion. The image was acquired using the R48 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/R0 = 0.57, (c) is a side view, and (d) is a three-dimensional reconstruction of the act iv i ty d is tr ibut ion 26 4.3 Four views of the act iv i ty profile for the case i n which the tracer fibres were wel l -mixed, or f luidized, after the expansion. The image was acquired using the R48 tracer fibres w i t h the upstream velocity set at 0.7 m/s (see Table 3.1). Image (a) represents the concentration profile i n the rad ia l direction for a central horizontal slice at four different ax ia l positions. Image (b) represents the concentration profile i n the radia l direct ion from a central vertical slice at four different axia l positions. Image (c) is an estimate of the growth of the depletion zone. Estimates are given of the inner and outer rad i i of the depletion zone and their difference. Image (d) is an estimate of the average concentration of the jet as a function of ax ia l posit ion 27 4.4 Four views of the act ivity profile for the case i n which the tracer fibres were not wel l -mixed after the expansion. We define this case as p lug flow type behavior. The image was acquired using the R48 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/RQ — 1.00, (c) is a side view, and (d) is a three-dimensional reconstruction of the act iv i ty d is tr ibut ion 29 4.5 A view of the act iv i ty profile and jet expansion for the case i n which the tracer fibres were not well -mixed after the expansion. T h e figures are from the image acquired using the R48 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 rad ia l direction from a central horizontal slice at four different axia l positions. Image (b) represents the w i d t h of the act iv i ty profile as a function of ax ia l posit ion 30 4.6 Two cases i n which the tracer fibres were well -mixed after the expansion. In images (a)-(c) we examine the case i n which the upstream velocity was set at 0.9 m/s w i t h the R100 tracer fibres. Images (d)-(f) represent the case w i t h the R14 tracer fibres. The images i n the first co lumn ((a) and ((d)) represent the top view. The images i n the second column represent the cross-sectional view at x/R0 = 0.57. The remaining two images represent the side view. . . 31 4.7 Four views of the act iv i ty profile for the case i n which the tracer fibres were part ia l ly -mixed after the expansion. The image was acquired using the R14 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/R0 = 0.57, (c) is a side view, and (d) is a three-dimensional reconstruction of the act ivity d is tr ibut ion 32 4.8 The velocity profile of the suspension as measured using U D V . Six exper-iments were conducted i n which the upstream velocity Vi set i n the range from 0.5 to 1.0 m/s. The error bars represents the standard deviation de-termined from the average of this data set 33 4.9 The concentration profile downstream of the expansion, (a) Scan 1, (b) Scan 2, (c) Scan 5 and (d) Scan 6. Details of each experiment are given i n Table 3.1 34 A . l Pro ject ion Geometry 44 B . l Scan 1 2D Slices 47 B.2 Profiles for Scan 1. Horizontal Profiles (a), Jet Expans ion (b), and Down-stream Concentrat ion (c) 48 B.3 Scan 2 2D Slices 49 B.4 Profiles for Scan 2. Horizontal Profiles (a), Vert i ca l Profiles (b), Deplet ion Zone D a t a (c), Jet Densification (d), and Downstream Concentrat ion (e). . 50 B.5 Scan 3 2D Slices 51 B.6 Profiles for Scan 3. Hor izontal Profiles (a), Vert i ca l Profiles (b), Deplet ion Zone D a t a (c), Jet Densification (d), and Downstream Concentrat ion (e). . 52 B.7 Scan 4 2D Slices 53 B.8 Profiles for Scan 4. Downstream Concentration (a), and Jet Densif ication (b). 54 B.9 Scan 5 2D Slices 55 B.10 Profiles for Scan 5. Hor izontal Profiles (a), Vert i ca l Profiles (b), Deplet ion Zone D a t a (c), Jet Densification (d), and Downstream Concentrat ion (e). . 56 B . l l Scan 6 2D Slices 57 B.12 Profiles for Scan 6. Hor izontal Profiles (a), Vert i ca l Profiles (b), Deplet ion Zone D a t a (c), Jet Densification (d), and Downstream Concentrat ion (e). . 58 Acknowledgements I would like to thank K e n Buckley, Suzy L a p i , and Thomas R u t h from the T R I U M F Life Sciences program for a l l their hard work, my supervisors M a r k M a r t i n e z and James Olson for their guidance and insight, and a l l my friends at the P u l p and Paper Centre. F inanc ia l support from the N a t u r a l Sciences and Engineering Research C o u n c i l of C a n a d a and the T R I U M F Life Sciences program are gratefully acknowledged. S T U A R T J A M E S H E A T H The University of British Columbia August 2006 To my family, Robert , Mar ie -C laude , Alexandre, and Nathal ie for a l l their support and encouragement. Chapter 1 Introduction The focus of the present work is an experimental study of the concentration distr ibut ion of a semi-dilute fibre suspension-undergoing steady flow i n an abrupt 1:5 sudden expan-sion. A l t h o u g h the flow of multiphase fluids through sudden expansions is found i n many industr ia l and natura l settings, there are s t i l l many unanswered questions regarding the mechanism of particle dispersion (fluidization) or c lumping (flocculation). T h e motivation for the present work stems from an interest i n the papermaking process. Under normal processing conditions, papermaking suspensions mechanically entangle to form a network, which possesses a measurable y ie ld stress (Duffy & Titchener (1975), T h a l e n & Wahren (1964a,b)). 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 an abrupt expansion. Th i s aids i n evenly dispersing the suspension. In a paper machine sudden expansions are found i n the cross-flow distr ibutor , the turbulence generator, and the headbox nozzle. Sus-pensions which are evenly dispersed reduce the cloudiness or graininess of the sheet and aid in the e l iminat ion of grammage (areal density) variations, a property that is important for a l l paper grades. W h i l e the objectives of the papermaking process are clear, and the equipment on the paper machine well known, the exact mechanism by which f luidization and material redistr ibut ion occurs remains obscure. O u r objective is to provide insight into this phenomenon by visual iz ing the flow of the papermaking suspension through a sudden expansion using Pos i t ron Emiss ion Tomography ( P E T ) . Understanding the mot ion of an aqueous fibre suspension flowing through a sudden expan-sion is difficult. Insight into this phenomena can be gained by first examining the simpler case of the flow of single-phase f luid. For Newtonian fluids, Macagno & H u n g (1967) indicate that over a l l Reynolds numbers, a vortex exists immediately downstream of the expansion. There is general agreement that at low Re the size of the vortex increases linearly w i t h Reynolds number and then decreases w i t h Re > 635 (Latornel l & Po l lard (1986)) . In contrast w i t h this, Non-Newtonian fluids exhibit vortex lengths that differ significantly from Newtonian fluids. W i t h yie ld stress fluids, the recirculation lengths were found to be smaller when compared to Newtonian fluids at a comparable Reynolds number (Hammad et a l (1999), Jossic et a l (2002)). The case of expansion flows w i t h particle suspensions remains largely unexplored. Th is class of flow is str ikingly different than single-phase flows as particle concentration i n -homogeneities are generated through particle collisions, shear induced particle migration (Leighton & Acrivos (1987), Ph i l l i p s et a l (1992)), density differences between the particles and the carrier f lu id , inert ia ; and, i n the case of papermaking fibres, mechanical floccula-t ion of the particles fibres (Kerekes et a l (1985)). To help i l lustrate this complexity, there is evidence that w i t h suspensions of neutrally buoyant monodisperse spheres, particle ac-cumulation or depletion is evident in the vortex depending upon the ratio of the upstream tube to particle diameters (Al tobe l l i et a l (1997a,b), K a r i n o & G o l d s m i t h (1977)). In a recent study, Moraczewski et a l (2005) observe that a low concentration region exists which divides the central jet and the recirculation region. T h e y attr ibute this to inhomo-geneities in the inlet concentration that were convected downstream. The concentration profiles i n this case were measured using N M R after the flow had been stopped. W i t h regards to papermaking suspensions, there is evidence of seemingly two different be-haviours. A r o l a et a l (1998), for example, imaged the ax ia l velocity profile of a 0.5% (wt) wood pulp suspension flowing through a 1:1.7 sudden expansion using nuclear magnetic resonance imaging ( N M R ) . These authors report that the pulp suspension exhibited be-havior similar to that of a confined jet. In recent work, Salmela & K a t a j a (2005) used an optical technique to measure the floe size and fibre flow field of a semi-dilute suspension after the expansion. They 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 Newtonian fluid. Our work is focused on complimenting these previous studies by measuring the steady-state concentration profiles of papermaking fibres as they pass through a sudden expansion. Here, the behaviour of Fluorine-18 ( 1 8 F ) labeled papermaking fibres flowing i n the midst of non-radioactive fibres are studied using P E T . We measure the radioact iv i ty distr ibut ion, three-dimensionally, near the expansion plane and then far downstream of the expansion plane. The experimental conditions were such that the bulk concentration of the suspension was fixed while we varied the volumetric flowrate and size of the 1 8 F labeled fibres. The key advantage of this measurement technique is that the concentration profile can be determined for each particle fraction directly without stopping the flow. In addit ion, we measure the ax ia l velocity of the suspension far downstream of the expansion plane using pulsed ultrasound Doppler anemometry ( U D V ) . In chapter 2 the technique behind P E T is quickly reviewed. Chapter 3 describes the exper-imental apparatus and experimental protocol used i n this work. T h e results from the six successful scans are discussed i n chapter 4. The results are presented i n three subsections. F irs t the effect of velocity is examined, followed by the effect of particle size; and, finally, the observations made far downstream of the expansion. Chapter 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 algorithm 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 and associated figures for each of the cases examined i n this study. Chapter 2 Positron Emission Tomography Positron emission tomography ( P E T ) is a non-invasive imaging technique developed i n the medical field for measuring the metabolic act ivity of cells in-vivo. P E T is unique because it produces images of basic biochemistry or function, rather than other diagnostic imaging techniques such as x-rays, C T scans or magnetic resonance imaging ( M R I ) , which produce images of anatomy or structure (Ollinger & Fessler (1997)). Pos i t ron emit t ing isotopes of carbon ( 1 1 C ) , nitrogen ( 1 3 N ) , oxygen ( 1 5 0 ) and fluorine ( 1 8 F ) are produced i n an on-cite cyclotron and are incorporated into compounds of biological interest. These isotopes have relatively short half-lives of 20.3 minutes, 9.97 minutes, 2.03 minutes and 109.8 minutes respectively. A P E T study begins w i t h the injection of a radiopharmaceutical . To allow for transport to, and 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 positron, which travels a short distance before annihi lat ing w i t h an electron. Th i s annihi lat ion produces two high-energy (511 keV) photons propagating i n nearly opposite directions. If two photons are Front View Side View Figure 2.1: Schematic of the detector arrangement i n a P E T tomograph. detected in a short (~10 ns) t iming window, an event is recorded along the line connecting the two detectors, sometime referred to as the line of response ( L O R ) . Summing many such events results i n quantities that approximate line integrals through the radio-isotope distr ibut ion (Ollinger & Fessler (1997)). For 2D imaging these line integrals form a discrete approximation of the R a d o n transform (Deans (1983)) of a cross-section of the radio-isotope concentration, and can be inverted to form an image of the radioisotope distr ibution. A schematic diagram of a P E T tomograph is shown i n Figure 2.1. 2.1 Detectors The most cr i t i ca l components i n a P E T tomograph are the detectors (Dahlbom & Hoff-man (1988)). Detectors are arranged i n blocks as shown i n Figure 2.2. Detector blocks are formed by optical ly coupling a rectangular bundle of crystals to one or more photo-multipl ier tubes ( P M T s ) . W h e n a photon is incident on the crystal , electrons are moved from the valence band to the conduction band. Light is emitted as these electrons return to Readout Boards <?J3- 'J1//XPhoto-mult ip l ier Tube Figure 2.2: A schematic of a detector from the m i c r o P E T R 4 system. Consist ing of an 8 x 8 L S O array, 10 cm fibre optic bundle, and a Hamamatsu R5900-C8 posit ion sensitive photo-multipl ier tube ( P S - P M T ) . W i t h the readout boards attached to the rear of the tube. the valence band at impurit ies i n the crystal . Since impurit ies i n the crystal usually have meta-stable excited states, the light output decays exponentially at a rate characteristic to the crystal . The ideal crystal has: (1) high density, so that a large fraction of incident photons scintil late, (2) high light output, for posit ioning accuracy, (3) fast rise t ime, for accurate t iming , and (4) a short decay time, so that high counting rates can be handled (Ollinger h Fessler (1997)). The block is fabricated i n such a way that the amount of light collected by each P M T varies uniquely depending on the crystal i n which the scinti l lat ion occurred (Dahlbom & Hoffman (1988)). Hence, integrals of the P M T outputs can be decoded to yie ld the posit ion of each scinti l lation. The sum of the integrated P M T outputs is proport ional 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 lu te t ium oxyorthosilicate (LSO) crystals, arranged i n an 8 x 8 array, fibre-optically coupled to a single posit ion sensitive photomult ipl ier tube ( P S - P M T ) . Each crystal is 2.1 mm wide i n the transverse plane, 2.1 mm wide i n the ax ia l dimension and 10 m m deep. Figure 2.3: Sampl ing pattern i n the transaxial plane for a P E T tomograph. E a c h segment i n the detector r ing represents one crystal . The solid lines show the paral lel projections for the first angle, the dotted lines for the second angle, and the dashed lines for the th i rd angle. 2.2 Resolution W h e n data is acquired i n the 2D slice-collimated mode, the L O R s connecting crystals can be binned into sets of paral le l projections at evenly spaced angles, as shown i n Figure 2.3. Two 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 than at the center. Second, the samples along the heavy line at angles one (#i) and 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 yield one-half the number of projection angles w i t h a sampling distance of one-half the detector w i d t h (Ollinger & Fessler (1997)). The Nyquist criterion states that the sampling distance be one-half the spatial resolution, expressed as the fu l l -width-at -hal f -maximum ( F W H M ) . T h e fu l l -width-at -hal f -maximum is defined as the distance between the half-value points of the impulse response. Th 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 would support a spatial resolution of 2.1 m m . In fact, a tomograph w i t h this crystal size has a measured resolution that is somewhat worse; varying from 1.8 mm at the center of the field of view, to 2.5 mm at the edge. The 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 than 5 m m , the f inal resolution of the image usually being greater than 8 m m . T h i s is because the reconstruction algorithms trade off resolution for reduced image variance. Th i s final resolution is called the reconstructed resolution. Therefore, the resolution of P E T images as they are typical ly used is not determined by the detectors, but by the degree to which resolution must be degraded to achieve an acceptable image variance. Since the variance is determined by the number of counts that can be collected dur ing the scan, the constraints that govern the resolution of P E T images are the amount of radioact iv i ty used, the scan durat ion, the sensitivity of the tomograph, and the count-rate capabil i ty of the tomograph (Ollinger & Fessler (1997)). 2.3 Attenuation For an incident 511 keV photon there are two possible interactions; photoelectric ab-sorption, and C o m p t o n scatter. In materials w i t h low atomic numbers the incidence of photoelectric absorption, for 511 keV photons, is negligible. In a C o m p t o n interaction the photon interacts w i t h an outer shell electron. In doing so its pa th is deflected, and it loses some of its energy. Most scattered photons are scattered out of the field of view and are never detected. T h e effect of these interactions is termed attenuation. T h e probabi l i ty of a photon not interacting as it propagates along the line £, at transverse distance d, and Scattered Event Accidental Coincindence Figure 2.4: D i a g r a m of a scattered event (left) and an accidental coincidence (right). Photons shown leaving the r ing are scattered through an oblique angle such that their paths do not intersect a detector. angle 9 is termed the survival probabil i ty ; and is given by where pi is the linear attenuation coefficient at position x. The significance of equation (2.1) is that the attenuation experienced by a given pair of annihi lat ion photons is independent of the posit ion of their annihi lat ion along the L O R . Th i s makes possible a simple pre-correction of the data (Ollinger & Fessler (1997)). 2.4 Scattered Events Annihi lat ions i n which one or bo th photons are scattered, but 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 Figure 2.4. These events are incorrectly positioned because the photons' paths are no longer collinear. The overall effect is to add an error signal to the data at low spatial frequencies. Since photons lose some of their energy when they undergo C o m p t o n interaction, they can be be discr iminated from un-scattered photons by measuring the energy they deposit (2.1) i n the crystal . A l t h o u g h this measurement is only accurate to w i t h i n + / - 10% on 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 many scattered photons and the relatively smal l solid angle presented by the detector r ing , 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 from separate annihilations are detected w i t h i n the same coincidence t iming window, they w i l l be recorded as shown on the right side of F igure 2.4. These events are termed accidental coincidences, or randoms. The rate of accidental coincidences can be related to the singles rate by noting that for each single detected at detector i, on average TRJ singles occur at detector j during the coincidence t i m i n g window r ; where Rj is the singles rate at detector j. Since each of these TRJ singles results i n a coincidence, there are TR^RJ coincidences per unit t ime for which the first detected photon is incident on detector i. The tota l number of accidental coincidences is the sum of those for which the first photon is detected at detector j and those for which the first photon is detected at detector i. Hence, the rate of random coincidences along the L O R connecting detectors i and j is given by Rr = 2TRIRJ (2.2) Examinat i on of equation (2.2) shows that reducing the coincidence t i m i n g window reduces the counting rate of accidental coincidences. However, t iming inaccuracies due to variations i n the rise-time of the crystal light output require a t iming window of 6-8 ns for L S O . Since the incident singles rates are proport ional to the amount of radioactivity, the accidental coincidence rate increases as the square of the amount of radioact iv i ty i n the field of view (for counting rates that do not saturate the detectors). Th i s count-rate l imi tat ion , along w i t h detector deadtime, determines the upper l imit on the radioact iv i ty used for many studies. 2.6 Detector Deadtime The time required to process a single event l imits the counting rate of a P E T scanner (Hoffman et a l (1989)). Event processing begins w i t h the ris ing edge of the pulse for the first detector involved. The pulse is integrated for some t ime interval , then posit ion calculations and energy discr iminat ion are performed. The detector is "dead" to new events during this t ime. A t very low counting rates, randoms are negligible and the number of true events is l inearly related to the amount of act ivity i n the field of view. The number of randoms increase as the square of the radioactivity in 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 and randoms peak and then decline because of detector saturation. Deadtime is the dominant effect that l imits the amount radioactivity used. 2.7 Physical Model If statistical effects are ignored, these factors can be incorporated into a model for the total number of recorded events to y ie ld YM = 7de[Vdepd9Mdg + rfdgrde + r}sdgsd6\ (2.3) 12 where MdQ is the number of annihilations w i t h photons emitted along a L O R (specified by (d, 0) i n Figure 2.3), Pde is the survival probability, T&Q is the number of accidental coincidences, sae is the number of scattered events, rfd6 is the probabi l i ty of detection for true events, rfdg is the probabi l i ty of detection for accidental coincidences, r)dd is the probabil i ty of detection for scattered events, and 7^0 is the probabi l i ty of an event not being lost due to deadtime. P r i o r to the emission scan, a transmission scan is performed to characterize the effects of attenuation from the subject. Here a point source containing 5 7 C o rotates around the subject to provide a flux of photons along each line of response. The measured data y ie ld the number of transmitted events, Tde, along each L O R . Every morning a blank scan , i.e. a transmission scan w i t h nothing i n the tomograph, is performed to yield a data set, E>de- T h e survival probabil i ty is approximated by their ratio : P = This estimate of survival probabilities would be exact i f the data were noiseless. However, they are not noiseless, so they contribute significantly to the overall image variance unless noise reduction algorithms are applied. These algorithms uti l ize smoothing (Palmer et a l (1986)), segmentation and re-projection (Meikle et a l (1993), X u et a l (1994)), or statistical image reconstruction and re-projection (Bouman & Sauer (1996), Fessler (1995), Fessler et a l (1996), Oll inger (1992)). A simple way to estimate the accidental coincidences is to note that the arr iva l times of the photons due to randoms are uniformly distributed i n t ime while those of true coincidences fal l w i t h i n the t i m i n g window. Col lect ing data i n a second coincidence t i m i n g window that is offset in t ime, such that it collects no true coincidences, yields data w i t h nearly the same mean as that of the accidental coincidences fal l ing i n the trues t i m i n g window. The measured data are given by the product Id6'ifd0rd6i so the detector efficiencies for accidental coincidences, r]de, do not have to estimated. Therefore, not only is the method simple to implement, but it can be performed i n hardware before the data are stored. The major drawback of this approach is that the variance of the estimate is of the same order of magnitude as the variance of the data, i f a significant fraction of detected events are accidental coincidences. In this case, the subtraction can lead to a significant increase i n the variance of the data unless noise reduction methods are used (Casey &; Hoffman (1986)). T h i s variance increase can be avoided by counting the number of singles at each detector and using the rate of random coincidences, Rr. Since there are many more singles than true coincidences, the effect on variance is relatively minor. T h i s approach is not widely used because of the addit ional requirements placed on the acquisit ion hardware, and because singles rates often vary over the course of an acquisit ion. The detector efficiencies for true and scattered events are estimated from a scan of a cal i -bration source w i t h known characteristics (Hoffman et a l (1989)). Deadtime is dependent on many factors related to the architecture and design of a specific machine, so its esti-mation is tai lored to the tomograph (Daube-Witherspoon & Carson (1991)). It is usually assumed to be constant over the durat ion of the scan. These parameters can be used to estimate the number of emitted photons by using the expression Mde = ^ - R d e ) - S d g { 2 A ) 1deVdePde where we assume that Rdg = jderidgE[rdg], Sdg = rYd9VdgE[sde], and E[-] denotes expecta-t ion. The data modeled i n equation (2.4) are often stored i n 2D arrays w i t h the columns indexed by d and the rows by 9. These data 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 impli fy the problem is to ignore the measurement noise altogether, and to assume that the measured data approximate line integrals through the radioisotope d is tr i -bution. T h i s leads to the classical filtered-backprojection ( F B P ) method for tomographic image reconstruction (Kak & Slaney (1988)). T h i s method is routinely used for x-ray C T , as well as for P E T and S P E C T (Single Photon Emiss ion C o m p u t e d Tomography). Its popularity stems from its computational simplicity, and not because of any advantage i n image quality. A mathematical description of filtered backprojection is provided i n Append ix A . Chapter 3 Materials and Methods The portable closed-loop system used for these experiments consists of a 120 L tank, a centrifugal pump, a bypass loop, two magnetic flow meters, two pressure transducers, a test section and valves for control. The test section is made of clear polycarbonate pipe 70 mm i n diameter and 1.1 m i n length. The inlet pipe is 14 mm i n diameter, forming a 1:5 axisymmetric sudden expansion. Figure 3.1 shows a cross-sectional view of the abrupt expansion. F low reaches and leaves the test section through 4.5 m of reinforced hose, which ensures r mm L L mm FLOW Figure 3.1: A schematic of the sudden expansion. Full Bore Quick Disconnect Lead Shielding Linear Stage Lead Shielding Full Bore Quick Disconnect Test Section Linear Stage Figure 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 terminated w i t h a pair of full-bore ba l l valves and a full-bore quick-disconnect coupling to facilitate placement and removal of the test section into the gantry of the tomograph. The test section is mounted to 760 mm linear stages on either side of the tomograph so that the test section can be moved along its axis. To shield the detector blocks from radiat ion originating outside the tomograph 19 mm thick lead shielding is positioned concentrically w i t h the test section, butted up against the camera. Th i s thickness of lead stops ~ 95% of incoming 511 keV gamma photons. Figure 3.2 is a schematic of the experimental setup and Figure 3.3 represents the portable flow loop. The experiments were conducted by first radioactively labell ing 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 and introducing them into a non-radioactive pulp suspen-sion. The fibre length of each fraction of fibres and the whole suspension, as determined through use of an opt ical fibre analyzer, are provided i n Figure 3.4. A s shown, the names of the fibre fractions are defined using the screen sizes by which they were retained i n the Bauer -McNet t device. T h i s is the tradi t ional method of defining fibre fractions i n the pulp Bypass Loop »- To Test Section "• Return Line Tank LT" Pressure Transducers Transmitters Power Box Figure 3.3: A schematic of the portable flow loop. and paper l iterature. T h e fibres were labeled by suspending them i n a solution of acetic acid while 1 8 F — F2 was bubbled through the suspension at 10 m l / m i n w i t h constant st ir -r ing. After the addit ion of the fluorine the fibres were filtered and washed w i t h disti l led water. A t this point the fibres were labeled w i t h 1 8 F w i t h a 10% yie ld based upon the total radioactivity introduced. 1 8 F has been chosen here in preference to other positron emitt ing tracers such as 1 5 0 , n C , or 1 3 N because of its reasonably long half-life of 110 minutes and its reactivity w i t h T M P pulp fibres. The tomograph used i n this study is the Concorde Microsystems m i c r o P E T R4 . The m i c r o P E T Rodent 4-ring system (R4) has a 7.8 c m axia l extent, a 10 c m transaxial field of view ( F O V ) and a 12 cm gantry aperture. The system is composed of 96 detector modules, each w i t h an 8 x 8 array of 2.1 x 2.1 x 10 m m lutet ium oxyorthosilicate (LSO) crystals, arranged i n 32 crystal rings 14.8 cm i n diameter. Each of the detector crystals are coupled to a Hamamatsu R5900-C8 posit ion sensitive photomultipl ier tube ( P S - P M T ) v ia a 10 cm long optical fibre bundle. T h e detectors have a t iming resolution of 3.2 ns , an average Fibre Length (mm) Figure 3.4: T h e fibre length distributions of the S B K pulp and the three tracer fibres used (labeled R14,R48 and R100). The nomenclature for the fibre fractions is retained from the Bauer Mcnett device used to separate the fibre suspension. energy resolution of 18.45%, and an average intrinsic spatial resolution of 1.75 m m . The system operates i n 3D mode without inter-plane septa, acquiring data i n list mode. Us ing the 2D filtered back projection reconstruction algorithm, 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 direct ion (horizontal direction of the F O V ) , and 2.07 m m F W H M i n the radia l direction (vertical direct ion of the F O V ) . The tangential resolution slowly increases to 3.38 m m F W H M at the edge of the F O V . The radial resolution increases to 3.00 m m F W H M at 25 m m radia l offset and then deteriorates l inearly to 3.68 m m F W H M at the edge of the F O V (Mok et a l (2003)). A l l images i n this study were reconstructed using the 2D filtered back projection algor i thm. Before beginning a series of scans w i t h pulp we started w i t h three test scans using water and 1 1 Cfmethyl-iodide]. T h e purpose of the prel iminary scans were two fold. F i rs t ly , we wanted to characterize the effect of having ~ 95% of the act ivity located outside of the tomograph's field of view; and secondly, to find out the count-rate capabil i ty of the tomograph w i t h our particular apparatus. In the first scan we set up the flow loop and filled the tank w i t h 40 L of water and added 3880 MBq o f 1 1 C[methy l - i od ide ] . The acquisit ion was r u n for 2.5 hours ( 7 half-lives). Af ter reconstructing the data we found that the act iv i ty had decayed much faster than expected. We discovered that the boi l ing point of 1 1 C[methyl - iod ide] was ~26 °C and that the act iv i ty was volatile. In our second scan we set up a sealed sudden pipe expansion w i t h the same diametr ical dimensions as the test section, but shortened to 30 cm i n length. The new phantom was filled w i t h water and 330 MBq of n C[methy l - i od ide ] and scanned for 2.5 hours. The results indicated that the tomograph had trouble w i t h very high and very low count rates. After 3.3 half-lives the the tomograph was able to measure the act ivity i n the phantom, and thereafter for a period of 3.3 half-lives. For 40 L of pulp suspension this translates into a m a x i m u m activity of 1110 MBq and a m i n i m u m of 110 MBq. W i t h this data we repeated the first scan w i t h 1000 MBq of 1 1 C[methyl - iodide] i n solution w i t h a sod ium bicarbonate buffer to prevent the act iv i ty from becoming volatile. The results were good and the act ivity levels confirmed for the subsequent series of pulp scans. Table 3.1 provides the fraction labeled, the upstream bulk flow rate, durat ion of scans, and act ivity added for each image successfully obtained i n this study. For each case an image was captured at the step and another 70 c m downstream of the expansion. Each scan begins w i t h the product ion of the labeled fraction. Meanwhi le , the flow-loop is set up around the tomograph. After the test section has been placed i n the gantry aperture of the tomograph and p lumbed i n , it is positioned i n the tomograph's field of view w i t h the aid of a laser line. P r i o r to the emission scan a transmission scan is performed to characterize the effects of attenuation of the photons, due to the test section and its contents. Here Scan Tracer Upstream Vel. Activity Image Dur. Image Dur . Phenom. (fraction) (m/s) (MBq) Step (s) Downstream (s) Behaviour 1 R48 0.5 585 3599 2736 plug 2 R48 0.7 480 2212 2922 fluidized 3 R48 0.8 1125 2708 2289 fluidized 4 R14 0.5 1015 2573 2197 partially-fluidized 5 R14 0.9 1045 1220 1134 fluidized 6 R100 0.9 475 2772 3072 fluidized Table 3.1: A summary of the scans conducted. a rotat ing point source containing C o rotates around the object to provide a flux of photons along each line of response. Before the activity is introduced, the p u m p is turned on and the system allowed to r u n for several minutes. Before the labeled fibres are added to the tank the flow rate is adjusted and the camera set to acquire data. The radioactive fibres are then added to the tank and allowed to be pumped though the system. A t this point data acquisit ion 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 The concentration profiles were obtained at various upstream velocities for suspensions w i t h a bulk concentration of 0.4 %( wt) w i t h three different tracer fibre sizes. A s the number of trials conducted was smal l , most figures have been included. T h e discussion of the results w i l l be conducted i n three subsections. In the first subsection we w i l l discuss the impact of velocity on the concentration distr ibution. In the second subsection we examine the effect of tracer particle size on the measured profile. F ina l ly , 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 which the tracer fibres were well mixed after the expansion. We defined these cases as " f lu id ized" , and the observed 8.5 CD E <I R14 o R48 a R100 504 0.6 0.7 0.8 Upstream Velocity (m/s) Figure 4.1: Est imates of the size of the recirculation zone for the cases i n which the suspension was fluidized after the expansion. hs is the step height. flow was qualitatively s imi lar to that reported by Salmela & K a t a j a (2005). The size of the recirculation zone is shown in Figure 4.1 for these cases only. It should be noted that the length of the recirculation zone L R has been scaled to the height of the step hs. It is difficult to compare our results quantitatively to either those reported by Salmela & K a t a j a (2005), as we have a different step size and 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 examining Scan 2, see Table 3.1 and Figure 4.2. A s w i t h a l l cases presented three views are provided; a top view, a side view, and a cross-sectional view. The top and side views are slices from the centre of the test section i n planes orthogonal to the viewing direction. A s shown, these images are qualitatively s imilar to those reported by Moraczewski et a l (2005) i n that we see a central jet surrounded by a recirculating zone. There are three observations that can be made immediately from this result. F i r s t , the fibres are not d istr ibuted evenly through the imaged-volume. A n asymmetry is apparent i n Figure 4.2(c) i n which there is a higher concentration of tracer fibres at the bot tom of the tube when compared to the top. This feature was found i n a l l fluidized cases examined. A s the images were integrated over some time the temporal evolution of this concentration asymmetry is not known. Rebinning the datasets into shorter durat ion frames may provide insight into this particle accumulation. However, w i t h a reduced number of events i n each frame much higher levels of noise are to be expected. Second, there is an annular region between the jet and the recirculation zone w i t h a concentration that is lower than the average concentration of the suspension. In other words, we observe a region w i t h particle depletion. F ina l ly , i n the centre of the recirculation zones the concentration of tracer fibres is significantly larger than the average concentration. The last two observations have been reported by Moraczewski et a l (2005). We attempt to quantify these features by examining the rad ia l concentration profiles at different ax ia l positions, see Figure 4.3 (a) and (b). In these figures we have normalized the concentration to the bulk concentration of the suspension, i.e. 0.4 w t % . R a d i a l and axia l distances have been normalized by the radius of the larger pipe; and the ax ia l origin, x = 0, is set at the step. T h e horizontal and vertical axes are y and z, respectively, and their origin is placed along the central axis of the test section. We speculate that the concentration-depletion zone results from the water layer formed i n the upstream tube. It is commonly understood that a water layer forms on the periphery of a pipe w i t h a flowing pulp fibre suspension. We have characterized both the radius of the jet and thickness of the concentration-depletion layer i n Figure 4.3(c) as a function of ax ia l posit ion. 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 Figure 4.3(d) the average concentration of the jet increases w i t h ax ia l posit ion. In this case we see a 50% increase i n the concentration i n the central port ion of the tube and advance the argument that this results from 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. p lug flow. We observed this i n one of the scans conducted; namely Scan 1, which was conducted at an upstream velocity of 0.5 m/s. A s shown i n Figure 4.4, the tracer fibres were not well distr ibuted after the expansion plane and traveled through the region visualized as a plug. Clear ly at this lower velocity the shear imparted by the fluid is insufficient to disrupt the fibre network. It must be noted that dur ing imaging, the central jet may be slowly meandering or folding as it travels down the length of the tube. T h i s feature can not be captured as the image acquired is averaged over some t ime. We characterize these curves by showing the rad ia l concentrations and the size of the central jet i n Figure 4.5. A s shown, we see that the tracer fibres spread radial ly w i t h increasing ax ia l distance. The mechanism by which these fibres spread is difficult, i f not impossible, to ascertain from these figures alone as the jet may meander during the imaging period. In other words, we can not ascertain i f the tracer fibre mix ing results from shear induced migrat ion or from the stabil ity of the jet. One of the s tr ik ing features i n this image is the relatively large localized concentration of act iv i ty near the top of the tube i n Figure 4.5(c). We are uncertain of the origin of this but its presence confirms the fact that mix ing does not take place at this velocity. T h e tracer fibres that accumulated at the top of the tube may have done so gradually, or may have arrived as a floe. Rebinning the dataset into frames of shorter duration would allow the evolution of the accumulation to be examined. T h i s , however, would result i n images and data w i t h higher noise content due to the reduced number of events i n each rebihned frame. (b) 8 (c) Figure 4.2: Four views of the act ivity profile for the case i n which the tracer fibres were well-mixed, or f luidized, after the expansion. The image was acquired using the R48 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/R0 = 0.57, (c) is a side view, and (d) is a three-dimensional reconstruction of the activity distr ibut ion. (a) 0.2 I 0 -0.2 -0.4 -0.6 -0.6 x/RQ=0.03 _ _ x/R°=0.20 x/R°=0.35 . . . . x/R°=0.68 (C) Normalized Concentration o Outer Region 0 Inner Region < Depleted Regio 0 0 0 0 0 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Distance Downstream of Expansion x/R 0 0.2 0.4 0.6 0.8 1 1.2 Distance Downstream of Expansion x/R Figure 4.3: Four views of the act iv i ty profile for the case i n which the tracer fibres were well -mixed, or f luidized, after the expansion. The image was acquired using the R48 tracer fibres w i t h the upstream velocity set at 0.7 m/s (see Table 3.1). Image (a) represents the concentration profile i n the radia l direction for a central horizontal slice at four different axia l positions. Image (b) represents the concentration profile i n the rad ia l direction from a central vertical slice at four different axia l positions. Image (c) is an estimate of the growth of the depletion zone. Est imates are given of the inner and outer rad i i of the depletion zone and their difference. Image (d) is an estimate of the average concentration of the jet as a function of ax ia l posit ion. (b) • (c) • Figure 4.4: Four views of the act ivity profile for the case i n which the tracer fibres were not well-mixed after the expansion. We define this case as plug flow type behavior. The image was acquired using the R48 tracer fibres wi th 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/R0 — 1.00, (c) is a side view, and (d) is a three-dimensional reconstruction of the act iv i ty d istr ibut ion. (a) 08 0.6 V V - . Cv-.. x/R =0.03 _ _ x/R°=0.35 . . . x/R°=0.68 . . . . x/R°=1.00 0.4 02 o a= ° -0.4 c ( % - - " -0.6 -O.B -1 Normalized Concentration Distance Downstream ot Expansion x/R Figure 4.5: A view of the act iv i ty profile and jet expansion for the case i n which the tracer fibres were not wel l -mixed after the expansion. The figures are from the image acquired using the R48 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 radia l direction from a central horizontal slice at four different ax ia l positions. Image (b) represents the w i d t h of the act iv i ty profile as a function of ax ia l posit ion. 4.2 Effect of Particle Size A t this point we compare the effect of particle size by examining two cases conducted at the same velocities using different length fractions. In the first comparison, we examine scans 5 and 6 i n which we compare the distr ibution of the R100 and R14 tracer fibres at 0.9 m/s (see Figure 4.6). A s shown, the act ivity d istr ibut ion of these tracer fibres appear somewhat similar. In both cases the features reported i n the previous section are apparent; that is an asymmetry i n the vertical direction, a depletion layer between the central jet and the recirculation zone, and particle accumulation i n the lower recirculat ing zone. We 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 R48 and R14 fibres at 0.5 m/s as given as scans 1 and 4 i n Table 3.1. The results for scan 1 have been shown earlier as Figure 4.4. Here it is apparent that the tracer fibres are not wel l -mixed after the expansion. W i t h the R14 fibres however, we find that the tracer fibres are well -mixed i n the upper port ion of the channel only (see Figure 4.7 ). In the upper por t ion of the channel we observe the depletion zone between the central jet and the outer portions of the channel. It should be noted that during this t r i a l it was visual ly observed that the lower port ion of the channel was static. We do not interpret these results as quantitative evidence that their is a difference between the motion of these two classes of fibres. We speculate that this result occurred due to the fact that the experimental protocol was not the same as the other scans. T h e pulp i n the test section was allowed to settle over several days prior to the scan and was not well mixed prior to the start of the scan. Furthermore, when the scan was conducted the flow rate began high and was lowered, as opposed to being ramped up to the targeted flow rate as in subsequent scans. T h i s resulted i n the region w i t h lower fibre concentration being fluidized first, and when the flow rate was dropped this region remained i n motion. We have included this result as it is interesting to report the possibil ity of a stable, static region i n this type of device. 4.3 Far Downstream Far downstream of the expansion (x/R0 = 20) we were able to measure b o t h the concentra-t ion d istr ibut ion using P E T and the velocity profile using ultrasound Doppler velocimetry ( U D V ) , a commercial device obtained from Signal Processing S A . For a l l flowrates tested, the velocity profiles at this point were similar and displayed p lug like behaviour. A s shown i n Figure 4.8,. a velocity boundary layer exists near the walls of the pipe i n the region \y/R0\ > 0.8. We display the concentration profiles at this point for four different cases (a) 1 (b) 1 (c) (d) 1(e) 1(f) Figure 4.6: Two cases i n which the tracer fibres were wel l -mixed after the expansion. In images (a)-(c) we examine the case in which the upstream velocity was set at 0.9 m/s w i t h the R100 tracer fibres. Images (d)-(f) represent the case w i t h the R14 tracer fibres. T h e images i n the first co lumn ((a) and ((d)) represent the top view. T h e images i n the second column represent the cross-sectional view at x/R0 = 0.57. T h e remaining two images represent the side view. Figure 4.7: Four views of the act ivity profile for the case i n which the tracer fibres were part ia l ly -mixed after the expansion. The image was acquired using the R14 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/R0 — 0.57, (c) is a side view, and (d) is a three-dimensional reconstruction of the activity distr ibution. 0.05 .0.01 1 ' ' ' ' ' 1 1 ' ' 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 y / R 0 Figure 4.8: T h e velocity profile of the suspension as measured using U D V . S ix experiments were conducted i n which the upstream velocity Vi set i n the range from 0.5 to 1.0 m/s. The error bars represents the standard deviation determined from the average of this data set. (see Figure 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 that the tracer fibres, trapped i n smal l floes at the inlet, are not sufficiently disrupted by the shear at the step to re-distribute themselves evenly downstream. A s the flow rate is increased more mix ing is induced and a more even rad ia l d istr ibut ion is evident. (a) (b ) -0.2 -0.4 -0.6 -0.8 • 0.8 0.6 0.4 s 0.2 g ° 0 -0.2 -0.4 -0.6 -0.8 i Normalized Concentration Normalized Concentration Figure 4.9: The concentration profile downstream of the expansion, (a) Scan 1, (b) Scan 2, (c) Scan 5 and (d) Scan 6. Details of each experiment are given i n Table 3.1. Chapter 5 Summary and Conclusion Positron Emiss ion Tomography ( P E T ) was used to investigate the dynamics of a 0.4% (wt) fibre suspension flowing through an axisymmetric 1:5 sudden expansion. S ix scans were conducted i n which bo th the upstream velocity and the size of tracer fibres labelled were varied. Images were taken upstream and 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 imparts shear that disrupts the fibre network causing measurable changes i n the local fibre concentration. Two distinct regions were clearly distinguished: plug like, i n which the tracer fibres were not mixed through the entire volume of the expansion; and fluidized, i n which the tracer fibres were well mixed. O u r results for the fluidized case are worth highlighting as we found that concentration inhomogeneities exist. 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S O S Z Y S N K I , R . M . & T A M D O O , P . M . 1985 T h e f locculation of pulp fibres Fund. Res. Syrnp., 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 Three-Dimensional Image Reconstruc-t ion Us ing 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 on the E v o l u t i o n of Shear Layer Instabilities i n L a m i n a r F low Through Ax i symmetr i c Sudden Expansions Phys. Fluids 2 9 , 2828-2835 L E I G H T O N , D . & A C R I V O S , A . 1987 The shear-induced migrat ion of particles i n concen-trated 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 Computat ional and Exper imenta l Study of a Capt ive Annular E d d y J. Fluid Mech. 2 8 , 43-64 M I E K L E , S . R . , D A H L B O M , M . , $ C H E R R Y , S . R , 1993 At tenuat i on Correct ion Us ing C o u n t - L i m i t e d Transmission D a t a i n Pos i tron Emiss ion Tomography 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 Eva luat i on of the H i g h Resolution S m a l l A n i m a l P E T Scanner Biomed. Eng. Appl Basis Comm. 1 5 ( 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 low of a concentrated suspension through an abrupt axisymmetric 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 Mathematics of Computerized Tomography New York: Wiley O L L I N G E R , 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 Likel ihood Transmission Images For Attenuat ion Correct ion In P E T Proc. 1992 IEEE Med. Imag. Conf. 1185-1187 O L L I N G E R , J . M . , & F E S S L E R , J . A . , 1997 Posi tron-Emiss ion Tomography 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 . , 1986 Transmission Profi le F i l t e r ing for Positron Emiss ion Tomography IEEE Trans. Nucl. Sci. 3 3 , 478-481 P H I L L I P S , 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 . 1992 A constitutive equation for concentrated suspensions that accounts for shear-induced particle migrat ion Phys. Fluids 4 , 30-40 S A L M E L A , J . , & K A T A J A , M . , 2005 Floe Rupture and Re -F loccu la t i on i n Turbulent Shear F low Fund. Res. Symp., Cambridge 1 , 35-50 S H E P P , L . A . , & L O G A N , B . F . , 1974 The Fourier Reconstruction of a Head Section IEEE Trans. Nucl. Sci. 2 1 , 21-43 T H A L E N , N . & W A H R E N , D . 1964 Shear Modulus and U l t i m a t e Shear Strength of Some P u l p F ibre Networks Svensk. Papperstidn. 67(7), 259-264 T H A L E N , N . & W A H R E N , D . 1964 A n Exper imental Investigation of the Shear Modulus of M o d e l F ibre Networks Svensk. Papperstidn. 67(11), 474-480 X u , M . , L U K , W . K . , C U T L E R , P . D . , & D I G B Y , W . M . , 1994 L o c a l Threshold for Seg-mented Attenuat ion Correct ion of P E T Imaging of the T h o r a x IEEE Trans. Nucl. Sci. 41, 1532-1537 Appendix A Filtered Backprojection Fi l tered backprojection was first applied to P E T by Shepp & Logan (1974). Introductory treatments of the a lgor i thm can be found in K a k & Slaney (1988), and H e r m a n (1980). W i t h more comprehensive treatments in Deans (1983), and Natterer (1986). The dis-tr ibut ion of the radioisotope is modeled by the function X(x,y, z) G L2. For a given 2D slice, we assume that the mean of an indiv idual measurement Y^g (equation (2.3)) is given by ge{d) = / X(x,y,z)dxdydz ( A . l ) Jl{d,9) where £(d,6) 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 data , M^g. In the rotated coordinate system of figure A . l , d — xg, so the line integral can be expressed as /oo X(xe,yg)dyg,e e (0,n),xg,yg,e (A.2) -oo 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 function (and the data that it approximates) as a projection. The Fourier transform of each projection is given by Th i s result, known as the projection-slice theorem, has two implications. F i r s t , the Fourier transform of a projection yields samples of the 2D Fourier transform of the image; and second, these samples lie along a line at the same angle, 6, i n the frequency domain as that of the projection in the spatial domain. This result can be wr i t ten i n more standard notation as where the Fourier transform of the image is now expressed i n polar coordinates (p, 7). Equat ion (A.3) can be used to reconstruct the image by constructing the Fourier transform i n polar coordinates, interpolat ing to rectangular coordinates, and then taking the inverse transform. A more efficient method can be derived as follows. The image \(x, y) is given by = k{ug,V0) 1^=0 (A.3) Ge(p) = A(p,7) \y=e,Pm (A.4) ( A . 5 ) xe=rcos((j)-0) Figure A . l : Pro ject ion Geometry. v = psia.0, x — rcos(/>, and y — rs'mcj) yields r-27r roo \(r,<j>) = / / pA{pcos9,psmB)ej27rprcos{,l>-e)dPde Jo Jo ( A . 6 ) Rewri t ing A(p cos 6,p s in 9) as A(p , 0) and using the facts that cos(<^ > — 6) = — cos(</> — 0 + ir) and Ge(p) = Gg+V(~p), this can be rewritten as P7T poo X(r,cp)= / \ p\A{p,6)ej27Tprc08tt-eUpdd Jo J-oo '0 J  A p p l y i n g the projection-slice theorem leads to | p | Ge{p)ej2lxprcos{<i'-e)dpde 0 J-oo * ge(rcos(<l>-0))d6 o (A.7) (A.8) cretizing leads to the expression Ne-i M r ^ ) = ^ X > ( r c o s ( 0 - 0 , ) ) ( A . 9 ) Nf> 1=0 Equat ion (A.9) shows that the value of the image at a point (r cos </>, r s in <f>) i n figure A . l can be found by first filtering the projections w i t h a ramp filter, then summing the filtered values at the coordinate xgx = r cos(9l — <f>) over a l l projection angles 0, . Note that the value at xgt w i l l contribute to a l l pixels along the L O R s that contributed to the measurement at this point. The algor i thm can be efficiently implemented by filtering each estimated projection, gg(d) = M^g, w i t h a ramp filter to yie ld gg(d) and then adding 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 . The latter operation is called backprojection, so the a lgor i thm is unsurprisingly called filtered-backprojection. T h i s a lgor i thm and 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 identical to the algor i thm 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 including each of the 63 image slices, central horizontal and vertical concentration profiles, jet and depletion zone data, jet densification data, and downstream concentration profiles. Scan Tracer (Fraction) Upstream Vel . (m/s) Posit ion of Step (Slice) Pos i t ion of Centre ( X , Y ) Image Centre (Past Expansion) 1 R48 0.5 50 (68,59) 2 cm 2 R48 0.7 50 ( 6 8 , 5 9 ) 2 cm 3 R48 0.8 33 ( 6 0 , 6 1 ) 0 cm 4 R14 0.5 33 (67,62) 0 cm 5 R14 0.9 50 (66,59) 2 cm 6 R100 0.9 50 (66,59) 2 c m 1 9 Figure B . l : Scan 1 2D Slices. Figure B.2 : Profiles for Scan 1. Hor izontal Profiles (a), Jet Expans ion (b), and Downstream Concentration (c). Figure B.3 : Scan 2 2D Slices. (a) 0.4 0.2 o or 0j >s -0.2 -0.4 -0.6 - . x/R°=0.20 . . . x/R°=0.35 . . . . x/R°=0.68 (c) 1 1.5 2 2.5 Normalized Concentration o Outer Region 0 Inner Region < Depleted Region 0 0 0 0 0 0 H » « « « ° o o ° 0 < > < > o o0 o 0 o o o ' o < < <<.<, <<<<<< < 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Distance Downstream of Expansion x/R -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Distance Downstream of Expansion x/R Normalized Concentration Figure B.5: Scan 3 2D Slices. Jl , , , , , , , 1 „ . 7 I , , , , , , , 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Distance Downstream of Expansion x/R, D i s t a n c e D o w n s t r e a m o f E x p a n s i o n m (e) 0.8 7 V -0.6 -0.4 0.2 > 0 0.2 1 0.4 f -0.6 -0.8 -1 1—v— Normalized Concentration Figure B.7: Scan 4 2D Slices. (a) (b) Normalized Concentration Distance Downstream of Expansion x/Ro Figure B.8: Profiles for Scan 4. Downstream Concentration (a), and Jet Densification (b). Figure B.9: Scan 5 2D Slices. 0.5 1 Normalized Concentration Figure B.12: Profiles for Scan 6. Horizontal Profiles (a), Ver t i ca l Profiles (b), Deplet ion Zone D a t a (c), Jet Densif ication (d), and Downstream Concentrat ion (e). 58 

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