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Time-Resolved γ-Densitometry Imaging within Fluidized Beds R. F. Mudde,*,† P. R. P. Bruneau,† and T. H. J. J. van der Hagen‡ Kramers Laboratorium voor Fysische Technologie, Delft University of Technology, Pr. Bernhardlaan 6, 2628 BW Delft, The Netherlands, and Reactor Physics Department, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands
γ-Densitometry is a nonintrusive measuring technique applicable to mulitiphase flows. In most cases, the time-averaged spatial distribution of the phases is measured. A combination of high spatial and temporal resolution simultaneously would provide valuable information of flowing multiphase systems. In this work, γ-ray densitometry is reexamined, specifically the reduction of sampling times and the ability to visualize evolving structures within fluidized beds. A multiple fan beam configuration using static 137Cs sources is evaluated by performing simulated reconstructions of the two-dimensional void fraction distribution within a 400 mm diameter fluidized bed. Results for both three and five source systems are presented, which indicate that spatial and temporal resolutions of 10 mm and 10 ms are achievable. Introduction Multiphase flows are frequently encountered in many industrial applications and are for the most part highly complex. Consequently, multiphase research provides scientists and engineers with a rich set of challenging problems relating to hydrodynamic and transport phenomena. In particular, it is the expertimental measurement aspects of multiphase flows that pose some of the more intriguing problems since many of the existing techniques developed for single phase flows are rendered ineffectual in the multiphase environment. By way of example, laser-based methods, which have made it possible to study single phase flows in great detail, rely on the medium under investigation being transparent to visible light. Multiphase flows are (with few exceptions) opaque to visible light thereby precluding the technique. The nett result is that experiments on local phenomena within multiphase flows are seriously handicapped by the lack of reliable, high resolution measuring equipment. In contrast, most solids are to some extend transparent to ionizing radiation such as X- or γ-rays and a variety of nuclear techniques have been developed during the past decade. Two quantities are of key importance: the velocity fields of the phases and the distribution of the phases, i.e., the volume fraction or hold up. Nuclear techniques, which assess the velocity fields, can make use of a flow tracer that emits or absorbs radiation. Several different approaches exist. (i) Positron emission particle tracking: The particle that is followed emits in a nuclear decay process a positron, which annihilates with an electron. In this process, two 512 keV photons traveling in the opposite are generated; see, for example, refs 1 and 2. (ii) X-ray particle tracking velocimetry: Several X-ray absorbing particles are tracked simultaneously in the multiphase flow; see, for example, ref 3. (iii) Computer automated radioactive particle tracking: A radioactive particle is tracked by measuring the spatial distribution of the intensity of * To whom correspondence should be addressed. E-mail:
[email protected]. † Kramers Laboratorium voor Fysische Technologie. ‡ Reactor Physics Department.
the emitted photons outside the reactor. In the group of Mike Dudukovic´, this technique has been developed and applied to many multiphase systems; see, for example, refs 4-6. Nuclear methods are also used for volume fraction measurements. In this case, the objective is to obtain the volume fraction distribution in a cross-section of the multiphase system via tomographic techniques. The nuclear methods compete with other tomographic modalities such as electrical impedance tomography. Although the nuclear approaches are slow as compared to their electrical counterparts, they do not suffer from the soft field effects characteristic of electrical methods. A distinct advantage of the impedance techniques is the easily achieved high temporal resolution, e.g., several hundred reconstructed frames per second (see, for example, refs 7-9). Nevertheless, the difficulties pertaining to resolution in the center region of the imaging plane remain. In contrast, nuclear techniques rely on hard fields; they do not in principle have the same difficulties in the central regions of the flow domain. The primary disadvantage of the nuclear techniques is the low frame rate (temporal resolution) that can be obtained due to inherent noise. High spatial resolution requires relatively long measuring times as compared to the impedance techniques. The nuclear technique is frequently referred to as “nuclear densitometry” since it is based on the measurement of the density of a given material via absorption of high-energy photons. Again, the group of Dudukovic´ has made a significant contribution to the development of the method; see, for example, refs 10-12. An extensive review of the use of nuclear techniques in opaque multiphase systems can be found in ref 13. Several examples of the use of experimental nuclear densitometry (with a wide range in terms of equipment sophistication) have been reported14 and used a single beam detector pair to perform horizontal, unidirectional scans over a cross-sectional plane within a 48 cm diameter bubble column. Assuming cylinder symmetry, this approach provides enough information to reconstruct the (“inifinite”) time-averaged volume fraction of the bubble phase. A similar procedure was followed in ref 15 to measure the solids volume fraction in a turbulent
10.1021/ie049091p CCC: $30.25 © 2005 American Chemical Society Published on Web 02/16/2005
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case of numerical methods such as CFD, the integrity of simulations is compromised since closure models cannot be validated. This paper addresses the question of whether it is possible to find the size and location of individual bubbles using nuclear densitometry. Our specific goals are to answer the following questions: (i) Is it possible to develop a densitometer that can use sampling times as small as 10 ms, and (ii) what is the minimum number of sources that needs to be used for reconstruction of sufficient quality? The latter question is assessed by inspecting the resolution in terms of the ability to identify two bubbles as two distinct objects when the bubbles are separated by 1 cm (in a 40 cm diameter fluidized bed). Tomographic γ-Densitometry Figure 1. Fluidized bed with two bubbles in the measuring plane of the γ-densitometer.
fluidized bed. Dudukovic´ and co-workers used a fan beam from a single γ-source with multiple detectors.10 By rotating both sources and detectors, some 4000 projections were taken to yield a spatial resolution of 5 mm. Because the total measurement time is of the order of 1 h, only a time-averaged volume fraction distribution is reconstructed. This type of tomographic reconstruction is referred to as computed tomography or CT. Fluidized Bed. Fluidization is among the most frequently used operations in chemical engineering (see, for example, ref 16). Proper design and operation of fluidized beds are hampered by the inability to use the laser-based experimental techniques, well-known from single phase research. Computational fluid dynamics applied to fluidization has, as a consequence, a lack of validation material. Furthermore, guidance from dedicated experiments on several modeling issues is, for the same reasons, rather limited. The focus of our work is on setting up nuclear densitometry that can provide some of the experimental data. Our main interest is in the bubbling and churn turbulent regime. In a bubbling fludized bed, gas pockets or voids move upward through the bed. There, the characteristic size ranges from centimeters to decimeters and their velocities are on the order of 0.1-1 m/s. Especially in the case of the bubbling fluidized bed, time-averaged volume fraction measurements represent only part of the information required for a satisfactory description of the phase distribution. Bubbles can be rather large, and information on their exact location, size, and velocity is required. A schematic of a fluidized bed with two bubbles in the measuring plane of the densitometer is given in Figure 1. A bubble of 5 cm diameter at a velocity of 0.5 m/s will be in the measuring plane for a time interval of 0.1 s. To measure properties for such a bubble, we require a spatial resolution well below 5 cm and we need a temporal resolution much smaller than the time the bubble is in the measuring plane. Thus, the aim of the densitometer is to be able to measure at a spatial accuracy of about 1 cm at a sampling time of 10 ms. In summary, the hydrodynamics of fluidization is still poorly understood, due in large part to the inability of measurement techniques to acquire data at the requisite length and time scales simultaneously. As a result, methodologies for the design and scale-up of processing equipment remain predominantly empirical, and in the
Configuration. To capture the bubbles in a fluidized bed, a good resolution in space and in time is required. To focus the work, we chose a particular example that is related to the research of the possibilities of a nuclear fission reactor based on fluidizing a powder that contains a uranium core (see refs 17 and 18). The requirements for the densitometer derived from this example essentially imply the following question: Is it possible to achieve a spatial resolution of 1 cm and a temporal resolution of 10 ms within a 40 cm fluidized bed? A bubble of 5 cm diameter travelling with a velocity of about 50 cm/s moves 5 mm in this 10 ms measurement time. If the specified resolution is indeed possible, then motion blur will be limited and the passage of such a bubble can be recorded in some 10 consecutive tomograms. To answer this question, we have performed simulations that mimic the measurement with the densitometer through the cross-section of the fluidized bed in which two bubbles are present. Two different configurations using three and five sources (illustrated in Figure 2) are investigated, Measuring Principle and Reconstruction Algorithm. When a narrow, parallel beam of monoenergetic γ- or X-rays is transmitted through a closed system containing a two-phase mixture, the number of photons registered per second, R, follows from the LambertBeer law and can be written as:
R ) R0 exp{- [(1 - R)µp + Rµg]d}
(1)
where R0 is the number of photons registered per second when the system is in a vacuum; µp and µg denote the linear absorption coefficient of the particle and gas phase; R is the volume fraction of the gas phase; and d is the inner diameter of the system. It should be noted that the attenuation characteristics of the fluidized bed wall are incorporated in R0. Further note that both µp and µg are in principle functions of the photon energy E. The average gas fraction along the photons beam follows from the measured count rate, R, and two calibration measurements: Rg is the count rate when the system is filled with gas only, and Rpb denotes the count rate for the system filled with powder. For Geldart B powders, the packed bed porosity is almost equal to the minimum fluidization one. We will assume that for a bubbling bed, the powder will be at minimal fluidization porisity and the bubbles are completely empty of particles. The beam will pass through both the powder and the bubbles, and we can conveniently define the
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Figure 2. Three (top left) and five (right) source fan beam configurations.
(chordal average) bubble volume fraction Rb. Then, eq 1 can be rearranged to provide the chordal average of the bubble fraction:
Rb ) 1 -
lnR - lnRg lnRpb - lnRg
(2)
By measuring along many independent beam paths, sufficient information can be gathered for a tomographic reconstruction of the actual spatial bubble void fraction. Two complications arise. Clearly, the finite number of measurements will influence the achievable spatial accuracy. Exactly how many measurements can be taken will be determined by both the number of sources and the detectors that can be used. In the interests of costs and safety, it is desirable to minimize the number of sources whereas the number of detectors used will be dependent (costs aside) on geometric issues. For detectors of a given size, increasing the detector count implies that they are further away from the source. Obviously, this means that the measured count rate decreases and measurement noise becomes more problematic. Noise is the second complication, consequent to the statistical nature of nuclear decay. This means (especially with γ-rays) that inherent Poisson noise is present. For a given sampling time δt, the average number of counts is N. Because of the Poisson nature, the standard deviation in N is xN. Hence, the standard deviation in the count rate R ≡ (N/δt) is given by σR ) x(R/δt). It is clear that this imposes limitations on the accuracy: Short sampling times are required for good temporal resolution, but reconstruction will be adversely affected from the inherent noise. In the simulations, this effect of noise is explicitely investigated (see below). Tomographic Reconstruction. Reconstruction methods can be broadly classified into two groups: analytic and algebraic. The analytic methods are based on the Fourier transform, and while accurate and fast, they require uniform and closely spaced attenuation data. In contrast, algebraic techiques are iterative and reconstruct the object function on a discretized domain. Although significantly slower, algebraic methods offer more flexiblity in terms of limited data sets and are
Figure 3. Nomenclature for algebraic reconstruction methods using a fan beam configuration.
more appropriate for the CT configuration system under consideration here. (Detailed accounts of reconstruction techniques can be found in refs 19, 20, or 21.) In algebraic reconstruction techniques (ART; see, for example, ref 19), the object to be imaged is modeled as an array of discrete unknowns, resulting in a system of equations. In Figure 3, a square grid is superimposed on the image f(x,y) forming an array of pixels (or cells). In the present case, f(x,y) is the desired void fraction distribution Rg(x,y). The square pixel array has dimension m × m, providing a total number of pixels M ) m2. Each pixel is given a single index representation with the numbering sequence starting in the upper left-hand corner of the pixel array. Within each pixel, the function f(x,y) is assumed to be constant with a value of the j-th pixel designated fj. For the i-th ray, pi is the measured value of Rb along the path. The relationship is expressed as M
pi )
wij fj ∑ j)1
i ) 1, 2 ... K
(3)
where wij is the weighting factor, representing the contribution of the j-th cell to the i-th measurement. The numerical value of the weighting factor wij is equal to the fractional area of the j-th image cell intercepted by the i-th ray. If there are p sources and q rays per source, then the total number of rays in all of the projections is given by K ) p × q. The summation in eq 3 is taken over the total number of pixels within the grid M and
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Figure 4. Chordal gas fraction for the five source geometry. Left noiseless, right with noise.
for all raysums of the scan, K, which can be written as
[
w11 f1 + w12 f2 + ... + w21 f1 + w22 f2 + ... + l l wK1 f1 + wK2 f2 + ... +
w1M fM ) p1 w2M fM ) p2 l l wKM fM ) pK
]
(4)
In general, the system of equations is not square and normal inversion techniques cannot be used. The principle of iterative reconstruction methods is to first guess an initial solution to the vector f. Denoting this initial guess as [fh], a corresponding estimated projection vector, [p j ] can be computed. The difference between the measured p and the current approximation [p j ] is then used to update the estimate [fh]. In summary, for each iteration k: (i) For each ray i, calculate the correction term numerator ∆pi wij ) (pi - pi)wij . (ii) Compute the correction term denominator, ∑j)1M wij2. (iii) Apply the correction to each pixel, j, through which the current ray passes, i.e.,
hf j(k) ) hf j(k-1) +
∆pi wij M
(5)
wij ∑ j)1
2
The above algorithm is of the family of so-called rowaction methods. These have the properties that (i) no changes are made to the matrix w, (ii) no operations are performed on the system as a whole, (iii) in each iteration step only one row of the matrix w is involved, and (iv) for calculating f k only f k-1 is required. All algebraic methods are essentially based on the above algorithm; the differences between the various methods such as ART (solves Wf ) p with constraint minx||x||22), SART (simulteneous ART), MART (multiplicative ART; this technique is a maximum entropy algorithm, satisfying the constrained optimization minx ∑i xi ln xi such that Wf ) p), etc. are the formulation of the correction term and pixel update sequencing. In the present work, the void fraction R is bounded between the value of the unfluidized powder, Rmin ) 0.4, and the maximum value, which is obviously 1. In case, during iteration, the value of a reconstructed pixel falls out of this range, its value is set back at the closest limit: R(i,j) < Rmin f R(i,j) ) Rmin, R(i,j) > 1 f R(i,j) ) 1.
coefficient µp ) 9.207 m-1), and the void fraction of the powder is set at 0.4. The diameter of the fluidized bed is 40 cm. Because the absorption of air is very small as compared to polystyrene, it is assumed that µg ≈ 0. The wall of the fluidized bed is assumed to be infinitely thin. For both configurations, idential sources are used as follows: 100 mCi, 137Cs. In the simulations, the transmitted photons are detected by 31 detectors per source. The distance from source to detectors is 1.67 and 2.74 m for the three and five source geometries, repectively. The sampling time is set at δt ) 10 ms. An overall efficiency of 0.4 on the detecting side is taken into account to calculate the count rate. Note that the artifical data are generated from a binary system: The void fraction is either 0.4 (powder) or 1 (gas bubble). However, in the reconstruction, all values of R between 0.4 and 1 are allowed. Hence, a gray value reconstruction is generated that later, via appropriate thresholding, is turned into a binary reconstruction (see, for example, Figure 5). In fluidization, both systems that (almost) have a binary distribution or that have a more gradual change of R around the bubbles are found. In principle, the reconstruction technique can handle both as is well-known from medical applications (see, for example, ref 19). However, for the present study, it is, due to the small sampling times, not clear how the smearing of the reconstructed image will affect the possibility to confidently reconstruct gradual changes in the void fraction. This point needs further attention in future work. In the simulations presented here, both the SART and the MART reconstruction algorithms have been used. The SART reconstruction algorithm (see ref 21) represents a reasonable compromise between convergence rate and accuracy, while being more tolerant to noisy data than the standard ART algorithm. The modified MART algorithm (due to refs 22 and 23) provides similiar accuracy to that of SART but is significantly faster (of the order three times for our implementations) and is now the method of choice. In both three and five source cases, a 64 × 64 grid is used, providing a minimum resolution of 6.25 mm. Noise. Simulations are performed for both ideal (i.e., noiseless) and noisy data, to assess the reconstruction degradation due to the Poisson noise. The mean beam count rate, Rm, is first calculated from the beam geometry and the known source strength of 100 mCi. Subsequently, the noise is added according to the standard deviation described above:
Simulations Artificial data are generated for two configurations using three and five sources (see Figure 2). The particle phase consists of polystyrene particles (linear absorption
R* ) Rm + XN
xδtR
(6)
where XN is a normally distributed random number in
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Figure 5. Comparison of reconstruction of two bubbles: three and five sources, noiseless, and with noise.
the range [-1, 1]. Finally, R* is substituted into eq 2 to get the (noisy) measured value of the chordal averaged bubble fraction, Rb, which is fet to the reconstruction algorithm. As a first test, two bubbles, each with a diameter of 5 cm, are placed in the reconstruction plane. The first bubble is located at the fluidized bed axis origin, while the other is centered 6 cm from the column axis. The influence of noise on the generated chordal gas fraction is shown in Figure 4 where the integral of the gas fraction along a γ-ray normalized by the column diameter (ordinate) is plotted against ray number (abcissa). This particular plot is for the five source geometry, using a total of 5 × 31 (155) detectors. In the noisless case, one can clearly see the peaks corresponding to the two bubbles. Three fan beams show two distinct peaks: Each of their beams passes through no or one bubble. The two other fan beams generate only one peak as some of the individual beams pass through both bubbles. The noisy case shows the same features, but the influence of the noise makes the distinction of the two separate peaks more difficult. Reconstructions of the two bubble phantom are presented in Figure 5, where the central image is the original (reference) phantom. The top row images correspond to the three source configuration (noiseless and noisy) while the lower images are for the five source case. A typical reconstruction requires 250 iterations taking 3 min on a 1.2 GHz PC. For smaller bubbles and/or shorter sampling times, the number of iterations needed increases to well above 300 with corrsepondingly longer reconstruction times. It is clear from these images that the reconstruction using only three sources is poor, especially when noise
is included. The five source configuration performs significantly better and was selected for further testing. Two bubbles of equal diameter are placed with their centers on a diameteral line of the fluidized bed where the distance from their center to the column axis is 10 cm. The bubble diameter is varied between 10 and 0 cm, decremented in 1 cm steps. For each case, a noiseless and a noisy simulation are performed. From the reconstructions, after some noise filtering, the area of the reconstructed bubbles is measured. This area is compared with the original area and with the area that the bubbles have on the grid. A pixel is part of the bubble if 50% of its area is occupied by the bubble. This thresholding obviously reduces the area, but this effect is also present in the calculation of the reconstructed bubbles, where thresholding was also necessary. The results are shown in Figure 6. In the figure, the solid line represents the true area and the dotted line represents the area on the grid after thresholding. The circles indicate the noiseless reconstruction; the square and diamond correspond to the noisy case. The noiseless case yields slightly greater areas than that with noise. Furthermore, the reconstructed areas are systematically shifted as compared to the areas obtained from thresholding the grid pixels. The smallest bubble size still discernible is 2 cmsbubbles of a diameter of 1 cm are no longer visible. Thus, the resolution is of the order 1 cm, i.e., two pixels. A second example illustrates the ability to distinguish two objects in close proximity: a 10 cm diameter bubble centered on the bed axes and a smaller 4 cm diameter bubble separated by a distance of 1 cm; see Figure 7. The interbubble separation distance is less than two pixels. Note that the original bubbles are only separated
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Figure 6. Comparison of reconstructed bubble area. The full line represents the true bubble area as a function of the square of the bubble diameter; the dashed line shows the area of the bubble on the grid. The area of the reconstructed bubbles is given by the symbols: noiseless (O) and with noise (0, ]).
Figure 8. Reconstruction of bubbles close to the wall: db ) 6 cm; five source case.
Figure 7. Reconstruction of a large (10 cm) and small (4 cm) bubble separated by 1 cm: (left) original bubbles on the grid and (right) reconstructed bubbles; five source case.
by one pixel. The figure shows an unfiltered, unthresholded reconstruction. The reconstruction might be considered acceptable given that the two bubbles are clearly separated and discernible; that is, noise effects have not resulted in blurring to a single object. Notwithstanding, this example does indicate that an accurate reconstruction of the size and shape is difficult to obtain. Sensitivity at Wall. As shown above, γ-densitometry is capable of measuring at the center of the fluidized bed with good resolution, a distinct advantage over the competing (albeit much faster) electrical impedance techniques. Nonetheless, electrical techniques do provide good resolution in the near-wall region. To assess the corresponding near-wall capabilities of the γ-densitometer, additional simulations have been performed with bubbles close to the wall. A 6 cm bubble is positioned with its center 16 cm from the column axis. Furthermore, the influence of bubble angular position within the fluidized bed was investigated by varying the relative position of the bubble with respect to the sources. In Figure 8, the reconstructions are shown. The fivesource configuration (see Figure 2) is used, with one of the sources located on the negative vertical axis. The distance from the bubble edge to the column wall is 1 cm, i.e., less than two pixels of the 64 × 64 grid. The reconstructions show that when the bubble is directly facing a source (Figure 8d), the reconstructed image is smeared with the wall. In those cases where a bubble is offset from this position, this effect is dramatically reduced (Figure 8a-c). Several simulations have been performed to test whether this could be a peculiarity attributable to the random nature of the noise. It was found, however, that the bubble facing the source was always attached to the wall (see also Figure
Figure 9. Three-dimensional representation of two reconstructed bubbles close to the wall: db ) 6 cm; five source case.
8e). Finally, simulations were performed to see whether two bubbles close to the wall could be reconstructed with confidence. Also, these bubbles had a diameter of 6 cm and their center was located 16 cm from the column axis. Figure 8e,f shows two of these cases. Reconstruction of two bubbles that are separated by a large distance gives comparable results as for single bubbles. When the two bubbles are close together (their spacing in Figure 8f is only 1 cm), satisfactory reconstructions are still possible. A three-dimensional representation of the (unfiltered) reconstruction is provided in Figure 9. In the plot, the gas fraction is represented on the vertical axis and shows that the two bubbles “rise out” of the fluidized background relatively sharply. The smaller, isolated peaks are all due to noise. A simple thresholding technique and “salt and pepper removal” removes these peaks from the reconstructed image, leaving the bubbles almost unaffected. We can conclude that the reconstruction performance of the densitometer is not a function of the position of the objects in the reconstruction plane, neglecting some blurring with the wall for bubbles directly facing a source. Sampling Time. The time scale of the bubbles in a fluidized bed requires a sampling time well below a second. As mentioned, we aim at sampling times of 10 ms. The simulations discussed above indicate that this is possible. The sensitivity of the reconstructions with respect to the sampling time should be small. This is investigated by simulating the measurement of a single, 5 cm diameter bubble, for the five source case. The bubble is placed in the center, and 10 simulations are done with a fixed sampling time. The sampling time,
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Figure 10. Sensitivity of the reconstructed bubble area to the sampling time. Circles, 5 cm diameter bubble in center of column; squares, 5 cm diameter bubble located at (r/R) ) (1/2).
∆Ts, is varied logarithmically from 0.5 s, where the influence of the Poisson noise is small, to 1 ms. Similar runs are made for a 5 cm bubble located halfway the column radius. The reconstructed bubble area (normalized by the area of the bubble on the grid) is shown in Figure 10. This graph shows the mean dimensionless area (full symbols) and the standard deviation (error bars) for each of the sampling times. As can be seen from the figure, the standard deviation obviously increases at decreasing sampling time due to the increasing contribution of the Poisson noise. The figure also shows that the reconstruction starts to deteriorate for sampling times below 5 ms. This can also be seen by inspecting the shape of the reconstructed bubbles: At sampling times below 5 ms, this shape becomes eratic and no longer reflects the circular shape of the original bubble. Concluding Remarks Simulations of γ-ray CT imaging within a 400 mm model fluidized bed indicate that the modality is feasible in the sense that idealized void structures can be discriminated with temporal and spatial resolutions of 10 ms and 10 mm using five static sources. For the same fluidized bed geometry and system parameters, a three source system cannot provide the desired resolution and discrimination. Noise due to the short sampling time (10 ms) has a significant influence on the resolution and reconstruction quality. Although the idealized conditions presented here provide a general indication of the proposed system capability, several issues not addressed here are currently the focus of further research. These include void motion blur, which is expected to have a significant effect on image quality, and the optimization of geometric parameters such as source detector spacing, fan beam angles, and grid size. Acknowledgment This work was made possible via Grant 99MFS11 of the FOM/STW program “Dispersed Multiphase Flows”.
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Received for review September 17, 2004 Revised manuscript received December 9, 2004 Accepted December 10, 2004 IE049091P
