Ind. Eng. Chem. Res. 2010, 49, 5061–5065
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Double X-ray Tomography of a Bubbling Fluidized Bed R. F. Mudde* Department of Multi Scale Physics, Delft UniVersity of Technology, Pr. Bernhardlaan 6, 2628 BW Delft, The Netherlands
This paper discusses the measurement of bubbles moving through a fluidized bed imaged with a double X-ray tomographic scanner. The scanner is made of 3 X-ray sources equipped with two rings of 90 CdWO4 detectors. The fluidized bed has a diameter of 23 cm and is filled with a Geldart B powder. The scanner measures the attenuation in two thin parallel slices separated by a small vertical distance. This allows the estimate of the velocity of individual bubbles rising through the bed. Data are collected at a frequency of 2500 Hz. To remove the noise, data are averaged over 10 samples before tomographic reconstruction is performed, resulting in 250 independent frames per second. From the measured bubble velocity and the passage time through the reconstruction planes, the vertical dimensions of the bubbles are found. This allows imaging of the real shape of the bubbles and calculation of the volume of each individual bubble. Introduction Like many reactors used in chemical engineering, fluidized beds are multiphase systems. Owing to the relatively high solids volume fraction and the small size of the particles, these systems are opaque even if the wall of the fluidized bed is made of transparent material. Consequently, the laser based techniques like laser doppler anemometry (LDA), particle image velocimetry (PIV), or light induced fluorescence (LIF), that are so successful in single phase research are of rather limited use. This is a handicap that hampers progress in the research on multiphase reactors used in chemical engineering. As alternatives for the laser based techniques various measuring principles have been put forward. In the field of tomography there are three different approaches: (1) electrical capacitance tomography (ECT), (2) nuclear magnetic imaging (MRI), and (3) X-ray or γ densitometry. ECT relies on the measurement of the capacitance between two plates mounted on the outside of the fluidized bed. By placing 12 (8 or 16 are also used) sensor plates in a ring, sufficient information can be obtained to reconstruct the solids and gas distribution inside the measuring plane formed by the sensors. These methods are fast, giving a high temporal resolution. Images can be produced at rates of several hundreds per second (see, e.g., refs 1-3). However, ECT works with socalled soft fields: a change in the electro-magnetic properties in one location changes the field everywhere in the domain. This makes reconstruction difficult and the resolution in the center of the object is usually relatively poor. In recent work L.-S. Fan and co-workers4-6 have used a double ring of sensors, allowing a three-dimensional reconstruction. MRI is used by the group of Gladden.7,8 These researchers report that they can measure at a frequency of 1000 Hz and a spatial resolution of 400 µm. However, they are restricted to diameters of some 5 cm which is too small for bubbling fluidized beds. Nuclear densitometry measures the attenuation of high-energy photons through the fluidized bed. Radiation can be used to measure the volume fraction of gas and solids in a fluidized bed. If tomographic techniques are used, the volume fraction distribution in a cross section of the fluidized bed can be found. Several papers have appeared in literature on the use of nuclear densitometry (see refs 9-12). Most of them deal with time* To whom correspondence should be addressed. E-mail:
[email protected].
averaged information. The reason for this is the inherent presence of noise in the generation of X-ray and γ radiation. Moreover, the sources need to be of modest intensity due to safety requirements, but also most detectors cannot deal with high dose rates. As a practical solution, the tomograms are made by rotating the source and detector array around the object. This makes the data acquisition slow, on the order of a second for a tomogram. Obviously, when studying bubbles in a fluidized bed, this is at least an order of magnitude too slow. A very fast X-ray tomographic system has been developed by Hampel and co-workers (see refs 13, 14). They generate the X-rays by moving an electron beam over a Tungsten element. Tomographic images up to a rate of 10000 frames/s are created, showing that nuclear techniques can also be very fast. In the present paper, we focus on the use of X-rays for tomographic imaging of bubbles in a 23 cm fluidized bed. Our system consists of three X-ray sources that simultaneously radiate each a fan beam through the bubbling bed. The radiation is detected by two rings of 3 × 30 detectors, that is, 30 detectors per source per plane. The vertical distance between the measuring planes is small, namely 2 cm. From the data collected, the solids distribution is reconstructed in the imaging planes. Consecutive images are stacked, giving an impression of the bubbles, their shape, and spatial distribution. By comparing the information gathered in the two planes, the velocity of individual bubbles can be estimated. Once this is known, the vertical size of a bubble can also be calculated. Thus, we can obtain the volume of a bubble. Experimental Setup The fluidized bed is a 23 cm inner-diameter perspex tube (wall thickness 5 mm). The powder is 0.56 mm sized polystyrene particles (standard deviation of particle size distribution is 0.16 mm). This is a Geldart type B powder. The polystyrene has a density of 1.102 · 103kg/m3. The mean density of loosely packed polystyrene particles is 625 kg/m3. The minimum fluidization and minimum bubbling velocity are virtually identical for this powder: 0.12 m/s. In the bubbling regime of fluidization, voids travel upward through the powder mass. The ungassed height of the bed is 90 cm. Air at room temperature is supplied to the bed via a wind box through a porous plate (sintered bronze, pore size 30 to 70 µm, plate thickness 7 mm). A schematic of a fluidized bed with two bubbles in one of the measuring planes of the densitometer is given in Figure 1.
10.1021/ie901537z 2010 American Chemical Society Published on Web 01/19/2010
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as a thin line. For monoenergetic γ - or X-rays transmitted through a closed system containing a particle-gas two-phase mixture, the number of photons registered per second, R, follows from the Lambert-Beer law: R ) R0 exp[-((1 - R)µp + Rµg)d]
Figure 1. Fluidized bed with two bubbles in one of the measuring planes of the X-ray densitometer.
The three X-ray sources are placed at 120° around the fluidized bed. Two rings of three arrays of sensors are placed opposite of the sources. The distance from the X-ray-target to the center of the fluidized bed is 68.5 cm, from X-ray target to the detectors is 138.6 cm. The measuring plane is 52 cm above the distributor plate. The distance between the two measuring planes is 2 cm; the thickness of each of the planes is determined experimentally and found to be 4.4 mm. A schematic of a side and top view is given in Figure 2. The X-ray sources used are manufactured by Yxlon International GmbH. The maximum X-ray energy is 150 keV. We operate the sources at a low energy flux with a tube current less than 1 mA. The detectors all consist of a CdWO4 scintillation crystal optically coupled to a PIN photo diode. They are manufactured by Hamamatsu (type: S1337-1010BR). Their crystal size is 10 mm × 10 mm × 10 mm. A plastic casing in the form of an arc holds two horizontal arrays of 32 detectors. The curvature of the array is such that the distance to the focal point of the source is equal for all detectors. Of these detectors 30 are used to produce data for tomographic reconstruction, the two outermost detectors are used to monitor the alignment of the object vessel and proper functioning of the tomographic system. The data are collected simultaneously at a sampling frequency of 2500 Hz, both for the upper and lower ring of detectors. The measured data are read out using a 12 bit ADC-card. The entire process is controlled via a workstation that sends out the trigger signals to the sources and reads out the detectors. Tomographic Reconstruction Measuring Principle. Each detector measures the attenuation of the X-rays in a thin cone. In the reconstruction this is treated
(1)
where R0 is the number of photons registered per second when the fluidized bed would be completely empty, that is, vacuum; µp and µg are the linear absorption coefficients of the particle and gas phase; R is the volume fraction of the gas phase; d is the path length of the photon beam through the system. It should be noted that the attenuation characteristics of the fluidized bed wall are incorporated in R0. For monoenergetic radiation a two point calibration is sufficient, for example, an empty column filled with air and a packed bed could be used. However, most X-ray sources generate a wide spectrum of X-ray energies. The attenuation coefficients, µp and µg, are functions of the photon energy E. Hence, a two-point calibration cannot be used: the absorption of the photons of lower energy is much stronger. Consequently, the detected number of photons does not obey Lambert-Beer’s law. Instead, for an increasing amount of powder on the photon beam the relative number of high-energy photons increases, the so-called beam hardening. Calibration. To circumvent the effects of beam hardening, we calibrate all detectors individually. We put different amounts of packed powder inside the fluidized bed in between a source and its detectors. The calibration includes a completely empty and a completely filled bed. These two calibration points provide the upper and lower limit of the signal. Seven different calibration points are obtained for each calibration. As the attenuation of the X-ray beam is only a function of the amount of powder on the beam, the spatial distribution of the powder is irrelevant, only the integral effect is measured. We fit a smooth function of the form Acal + Bcal exp(-x/Ccal) to the data, with x being the distance traveled by the beam through the powder phase. For every detector, an individual curve is obtained. It provides a one-to-one relation between the measured attenuation and the amount of powder on the X-ray beam as measured by the given detector. Using this calibration curve, the data measured in an experiment can be converted to a measured lineaveraged particle fraction, Rmeas, the beam encounters on its path through the column. Note that the exponential function in itself is not relevant. All we need is a smooth function that describes the data points accurately. Figure 3 shows an example of a calibration curve and the corresponding data. The only purpose of the fit is to convert the measured data into a particle fraction on a given X-ray beam.
Figure 2. Side view (left: not to scale) and top view (right: to scale) of the X-ray tomographic system.
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Figure 3. Calibration curve of one of the detectors with its corresponding beam going almost through the bed center. The horizontal axis gives the lengths of the packed bed that is put into the beam; the vertical axis denotes the detector output in a 12 bit ADC representation.
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for our CT scanner. Detailed discussion of reconstruction techniques can be found in refs 16-18. The raw data measured are converted from counts (or voltage) to a line averaged solids fraction, that is, centimeters of a particular beam filled with powder compared to the total length of that beam through the fluid bed. The solids fraction R(x, y) is reconstructed on a grid of square pixels representing the cross-section of the fluidized bed. For this paper, we have reconstructed on a pixel array of 55 × 55 pixels; that is, the linear size of each pixel is 4.18 mm. The circular cross-section of the fluidized bed exactly fits in this square. During the reconstruction, all pixels outside the circle are set to a solids fraction of zero. A given beam travels through a known collection of the pixels. If we denote the length of ray i through pixel j (the pixels are numbered as a vector, rather than a matrix) by Wij and denote the solids fraction of pixel j by Rj, we can estimate the total relative amount of solids pi (called the ray sum) on the line of beam i from N×N
p˜i )
∑W R
ij j
(2)
j)1
where N × N is the total number of pixels (55 × 55 ) 3025 in the present study). We use a linear weighing matrix W. Hence, the weighing factor Wij is the length of ray i through pixel j, ignoring the finite width of the beam (see Figure 4). A reconstruction means solving the unknown pixel-based solids fraction Rj from eq 2 for M different rays on N2 pixels. The number of independent measurements is only 3 × 30 ) 90. Hence, the problem is ill-posed. Moreover, there will be measuring noise in the data. We write formally: f + fε b p )W·R Figure 4. The measuring plane is divided in N × N pixels. The weighing factor, Wij, is the length of ray i through pixel j.
Tomographic Reconstruction. For performing tomographic reconstructions a variety of algorithms exists. The most simple one is the Filtered Back Projection algorithm. It is easy and fast. However, it is known to blur excessively and is not suitable for reconstructing bubble-like objects.15 A second class is formed by the algebraic techniques, which are iterative and reconstruct the object function on a discretized domain. Although significantly slower, algebraic methods offer more flexibility in terms of limited data sets and are more appropriate
(3)
where b ε contains all errors. The algebraic reconstruction techniques (ART, see ref 16) are designed to minimize the mismatch between the data b p and W · b R. We use the simultaneous algebraic reconstruction technique (SART).19 SART simultaneously applies to a pixel the average of the corrections generated by all rays, rather than on a ray by ray base. The SART algorithm is given by ) Rnj + Rn+1 j
1 pi - p˜i W W+,j Wi,+ ij
(4)
N×N where W+, j ) ΣiN×N ) 1 Wij, Wi,+ ) Σj ) 1 Wij. Recently Jiang and 20 Wang have shown that SART is a maximum likelihood
Figure 5. Raw signal of two of the detectors (Usup ) 1.8Umf). The corresponding X-rays travel almost through the center of the bed. The vertical axis denotes the detector output in a 12 bit ADC representation. The lower line is for a detector in the lower detection plane, the dashed upper curve for a parallel line in the upper detection plane. Note that the upper curve is vertically shifted over 500 units for clarity.
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estimator for the mapping of R in case the error in eq 3 is purely stochastic with a Gaussian distribution. In the reconstructed images, pepper and salt noise will show up. This can be reduced by using a so-called one-step-late algorithm (see ref 21). We invoked an algorithm based on the median root function (suggested first by Alenius and Ruotsalainen22). It effectively removes pepper and salt noise, but keeps the edges of larger objects sharp enough. In the reconstruction algorithm an extra step is added to eq 4: n+1 ROSL, j )
1+β
n ROSL, j
1 Rn+1 j n - Med(ROSL, ) j
(5)
n Med(ROSL, j)
where Rjn is the value of pixel j after the nth SART step and n ROSL, j is the same after the nth one-step-late correction. The median filter Med(Rj) replaces the value of the jth pixel by the median of the pixel values contained in the jth pixel neighborhood. The definition of this pixel neighborhood is the only parameter of the median filter. We use a 3 × 3 neighborhood; a 5 × 5 neighborhood causes too much smearing. The parameter β in eq 5 controls the weight of the correction. We have set it to β ) 0.1. Each new reconstruction is started from a minimal fluidized bed, that is, the solids fraction inside the bed is set to 1. Outside the bed, it is set to zero. In principle, it would be faster to start the next reconstruction from the previous, but speed is not an issue here as the data are processed of line. In this paper for each reconstructed image 400 iterations are used. In previous work23,24 we showed that the minimum in the error, defined as the sum of the absolute difference of all pixels between two iteration steps, is reached at about 1200-1500 iterations. However, the difference with 400 iterations is small: only small changes in the distribution of pepper and salt noise are observed, the reconstructed bubbles do not change anymore. Experimental Results Results. A fluidization experiment is run in the bubbling regime. Experiments are performed at a superficial gas velocity of 22 cm/s, that is, 1.8Umf. The measuring plane is 52 cm above the gas distributor plate. Data are taken for a period of about 5 s. The number of data points per detector is 12523. For a single tomogram 10 data points are averaged to reduce the influence of the Poisson noise. The time interval between two consecutive frames is thus 4 ms. We processed 999 images per measurement. In Figure 5 the response of two detectors is shown. The corresponding beams go through the center of the fluidized bed. The lower curve is from detector 15 in the lower ring, the upper dashed curve (which is shifted vertically by 500 units for clarity) is detector 15 from the upper ring. In this example 10 clear peaks are seen in both curves, indicating that the corresponding beams have been “interrupted” by 10 bubbles (at least, as bubbles seen by a beam may overlap). The vertical component of the bubble velocity can be estimated for each bubble by calculating the time lag between two corresponding peaks in Figure 5. To this end, first the start of the bubble has to be found from the raw data. In the present paper this is done for each of the sources with its 30 detectors. Thus, we obtain three velocity estimates. In most cases the spread in these values is less than 5%. The velocity ranges between 0.5 and 1.2 m/s. The raw signals are averaged over 10 samples and fed to the reconstruction algorithm for reconstruction of the gas and solids distribution on a square of 55 × 55 pixels. We processed the
Figure 6. Semi-3D images of the measured bubbles. Left, lower detector plane; right, upper detector plane.
first 10000 data points of the series shown in Figure 5, that is, 4 seconds of continuous measurement. This generates 999 reconstructed gray-value images. These image were thresholded by a threshold value of 100 (0 is black, 255 is white) removing noise leaving only objects with a minimum diameter of at least 3 cm. These 999 images are stacked to form a quasi 3D image of the bubbles. This is done for both the lower and the upper plane of detectors. The result is shown in Figure 6. As expected,
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supplement with a one-step-late approach. From the time-offlight between the two measuring planes, that are separated by 2 cm, the bubble velocities has been estimated. Once the bubble velocity is known, the vertical dimensions of the bubbles can also be obtained and the bubble volume is calculated. We observed bubbles of equivalent diameters ranging from 6 to 15 cm. The data obtained show that with X-ray tomography properties of the bubbles, like volume and velocity can be obtained with reasonable accuracy.
Table 1. Some Properties of the Bubbles in Figure 6 t (s)
V (px2 · s)
Vb (m/s)
deq (cm)
0.08 0.36 0.74 0.97 1.16 1.58 1.93 2.12 2.52 2.64 2.78 3.34 3.75 3.88
68 65 22 50 49 59 19 50 35 20 27 197 44 13
0.66 0.63 1.00 0.53 0.63 0.71 1.25 0.76 0.72 0.82 0.62 0.65 1.19 0.53
10.0 11.1 9.5 10.3 10.0 11.7 7.6 10.8 11.3 7.8 8.2 15.1 11.4 6.1
Literature Cited
both detector-planes see the same bubbles passing by, with almost the same shape, but slightly shifted in time. Note that the two horizontal axis are in pixels (each pixel side is 4.18 mm) whereas time is the coordinate of the vertical axis. Here the unit is 4 ms. As different bubbles have different velocities, the vertical axis cannot be globally transformed into a length scale. This has to be done for each bubble individually using the estimated velocity of that bubble. As the range of velocities is rather broad, care should be taken by drawing conclusions about the size of the bubbles in Figure 6. Table 1 provides some characteristic data of the 14 bubbles seen in Figure 6: the time, t, at which the center of gravity passes the lowest measuring plane, the size, V, in (px2 · s), the bubble velocity, Vb, (in m/s) and the equivalent bubble diameter, deq (in cm). The measured equivalent bubble size is on the order of 10-11 cm. This is in good agreement with, for instance, the correlation of Rowe25 for porous plate spargers: db ) (U - Umf)1/2z3/4g-1/4
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(6)
where z is the height above the distributor and g is the acceleration of gravity. For our experiments this relation predicts bubbles of 11 cm, whereas the mean of the bubble sizes in Table 1 is equal to 10.1 cm. The velocity of the bubbles is in the range of the predictions from Davidson’s equation26 for bubbles in a fluidized bed: Vb ) 0.66(gdeq)1/2, which gives for a bubble of 10 cm a velocity of 0.65 m/s. Concluding Remarks In this paper, we present measurements of the bubbles in a fluidized bed using an X-ray based tomographic scanner. The scanner consists of three industrial X-ray tubes with two arrays of 30 detectors per tube containing CdWO4 scintillator crystals. The data are collected at a frequency of 2500 Hz. For reconstruction of the particle and gas distribution in the measuring planes, 10 data points are averaged, giving a frequency of 250 frames/sec. The bed has a diameter of 23 cm and the reconstructions are carried on a 55 × 55 square pixel grid. The reconstructions are based on the SART algorithm
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ReceiVed for reView September 30, 2009 ReVised manuscript receiVed December 18, 2009 Accepted December 21, 2009 IE901537Z
