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Charge distribution around a rising bubble in a two-dimensional fluidized bed by signal reconstruction
Powder Technology 177 (2007) 113 – 124 www.elsevier.com/locate/powtec
Charge distribution around a rising bubble in a two-dimensional fluidized bed by signal reconstruction Aihua Chen, Hsiaotao T. Bi ⁎, John R. Grace Fluidization Research Centre, Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, Canada V6T 1Z3 Received 25 January 2006; received in revised form 22 December 2006; accepted 23 February 2007 Available online 6 March 2007
Abstract A signal reconstruction technique was developed recently to measure the electrostatic charge density distribution surrounding a bubble rising in a two-dimensional gas–solid fluidized bed using four induction probes flush with the outside wall of a two-dimensional fluidized bed and connected to charge amplifiers for recording induced charge signals as bubbles pass by the probes. The model was implemented in Visual FORTRAN using the LSGRR Subroutine to invert the matrix for the signal reconstruction. However, if the matrix was mathematically singular or ill-conditioned, a least-squares routine or the singular value decomposition routine provided had to be used to give approximate results. If the dimension of the matrix exceeds the permissible limit, or the number of pixels is much larger than the number of measurements, the matrix cannot be inverted using the LSGRR subroutine. This paper shows that an iterative linear back projection algorithm (LBP), often used in linear tomography, can be applied to improve the reconstruction resolution of the charge distribution around rising single bubbles in a gas–solid fluidized bed. The experimentally measured induced charges from four induction probes were then applied to reconstruct the real charge distribution around a single rising gas bubble injected in a two-dimensional column as it passed the probes. The results reconstructed using the LBP method are found to be smoother and more accurate than those obtained from the LSGRR subroutine. © 2007 Elsevier B.V. All rights reserved. Keywords: Induction probe; Electrostatic charges; Charge distribution; Fluidized bed; Signal reconstruction
1. Introduction Electrostatic charging always exists in industrial gas–solid fluidized bed reactors. This can induce particle agglomeration, changes in hydrodynamic behaviour, adhesion of particles to the walls, interference with instruments, nuisance discharges or explosions, and lead to undesirable by-products. The electrostatic charge can be generated from particles contacting with or separating from each other, and particle contacting with the fluidizing gas and the reactor vessel in which they are contained. A bubbling fluidized bed consists of two distinct phases, a dense phase (sometimes called the emulsion phase) and a dispersed or bubble phase. The reactant concentration is usually higher in bubbles and the region surrounding bubbles than in ⁎ Corresponding author. E-mail address: [email protected] (H.T. Bi). 0032-5910/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2007.02.036
the remote dense phase, leading to faster reaction near the bubbles than elsewhere in the dense phase. Particles close to bubbles also tend to move faster than more remote particles in the dense phase region. Particle–particle collisions are therefore more vigorous for particles close to bubbles, potentially leading to higher rates of charge generation due to particle–particle contact charging. It is therefore expected that the particles adjacent to bubbles may generate and carry more charges than those further away. A collision probe inside a two-dimensional column was employed by Park et al. [1] to explore charge induction and transfer when a single bubble passed. Park et al. [2] and Chen et al. [3] developed a theoretical model to explain the electrical current signals due to the passing of isolated gas bubbles. The model, which assumes that there is a charge distribution close to the bubbles and much higher charge density in the wake region of the same sign as in the region at the front of the bubble, correctly predicted the trend of charge
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Fig. 1. Front view of the two-dimensional fluidization column, showing the optimal location and size of a bubble rising by the induction probes.
signals registered by a sensitive collision probe when bubbles passed. Verification of the quantitative predictions from this model requires information on the distribution of specific charge density surrounding rising gas bubbles in gas–solid fluidized beds. Examining the distribution of charges near rising bubbles can assist in understanding electrostatic charge generation, charge separation, dissipation and accumulation in fluidized bed reactors. Chen et al. [4] developed a technique to measure the charge distribution surrounding a single bubble rising in a two-dimensional gas–solid fluidized bed. Four induction probes, flush
with the outside wall of the column and connected to charge amplifiers, recorded induced charge signals as bubbles passed. The charge distribution surrounding the bubble was then reconstructed assuming the bubble to be symmetrical and the charge around the bubble to remain constant as it rises. The emulsion phase far from the bubble in a two-dimensional fluidized bed of glass beads was found to be negatively charged, whereas the magnitude of the charge density decreased gradually toward the bubble–dense phase interface, with a nearly zero charge density inside the bubble. The wake of the bubble was more negatively charged than the emulsion phase, supporting previous speculations [3]. Chen et al. [4] implemented the model in FORTRAN using the LSGRR subroutine to invert the matrix needed in the reconstruction. If the matrix was mathematically singular or ill-conditioned, a leastsquares routine or the singular value decomposition routine provided by Visual FORTRAN Professional Version 5.0 was implemented to give approximate results. If the dimension of the matrix exceeds the permit limit, or the number of the pixels is much larger than the number of measurements, the matrix is difficult to invert using the LSGRR subroutine. This paper uses the same measurement technique as in [4] to measure the charge distribution surrounding a single rising gas bubble in a two-dimensional gas–solid fluidized bed. However, an iterative linear back projection algorithm (LBP) is applied to improve the reconstruction resolution. The effects of probe size are also investigated. 2. Measurement technique Tests on the static charge around bubbles were conducted with induction probes in a two-dimensional fluidized bed. The column, made of Plexiglas, has an inner thickness of 14 mm, a width of 280 mm, and a height of 1.24 m. The windbox volume is 800 cm3. Glass beads of average diameter 560 μm and density 2500 kg/m3 served as the bed material. The settled bed height was approximately 700 mm. The bed was fluidized by air at a flow rate of 0.002 m3/s, leading to a regular pattern of independent bubbles
Fig. 2. Schematic of induction probe. It is flush with the outside wall of the column (as shown). Around the probe and on the column wall is the grounded shielding. When a bubble is directly in front of the probe, little or no charge is induced. When the bubble rises past the probe, the charge of the emulsion phase is measured.
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Fig. 3. Positions of probes and pixel grid used for reconstruction.
Fig. 4. Normalized sensitivity distribution for single probes of different size at x = 26 mm, y = 0.
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with few particles raining through them. These bubbles were approximately 80 mm in diameter and had rise velocities of about 0.26 m/s. Fig. 1 shows a schematic of the column and probes. Copper tape on both the front and rear walls was connected to ground to maintain a zero potential on the column wall. Four induction probes were placed 520 mm above the perforated plate distributor. The distance between adjacent probes was 13 mm. A Canon digital video camera operating at 30 frames/s recorded the rise of bubbles through the column. The images were used to ensure that measurements were performed only on single bubbles not directly influenced by others. Bubbles whose centre passes probe 1 and whose edge passes probe 4 were typically selected for further analysis. The video was therefore always synchronized to the charge measurements. Induction probes (Fig. 2) constitute the simplest type of field meters. Each induction probe consists of an insulated, circular
Table 1 Parameters used in the reconstruction
dB dprobe Npixel, x × Npixel, z Δxpixel × Δzpixel Mx × Mz Δdprobe, x × Δdprobe, z
LBP method
[4]
80 mm 10 mm 21 × 157 3.3 mm × 1.3 mm 4 × 137 13 mm × 1.3 mm
80 mm 10 mm 12 × 48 5 mm × 5 mm 4 × 137 13 mm × 1.3 mm
Charge density distribution around a gas bubble in a fluidized bed of dielectric particles was reconstructed using the iteration linear back projection (LBP) algorithm based on induced charge signals from 4 dynamic probes. The probe size does not influence the reconstruction resolution greatly, with an optimal diameter ranging from 3 to 6 mm for the probe arrangement used in the current study.
copper disc (10 mm in diameter), connected to the core of a coaxial cable, embedded in a Teflon cylinder, and wrapped with a single strip of conducting copper tape. The shielding of the coaxial cable is connected to the shielding around the probe and to a common ground. Together with the coated and grounded walls of the column, this minimizes distortion of the signal from charges above, below, or at the sides of the probe. The metal sensor is charged by induction due to the electric field generated by the charged particles. Each induction probe is connected to a Kistler charge amplifier (model 5011B or 5015) by a coaxial cable. The sampling rate was 300 Hz. The four probes can measure induced charges during the passage of bubbles using the technique described above. 3. Reconstruction of charge distribution The region around the bubble is first divided into a number of pixels, as shown in Fig. 3. Combining Gauss's law with the definition of electric field and potential [5], the electrostatic field E and potential V are given by: div E ¼ q=P
ð1Þ
and E ¼ −grad V
ð2Þ
where q is the volume charge density and Π is the permittivity of the medium, assumed to be uniform in the region of interest. From Eqs. (1) and (2) divðgrad V Þ ¼ j2 j ¼
A2 V A2 V A2 V q þ 2 þ 2 ¼− P A2 x A y A z
ð3Þ
It is assumed that there are charges only in pixel i with constant charge density, qi, in the y-direction, and no charges in other regions. Hence
Fig. 5. Flowchart of the iterative linear back projection algorithm for reconstruction.
q ¼ qi at x ¼ xi ; 0bybdB and z ¼ zi ; q ¼ 0 in all other regions:
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117
Fig. 6. Simulation of rising bubble with uniform charge of − 1pC/mm3 in the emulsion phase.
The permittivity, Π, of the medium in the bed is assumed to be uniform and isotropic in the dense phase and in the bubble phase, respectively, in order to calculate the charge distribution around the bubble. If we let vV ¼ − LPV 2 q , x′ = x / Lp, p 1 y′ = y / Lp and z′ = z / Lp, then the dimensionless form of Eq. (3) is
For an infinitesimal element of the probe at (xj, yj = 0, zj), the electric field perpendicular to the wall can be estimated by
A2 vV A2 vV A2 vV þ þ ¼ qV A2 xV A2 yV A2 zV
The induced charge on an infinitesimal element of the probe is then
ð4Þ
E8 ¼ Ey ¼ −
AV j Ay x¼xj ;y¼0;z¼zj
where
rij ¼ PE8 ds ¼ −P
qV ¼ 1 at xV ¼ xi =Lp ; 0byVbyi =Lp and zV ¼ zi =Lp ; qV ¼ 0 in all other regions
ds ¼ qi Lp
Boundary conditions: At grounded wall, y′ = 0 and y′ = δB / Lp, v′ = 0; at x′ = − W / 2Lp and x′ = W / 2Lp, v′ = 0; at z = 0 and z = Hbed /Lp, AAvV ¼ 0; Eq. (4) can be solved to obtain v′ and hence 2 xV V.
AV j Ay x¼xj ;y¼0;z¼zj
ð5Þ
ð6Þ
AV V j ds AyV xV¼xj =Lp ;y¼0;z¼zj =Lp
The total charge on probe j is rij
total
¼
RR
rij ds surface of probe j
ð7Þ
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Fig. 7. Second simulation of a rising bubble with a uniform charge of −1 pC/mm3 in the emulsion phase, but now with a thin layer of highly charged particles (− 1.5 pC/ mm3) around the bubble.
If the probe is a circular plate of radius rprobe centred at (xj0, 0, zj0), the total charge of probe j from the ith pixel can be expressed as rij
total
RR
¼
Qmj ¼
rij ds
xj0 −rprobe
2 zj0 −½rprobe −ðxj −xj0 Þ2 0:5
AV V j dzj dxj AyV xV¼xj =Lp ;yV¼0;zV¼zj =Lp
ð8Þ Let Z xij ¼ Lp
2 xj0 þrprobeZ zj0 þ½rprobe −ðxj −xj0 Þ2 0:5
xj0 −rprobe
N X i¼1
surface of probe j Z xj0 þrprobeZ zj0 þ½r2 −ðxj −xj0 Þ2 0:5 probe
¼ qi Lp
Then the induced charge from all pixels to probe j is obtained by
2 zj0 −½rprobe −ðxj −xj0 Þ2 0:5
AV V dzj dxj j AyV xV¼xj =Lp ;yV¼0;zV¼zj =Lp
ð9Þ Eq. (8) is used to calculate the induced charge on probe j due to only pixel i having a charge density qi.
rij
total
¼
N X
xij qi ð j ¼ 1; 2; N M Þ
ð10Þ
i¼1
Theoretically, if the charge induced on the probe, Qmj, can be measured, qi can be reconstructed for each pixel using multiple probes. The quality of the reconstruction depends on the number of probes and the number of measurements of each probe. For good reconstruction, measurements should be taken inside, on the border and outside the bubble. The greater the number of probes, the better the reconstruction. In our experiment, only four probes were fixed horizontally on the wall and many pixels had to be reconstructed (see Fig. 3). Ideally, one should install many probes surrounding the rising bubble to reconstruct the charge density distribution. For the current case with only four fixed probes, it is assumed that each bubble is symmetrical and that the charge around the bubble
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Fig. 8. Third simulation of a rising bubble, with a uniform charge of − 1 pC/mm3 in the emulsion phase and a highly charged wake (− 1.5 pC/mm3).
does not vary as it passes through the measurement region. Multiple measurements can thus be taken as a function of distance between the probe and bubble as the bubble rises vertically through the array of probes. Although there were only four probes, measured induced charge signals at different vertical locations relative to the bubble can be obtained and used to solve Eq. (10) and produce the charge distribution around the rising bubble at a reasonable resolution. More details are provided in [4]. The Iterative Linear Back Projection (LBP) [6], often used in linear tomography, is applied in the present work. The function relating the charge distribution inside the column and the induced charge to the probe is linearized using sensitivity maps. The normalized sensitivity of a probe j to pixel i with charge qi = 1 C is defined as Si; j
xi; j ¼X xi; j
ð11Þ
i
Fig. 4 shows that the normalized sensitivity distribution for single probes with different diameters, dprobe = 10, 8, 6, 4, 2 mm
at x = 26 mm and y = 0. It can be seen that the range of the charged zone sensed by a probe flush with the wall does not change with probe size, because the sensitivity is normalized. However, the smaller the probe, the higher the maximum normalized sensitivity. In the LBP method, the measured induced charge Qmj is transformed to X 0 1 xi; j qi X Qmj xi; j BX C X i Qj ¼ X ¼ X qi A ¼ ðSi; j qi Þ @ xi; j xi; j i¼1 xi; j i i
i
i
ð12Þ Using the normalized sensitivity map, the charge for a pixel i can be calculated by linear back projection from X Si; j Qj 1 ⁎ qi ¼ X ð13Þ Si; j j
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Fig. 9. Reconstructed charge distribution (Case 1, Hmf = 0.7 m, dB = 0.08 m, uB = 0.2 m/s, dp = 565 μm, ρp = 2500 kg/m3, Qair = 1.78 m3/s).
For the reconstruction, Eq. (13) is solved once for every pixel of the reconstruction image. The induced charge on the probe j is then estimated by Q⁎j ¼
X
Si; j q⁎i
ð14Þ
i
Eqs. (13) and (14) were applied iteratively. For every iteration step, a charge difference is calculated for every pixel of the reconstructed image according to X Si; j DQ⁎j j Dq⁎1 ¼ X ð15Þ Si; j j
where ΔQj* = |Qj − Qj*| is the difference between the calculated and measured charges. For each iteration, a new charge distribution is obtained from ⁎ qnew ¼ qold i i þ kDqi
ð16Þ
where λ is a relaxation factor. This gives the additional possibility of using under-relaxation (λ b 1) or over-relaxation (λ N 1) in the reconstruction. With under-relaxation the convergence is usually smoother and better. The use of over-relaxation, on the other hand, can increase the speed of reconstruction. Eq. (15) is applied repeatedly until the difference of induced charges on all probes has been minimized. or until a specified maximum number of iteration steps has been reached. Fig. 5 presents the flowchart of the back projection algorithm for the reconstruction of the charge distribution around a single bubble. 4. Results and discussions A single circular bubble of diameter 80 mm with its associated charge distribution was assumed to pass the probes, and the induced charge was used to reconstruct the original charge distribution. There were four probes of diameter 10 mm located horizontally at 13 mm intervals, as shown in Fig. 3 and as in the earlier work [4]. Three assumed charge distributions were first used to test the LBP reconstruction method and to compare
A. Chen et al. / Powder Technology 177 (2007) 113–124
with simulation results from [4]. The parameters used in the reconstruction are listed in Table 1. The number of pixels, 21× 157, which can be used by the LBP method, is much larger than the number of 12 × 48, which could be utilized by Chen et al. with the LSGRR subroutine, whereas the number of probes, M = 4, is the same. The reconstructed profiles are presented in Figs. 6–8. It was first assumed that there is a uniform charge distribution in the emulsion phase (around the bubble) of q = − 1 pC/mm3 and that the inside of the bubble does not carry charges. The charges induced on the four probes are shown in Fig. 6(a) based on the assumed charge distribution in Fig. 6(b). Using these four series of calculated induced charges, the charge distribution around the bubble is reconstructed based on the above model. The results from Chen et al. [4] and from the LBP method are shown in Fig. 6(c) and (d) respectively. Fig. 7 shows the induced charges for the four probes (7a), the reconstructed charge profiles from [4] (7c), and the reconstructed profiles from the LBP method (7d) with a bubble assumed to have a thin mantle of highly charged particles around its exterior surface (7b).
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A higher charge density is then assumed in a wake region occupying the bottom portion of the bubble, as shown in Fig. 8 (b). The resulting induced charges from the four probes appear in Fig. 8(a). The higher charge density in the bubble wake region is captured in the reconstructed profile in Fig. 8(c) from [4] and Fig. 8(d) from the LBP method. The experimental data on induced charges measured from the four induction probes by [4], shown in Figs. 9(a), 10(a), 11 (b), were then used to reconstruct the charge distribution around a single bubble passing through the two-dimensional column. The reconstructed results from the LBP methods are presented in Figs. 9(c), 10(c) and 11(c), with the corresponding results from [4] shown for comparison in Figs. 9(b), 10(b), 11(c). From Figs. 6–11, it is seen that the charge density distributions reconstructed using the LBP method are much smoother and the standard deviation is smaller than those presented by [4] based on the LSGRR subroutine. It can thus be concluded that the LBP method can improve reconstruction of charge distributions surrounding a single bubble in a gas–solid fluidized bed. It should be pointed out that the LBP method will
Fig. 10. Reconstructed charge distribution (Case 2, Hmf = 0.7 m, dB = 0.08 m, uB = 0.26 m/s, dp = 565 μm, ρp = 2500 kg/m3, Qair = 1.78 m3/s).
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Fig. 11. Reconstructed charge distribution (Case 3, Hmf = 0.7 m, dB = 0.08 m, uB = 0.26 m/s, dp = 565 μm, ρp = 2500 kg/m3, Qair = 1.78 m3/s).
require much more calculation time if the number of pixels is large. Finally the effect of probe size is examined for the current array of four probes as shown in Fig. 3, with the distance between the centres of adjacent probes fixed at Δdprobe, x = 13 mm. In reconstruction calculations, the number of pixels, Npixel, x × Npixel, z = 21 × 157, and the size of pixels, Δxpixel × Δzpixel = 3.3 mm × 1.3 mm, were maintained constant. Fig. 12 shows the reconstructed results with the same number of probes, M = 4, but different probe diameters, dprobe = 10, 8, 6, 4, and 2 mm. It is seen that the size of probes does not influence the reconstruction resolution greatly. This may be because the normalized sensitivity map was applied in the LBP reconstruction method. It is also found that the standard deviation decreases slightly with decreasing diameter of the probes. However, if the probes are too small, e.g. 2 mm, the standard deviation increases slightly again. As shown in Fig. 4, the probe can only receive information from the front of the probe itself, a very narrow region. The smaller the probe, the higher the maximum normalized sensitivity. Although the area of the charged zone
covered by a smaller probe is the same as for a larger probe, the edge of the zone is not as well sensed by too small a probe relative to the high maximum sensitivity. In the reconstruction, the distance between the centres of adjacent probes is set to be the same for different probe diameters. However, the distance between the edges of adjacent probes becomes larger with decreasing probe diameter. Therefore, if the edges of adjacent probes are far apart, there will exist a region between them that can be less well sensed by the probes, resulting in poor results. The probe diameter of 10 mm currently used in our experiments appears to be reasonable, although the quality of the reconstructed charge distribution can be slightly improved by reducing the probe diameter to 3–6 mm in the future. 5. Conclusions The iteration linear back projection (LBP) algorithm using sensitivity maps can appreciably improve the reconstruction resolution of charge density distributions surrounding a single gas bubble in two-dimensional fluidized beds. The size of the probes does not influence the reconstruction resolution greatly,
A. Chen et al. / Powder Technology 177 (2007) 113–124
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Fig. 12. Reconstruction results with different probes sizes for three assumed charge distribution (Npixel, x × Npixel, z = 21 × 157, Δxpixel × Δzpixel = 3.3 mm × 1.3 mm, Mx × Mz = 4 × 137, Δdprobe, x = 13 mm, Δdprobe, z = 1.3 mm).
with an optimal diameter ranging from 3 to 6 mm for the probe arrangement used in the current experimental study. Nomenclature d diameter, m E electric field, V/m Hbed fluidized bed height, m Hmf bed height at minimum fluidization, m Lp distance between probes 1 and 4 (see Fig. 3), m M number of probes, – Mz Number of measurement along the vertical bubble path N number of pixels, – q volumetric charge density, C/m3 Q induced charge on probe from all pixels, C Qm measured induced charge on probe, C Qair air flow rate, m3/s r radial coordinate, m
area, m2 normalized sensitivity, – velocity, m/s PV dimensionless electrical potential, − 2 , – L p q1 electric potential, V width of two-dimensional column, m Cartesian coordinates, x horizontal in pane of interest, y normal to two-dimensional plane, z vertical, m x′, y′, z′ normalized coordinates, x′ = x / Lp, y′ = y / Lp, z′ = z /Lp, –
s S u v′ V W x, y, z,
Greek letters δB thickness of two-dimensional column, m σ induced charge on probe from one pixel, C Π permittivity of medium, F/m λ relaxation factor, – ρ density, kg/m3 ω matrix in Eq. (10)
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Subscript B bubble i ith pixel j jth probe x, y, z x, y, z directions p particle probe probe pixel pixel total total References [1] A.-H. Park, H.T. Bi, J.R. Grace, Reduction of electrostatic charges in gas– solid fluidized beds, Chem. Eng. Sci. 57 (2002) 153–162.
[2] A.H. Park, H.T. Bi, J.R. Grace, A. Chen, Modeling charge transfer and induction in gas–solid fluidized beds, J. Electrost. 55 (2002) 135–168. [3] A. Chen, H.T. Bi, J.R. Grace, Effects of charge distribution around bubbles on charge induction and transfer to a ball probe in gas–solid fluidized beds, J. Electrost. 58 (2003) 91–115. [4] A. Chen, F.K. van Willigen, J.R. van Ommen, H.T. Bi, J.R. Grace, Charge distribution around a rising bubble in a two-dimensional fluidized bed, AIChE J. 52 (2006) 174–184. [5] J. Cross, Electrostatics: Principles, Problems and Applications, Hilger, Bristol, 1987. [6] T. Loser, R. Wajman, D. Mewes, Electrical Capacitance Tomography: image reconstruction along electrical field lines, Meas. Sci. Technol. 12 (2002) 1083–1091.
Charge distribution around a rising bubble in a two-dimensional fluidized bed by signal reconstruction Aihua Chen, Hsiaotao T. Bi ⁎, John R. Grace Fluidization Research Centre, Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, Canada V6T 1Z3 Received 25 January 2006; received in revised form 22 December 2006; accepted 23 February 2007 Available online 6 March 2007
Abstract A signal reconstruction technique was developed recently to measure the electrostatic charge density distribution surrounding a bubble rising in a two-dimensional gas–solid fluidized bed using four induction probes flush with the outside wall of a two-dimensional fluidized bed and connected to charge amplifiers for recording induced charge signals as bubbles pass by the probes. The model was implemented in Visual FORTRAN using the LSGRR Subroutine to invert the matrix for the signal reconstruction. However, if the matrix was mathematically singular or ill-conditioned, a least-squares routine or the singular value decomposition routine provided had to be used to give approximate results. If the dimension of the matrix exceeds the permissible limit, or the number of pixels is much larger than the number of measurements, the matrix cannot be inverted using the LSGRR subroutine. This paper shows that an iterative linear back projection algorithm (LBP), often used in linear tomography, can be applied to improve the reconstruction resolution of the charge distribution around rising single bubbles in a gas–solid fluidized bed. The experimentally measured induced charges from four induction probes were then applied to reconstruct the real charge distribution around a single rising gas bubble injected in a two-dimensional column as it passed the probes. The results reconstructed using the LBP method are found to be smoother and more accurate than those obtained from the LSGRR subroutine. © 2007 Elsevier B.V. All rights reserved. Keywords: Induction probe; Electrostatic charges; Charge distribution; Fluidized bed; Signal reconstruction
1. Introduction Electrostatic charging always exists in industrial gas–solid fluidized bed reactors. This can induce particle agglomeration, changes in hydrodynamic behaviour, adhesion of particles to the walls, interference with instruments, nuisance discharges or explosions, and lead to undesirable by-products. The electrostatic charge can be generated from particles contacting with or separating from each other, and particle contacting with the fluidizing gas and the reactor vessel in which they are contained. A bubbling fluidized bed consists of two distinct phases, a dense phase (sometimes called the emulsion phase) and a dispersed or bubble phase. The reactant concentration is usually higher in bubbles and the region surrounding bubbles than in ⁎ Corresponding author. E-mail address: [email protected] (H.T. Bi). 0032-5910/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2007.02.036
the remote dense phase, leading to faster reaction near the bubbles than elsewhere in the dense phase. Particles close to bubbles also tend to move faster than more remote particles in the dense phase region. Particle–particle collisions are therefore more vigorous for particles close to bubbles, potentially leading to higher rates of charge generation due to particle–particle contact charging. It is therefore expected that the particles adjacent to bubbles may generate and carry more charges than those further away. A collision probe inside a two-dimensional column was employed by Park et al. [1] to explore charge induction and transfer when a single bubble passed. Park et al. [2] and Chen et al. [3] developed a theoretical model to explain the electrical current signals due to the passing of isolated gas bubbles. The model, which assumes that there is a charge distribution close to the bubbles and much higher charge density in the wake region of the same sign as in the region at the front of the bubble, correctly predicted the trend of charge
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A. Chen et al. / Powder Technology 177 (2007) 113–124
Fig. 1. Front view of the two-dimensional fluidization column, showing the optimal location and size of a bubble rising by the induction probes.
signals registered by a sensitive collision probe when bubbles passed. Verification of the quantitative predictions from this model requires information on the distribution of specific charge density surrounding rising gas bubbles in gas–solid fluidized beds. Examining the distribution of charges near rising bubbles can assist in understanding electrostatic charge generation, charge separation, dissipation and accumulation in fluidized bed reactors. Chen et al. [4] developed a technique to measure the charge distribution surrounding a single bubble rising in a two-dimensional gas–solid fluidized bed. Four induction probes, flush
with the outside wall of the column and connected to charge amplifiers, recorded induced charge signals as bubbles passed. The charge distribution surrounding the bubble was then reconstructed assuming the bubble to be symmetrical and the charge around the bubble to remain constant as it rises. The emulsion phase far from the bubble in a two-dimensional fluidized bed of glass beads was found to be negatively charged, whereas the magnitude of the charge density decreased gradually toward the bubble–dense phase interface, with a nearly zero charge density inside the bubble. The wake of the bubble was more negatively charged than the emulsion phase, supporting previous speculations [3]. Chen et al. [4] implemented the model in FORTRAN using the LSGRR subroutine to invert the matrix needed in the reconstruction. If the matrix was mathematically singular or ill-conditioned, a leastsquares routine or the singular value decomposition routine provided by Visual FORTRAN Professional Version 5.0 was implemented to give approximate results. If the dimension of the matrix exceeds the permit limit, or the number of the pixels is much larger than the number of measurements, the matrix is difficult to invert using the LSGRR subroutine. This paper uses the same measurement technique as in [4] to measure the charge distribution surrounding a single rising gas bubble in a two-dimensional gas–solid fluidized bed. However, an iterative linear back projection algorithm (LBP) is applied to improve the reconstruction resolution. The effects of probe size are also investigated. 2. Measurement technique Tests on the static charge around bubbles were conducted with induction probes in a two-dimensional fluidized bed. The column, made of Plexiglas, has an inner thickness of 14 mm, a width of 280 mm, and a height of 1.24 m. The windbox volume is 800 cm3. Glass beads of average diameter 560 μm and density 2500 kg/m3 served as the bed material. The settled bed height was approximately 700 mm. The bed was fluidized by air at a flow rate of 0.002 m3/s, leading to a regular pattern of independent bubbles
Fig. 2. Schematic of induction probe. It is flush with the outside wall of the column (as shown). Around the probe and on the column wall is the grounded shielding. When a bubble is directly in front of the probe, little or no charge is induced. When the bubble rises past the probe, the charge of the emulsion phase is measured.
A. Chen et al. / Powder Technology 177 (2007) 113–124
Fig. 3. Positions of probes and pixel grid used for reconstruction.
Fig. 4. Normalized sensitivity distribution for single probes of different size at x = 26 mm, y = 0.
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A. Chen et al. / Powder Technology 177 (2007) 113–124
with few particles raining through them. These bubbles were approximately 80 mm in diameter and had rise velocities of about 0.26 m/s. Fig. 1 shows a schematic of the column and probes. Copper tape on both the front and rear walls was connected to ground to maintain a zero potential on the column wall. Four induction probes were placed 520 mm above the perforated plate distributor. The distance between adjacent probes was 13 mm. A Canon digital video camera operating at 30 frames/s recorded the rise of bubbles through the column. The images were used to ensure that measurements were performed only on single bubbles not directly influenced by others. Bubbles whose centre passes probe 1 and whose edge passes probe 4 were typically selected for further analysis. The video was therefore always synchronized to the charge measurements. Induction probes (Fig. 2) constitute the simplest type of field meters. Each induction probe consists of an insulated, circular
Table 1 Parameters used in the reconstruction
dB dprobe Npixel, x × Npixel, z Δxpixel × Δzpixel Mx × Mz Δdprobe, x × Δdprobe, z
LBP method
[4]
80 mm 10 mm 21 × 157 3.3 mm × 1.3 mm 4 × 137 13 mm × 1.3 mm
80 mm 10 mm 12 × 48 5 mm × 5 mm 4 × 137 13 mm × 1.3 mm
Charge density distribution around a gas bubble in a fluidized bed of dielectric particles was reconstructed using the iteration linear back projection (LBP) algorithm based on induced charge signals from 4 dynamic probes. The probe size does not influence the reconstruction resolution greatly, with an optimal diameter ranging from 3 to 6 mm for the probe arrangement used in the current study.
copper disc (10 mm in diameter), connected to the core of a coaxial cable, embedded in a Teflon cylinder, and wrapped with a single strip of conducting copper tape. The shielding of the coaxial cable is connected to the shielding around the probe and to a common ground. Together with the coated and grounded walls of the column, this minimizes distortion of the signal from charges above, below, or at the sides of the probe. The metal sensor is charged by induction due to the electric field generated by the charged particles. Each induction probe is connected to a Kistler charge amplifier (model 5011B or 5015) by a coaxial cable. The sampling rate was 300 Hz. The four probes can measure induced charges during the passage of bubbles using the technique described above. 3. Reconstruction of charge distribution The region around the bubble is first divided into a number of pixels, as shown in Fig. 3. Combining Gauss's law with the definition of electric field and potential [5], the electrostatic field E and potential V are given by: div E ¼ q=P
ð1Þ
and E ¼ −grad V
ð2Þ
where q is the volume charge density and Π is the permittivity of the medium, assumed to be uniform in the region of interest. From Eqs. (1) and (2) divðgrad V Þ ¼ j2 j ¼
A2 V A2 V A2 V q þ 2 þ 2 ¼− P A2 x A y A z
ð3Þ
It is assumed that there are charges only in pixel i with constant charge density, qi, in the y-direction, and no charges in other regions. Hence
Fig. 5. Flowchart of the iterative linear back projection algorithm for reconstruction.
q ¼ qi at x ¼ xi ; 0bybdB and z ¼ zi ; q ¼ 0 in all other regions:
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Fig. 6. Simulation of rising bubble with uniform charge of − 1pC/mm3 in the emulsion phase.
The permittivity, Π, of the medium in the bed is assumed to be uniform and isotropic in the dense phase and in the bubble phase, respectively, in order to calculate the charge distribution around the bubble. If we let vV ¼ − LPV 2 q , x′ = x / Lp, p 1 y′ = y / Lp and z′ = z / Lp, then the dimensionless form of Eq. (3) is
For an infinitesimal element of the probe at (xj, yj = 0, zj), the electric field perpendicular to the wall can be estimated by
A2 vV A2 vV A2 vV þ þ ¼ qV A2 xV A2 yV A2 zV
The induced charge on an infinitesimal element of the probe is then
ð4Þ
E8 ¼ Ey ¼ −
AV j Ay x¼xj ;y¼0;z¼zj
where
rij ¼ PE8 ds ¼ −P
qV ¼ 1 at xV ¼ xi =Lp ; 0byVbyi =Lp and zV ¼ zi =Lp ; qV ¼ 0 in all other regions
ds ¼ qi Lp
Boundary conditions: At grounded wall, y′ = 0 and y′ = δB / Lp, v′ = 0; at x′ = − W / 2Lp and x′ = W / 2Lp, v′ = 0; at z = 0 and z = Hbed /Lp, AAvV ¼ 0; Eq. (4) can be solved to obtain v′ and hence 2 xV V.
AV j Ay x¼xj ;y¼0;z¼zj
ð5Þ
ð6Þ
AV V j ds AyV xV¼xj =Lp ;y¼0;z¼zj =Lp
The total charge on probe j is rij
total
¼
RR
rij ds surface of probe j
ð7Þ
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A. Chen et al. / Powder Technology 177 (2007) 113–124
Fig. 7. Second simulation of a rising bubble with a uniform charge of −1 pC/mm3 in the emulsion phase, but now with a thin layer of highly charged particles (− 1.5 pC/ mm3) around the bubble.
If the probe is a circular plate of radius rprobe centred at (xj0, 0, zj0), the total charge of probe j from the ith pixel can be expressed as rij
total
RR
¼
Qmj ¼
rij ds
xj0 −rprobe
2 zj0 −½rprobe −ðxj −xj0 Þ2 0:5
AV V j dzj dxj AyV xV¼xj =Lp ;yV¼0;zV¼zj =Lp
ð8Þ Let Z xij ¼ Lp
2 xj0 þrprobeZ zj0 þ½rprobe −ðxj −xj0 Þ2 0:5
xj0 −rprobe
N X i¼1
surface of probe j Z xj0 þrprobeZ zj0 þ½r2 −ðxj −xj0 Þ2 0:5 probe
¼ qi Lp
Then the induced charge from all pixels to probe j is obtained by
2 zj0 −½rprobe −ðxj −xj0 Þ2 0:5
AV V dzj dxj j AyV xV¼xj =Lp ;yV¼0;zV¼zj =Lp
ð9Þ Eq. (8) is used to calculate the induced charge on probe j due to only pixel i having a charge density qi.
rij
total
¼
N X
xij qi ð j ¼ 1; 2; N M Þ
ð10Þ
i¼1
Theoretically, if the charge induced on the probe, Qmj, can be measured, qi can be reconstructed for each pixel using multiple probes. The quality of the reconstruction depends on the number of probes and the number of measurements of each probe. For good reconstruction, measurements should be taken inside, on the border and outside the bubble. The greater the number of probes, the better the reconstruction. In our experiment, only four probes were fixed horizontally on the wall and many pixels had to be reconstructed (see Fig. 3). Ideally, one should install many probes surrounding the rising bubble to reconstruct the charge density distribution. For the current case with only four fixed probes, it is assumed that each bubble is symmetrical and that the charge around the bubble
A. Chen et al. / Powder Technology 177 (2007) 113–124
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Fig. 8. Third simulation of a rising bubble, with a uniform charge of − 1 pC/mm3 in the emulsion phase and a highly charged wake (− 1.5 pC/mm3).
does not vary as it passes through the measurement region. Multiple measurements can thus be taken as a function of distance between the probe and bubble as the bubble rises vertically through the array of probes. Although there were only four probes, measured induced charge signals at different vertical locations relative to the bubble can be obtained and used to solve Eq. (10) and produce the charge distribution around the rising bubble at a reasonable resolution. More details are provided in [4]. The Iterative Linear Back Projection (LBP) [6], often used in linear tomography, is applied in the present work. The function relating the charge distribution inside the column and the induced charge to the probe is linearized using sensitivity maps. The normalized sensitivity of a probe j to pixel i with charge qi = 1 C is defined as Si; j
xi; j ¼X xi; j
ð11Þ
i
Fig. 4 shows that the normalized sensitivity distribution for single probes with different diameters, dprobe = 10, 8, 6, 4, 2 mm
at x = 26 mm and y = 0. It can be seen that the range of the charged zone sensed by a probe flush with the wall does not change with probe size, because the sensitivity is normalized. However, the smaller the probe, the higher the maximum normalized sensitivity. In the LBP method, the measured induced charge Qmj is transformed to X 0 1 xi; j qi X Qmj xi; j BX C X i Qj ¼ X ¼ X qi A ¼ ðSi; j qi Þ @ xi; j xi; j i¼1 xi; j i i
i
i
ð12Þ Using the normalized sensitivity map, the charge for a pixel i can be calculated by linear back projection from X Si; j Qj 1 ⁎ qi ¼ X ð13Þ Si; j j
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Fig. 9. Reconstructed charge distribution (Case 1, Hmf = 0.7 m, dB = 0.08 m, uB = 0.2 m/s, dp = 565 μm, ρp = 2500 kg/m3, Qair = 1.78 m3/s).
For the reconstruction, Eq. (13) is solved once for every pixel of the reconstruction image. The induced charge on the probe j is then estimated by Q⁎j ¼
X
Si; j q⁎i
ð14Þ
i
Eqs. (13) and (14) were applied iteratively. For every iteration step, a charge difference is calculated for every pixel of the reconstructed image according to X Si; j DQ⁎j j Dq⁎1 ¼ X ð15Þ Si; j j
where ΔQj* = |Qj − Qj*| is the difference between the calculated and measured charges. For each iteration, a new charge distribution is obtained from ⁎ qnew ¼ qold i i þ kDqi
ð16Þ
where λ is a relaxation factor. This gives the additional possibility of using under-relaxation (λ b 1) or over-relaxation (λ N 1) in the reconstruction. With under-relaxation the convergence is usually smoother and better. The use of over-relaxation, on the other hand, can increase the speed of reconstruction. Eq. (15) is applied repeatedly until the difference of induced charges on all probes has been minimized. or until a specified maximum number of iteration steps has been reached. Fig. 5 presents the flowchart of the back projection algorithm for the reconstruction of the charge distribution around a single bubble. 4. Results and discussions A single circular bubble of diameter 80 mm with its associated charge distribution was assumed to pass the probes, and the induced charge was used to reconstruct the original charge distribution. There were four probes of diameter 10 mm located horizontally at 13 mm intervals, as shown in Fig. 3 and as in the earlier work [4]. Three assumed charge distributions were first used to test the LBP reconstruction method and to compare
A. Chen et al. / Powder Technology 177 (2007) 113–124
with simulation results from [4]. The parameters used in the reconstruction are listed in Table 1. The number of pixels, 21× 157, which can be used by the LBP method, is much larger than the number of 12 × 48, which could be utilized by Chen et al. with the LSGRR subroutine, whereas the number of probes, M = 4, is the same. The reconstructed profiles are presented in Figs. 6–8. It was first assumed that there is a uniform charge distribution in the emulsion phase (around the bubble) of q = − 1 pC/mm3 and that the inside of the bubble does not carry charges. The charges induced on the four probes are shown in Fig. 6(a) based on the assumed charge distribution in Fig. 6(b). Using these four series of calculated induced charges, the charge distribution around the bubble is reconstructed based on the above model. The results from Chen et al. [4] and from the LBP method are shown in Fig. 6(c) and (d) respectively. Fig. 7 shows the induced charges for the four probes (7a), the reconstructed charge profiles from [4] (7c), and the reconstructed profiles from the LBP method (7d) with a bubble assumed to have a thin mantle of highly charged particles around its exterior surface (7b).
121
A higher charge density is then assumed in a wake region occupying the bottom portion of the bubble, as shown in Fig. 8 (b). The resulting induced charges from the four probes appear in Fig. 8(a). The higher charge density in the bubble wake region is captured in the reconstructed profile in Fig. 8(c) from [4] and Fig. 8(d) from the LBP method. The experimental data on induced charges measured from the four induction probes by [4], shown in Figs. 9(a), 10(a), 11 (b), were then used to reconstruct the charge distribution around a single bubble passing through the two-dimensional column. The reconstructed results from the LBP methods are presented in Figs. 9(c), 10(c) and 11(c), with the corresponding results from [4] shown for comparison in Figs. 9(b), 10(b), 11(c). From Figs. 6–11, it is seen that the charge density distributions reconstructed using the LBP method are much smoother and the standard deviation is smaller than those presented by [4] based on the LSGRR subroutine. It can thus be concluded that the LBP method can improve reconstruction of charge distributions surrounding a single bubble in a gas–solid fluidized bed. It should be pointed out that the LBP method will
Fig. 10. Reconstructed charge distribution (Case 2, Hmf = 0.7 m, dB = 0.08 m, uB = 0.26 m/s, dp = 565 μm, ρp = 2500 kg/m3, Qair = 1.78 m3/s).
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Fig. 11. Reconstructed charge distribution (Case 3, Hmf = 0.7 m, dB = 0.08 m, uB = 0.26 m/s, dp = 565 μm, ρp = 2500 kg/m3, Qair = 1.78 m3/s).
require much more calculation time if the number of pixels is large. Finally the effect of probe size is examined for the current array of four probes as shown in Fig. 3, with the distance between the centres of adjacent probes fixed at Δdprobe, x = 13 mm. In reconstruction calculations, the number of pixels, Npixel, x × Npixel, z = 21 × 157, and the size of pixels, Δxpixel × Δzpixel = 3.3 mm × 1.3 mm, were maintained constant. Fig. 12 shows the reconstructed results with the same number of probes, M = 4, but different probe diameters, dprobe = 10, 8, 6, 4, and 2 mm. It is seen that the size of probes does not influence the reconstruction resolution greatly. This may be because the normalized sensitivity map was applied in the LBP reconstruction method. It is also found that the standard deviation decreases slightly with decreasing diameter of the probes. However, if the probes are too small, e.g. 2 mm, the standard deviation increases slightly again. As shown in Fig. 4, the probe can only receive information from the front of the probe itself, a very narrow region. The smaller the probe, the higher the maximum normalized sensitivity. Although the area of the charged zone
covered by a smaller probe is the same as for a larger probe, the edge of the zone is not as well sensed by too small a probe relative to the high maximum sensitivity. In the reconstruction, the distance between the centres of adjacent probes is set to be the same for different probe diameters. However, the distance between the edges of adjacent probes becomes larger with decreasing probe diameter. Therefore, if the edges of adjacent probes are far apart, there will exist a region between them that can be less well sensed by the probes, resulting in poor results. The probe diameter of 10 mm currently used in our experiments appears to be reasonable, although the quality of the reconstructed charge distribution can be slightly improved by reducing the probe diameter to 3–6 mm in the future. 5. Conclusions The iteration linear back projection (LBP) algorithm using sensitivity maps can appreciably improve the reconstruction resolution of charge density distributions surrounding a single gas bubble in two-dimensional fluidized beds. The size of the probes does not influence the reconstruction resolution greatly,
A. Chen et al. / Powder Technology 177 (2007) 113–124
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Fig. 12. Reconstruction results with different probes sizes for three assumed charge distribution (Npixel, x × Npixel, z = 21 × 157, Δxpixel × Δzpixel = 3.3 mm × 1.3 mm, Mx × Mz = 4 × 137, Δdprobe, x = 13 mm, Δdprobe, z = 1.3 mm).
with an optimal diameter ranging from 3 to 6 mm for the probe arrangement used in the current experimental study. Nomenclature d diameter, m E electric field, V/m Hbed fluidized bed height, m Hmf bed height at minimum fluidization, m Lp distance between probes 1 and 4 (see Fig. 3), m M number of probes, – Mz Number of measurement along the vertical bubble path N number of pixels, – q volumetric charge density, C/m3 Q induced charge on probe from all pixels, C Qm measured induced charge on probe, C Qair air flow rate, m3/s r radial coordinate, m
area, m2 normalized sensitivity, – velocity, m/s PV dimensionless electrical potential, − 2 , – L p q1 electric potential, V width of two-dimensional column, m Cartesian coordinates, x horizontal in pane of interest, y normal to two-dimensional plane, z vertical, m x′, y′, z′ normalized coordinates, x′ = x / Lp, y′ = y / Lp, z′ = z /Lp, –
s S u v′ V W x, y, z,
Greek letters δB thickness of two-dimensional column, m σ induced charge on probe from one pixel, C Π permittivity of medium, F/m λ relaxation factor, – ρ density, kg/m3 ω matrix in Eq. (10)
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Subscript B bubble i ith pixel j jth probe x, y, z x, y, z directions p particle probe probe pixel pixel total total References [1] A.-H. Park, H.T. Bi, J.R. Grace, Reduction of electrostatic charges in gas– solid fluidized beds, Chem. Eng. Sci. 57 (2002) 153–162.
[2] A.H. Park, H.T. Bi, J.R. Grace, A. Chen, Modeling charge transfer and induction in gas–solid fluidized beds, J. Electrost. 55 (2002) 135–168. [3] A. Chen, H.T. Bi, J.R. Grace, Effects of charge distribution around bubbles on charge induction and transfer to a ball probe in gas–solid fluidized beds, J. Electrost. 58 (2003) 91–115. [4] A. Chen, F.K. van Willigen, J.R. van Ommen, H.T. Bi, J.R. Grace, Charge distribution around a rising bubble in a two-dimensional fluidized bed, AIChE J. 52 (2006) 174–184. [5] J. Cross, Electrostatics: Principles, Problems and Applications, Hilger, Bristol, 1987. [6] T. Loser, R. Wajman, D. Mewes, Electrical Capacitance Tomography: image reconstruction along electrical field lines, Meas. Sci. Technol. 12 (2002) 1083–1091.