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Investigation of gas-solids bubbling fluidized bed by electrical capacitance tomography and modified Tikhonov regularization

Qiang Guo ab Shuanghe Meng a Dehu Wang a Yinfeng Zhao a Mao Ye a Wuqiang Yang c and Zhongmin Liu a

a National Engineering Laboratory for Methanol-To-Olefins Dalian Institute of Chemical Physics Chinese Academy of Sciences Dalian 116023 China

b University of Chinese Academy of Sciences Beijing 100049 Chinac School of Electrical and Electronic Engineering The University of Manchester Manchester M13 9PL UK

Corresponding Author Mao Ye Email maoyedicpaccn

Abstract Electrical capacitance tomography (ECT) provides a non-intrusive and non-invasive means to visualize the cross-sectional distribution of material by measuring capacitance between electrode pairs which are mounted around the periphery of a pipe or vessel such as a fluidized bed Successful application of ECT strongly depends on the image reconstruction algorithm used For on-line measurement of a fluidized bed it is necessary to employ an algorithm which can produce high-quality images without extensive computation Using the conventional Tikhonov method the image quality in the central area in a gas-solids bubbling fluidized bed is good but suffers from artifacts in the near wall region especially when bubbles are dispersed in a high permittivity emulsion phase To solve this problem a similar division operation learned from the linear back projection (LBP) method is introduced to modify the conventional Tikhonov regularization method To evaluate the proposed algorithm computational fluid dynamics (CFD) is used for numerical simulation and a bubbling fluidized bed for experiment Both simulation and experiment results indicate the artifacts can be effectively removed and the image quality which can be achieved by the new method is similar to Landweber iteration Furthermore the modified Tikhonov regularization method shows high accuracy in obtaining some most important hydrodynamic parameters in the gas-solids bubbling fluidized bed Finally it has been found that although different image reconstruction algorithms may show large difference in image quality they have only minor effect on the characterization of the dynamic behaviors based on the statistical analysis of the calculated overall solids concentration

Keywords gas-solids bubbling fluidized bed electrical capacitance tomography image reconstruction algorithm Tikhonov regularization computational fluid dynamics

1 IntroductionGas-solids bubbling fluidized bed reactors are used in many industrial processes such as coal gasification

power generation granulation and polymerization For measuring the hydrodynamic characteristics in these reactors numerous experimental techniques have been developed1-6 The intrusive methods include optical probe13 capacitance probe134 and pressure measurement15 It is easy to implement the intrusive methods However they can only provide local information on flow behavior In recent years non-intrusive techniques such as tomography2 have been developed which can be used to visualize the entire flow field without causing disturbance to the flow Compared with other industrial process tomography techniques Electrical Capacitance Tomography (ECT) has advantages fast imaging speed no radiation robust and low cost7 Considering the non-conductive nature of the materials in gas-solids bubbling fluidized beds ECT is a suitable measuring technique for investigation of their hydrodynamics68-12

1

In ECT the sensing electronics is used to measure variations in capacitance between pairs of electrodes which are placed around the periphery of the pipe or vessel under investigation These measurements are then used to reconstruct the permittivity distribution as the presentation of the material distribution inside the sensing area using an image reconstruction algorithm13-16 Two major difficulties are associated with the reconstruction process 14 First the number of the measured independent capacitance data is far less than the number of unknown image pixels and hence it is severely under-determined Secondly due to the ill-posed property of ECT the reconstruction results are sensitive to the raw capacitance measurement error To address these problems different reconstruction techniques including single step and iterative algorithms have been proposed 1417 But so far only linear back projection (LBP) Tikhonov regularization and Landweber iteration methods are popular18-20 As an example Figure 1 shows some typical results reconstructed by these three algorithms

Figure 1 Typical results reconstructed by three widely used image reconstruction algorithms

As can be seen in Figure 1 LBP can only give qualitative images for all distributions while it is widely used for on-line image reconstruction because of its simplicity and fast speed61021 Although LBP gives poor image quality in the central area there are no any artifacts in the near wall region (Figure 1) Peng et al25

validated that LBP could provide a good image quality for the annular and stratified distributions Tikhonov regularization can give satisfied results in the central area even with multiple objects like four rods and four bubbles The problem with Tikhonov regularization is the artifacts shown in the near wall region142223 especially in the cases of low-permittivity materials present in a high-permittivity background which is common in a gas-solids bubbling fluidized bed where the discrete bubble phase is dispersed in the continuous solids phase Landweber iteration can give best images in almost all cases14 However it can be applied as an off-line reconstruction method only because of the slow reconstruction speed

A generic sensitivity matrix is often used for both one-step and iterative algorithms142324 while it should change with different permittivity distributions24 because of the ldquosoft fieldrdquo nature of ECT It is found that the inhomogeneous distribution of the generic sensitivity map is the main reason for the artifacts in the near wall region in the images reconstructed by Tikhonov regularization Xue et al23 found that the image quality reconstructed by Tikhonov regularization could be improved and the artifacts in the reconstructed images

2

reduced by updating the sensitivity matrix However calculation of the updated sensitivity matrix is very time-consuming2324 and it is not practical for on-line imagereconstruction Ye et al16 proposed a sparsity reconstruction algorithm combined with a soft-threshold function according to the type of continuous phase and found that it can improve the image quality significantly for distributions with high permittivity as a continuous phase like a bubbling fluidized bed However it is difficult to determine the threshold

Considering the dynamics of gas-solids bubbling fluidized bed reactors a non-iterative algorithm has to be used for on-line measurement This paper proposes a modified Tikhonov regularization method to eliminate the artifacts and hence enhance the image quality by scaling each pixel grey in the imaging area using a division operation learned from LBP To evaluate this method experiment was carried out on a bubbling fluidized bed The results from numerical simulation based on the Computational Fluid Dynamics (CFD) are used as the permittivity distribution input The accuracy of the proposed algorithm to obtain some key hydrodynamic parameters like overall solids concentration bubble size and radial profile of solids concentration is verified

2 ECT model and image reconstruction algorithms21 ECT model

A circular 12-electrode ECT sensor is modelled with an electrode covering ratio of 09 as shown in Figure 2 To measure capacitance a defined potential is applied to one of the electrodes in turn from E1 to E11 and the others are used as detection electrodes to obtain the capacitance of all possible electrode pairs This measurement strategy can yield 66 independent capacitance measurements

The forward problem of ECT is to calculate inter-electrode capacitance from the predefined permittivity distribution and is expressed as

Figure 2 Simulation model of 12-electrode ECT sensor R1 = 70

mm R2 = 73 mm and R3 = 77 mm

CM=minusε01V ∬

Γ

ε r (x y )nablaφ ( x y ) d Γ (1)

where ε0 is the permittivity of vacuum V is the potential between the calculated electrode pair εr(xy) and φ(xy) are the relative permittivity distribution and the potential distribution in the sensing domain respectively and Γ is the electrode surface

Because Eq 1 is too complicated a simplified linear equation in a normalized form is usedλ=Sg (2)

where g is the normalized permittivity and λ is the normalized capacitances defined by

λ=CMminusCL

CHminusC L

(3)

where CM is the measured capacitance CH and CL are the capacitances when the sensor is full of higher and lower permittivity materials respectively

The parameter S in Eq 2 represents the normalized sensitivity matrix which refers to the change in capacitance of each electrode pair in response to a perturbation of the permittivity The construction of the sensitivity matrix needs discretization for implementation In this work the sensing area is subdivided into

3

64times64 grids resulting in 3228 pixels Eq 4 is commonly employed to calculate the sensitivity matrix

Sijiquest ( x y )=minus ∬

p (x y)

nablaφi(x y )V i

∙nablaφ j(x y )

V jdxdy (4)

where Sij

iquest defines the sensitivity between the ith and jth electrodes at the location of pixel p(xy) φi(xy) and

φj(xy) are the electrical potential distribution inside the sensing domain when the ith and jth electrodes are excited by applying a voltage of Vi and Vj respectively

S is normalized as

Smn=Smn

iquest

summ=1

M

Smniquest

(5)

where Smn and S mn are the entries in the mth row and the nth column of S and S respectively

Note that while S will change with permittivity distribution a generic S calculated based on a vacuum permittivity distribution2324 is commonly used for image reconstruction because (1) the true distribution is unknown in a real application and (2) it takes too long time to update the sensitivity matrix and therefore is not practical for on-line image reconstruction

22 Image reconstruction algorithmsTo solve the inverse problem in ECT the measured capacitance is transformed into a spatial permittivity

distribution using an image reconstruction algorithm Three commonly used algorithms are LBP Tikhonov regularization and projected Landweber iteration

221 LBPLBP was the first developed algorithm for ECT15 The main principle of LBP is to replace the inverse of S

which is non-existent with its transpose by the transpose of S as expressed by

g= ST λST uλ

(6)

where g is the reconstructed normalized permittivity uλ is the identity vector The division operation is

manipulated in a one-to-one mode

222 Tikhonov regularizationTikhonov regularization is one of the most popular methods to solve ill-posed problems 26 and have been

applied in ECT182227 The conventional Tikhonov regularization is

g=(ST S+μI )minus1 ST λ (7)

where μ is the regularization parameter and I is a unit matrix As well known it is crucial to choose μ to obtain the accurate estimation of the solution However it is

difficult to determine an appropriate value of μ in theory Therefore this is usually done by trial and error

223 Projected Landweber iterationLandweber iteration with a projection was introduced to ECT28 and is expressed as

4

gk=P ( gkminus1+α k ST ekminus1) (8)

P [ f (x )]= 0f (x)

1

if f (x )lt0if 0le f (x )le 1

if f (x )gt1

(9)

where ek-1 is the deviation between the measured capacitance and the capacitance calculated from the kth reconstructed permittivity distribution which is defined by

ekminus1=λminusS gkminus1 (10)The initial estimation g0in Eq 8 is calculated by LBP in Eq 6 A drawback of Landweber method is its

semi-convergence characteristic To improve its convergence speed Liu et al29 suggested an optimal step length αk during iteration as formulated by

α k=ST ekminus1S ST ekminus1

(11)

Since Landweber iteration can produce best images in most cases by simulation and experiment results based on stationary object tests14 it is the most popular among iterative algorithms 1114253031 Therefore it is used as a reference for assessment

224 Modified Tikhonov regularizationFigure 3 shows the average sensitivity of all electrode pairs

It can be seen that in the central area the sensitivity distribution is homogenous while in the near wall region a heterogeneous distribution is clearly seen More specifically the gaps between adjacent electrodes have the highest sensitivity and the areas closed to the electrodes have the lowest sensitivity Considering this feature together with the images reconstructed by Tikhonov regularization as shown in Figure 1 (later in Figure 8 and Figure 14) it is assumed that the reason for the artifacts in the near wall region is the inhomogeneous distribution of the sensitivity Therefore a scaling method like the division operation in LBP is proposed to modify Tikhonov regularization to be

Figure 3 Sensitivity map of the simulated ECT sensor

g= (ST S+μI )minus1 ST λ(ST S+μI )minus1 ST uλ

(12)

225 Evaluation criteriaThree criteria are used to compare different image reconstruction algorithms They are correlation

coefficient (CC)1430 average absolute deviation (AAD)13 and reconstruction speed14 CC and AAD are defined as

CC=sumi=1

N

( giminusg)(g iminusg)

radicsumi=1

N

(g iminus g)2sumi=1

N

(giminusg)2

(13)

5

AAD= 1N sum

i=1

N

|ϕsminusϕs| (14)

where N is the number of pixels (3228 in this case) ϕ s and

ϕ s are the true and reconstructed solids

concentration and g and g are the mean values of g and g

CC indicates the spatial similarity between the reference and reconstructed images while AAD refers to the accuracy of an algorithm for displaying the solids concentration distribution The best algorithm will give the maximum value of CC minimum value of AAD and shortest time

3 Simulation and experimental setup 31 CFD simulation results

To evaluate the performance of different image reconstruction algorithms it is common to carry out simulation andor experiment with stationary objects In this way only some simple permittivity distributions can be tested For gas-solids bubbling fluidized beds and other multiphase systems however the true permittivity distribution is complicated Because of mixing of gas and solids the normalized permittivity is not exactly 0 or 1 In addition the nature of the ldquosoft-filedrdquo sensing of ECT means that the electric flied is distorted by the materials Therefore it is necessary to take hydrodynamic characteristics of the two-phase flow into consideration Ye et al3031 reported a fluid-electrostatic filed coupling method in which the two-phase flow field and the electrostatic field are coupled by an additional electric force However the electric force has no obvious effect on the hydrodynamic characteristics for the normally used excitation voltage lower than 25 V 30 In this work the coupling of the flow filed and the electrostatic field is ignored and the CFD simulation results are only served as the input for image reconstruction as suggested by Banaei et al13

Figure 4 (a) shows the process for evaluation of image reconstruction algorithms using CFD simulation results as the permittivity distribution input In this method CFD simulation of a lab-scale bubbling fluidized bed of 14 cm in diameter and 28 m height was performed with the kinetic theory of granular flow (KTGF) based Eulerian granular model in Fluent 63 The physical properties of the gas phase and particles are listed in Table 1 Figure 4 (b) and (c) show

Figure 4 Procedure for evaluation of image reconstruction algorithms using CFD simulation as the input permittivity distribution

6

Table 1 Physical properties of air and particles

Air

Density ρg kgm3

Viscosity μg Pas

Relative permittivity εg

1225

17894times10-5

1

Particles

Density ρs kgm3

Diameter dp μm

Relative permittivity εs

2328

280

4

Figure 5 (a) shows the 3D geometry of the simulated bed where the gas enters the unit with a constant velocity of 09 ms from the bottom inlet and exit the unit from an atmospheric pressure outlet located at the top The hexahedral cells are generated in 3D as shown in Figure 5 (b) for axial view and Figure 5 (c) for cross-sectional view The total number of CFD grids for each cross-sectional plane is 845 The wall boundary conditions are defined following those by Sinclair and Jackson32 where a no-slip boundary condition is specified for the gas phase and the partial-slip boundary condition is used for the solids with the specularity coefficient of 06

The initial bed height is set as 70 cm with an average solids concentration of 42 More detailed simulation settings are summarized in Table 2 To fully capture the bubble dynamics CFD simulation lasted for 24 seconds and only the results in the last 18 seconds were extracted and sent to ECT for reconstruction

Figure 5 Geometry and mesh for CFD simulation of bubbling

fluidized bed

Table 2 Simulation settings in Fluent

Properties Setting

Unsteady formulation

Momentum amp Volume discretization

Drag coefficient

Granular temperature

Granular viscosity

Granular bulk viscosity

Frictional viscosity

Solids pressure

Radial distribution

Time step

Restitution coefficient

Packing limit ϕmax

First-order implicit

First-order upwind

Gidaspow

Algebraic

Gidaspow

Lun et al

Schaeffer

Lun et al

Lun et al

5times10-4s

09

063

The 3D ECT problem is commonly simplified to 2D7 by neglecting the axial hydrodynamics and reconstructing the cross-section of the bubbling fluidized bed only As shown in Figure 4 (a) the cross-sectional material distribution is first obtained from the slices of the bubbling bed Then the permittivity distribution is converted by the calculation of the relative permittivity εr as expressed by

ε r=εs ϕs+εg(1minusϕs) (15)where εg and εs are the relative permittivity of gas and the solids and listed in Table 1

7

For the forward problem the inter-electrode capacitance for a specified permittivity distribution was obtained using COMSOL Multiphysics and Matlab The high permittivity used in the full calibration process is calculated by Eq 15 where ϕs = ϕmax Once the inter-electrode capacitance has been obtained the permittivity

distribution can be reconstructed by solving the inverse problem Then the solids concentration ϕ s

is calculated

by the parallel model in which the normalized permittivity equals to the normalized solids concentration

ϕ s=ϕmax g (16)

As the cross-section of the CFD simulation was performed in 845 CFD grids as shown in Figure 5 (c) which is different from that of the 3228 ECT square pixels it is necessary to map the results of the CFD grids shown in Figure 4 (b) to the ECT pixels To do so the coordinates of all the ECT pixels were positioned in CFD grids and then the corresponding solids concentration in each position was extracted and used to construct to the ECT image as shown in Figure 4 (c) Finally the reconstructed images can be compared with the reference images

32 Experimental setupUnlike numerical simulation the measured capacitance data in experiment contain noise To verify the

feasibility as well as the noise immunity of the modified Tikhonov regularization method a cylindrical bubbling fluidized bed equipped with a 12-electrode ECT sensor was set up as shown in Figure 6 (a) The bed is made of quartz glass and has the height of 1 m the inner diameter of 60 cm and the outer diameter of 66 cm Airflow under ambient condition is introduced to the bed through a porous polypropylene plate with a mean pore size of 10 μm The bed material fluidized is fluid catalytic cracking (FCC) catalyst with the Sauter mean diameter of 65 μm and particle density of 1370 kgm3 The particle size distribution is shown in Figure 6 (d) which is typical Group A particles according to the Geldart classification33 Preliminary tests revealed the minimum fluidization velocity and the minimum bubbling velocity of the FCC particles are 275 mms and 995mms respectively

Figure 6 (b) shows the 12-electrode ECT sensor whose cross-sectional dimensions are detailed in Figure 6 (c) The sensor electrodes are 3 cm high and struck to the onside wall of the bed7 The width of the electrodes is specified to keep the covering ratio of the electrodes to be the same as the simulation ECT model As shown in Figure 6 (a) the mid-position of the sensor is located at 20 cm above the distributor ie the measurement region encompasses a height between 185 cm and 215 cm above the distributor and therefore each pixel in the reconstructed imaging area represents an axial average over this circular measurement volume An AC-based ECT system20 was applied for capacitance measurement Before the measurement began air and the unfluidized seeds with a static height of 33 cm were used to obtain CL and CH in Eq 3 to calibrate the ECT system To make the bed operate in the bubbling regime the superficial gas velocity was controlled between 1166 cms and 12831 cms with a step of 1166 cms For each velocity a total of 10000 sets of capacitance data were collected corresponding to 100 seconds as the data acquisition rate of the ECT system is 100 frames per second

8

Figure 6 Experimental setup

4 Results and Discussions41 Evaluation by simulation

After the CFD simulation finished 78 snapshots in the last 18 seconds and in the bottom zone of the bubbling fluidized bed were captured and used as the reference distribution for ECT These 78 distributions represent appearance of single bubble two bubbles and multiple bubbles located in different positions notable bubble dynamics including bubble growth bubble coalescence and bubble movement are also reflected All the following discussions are based on these chosen slices

411 Determination of regularization parameterThe quality of images reconstructed by Tikhonov regularization strongly depends on the regularization

parameter μ To find the optimal value of μ the dependency of CC is shown in Figure 7 (a) and AAD in Figure 7 (b) for both the conventional and the modified Tikhonov regularization on the regularization parameter Note that CC and AAD are the average of all the 78 extracted distributions as mentioned earlier For both algorithms with the increase of μ CC first increases to a maximum and then decreases until μ reaches 01 after which CC shows no significant change Regarding AAD both algorithms show a same change tendency but the tendency is just the reverse compared with that of CC In accordance with the evaluation criteria the most appropriate value of μ for the simulated ECT sensor is 00001 for both Tikhonov regularization methods The modified Tikhonov method improves CC and AAD with all tested values of μ In particular a higher value of μ in the magnitude of 100 which is too large in the conventional Tikhonov regularization is acceptable for the modified algorithm

9

Figure 7 Effect of the regularization parameter μ on (a) CC and (b) AAD

412 Image quality comparisonFigure 8 shows six typical cases By comparing the reconstructed distributions with the reference images it

can be seen that LBP can give high-quality images near the wall while the details in the central region are all distorted due to the lower sensitivity in the central region Only some simple distributions like the single bubble distribution in Case 1 and the gas slug in Case 5 can be reconstructed In contrast the conventional Tikhonov regularization can reproduce the material distribution with a good quality except for the artifacts in the near wall region with high solids concentration in the pixels in the gaps between adjacent electrodes and low solids concentration in the pixels near the electrode surface After the modification made by the division operation in the conventional Tikhonov method the artifacts disappeared as shown in the 5 th row in Figure 8 and acceptable results can be obtained in all tested cases A further comparison between the modified Tikhonov regularization and Landweber iteration indicates that almost all images reconstructed by the modified Tikhonov technique are similar to Landweber algorithm with 25 iterations In some cases like the complex multiple bubble distribution in Case 3 and Case 4 the modified Tikhonov technique can produce images as good as that by the Landweber algorithm with 200 iterations

10

Figure 8 Cross-sectional solids concentration distribution reconstructed by different algorithms using the CFD results as the inputs

In Figure 9 CC and AAD obtained by different image reconstruction algorithms are compared where CoTi and MoTi represent the conventional and modified Tikhonov regularization methods respectively and the alphabet L followed by a number means Landweber algorithm with a certain number of iterations From Figure 9 it can be concluded that LBP and conventional Tikhonov methods give the relatively low values of CC because of the poor quality of images reconstructed With Landweber iteration CC rises quickly in the first 50 iterations with the increase in the number of iterations and then shows a slow increase in the next 150 iterations As the iteration process continues a decrease in CC is noticed suggesting the semi-convergence characteristics of the Landweber method With the modified Tikhonov method a value of 09 which is as high as that obtained by Landweber with 25 iterations can be reached

Figure 9 also shows that a higher CC always gives a lower AAD or vice versa To evaluate the stability of the images reconstructed by different algorithms the error bars calculated from the variance of the tested 78 distributions are shown in Figure 9 A lower value of the variance implies the ability for an algorithm to produce images with consistent image quality From the error bars a remarkable vibration for CC occurs for LBP conventional Tikhonov algorithm and Landweber algorithm with iterations less than 50 while a less noticeable variance holds for the modified Tikhonov and Landweber algorithm with iterations more than 50 For AAD the vibration for all algorithms is small To sum up the results in Figure 8 and Figure 9 the modified Tikhonov method can improve not only the image quality but also the stability compared with the conventional Tikhonov method The modified Tikhonov method can produce similar images to Landweber method with 25 iterations

11

Figure 9 Quantitative evaluation of different image reconstruction algorithms The error bars represent the variance of the tested 78

distributions

413 Elapsed timeOne of the most advantages of ECT is its high speed With a twin-plane ECT sensor the flow velocity34 and

even the velocity profile35 in the fluidized bed can be measured To do so both a rapid data acquisition rate and a high-speed algorithm are necessary To compare the speed of different algorithms the elapsed time for image reconstruction with different algorithms is evaluated on a PC with an Intel Core i5 330 GHz as shown in Figure 10

As the matrix (STS+μI)-1ST can be calculated and stored in the stack memory in advance19 both the conventional and the modified Tikhonov methods have the same computational costs as LBP which is about 03 ms For the Landweber iteration the computation time increases linearly with the number of iterations and more specifically the time cost by Landweber method with 25 iterations and 200 iterations is about 100 and 1000 times that by the non-iterative algorithms respectively

Figure 10 Computational costs of different image reconstruction

algorithms

414 Accuracy of overall solids concentration measurementBy pixel averaging the overall solids concentration can be measured Figure 11 (a) shows the relative error

of the overall solids concentration measured by different algorithms As the profile of the overall solids concentration is a key indicator for characterizing fluidization regime transition in gas-solids fluidized beds the mean and the standard deviation (Sd) of the 78 measured values of overall solids concentration are computed as shown in Figure 11 (b)

Figure 11 (a) shows that all algorithms underestimate the overall solids concentration in all cases thus the average of the 78 measured overall solids concentration is also lower as shown in Figure 11 (b) Figure 7 given by Wei et al19 also indicates a lower volume fraction reconstructed by LBP Tikhonov regularization and Landweber iteration methods A possible reason is that the transformation from the permittivity distribution into the material distribution as defined in Eq 16 uses the parallel model in which the normalized permittivity and the normalized solids concentration take the same value Some other concentration models like the series model

12

and Maxwell model set the value of the normalized solids concentration higher than the normalized permittivity13 In this way the under-estimation problem may be solved Figure 11 shows that except for the conventional Tikhonov regularization other algorithms can give the overall concentration at the same level The absolute relative error is within 10 for all of the 78 cases and is approximately 4 in average Similarly according to Figure 11 (b) all algorithms give an accurate estimation of Sd of the overall solids concentration with the absolute relative error between 2 and 5

Figure 11 Evaluation of different image reconstruction algorithms for overall solids concentration measurement (a) individual case and (b)

statistical analysis of all 78 cases

415 Accuracy of the bubble size measurementIn a fluidized bed bubbles can affect the fluidization quality solids mixing and interphase mass or heat

transfer Therefore accurate quantification of the bubble size is important for the design and operation of a fluidized bed Figure 12 (a) shows the performance of different image reconstruction algorithms for measurement of the bubble size in the 78 cases The mean bubble diameter of these 78 cases is plotted in Figure 12 (b) The bubbles are defined as the surface area where the solids concentration is lower than a certain threshold and the threshold is taken as 02 according to the literatures 1236 As can be seen LBP under-estimates the bubble size in almost all cases and the reconstructed mean diameter is also lower than the true bubble with a relative error of -29 While the conventional Tikhonov method over-estimates in most cases and results in a relative error of 9 for the mean bubble size owing to existence of the low solids concentration in the pixels near the electrode surface as mentioned earlier Using Landweber iteration with different number of iterations the mean bubble sizes are scattered on both sides of the true margin in Figure 12 (b) Especially Landweber method with 50 and 100 iterations give the closest bubble sizes with a relative error of -05 and 08 respectively and the latter also provide the highest value of CC and the lowest value of AAD as shown in Figure 9 The modified Tikhonov method gives the mean bubble size very close to the true and the relative error is only -1 which is similar to Landweber method with 50 and 100 iterations Figure 12 (a) also shows that the individual bubble size determined by the modified Tikhonov method is accurate The bubble sizes in some cases reconstructed by LBP modified Tikhonov and Landweber iteration methods have a relative error as large as -100 compared with the true bubbles from CFD simulation These bubbles have the sizes smaller than 5 of the imaging area implying the spatial resolution of ECT is about 537

13

Figure 12 Evaluation of different image reconstruction algorithms for bubble size measurement (a) individual case and (b) average of all 78

cases

416 Accuracy of radial solids concentration profile measurement Two representative distributions are shown in Figure 13 (a) for a solids slug flow and Figure 13 (b) for a

core-annulus structure flow to compare the radial solids concentration profiles produced by different image reconstruction algorithms Figure 13 shows that except for LBP showing a smoothing effect on the sharp transition between the emulsion phase and the bubble phase all other algorithms can quantitatively reconstruct the profile The conventional Tikhonov method gives large deviation near the wall because of the artifacts as shown in Figure 8 The curves produced by the modified Tikhonov method and Landweber method with 25 and 200 iterations almost coincide with each other and approach to the curve of the true profile in Figure 13 showing the ability of these algorithms for providing the radial profile of the solids concentration Figure 13 also indicates that in most radial positions the measured solids concentration by the modified Tikhonov regularization and Landweber methods is slightly lower than the true solids concentration which is in line with Figure 11

Figure 13 Accuracy of different image reconstruction algorithms with respect to the radial profile of the solids concentration

42 Evaluation by experimentTo further verify the modified Tikhonov regularization method three key aspects obtained by different

image reconstruction algorithms are discussed They are image quality bubble size and the statistical analysis of

14

the overall solids concentration

421 Image quality comparisonFigure 14 shows two sets of typical images reconstructed by different algorithms using experimental data

The regularization parameter in both the conventional and the modified Tikhonov methods adopts the optimized value from simulation Although the true distributions are unknown in a real fluidized bed it is still clear from Figure 14 that there are artifacts distributed in the near wall region in the images reconstructed by the Tikhonov method and the distribution of the artifacts is the same as simulation With the modified Tikhonov method the artifacts disappear and the images are as good as Landweber method with 25 iterations (Case 1) or 200 iterations (Case 2) An additional comparison with the images reconstructed by LBP6101421 indicates that more details can be seen with the modified Tikhonov method without extensive computation

Figure 14 Cross-sectional solids concentration distribution reconstructed by different algorithms using experimental data

422 Bubble size measurementTo quantify the bubble size determined by

different image reconstruction algorithms Figure 15 shows the bubble diameter calculated at each gas velocity alongside the estimated bubble size obtained from the correlations of Darton et al38 and Werther39

The two correlations were originally developed in beds with a diameter larger than 10 cm and hence these correlations are qualitative rather than quantitative As shown in Figure 15 the bubble diameters calculated using different algorithms and the correlations have a similar variation tendency showing the increase in bubble size with the increase in the gas velocity Although there is some

Figure 15 Bubble size determined by different image reconstruction

algorithms and the correlations of Darton et al38 and Werther39

discrepancy between the correlations and ECT measurement the bubbles calculated by all techniques lie within the window of the correlations especially when the superficial gas velocity is larger than about five times of the minimum bubbling velocity Comparing Figure 15 with Figure 12 it can be seen that the experimental data is in good agreement with the simulation In both situations Tikhonov and LBP methods give the biggest and smallest estimation of the bubble diameters respectively while the bubble sizes calculated by the modified Tikhonov method fall in between that by Landweber iteration with 25 and 200 iterations

15

423 Statistical analysis of overall solids concentrationTime domain analysis of the measured overall solids concentration is useful for describing the dynamic

behaviors in gas-solids fluidized beds According to literatures researchers use different image reconstruction algorithms to calculate the overall solids concentration6103740 To illuminate the role of algorithm on the statistical analysis of the overall solids concentration the average and SD of the overall solids concentration of 10000 frames measurements at each gas velocity are compared and shown in Figure 16 As a reference the average and Sd of the mean inter-electrode capacitance are also plotted

Figure 16 (a) Average and (b) Sd of the overall solids concentration measured by different image reconstruction algorithms and raw

capacitance at each superficial gas velocity

It can be seen from Figure 16 (a) that the average of the overall solids concentration reconstructed by all algorithms shows the same tendency The average concentration first decreases to a local minimum and then increases and finally shows a monotone decreasing with the increase in the superficial gas velocity similar to the variation in the average raw capacitance The local minimum point in Figure 16 (a) corresponds to the so-called bed contraction phenomenon41 For Geldart A particles there is an interval of homogenous expansion regime before the onset of bubbling3341 In the vicinity of the minimum bubbling velocity the bed reaches a locally maximal height leading to a locally minimal overall solids concentration After the minimum bubbling point the overall solids concentration decreases monotonously with the increase in the bubble size For Sd of the overall solids concentration it follows the trend of Sd of the average raw capacitance and increases linearly with the increase in the superficial gas velocity as shown in Figure 16 (b) for all algorithms Makkawi and Wright6 suggested that as long as the bed is operated in the bubbling regime Sd of the solids fraction is linearly related to the gas velocity Although different image reconstruction algorithms are used as far as the statistical analysis like the average and Sd of the overall solids concentration are concerned all algorithms present the same occurrence The results demonstrate that although different techniques show discrepancy in the reconstructed image quality for hydrodynamic study by the statistical analysis of the overall solids concentration all algorithms give the same conclusion Figure 16 also shows that the relative values of the average and Sd of the overall solids concentration obtained from different image reconstruction algorithms at any velocity agree well with the simulation results as shown in Figure 11

5 ConclusionsIn this work a similar one-to-one division operation which is originally used in LBP is introduced to

modify the Tikhonov method for the use of ECT in a gas-solids bubbling fluidized bed Both numerical simulation and experiment were carried out to evaluate the proposed algorithm Simulation was performed by comparing the reconstructed images with the slices extracted from CFD simulation The accuracy of the

16

modified Tikhonov method in obtaining some hydrodynamic characteristics like overall solids concentration bubble size and radial solids concentration profile were systemically evaluated The role of image reconstruction algorithm in describing the dynamic behaviors based on the statistical analysis of the overall solids concentration was also investigated Based on this work the conclusions can be summarized as follows LBP Tikhonov regularization and Landweber iteration are three most popular algorithms for ECT LBP and

Tikhonov methods are good for on-line measurement due to their fast speed However the images reconstructed by LBP are usually blurred With Tikhonov method artifacts are always shown in the near wall region and the artifacts are more severely in the case of gas-solids bubbling fluidized bed Landweber iteration can produce the best images in most cases However the computational cost prevents it from on-line application

The main reason of the artifacts shown in images reconstructed by Tikhonov method is the inhomogeneous distribution of the sensitivity in the generic sensitivity matrix The division operation introduced to modify the Tikhonov method can rescale the grey level in each pixel and hence the artifacts can be effectively removed

The regularization parameter in Tikhonov method can be determined by obtaining the highest value of CC and the lowest value of AAD

The quality of images reconstructed by the modified Tikhonov method is similar to that by Landweber method with tens iterations while the computing cost by the modified Tikhonov method is only one percent or less of that by Landweber iteration

The modified Tikhonov method shows high accuracy or the same accuracy as Landweber iteration in obtaining the overall solids concentration bubble size and profile of the radial solids concentration in the gas-solids bubbling fluidized bed

The overall solids concentration is under-estimated by the parallel model for all test algorithms in this paper If the concentration model is changed to the series model or Maxwell model over-estimation may occur It suggests that a new concentration model based on the true distributions is needed To develop such a model numerical simulation based on the CFD results as the permittivity distribution input in which the reconstructed concentration can be directly compared with the true value

For measuring the bubble size the 12-electrode ECT sensor used in this work is limited to bubble size larger than 5 of the imaging area independent of image reconstruction algorithm used

Although different image reconstruction algorithms show large divergence in the reconstructed image quality all algorithms can give the trend of the average and Sd of the overall solids concentration with the change in superficial gas velocity

AcknowledgmentsThe authors thank the financial support from the Newton Advanced Fellowship of the Royal Society UK

(Grant No NA140308) and the National Natural Science Foundation of China (Grant No 91334205)

NomenclatureAbbreviationsECT = electrical capacitance tomographyLBP = linear back projectionCFD = computational fluid dynamicsKTGF = kinetic theory of granular flowCC = correlation coefficient

Roman lettersC = capacitance pFV = potential difference VS = normalized sensitivity matrixI = unit matrixe = capacitance residual

17

AAD = average absolute deviationFCC = fluid catalytic crackingSd = standard deviation

Greek lettersεr = relative permittivityφ = potential distribution Vλ = normalized capacitanceuλ = identity vectorμ = regularization parameterα = step lengthϕ = concentration

g = normalized permittivity

Subscriptsg = air phases = solids phaseL = low calibrationH = high calibrationM = measurement

Superscript = average value = reconstructed value

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6 Makkawi YT and Wright PC Fluidization regimes in a conventional fluidized bed characterized by means of electrical

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7 Yang W Design of electrical capacitance tomography sensors Meas SciTechnol 21 (4) 2010 042001

8 Chandrasekera TC Li Y Moody D Schnellmann MA Dennis JS and Holland DJ Measurement of bubble sizes in

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10 Zhang W Cheng Y Wang C Yang W and Wang C-H Investigation on Hydrodynamics of Triple-Bed Combined

Circulating Fluidized Bed Using Electrostatic Sensor and Electrical Capacitance Tomography Ind amp Eng Chem Res

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11 van Buijtenen MS Buist K Deen NG Kuipers JAM Leadbeater T and Parker DJ Numerical and experimental study

on spout elevation in spout-fluidized beds AIChE J 58 (8) 2012 2524-2535

12 McKeen T and Pugsley T Simulation and experimental validation of a freely bubbling bed of FCC catalyst Powder

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13 Banaei M van Sint Annaland M Kuipers JAM and Deen NG On the accuracy of Landweber and Tikhonov

reconstruction techniques in gas-solid fluidized bed applications AIChE J 61 (12) 2015 4102-4113

14 Yang WQ and Peng L Image reconstruction algorithms for electrical capacitance tomography Meas Sci Technol 14

(1) 2003 R1-R13

15 Xie C Huang S Hoyle B Thorn R Lenn C Snowden and D Beck M Electrical capacitance tomography for flow

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16 Ye J Wang H and Yang W A sparsity reconstruction algorithm for electrical capacitance tomography based on modified

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Landweber iteration Meas Sci Technol 25 (11) 2014 115402

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reconstruction algorithms for electrical capacitanceresistance tomography Sensor Review 36 (4) 2016 429-445

18 Jing L Liu S Zhihong L and Meng S An image reconstruction algorithm based on the extended Tikhonov

regularization method for electrical capacitance tomography Measurement 42 (3) 2009 368-376

19 Wei K Qiu C Soleimani M and Primrose K ITS Reconstruction Tool-Suite An inverse algorithm package for

industrial process tomography Flow Meas Instrum 46 2015 292-302

20 Yang W and York T New AC-based capacitance tomography system IEE Proc-Sci Meas Technol 146 (1) 1999 47-

53

21 Pugsley T Tanfara H Malcus S Cui H Chaouki J and Winters C Verification of fluidized bed electrical capacitance

tomography measurements with a fibre optic probe Chem Eng Sci 58 (17) 2003 3923-3934

22 Peng L Merkus H and Scarlett B Using regularization methods for image reconstruction of electrical capacitance

tomography Part amp Part Sys Charact 17 (3) 2000 96-104

23 Xue Q Wang H Cui Z and Yang C Electrical capacitance tomography using an accelerated proximal gradient

algorithm Review Sci Instruments 83 (4) 2012 043704

24 Li Y and Yang W Image reconstruction by nonlinear Landweber iteration for complicated distributions Meas Sci

Technol 19 (9) 2008 094014

25 Peng L Ye J Lu G and Yang W Evaluation of effect of number of electrodes in ECT sensors on image quality IEEE

Sensors J 12 (5) 2012 1554-1565

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28 Yang WQ Spink DM York TA and McCann H An image-reconstruction algorithm based on Landwebers iteration

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30 Ye J Wang H Li Y and Yang W Coupling of Fluid Field and Electrostatic Field for Electrical Capacitance

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31 Ye J Wang H and Yang W Evaluation of electrical capacitance tomography sensor based on the coupling of fluid field

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1473-1486

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35 Mosorov V Sankowski D Mazurkiewicz and Dyakowski T The best-correlated pixels method for solid mass flow

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36 Yates Y Cheesman D and Sergeev Y Experimental observations of voidage distribution around bubbles in a fluidized

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37 Warsito W and Fan LS Measurement of real-time flow structures in gasndashliquid and gasndashliquidndashsolid flow systems using

electrical capacitance tomography (ECT) Chem Eng Sci 56 (21-22) 2001 6455-6462

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40 Ge R Ye J Wang H and Yang W Measurement of particle concentration in a Wurster fluidized bed by electrical

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capacitance tomography sensors AIChE J 60 (12) 2014 4051-4064

41 Girimonte R and Formisani B The minimum bubbling velocity of fluidized beds operating at high temperature Powder

Technol 189 (1) 2009 74-81

20

In ECT the sensing electronics is used to measure variations in capacitance between pairs of electrodes which are placed around the periphery of the pipe or vessel under investigation These measurements are then used to reconstruct the permittivity distribution as the presentation of the material distribution inside the sensing area using an image reconstruction algorithm13-16 Two major difficulties are associated with the reconstruction process 14 First the number of the measured independent capacitance data is far less than the number of unknown image pixels and hence it is severely under-determined Secondly due to the ill-posed property of ECT the reconstruction results are sensitive to the raw capacitance measurement error To address these problems different reconstruction techniques including single step and iterative algorithms have been proposed 1417 But so far only linear back projection (LBP) Tikhonov regularization and Landweber iteration methods are popular18-20 As an example Figure 1 shows some typical results reconstructed by these three algorithms

Figure 1 Typical results reconstructed by three widely used image reconstruction algorithms

As can be seen in Figure 1 LBP can only give qualitative images for all distributions while it is widely used for on-line image reconstruction because of its simplicity and fast speed61021 Although LBP gives poor image quality in the central area there are no any artifacts in the near wall region (Figure 1) Peng et al25

validated that LBP could provide a good image quality for the annular and stratified distributions Tikhonov regularization can give satisfied results in the central area even with multiple objects like four rods and four bubbles The problem with Tikhonov regularization is the artifacts shown in the near wall region142223 especially in the cases of low-permittivity materials present in a high-permittivity background which is common in a gas-solids bubbling fluidized bed where the discrete bubble phase is dispersed in the continuous solids phase Landweber iteration can give best images in almost all cases14 However it can be applied as an off-line reconstruction method only because of the slow reconstruction speed

A generic sensitivity matrix is often used for both one-step and iterative algorithms142324 while it should change with different permittivity distributions24 because of the ldquosoft fieldrdquo nature of ECT It is found that the inhomogeneous distribution of the generic sensitivity map is the main reason for the artifacts in the near wall region in the images reconstructed by Tikhonov regularization Xue et al23 found that the image quality reconstructed by Tikhonov regularization could be improved and the artifacts in the reconstructed images

2







mm R2 = 73 mm and R3 = 77 mm

CM=minusε01V ∬

Γ





λ=CMminusCL

CHminusC L

(3)



3



p (x y)

nablaφi(x y )V i

∙nablaφ j(x y )

V jdxdy (4)

where Sij



S is normalized as

Smn=Smn

iquest

summ=1

M

Smniquest

(5)







g= ST λST uλ

(6)









4


P [ f (x )]= 0f (x)

1


if f (x )gt1

(9)





(11)






(12)



CC=sumi=1

N


radicsumi=1

N

(g iminus g)2sumi=1

N

(giminusg)2

(13)

5

AAD= 1N sum

i=1

N

|ϕsminusϕs| (14)









6


Air

Density ρg kgm3

Viscosity μg Pas


1225

17894times10-5

1

Particles

Density ρs kgm3

Diameter dp μm


2328

280

4




fluidized bed


Properties Setting



Drag coefficient


Granular viscosity



Solids pressure

Radial distribution

Time step


Packing limit ϕmax


First-order upwind

Gidaspow

Algebraic

Gidaspow

Lun et al

Schaeffer

Lun et al

Lun et al

5times10-4s

09

063



7



is calculated


ϕ s=ϕmax g (16)





8






9




10




11


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20







mm R2 = 73 mm and R3 = 77 mm

CM=minusε01V ∬

Γ





λ=CMminusCL

CHminusC L

(3)



3



p (x y)

nablaφi(x y )V i

∙nablaφ j(x y )

V jdxdy (4)

where Sij



S is normalized as

Smn=Smn

iquest

summ=1

M

Smniquest

(5)







g= ST λST uλ

(6)









4


P [ f (x )]= 0f (x)

1


if f (x )gt1

(9)





(11)






(12)



CC=sumi=1

N


radicsumi=1

N

(g iminus g)2sumi=1

N

(giminusg)2

(13)

5

AAD= 1N sum

i=1

N

|ϕsminusϕs| (14)









6


Air

Density ρg kgm3

Viscosity μg Pas


1225

17894times10-5

1

Particles

Density ρs kgm3

Diameter dp μm


2328

280

4




fluidized bed


Properties Setting



Drag coefficient


Granular viscosity



Solids pressure

Radial distribution

Time step


Packing limit ϕmax


First-order upwind

Gidaspow

Algebraic

Gidaspow

Lun et al

Schaeffer

Lun et al

Lun et al

5times10-4s

09

063



7



is calculated


ϕ s=ϕmax g (16)





8






9




10




11


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20



p (x y)

nablaφi(x y )V i

∙nablaφ j(x y )

V jdxdy (4)

where Sij



S is normalized as

Smn=Smn

iquest

summ=1

M

Smniquest

(5)







g= ST λST uλ

(6)









4


P [ f (x )]= 0f (x)

1


if f (x )gt1

(9)





(11)






(12)



CC=sumi=1

N


radicsumi=1

N

(g iminus g)2sumi=1

N

(giminusg)2

(13)

5

AAD= 1N sum

i=1

N

|ϕsminusϕs| (14)









6


Air

Density ρg kgm3

Viscosity μg Pas


1225

17894times10-5

1

Particles

Density ρs kgm3

Diameter dp μm


2328

280

4




fluidized bed


Properties Setting



Drag coefficient


Granular viscosity



Solids pressure

Radial distribution

Time step


Packing limit ϕmax


First-order upwind

Gidaspow

Algebraic

Gidaspow

Lun et al

Schaeffer

Lun et al

Lun et al

5times10-4s

09

063



7



is calculated


ϕ s=ϕmax g (16)





8






9




10




11


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20


P [ f (x )]= 0f (x)

1


if f (x )gt1

(9)





(11)






(12)



CC=sumi=1

N


radicsumi=1

N

(g iminus g)2sumi=1

N

(giminusg)2

(13)

5

AAD= 1N sum

i=1

N

|ϕsminusϕs| (14)









6


Air

Density ρg kgm3

Viscosity μg Pas


1225

17894times10-5

1

Particles

Density ρs kgm3

Diameter dp μm


2328

280

4




fluidized bed


Properties Setting



Drag coefficient


Granular viscosity



Solids pressure

Radial distribution

Time step


Packing limit ϕmax


First-order upwind

Gidaspow

Algebraic

Gidaspow

Lun et al

Schaeffer

Lun et al

Lun et al

5times10-4s

09

063



7



is calculated


ϕ s=ϕmax g (16)





8






9




10




11


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20

AAD= 1N sum

i=1

N

|ϕsminusϕs| (14)









6


Air

Density ρg kgm3

Viscosity μg Pas


1225

17894times10-5

1

Particles

Density ρs kgm3

Diameter dp μm


2328

280

4




fluidized bed


Properties Setting



Drag coefficient


Granular viscosity



Solids pressure

Radial distribution

Time step


Packing limit ϕmax


First-order upwind

Gidaspow

Algebraic

Gidaspow

Lun et al

Schaeffer

Lun et al

Lun et al

5times10-4s

09

063



7



is calculated


ϕ s=ϕmax g (16)





8






9




10




11


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20


Air

Density ρg kgm3

Viscosity μg Pas


1225

17894times10-5

1

Particles

Density ρs kgm3

Diameter dp μm


2328

280

4




fluidized bed


Properties Setting



Drag coefficient


Granular viscosity



Solids pressure

Radial distribution

Time step


Packing limit ϕmax


First-order upwind

Gidaspow

Algebraic

Gidaspow

Lun et al

Schaeffer

Lun et al

Lun et al

5times10-4s

09

063



7



is calculated


ϕ s=ϕmax g (16)





8






9




10




11


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20



is calculated


ϕ s=ϕmax g (16)





8






9




10




11


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20






9




10




11


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20




10




11


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20




11


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20


distributions





algorithms




12






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20






13


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20


cases






14











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20











15








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20








16














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20














17









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20









115 (1) 2001 13-26














52 (32) 2013 11198-11207




Technol 129 (1-3) 2003 139-152




(1) 2003 R1-R13





18









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20









53








Technol 19 (9) 2008 094014


Sensors J 12 (5) 2012 1554-1565













1473-1486







bed Chem Eng Sci 49 (12) 1994 1885-1895






19



Technol 189 (1) 2009 74-81

20



Technol 189 (1) 2009 74-81

20

Research Explorer | The University of Manchester · Web viewThe bed material fluidized is fluid...

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