Electrical Capacitance Tomography Measurements of Gravity-Driven

The conducted load cell and fiber optic probe measurements are in reasonable agreement with the ECT measurements. The density wave formation has a clo...
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Ind. Eng. Chem. Res. 1999, 38, 621-630

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Electrical Capacitance Tomography Measurements of Gravity-Driven Granular Flows Jinsong Hua and Chi-Hwa Wang* Department of Chemical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

An electrical capacitance tomography (ECT) system is used in this study to investigate the density waves generated in the gravity-driven granular flow through a vertical pipe. The experiments are conducted in two stages. First, the time-averaged quantities of particle concentration and velocity are measured and correlated with the mass flow rate of particles. Second, the power spectra of particle concentration fluctuations are examined to determine the condition for the formation of density waves. The present work finds that the time-averaged particle concentration is higher toward the centerline of the pipe and the particle velocity is relatively uniform over the cross section of the pipe. The conducted load cell and fiber optic probe measurements are in reasonable agreement with the ECT measurements. The density wave formation has a close correlation with the local particle concentration. The experimental results also indicate that the dominant frequency might vary while the particles pass through the vertical pipe. Introduction Granular materials are widely processed in industry in the forms of powders and beads. Despite the importance of such materials to the industry and everyday life, their unusual flow behavior, which is different from those of liquids and solids, has not been fully understood. These include size segregation, flow pattern formation and convection cells under vibration, anomalous sound propagation and avalanche, and other unusual motions in a rotating mill. Even in the simplest geometry such as in hoppers, slope chutes, and vertical pipes the dynamics of granular flows is still quite complicated. The mechanism of the density waves formation in the gravity-driven granular flows in vertical channels and pipes has attracted the interest of researchers in recent years. Previous theoretical studies showed that the formation of density waves is due to the inelastic particle collisions.1 In contrast, some experimental studies mentioned elsewhere found that the roughness of the channel wall and the existence of interstitial fluids also have influence on the formation of these density waves. Baxter and Behringer2 studied the formation of flow pattern while a granular material (sand) was passing through a flat hopper using digital subtraction radiography. They found that the formation and propagation of density waves in the hopper were controlled by the parameters of the mass flow rate of granular material and the opening angle of the hopper. In addition, the density wave was formed immediately after the removal of the plug at the bottom of the wedge. Veje and Dimon3 investigated the granular flow consisting of a single layer of uniform balls in a two-dimensional, smallangled slant funnel. The results of Veje and Dimon3 were similar to the findings of Baxter and Behringer,2 showing that the opening angle of the funnel had a strong effect on the flow patterns of the balls. Quasi* To whom correspondence is addressed. Fax: (65) 779-1936. Phone: (65) 874-5079. E-mail: [email protected].

periodic kinematic density waves (shock wave) were only found while the opening angle of the funnel was in the range from 0.1° to 1°. They also reported the effect of wall roughness on the formation of the density wave. In the case of a smooth wall, no shock wave formation was observed. Horikawa et al.4 experimentally investigated another similar problem in which granular materials (sands) were drained from a hopper through a vertical glass pipe. A cock was attached to the lower end of the glass pipe to control the interstitial air flow. When the cock was fully open, no density waves were observed. On the other hand, the density waves in the sand flow could be clearly visible when the cock was half-closed. Horikawa et al.4 concluded that the back flow of air in the vertical pipe might cause the formation of density waves in the granular flows. Similar to the work of Horikawa et al.,4 Nakahara and Isoda5 also investigated the density waves of metallic spheres falling through a vertical glass pipe filled with an interstitial fluid (water or silicone oil) with the collection end of the pipe completely closed. They have systematically examined two important parameters affecting the formation of density wave: the packing rate of metallic spheres inside the pipe and the kinematic viscosity of the interstitial liquid that fills the pipe. They found that the power spectra of the density fluctuations obeyed the power law (P(f) ∼1/fβ, where β has a positive value) over a wide range of frequency and only at an intermediate packing rate of metallic spheres. It was found that the nonuniform granular flow (density wave) corresponds to the packing rate at high Reynolds number (Re > 10) condition. In the low Reynolds number (Re < 10) range, high viscosity of the interstitial fluid actually prevented the growth of the density waves. Consequently, these studies have shown the possible effects of boundary conditions on the formation of density waves, such as the opening angle of the hopper and the wall roughness. Obviously, these discrete boundary conditions for specific experiments could not disclose a general principle for the formation of density waves.

10.1021/ie980375h CCC: $18.00 © 1999 American Chemical Society Published on Web 02/04/1999

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Figure 1. Schematic diagram of the experimental setup.

Concurrently, some simulation studies [molecular dynamics (MD), lattice-gas-automata (LGA), and the grain kinetic theory] have examined the importance of inelastic particle collision on the formation of density waves in granular flows. Peng and co-workers1,6,7 used the LGA algorithm and found that the formation of density waves is determined by three important parameters: energy dissipation, average density, and the roughness of pipe wall. Lee8 investigated the particle interaction effect on the granular flow using MD simulations and reproduced the formation of density waves. Wang et al.9 studied the density pattern formation in gravity-driven granular flow using the continuum rheological model of Lun et al.10 They found that the inelastic interparticle collisions, channel width, and the average solid concentration are the three key factors involved in the density pattern formation. Most of the earlier studies on the formation of density waves were focused on the effect of particle-particle and particle-wall interactions. The common quantities used to elucidate the particle-particle interactions in such systems were the concentration and velocity of particles. Since particle collisions in the granular flow depend significantly on the solid concentration, the dynamic measurements of particle concentration become the key technique. To serve this purpose in the present study, the granular flow in a vertical pipe is investigated using an electrical capacitance tomography (ECT) system. By cross-measuring the capacitance among a set of electrodes mounted around the test section of a pipe through which the granular materials flow, the permittivity distribution of the mixture in the test section is measured and hence the profiles of component concentration of the mixture are determined. The ECT system has recently been used to monitor the particle concentration variation in horizontal pneumatic conveying and fluidized beds.11 First, the present work appears to be the first application of the ECT

system to the gravity-driven flow of granular materials. A comparison to previous studies, the ECT application to dynamic measurements of particle concentration has a few advantages. Primarily, it is a nonintrusive measurement and hence no disturbance to the main flow. Second, this can give the particle concentration profiles over a cross section of the pipe. Third, it has a quick time response and its sampling frequency can be as high as 50 frames/s. Finally, this technique is applicable over a larger particle concentration range. In this paper, investigations are focused on the density wave formation in the gravity-driven granular flow. For this purpose, the application feasibility of the ECT system is first examined by measuring the timeaveraged quantity of particle concentration. Second, the dynamic features of particle concentration are measured by the ECT system and further analyzed using the standard fast Fourier transform (FFT) power spectrum algorithm. In all experiments, particles are loaded (by a wire mesh) at the entrance of the pipe and the density waves are observed after the particles enter the vertical pipe. The variation of dominant frequency of density waves with the particle concentration is investigated. Experimental Setup and Procedure The schematic diagram of the experimental setup is illustrated in Figure 1. This basically consists of the following four parts: supply hopper, vertical pipe, collection hopper, and the instrumentation system. Two types of granular materials (glass beads and plastic cylinders) are used, with their basic characteristic parameters listed in Table 1. The granular materials are held in the conical supply hopper, which is connected to the vertical pipe through a diaphragm valve (as shown in Figure 1). The opening status of the valve controls the mass flow rate of

Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 623 Table 1. Physical Parameters of the Granular Materials parameter

glass beads

plastic cylinders

r F Vta height shape

1.75 mm 1586 kg/m3 8.3 m/s not applicable sphere

1.25 mm 1287 kg/m3 5.6 m/s 1.5 mm cylinder

a The terminal velocity (V ) is estimated by the experimental t drag coefficient of the free-fall sphere (or cylinder) velocity in an expanse of stagnant fluid.20

granular materials flowing through the vertical pipe. The supply hopper facilitates the batch process of flow with adequate flow time. A wire mesh (of suitable gap dimension) is inserted beneath the valve. Since the diameter of the hopper outlet is much smaller than the tube, the wire mesh is used as a “particle distributor” to avoid the situation of the granular materials flowing through the exit as a granular jet. The vertical pipe is made of plexiglass with an inner diameter of 36 mm and a length of 1 m. The ratio of pipe inner diameter to the particle diameter is about 10, close to the previous works reported in the literature.3-5,12 In contrast, the aspect ratio of pipe length to diameter is smaller than that of these previous studies. In the present design the lasting of the density wave is relatively short because of some modifications. For instance, the exit end of vertical pipe is fully opened4,5,12 and its inner surface is kept smoother.3,6,8 Hence, the acceleration of granular materials is much faster than that found in previous studies, owing to the reduction in the flow resistance caused by no back flow of interstitial fluid and the smoother pipe wall. Alternatively, as the particles fall down over a short distance, the granular materials expand down the tube and reach a more dilute regime, where the density wave is absent. The effect of interstitial fluid (air) on the granular flow can be estimated by the characteristic Bagnold number, which measures the ratio of inertia to viscous stresses:3,13

Ba )

4λ0.5Fr2γ η

(1)

where η is the dynamic viscosity of the interstitial fluid; r and F are the radius and density of a particle, respectively; λ and γ are the linear concentration and shear rate, respectively. If Ba < 40, the flow is in the “macro-viscous” regime, the rheological behavior is dominated by the viscous effect. In contrast, if Ba > 440, the momentum balance between particles is primarily governed by the inertia effect. For the Ba value between 40 and 440, the flow is in the transition regime. In this study, the diameter and mass of glass beads are 3.5 mm and 5.8 × 10-5 kg, respectively. The maximum falling velocity of particles is about 2.6 m/s. The viscosity of air η ) 1.7 × 10-5 kg/ m‚s. If the characteristic shear rate is taken to be the centerline velocity divided by tube diameter, the Bagnold numbers for the glass beads and plastic cylinders are estimated to be much greater than 440, suggesting that the viscous drag between the interstitial fluids and particles may not introduce significant effect on the granular flow. Moreover, because the exit of the vertical pipe is fully opened, there is no counter flow of air, which might also minimize the effects of interstitial fluid.

It is noted that a different expression (Ba ) 6πηVr/ mg) was used in the work of Veje and Dimon.3 This expression may need corrections for high Reynolds number flows at which the boundary layer separation occurs readily behind the particle. The previous studies by Veje and Dimon3 and Baxter and Behringer2 showed that the opening angle of the channel wall could indeed affect the formation of density waves. When the opening angle of the channel wall is extremely small (i.e., the two side channel walls are almost parallel), no density wave was observed. In the present study, a vertical pipe is used so that the opening angle effects can be eliminated. The theoretical work by Peng and Herrmann6 examined the effects of wall roughness on the power spectra structure of density wave. No density wave was observed in the channel flow of granular materials with smooth wall, and in contrast, even a small wall roughness can lead to the formation of density waves. Furthermore, the experimental work of Veje and Dimon3 showed that the roughness of the channel wall is essential for the formation of density wave. In the current study, the inner wall of the vertical Plexiglas pipe is kept smoother so that the effect of wall roughness is minimized. While passing through the vertical pipe, the granular materials are collected in the collection hopper and weighed by a load cell unit. Subsequently, the actual flow rate of granular materials is obtained. Besides the load cell measurement, the particle concentration is measured simultaneously by the electrical capacitance tomography system (effective for two-component mixtures) which is mounted on the outer wall of the vertical pipe. To examine the development of density waves, concentration measurements are conducted at several axial stations (A, B, and C) along the pipe. They are located at 30, 50, and 70 cm downward form the pipe entrance, respectively. The principle of the ECT system is based on the cross measurements of capacitance among a set of electrodes that are mounted around the vertical pipe to obtain mixture permittivity distribution over the cross section of the pipe. The permittivity of a mixture corresponds to its composition, and hence the concentration profiles in the pipe can be obtained nonintrusively. In addition to particle concentrations, the traveling velocity of density pattern can be obtained by the correlation analysis of the concentration signals from a twin-plane ECT system (Process Tomography LTD, Cheshire, U.K.). It has a space resolution of 32 × 32 pixels over the pipe cross section. The sampling rate is 50 frames/s to ensure enough time and space resolutions to detect the density waves formation. The ECT measurements are sensitive to the electrostatic charge. Hence, the ECT sensor is shielded with a copper sheet which is connected to the ground. Furthermore, the ECT electrodes are surrounded with the guard electrodes that are grounded. The particle sizes used in the present study are sufficiently large enough that the electrostatic forces are relatively negligible as compared to other forces. The vertical spatial resolution depends on the length of the ECT sensing electrodes. In the present study, the length of the ECT electrodes is 70 mm. The system is applied to the measurement of particle concentration in a relatively low-concentration regime. The ECT sensitivity and precision in particle concentration measurements depend on the capacitance

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Figure 2. Temporal variation of particle concentration at the cross section of measuring station C. The flow rate of glass beads is 0.31 kg/s. The number at the upper left corner of each frame (at the time interval of 0.02 s) refers to the corresponding time sequence.

measuring capability, size of the tube, length of the sensors, and the type of particles. The capacitance measurements in the current system range from 0.0003 to 2 pF. In previous studies,3-5,12 the density waves formed in the granular flows were measured by photodetecting systems. One limitation of these techniques is that the detected signal attenuates at high concentrations so that the particle concentration cannot be too high, otherwise the light beam cannot pass through the granular flow. This feature has actually limited the particle concentration measurements in previous studies to the dilute regime where the boundary effects might be the dominant factors. In contrast, the ECT system can be applied to dense systems where the effects of particle collisions are more important. The experimental procedures comprise two major steps. First, measurements of the time- and spaceaveraged quantities are conducted to examine the baseline features of flow and to check the accuracy of the ECT system. This is followed by the dynamic measurements of particle concentrations to examine the mechanism of density wave formation.

Results and Discussion Observation of Density Waves. Figure 2 shows the temporal variations of particle concentration profiles at the measuring station (C) over a period of time. The mass flow rate of glass beads is 0.31 kg/s. In this study, the software for data processing automatically scales the solid concentration of a closely packed bed to the value of 1.0. For a monodisperse system, this corresponds to a solid volume fraction of 65%. For other nonspherical or polydisperse systems, the values of random close packing solid fractions are different. To

avoid confusion, it is noted that the concentration value 1.0 indicates a fully packed bed. The red color indicates the highest particle concentration (0.25) in the given flow system, while the blue color refers to the lowest concentration (0). The time sequence is indicated at the upper left corner of each frame. It is observed that the particle concentration profiles change almost periodically. For example, in frames 1, 13, 28, 37, and 43, the particles are most concentrated at the center of the pipe. On the contrary, particles are more uniformly distributed over the cross section of pipe in frames 4, 25, and 40. These transient variations of particle concentration profiles in the vertical pipe are known as “density waves”. Raafat et al.12 showed schematically the pattern of density waves formed in the dry granular media falling through a vertical pipe. In terms of the geometry, the particles occupied either the center of the pipe (known as the bubble phase) or the whole cross section of the pipe (known as the clog phase). The comparison of the present work with Raafat et al.12 revealed a significant difference in the granular flow patterns in vertical pipes. These are potentially due to different mechanisms of density wave formation. Raafat et al.12 observed density wave in rather dense beds. On the contrary, the flow regime remains relatively dilute in the present work. A particular aspect of the present study is that the particles do not extend across the whole section, even in the case of a dilute bed. In terms of particle distribution, the ECT measurements clearly demonstrate that the peak particle concentration is higher when particles occupy the pipe center, while it is lower when particles occupy the whole cross section of the pipe. In the present study, particle collision is anticipated to be the major factor for density wave formation. This differs from the results of Raafat et al.,12 primarily in the experimental configuration. In the work of Raafat et al.,12 the control

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Figure 3. Schematic diagram of the density-wave pattern formation. Panels A, B, C, and D show the snapshots of particle distribution at different times.

valve was attached at the end of the pipe and hence the interstitial fluid (back flow) plays a more important role in the density wave formation. The schematic diagram shown in Figure 3 illustrates the density wave formation mechanism when particle collision is the key factor. When the particles collide with the pipe wall, they bounce back toward the center of the pipe. Strong particle collisions occur when the particles meet in the center, where a dense phase is formed. Further collisions redistribute the particles over the cross section of the pipe and result in a more dilute phase. Such behaviors are repeatedly observed when particles fall down, and hence the density wave is formed. This is indicated by a series of snapshots (panels A, B, C, and D of Figure 3) taken at a specified zone of the pipe. Averaged Characteristics. We now restrict attention to the coordinate system indicated in Figure 1, where the particle concentration R(x,y,z,t) varies with space [x and y are the coordinates at a given cross section whose axial coordinate is z] and time t. The timeaveraged particle concentration profile is defined as follows:

R j t(x,y,z) )

1 T

∫0TR(x,y,z,t) dt

(2)

where T is the period of time averaging. The period of time averaging for eq 2 is about 20 s. This averaging period is kept constant for every data set. Different time periods have been tried to make sure that the selection of time averaging will not affect the accuracy of results. Figure 4 shows a typical time-averaged particle concentration profile measured at station B of the pipe. In general, the particle concentration distribution is axissymmetric. The highest concentration is located at the center of the pipe and decreases toward the wall. This feature bears some similarities with the concentration profiles reported by some previous theoretical works for flows in vertical pipes and channels or inclined chutes.9,14-17 For “smoother wall” experiments, particles

collide on the wall and bounce back to the center of the pipe, thus resulting in a higher particle concentration in the center of the pipe. In another set of tests, the wall roughness is varied using No. 360 sandpaper through machinery. Veje and Dimon3 glued a transparent covering sheet on the chute walls to make them rougher. Since the diameter of the tube used in this work is small and the length is large, it is difficult to stick similar sheets (or sandpapers) on the tube’s inner wall. The wall roughness in the range of our investigations is found to have limited effect on the particle concentration, analogous to the dependence upon the wall boundary conditions in the theoretical investigations. The radial variation of time-averaged particle concentration profiles R j t (z ) zc) with the mass flow rate of particles is shown in Figure 5. Here, zc refers to the axial coordinate of the measuring station C. Normalized radial positions (0,1) and 0.5 refer to the wall and the center of pipe, respectively. R j t increases with the mass flow rate within the range 0.25-0.46 kg/s. The particle concentration in the center of pipe is significantly higher than that of the wall, and the radial nonuniformity increases with the mass flow rate of granular materials. Particles fall down more uniformly over the pipe cross section in the dilute flow, in contrast to the higher radial variation of particle concentration in the dense flow. The particle distribution patterns are dependent on the ratio of pipe-to-particle diameter. We found that the peak concentration occurred near the centerline of a pipe for the range of pipe diameters used in our investigation. When a larger diameter of pipe was introduced, the effective particle concentration was found to exceed our detection limit and therefore no data was reported for these cases. Alternatively, the particle size can be reduced for performing a similar test. But as the particle size is reduced, the effect of interstitial fluid and electrostatic forces will become more significant. It is anticipated that the peak concentration could show up at points other than the centerline if a much larger pipe is used.

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Figure 4. Time-averaged particle concentration profile R j t at the measuring station B with the flow rate of glass beads as 0.40 kg/s.

average concentration R j (z) is given by

∫∫ARjt(x,y,z) dx dy

1 A

R j (z) )

Figure 5. Variations of the radial glass beads concentration profiles R j t (Z ) ZC) with the mass flow rate.

Figure 6. Variation of the average glass beads concentration R j with the mass flow rate. Comparison of the ECT and FOP measurements. “SW” and “RW” refer to the pipes with the “smooth” and “rough” inner walls, respectively.

Figure 6 shows the variation of average particle concentration with the mass flow rate at the three measuring stations A, B, and C. The definition of the

(3)

where A is the cross-sectional area of the vertical pipe. Figure 6 shows the comparison between the ECT and FOP (fiber optic probes, PC-4 Powder Voidmeter, Triple Crown Hi-Tech, Vancouver, Canada)18 measurements for the average glass beads concentration. Some similarities in the particle concentration profile are observed for the ECT and FOP results of the “smooth” pipe. At the measuring station C, the ECT results of the “rough” and “smooth” pipes are not very different. Figure 6 also indicates that the average particle concentration almost increases linearly with the mass flow rate in the dilute regime. The “dilute regime” mentioned in this section is the regime in which the volume fraction of solid is less than 4%. The average solid concentration R j shown in Figure 6 ranges from 0.04 to 0.14 and therefore corresponds to the dilute regime mentioned above. This feature also appears to be in qualitative consistency with the results of channel flow reported in the literature,17,19 showing that the mass flux in vertical or slant channels reaches its highest at intermediate particle concentrations. In the dilute regime, in contrast, the average particle concentration increases almost linearly with the mass flow rate. The phase velocity of density wave is measured by correlating the signals between the two measuring stations B and D. The correlation algorithm is given by

C(d) )

1 T

∫0TRjs(zB,t)Rjs(zD,t-d) dt

R j s(z,t) )

∫∫AR(x,y,z,t) dx dy

1 A

(4) (5)

where d is the time-shift for the cross correlation and

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Figure 7. A temporal correlation between the signals of spaceaveraged particle concentrations measured by the ECT twin-plane system. The mass flow rates of glass beads and plastic cylinders are 0.344 and 0.152 kg/s, respectively.

R j s(zB,t) and R j s(zD,t) are the cross-section-averaged particle concentration at the sensing stations B and D, respectively. The variation of the correlation coefficient C with d is examined and the peak value in this plot (the maximum correlation Cmax) indicates the signal delay (dt) between the two sensing planes. The ECT twin-plane system allows for the simultaneous recording at two stations. Stations B and D were measured simultaneously and the recorded data were used in eq 4. An example cross-correlation between the two stations B and D is shown in Figure 7. Here, the distance L between the two stations is 40 cm, and the middle plane between the two ECT measuring planes is located 45 cm below the entrance of pipe. The time delay (dt) between the two planes is 11 frames (about 11 × 0.014 s). Hence, the propagating speed of the density wave (L/dt) is estimated as 2.58 m/s. This phase velocity is lower than the free falling velocity of objects in the specified zone, because of the energy dissipation caused by the interparticle collisions. The effect of particle type on the flow pattern is also examined. The phase velocity of density wave is found to show no significant difference for various granular materials with different packing rates (Figure 7). This feature may result from the insignificant resistance caused by the pipe wall and the interstitial fluid, corresponding to a large value of Bagnold number, as mentioned in the earlier sections. To check the accuracy of the results presented above, the mass flow rates measured by the twin-plane ECT system are compared with the load cell measurements. The mass flow rate of granular materials (ME) estimated by the ECT measurements is given by

ME )

1 T

∫0T∫∫Au(x,y,z,t) FR(x,y,z,t) dx dy dt

(6)

where u and F are the particle velocity and the bulk density of the closely packed materials, respectively. The ECT system gives both the particle concentration and phase velocity of the density pattern, with the latter quantity being close to the average velocity of the granular assembly:9

u j ≈ L/dt

(7)

Figure 8. Comparison of the mass flow rate estimated by the ECT system and load cell unit.

For the case of a smoother pipe, the transverse variation of particle velocity is not significant,14 and hence the mass flow rates can be estimated by the following expression:

ME ≈ u j FR jA

(8)

The comparison between the mass flow rates estimated by the ECT (ME) and load cell unit (ML) is shown in Figure 8. The insignificant difference between the two measurements indicates that the accuracy of ECT sensors is in the acceptable level. Axial Development of Flow. Results for the axial variation of particle concentration profiles for two mass flow rates (3.5 and 3.0 kg/s) are shown in Figure 9. The concentration profile measured at station A (near the wire mesh) is different from those measured at stations B and C. This indicates that the axial development of the solid concentration profile is relatively fast near the injection point. However, the data collected at stations B and C suggested that the flow has not yet been fully developed. There is a decrease in the averaged concentration from station B to C. In its role as a “particle distributor”, the wire mesh cannot eliminate the nonuniformity at the pipe entrance completely and hence the concentration profiles are formed because of preexisting nonuniformity at the wire mesh and are later modified by the particle redistribution along the tube. Radial Distribution of Particle Velocity. The correlation measurements over individual pixels show that the particle velocity evaluated at individual points is rather uniform across the vertical pipe (Figure 10). This seems to be in qualitative consistency with the experimental results reported by Savage.14 The particle velocities are in the range of 2.47-2.65 m/s for the normalized radial positions 0.17-0.48. Here, the dimensionless coordinates 0 and 0.5 refer to the wall and the centerline of the tube, respectively. Because of the significant noise near the wall, the corresponding pixel correlation (normalized radial positions 0.047 and 0.109) is not shown. In contrast, the radial particle concentration distribution is quite nonuniform. We agree with the point raised by the referee that if the flow is more concentrated toward the center of the pipe, the variation of grain velocity in the low concentration zone will not affect the accuracy of mass flow rate measurements significantly.

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Figure 10. The pixel correlation at different normalized radial positions. The positions 0 and 0.5 correspond to the wall and the centerline of the pipe, respectively.

Figure 9. Variations of particle concentration along the axial position at the mass flow rate of (a) 3.5 and (b) 3.0 kg/s.

Dynamic Characteristics of the Density Waves. Figure 11 shows the temporal variation of particle concentration measured at station C, indicating the formation of clusters (known as the density wave) when particles pass through the ECT sensor. The mass flow rate of glass beads is the same as that in Figure 2. To examine these density patterns, a standard fast Fourier transform (FFT) algorithm is used to analyze the power spectra P(f) of the concentration variations. A noteworthy point here is that a section-averaged signal (rather than the signal at a fixed pixel) is used. Some researchers4,5 reported that the power spectra of the density waves generated in granular flows obey the power law (P(f) ∼ 1/fβ, where β has a value ranging from 0.8 to 1.5, depending upon the experimental conditions) in the high frequency domain from 10 to 100 Hz. In previous studies, the density waves were monitored by the quick response photodetector system, which can capture signals lower than 1 kHz. In the present work, in contrast, the highest sampling frequency of the ECT system is about 50 Hz, with the capability of signal detection limited to lower than 25 Hz. Therefore, the results presented here are not sufficient to confirm the entire power law theory over the wide frequency range (from 10 to 100 Hz) as reported in the literature. However, the power spectrum in the frequency range between 10 and 25 Hz appears to be the low-frequency range of a power-law structure. The exponent parameter

Figure 11. Variation of the cross-section-averaged concentration R j s (Z ) ZC) with time. The flow rate of glass beads is 0.31 kg/s.

in the power law (β) roughly has a positive value of 0.2, which is smaller than the experimental value reported by Nakahara and Isoda5 and Horikawa et al.4 The noise in the present system is generally larger than that of Horikawa et al.4 and Nakahara and Isoda.5 In interpreting these results, one needs to recall the difference in the corresponding experiments. In the present study, the experimental power spectra are collected by analyzing the data at a full cross section of the tube. In contrast, the previous work in the literature involved the spectra measurements at only a fixed point (or the cumulative signals from a series of points on a straight line). Nakahara and Isoda5 studied the variation of power spectra structure with different packing rates for the granular flows through a vertical pipe. Their results showed that the power spectrum structure in the lowfrequency domain (f < 3 Hz) changed significantly with the particle packing rate and position along the axis of the pipe. This indicated that the power spectra in the frequency domain (3 < f < 100 Hz) correspond to the

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Figure 12. Power spectra variation of particle concentration fluctuations at the different measuring positions along the vertical pipe. The flow rate of glass beads is around 0.25 kg/s.

microscopic density fluctuation of particles, while the low-frequency domain (f < 3 Hz) describes the macroscopic movements of particle slugs (i.e., density waves). Hence, in an investigation of the dynamic characteristics of density waves using the ECT system, the focus of the present work is the behavior of the low-frequency component (