Electrical capacitance sensor array to measure density profiles of a vibrated granular bed

Powder Technology 270 (2015) 10–19

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Electrical capacitance sensor array to measure density profiles of a vibrated granular bed Karol Asencio a, W. Bramer-Escamilla b,⁎, Gustavo Gutiérrez c, Iván Sánchez a,⁎⁎ a b c

Laboratorio de Física de Estadística de Medios Desordenados, Instituto Venezolano de Investigaciones Científicas (IVIC), Caracas 1020-A, AP 20632, Venezuela Laboratorio de Física de la Materia Condensada, Instituto Venezolano de Investigaciones Científicas (IVIC), Caracas 1020-A, AP 20632, Venezuela Departamento de Física, Universidad Simón Bolívar, AP 1080-A, Venezuela

a r t i c l e

i n f o

Article history: Received 13 May 2014 Received in revised form 7 August 2014 Accepted 2 October 2014 Available online 13 October 2014 Keywords: Capacitive sensor Solids fraction Density Vibrated granular bed

a b s t r a c t Monitoring the dynamical evolution of the solids distribution within vibrating granular beds is of key importance to control a variety of industrial processes and to assist basic studies on phenomena like segregation and convection. One of the most used techniques is video analysis, however it is restricted to quasi-two dimensional beds and may need considerable postprocessing time. Here we explore an alternative technique based on a simple linear capacitor array that is able to resolve variations of the solid fraction along a single direction. This method has potential to be used in real-time control applications due to the simple signal processing needed and it’s suited to probe three dimensional beds. We show a comparison between spatiotemporal density profiles of a quasi two dimensional vertically vibrated bed obtained with video and capacitive methods, and discuss on the advantages and downsides of each one. We also present spatiotemporal density profiles of a three dimensional vertically vibrated bed using the capacitive technique, illustrating the distribution of material along the direction of vibration when the bed performs bouncing motion and when the forcing is enough to obtain a granular Leidenfrost effect. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Mechanical vibrations have important industrial applications in operations dealing with particulate and powdered materials. Vibrations are mostly used to consolidate (increase bulk density) a granular sample [1], they also have important applications in sieving, milling, discharge aid [2] and can improve transport rates in emulsification, drying and agglomeration operations [3]. Some characterization tests are based on mechanical vibrations, like fluidity tests for powders [4] and tests to determine the adhesion force between particles [5]. As a consequence of the widespread industrial use of vibrations, it is important to gain knowledge about the dynamical response of a vibrated granular bed (VGB)1. Many factors have an influence on the behavior of a VGB, among others we can list: the oscillation amplitude a, the angular frequency of vibration ω, single particle properties (size, density, shape, elastic restitution coefficient, friction coefficient), bed properties (height, size ratio, bulk density, permeability) and interstitial fluid properties (density, viscosity). The forcing is usually quantified using the maximum ⁎ Corresponding author. ⁎⁎ Corresponding author. E-mail addresses: [email protected] (W. Bramer-Escamilla), [email protected] (I. Sánchez). 1 We would use the term vibrated bed, since the commonly used term vibro-fluidized bed, may overlap with the term vibrated fluidized bed, referring to a fluidized bed (by fluid injection) that is also vibrated (also called aerated vibrated bes).

http://dx.doi.org/10.1016/j.powtec.2014.10.003 0032-5910/© 2014 Elsevier B.V. All rights reserved.

dimensionless acceleration Γ = aω2/g (where titg is the acceleration of gravity), or using also the maximum velocity aω. One of the most useful parameters characterizing a granular material is its bulk density, proportional to the packing fraction. For example an adequate distribution of solids can be critical in many industrial operations, to ensure the homogeneity of the process. In a vibrated granular bed the bulk density will vary with time and with the position within the bed. If we recall the relation of bulk density with the fluidity of the granular material (dilatancy), then we can envisage the importance of knowing the bulk solids distribution, and its time evolution even within a single oscillation cycle, in order to understand phenomena like convection or segregation. In this work, we show the temporal evolution of the bulk density of a vibrated bed, measured using an array of electrical capacitance probes, moving with the container, with enough temporal resolution to appreciate variations within a single oscillation cycle. We begin by discussing general experimental methods used to study the dynamical evolution of vibrated beds in Section 2, and then in Section 3 we offer a review of previous works reporting the packing density as a function of time or space in vibrated beds. In Section 4 we describe the experimental setup and electronics of the capacitance measurement system. In Section 5 we compare spatiotemporal density profiles obtained with the capacitive sensor array and with video analysis, and discuss on the advantages and downsides of both techniques. In Section 6 we use the capacitive sensor array to illustrate different regimes shown by a VGB at different values of Γ. Finally, we offer our concluding remarks.

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2. General experimental methods for probing vertically vibrated deep beds There are plenty of different probing techniques and strategies used to measure different parameters in processes with powders or granules, for example in fluidized beds [6,7], aerated vibrated beds [8–11] or pneumatic transport [12]. In this section we will discuss exclusively those that have been used to study VGBs. Visual observation, aided by stroboscopic lighting was one of the first techniques used [13–17] to gain insight on the dynamical behavior of deep vibrated beds2. The air gap formed at the bottom of the bed is perhaps the easiest observable that can be measured, for example using high speed photography or photography with synchronized lighting [13–15] or using parallel-plate capacitors [18]. The sole presence of the air gap has been detected in a bed of conductive particles, measuring the conductivity between the bed and the floor of the container [19]. The landing phase of the bed can be determined by a kink in the signal from an accelerometer monitoring the vibration of the bed container [19,20]. The vibration of a piezoelectric disc at the bottom of the column can be used to measure the flight time of the bed, and can be related to its degree of compaction [21]. The dilation of a vibrated granular column was measured using a stroboscope and following the motion of paper lids resting on top of the column [22]. The motion of particles on the surface of the bed was monitored using a laser displacement sensor in reference [23]. Rippie et al. [24] studied the capacitive response of a vibrated bed within a single cycle of oscillation, obtaining information on the flight time and the dilation of the column. Pressure sensors positioned at the lateral walls of a VGB have been used to study the distribution of inter-particle collisions [25]. Pressure sensor located at the bottom of a VGB have been used to study the subharmonic response of the column [26]. Also pressure sensors have been used to monitor the air pressure beneath the vibrating column [27,28,21]. The force exerted by the vibrating column on the container has been measured with strain gauges [29] and with an impedance head [30]. Detailed information on the motion of the grains within the bed is hard to obtain from video or photography, since only the grains at the boundaries can be observed. Many researchers use simplified experimental configurations that allow for individual particle tracking using video analysis techniques [31–38]. Convection currents can be studied with video analysis with the help of tracers or using layers of grains dyed with different colors [27,39,35,40–43]. If the granular material is partially transparent to visible light, some optical techniques can be used. Speckle visibility spectroscopy [44] and diffusing wave spectroscopy [45–47] have been used to measure grain mobility. The transmission of diffuse [48] or laser [49] light has been used to measure the packing density of a particulate sample. Other non invasive techniques like nuclear magnetic resonance (NMR) [50–54] and positron emission particle tracking [55,56] have also been used to observe convection and to measure dynamical granular temperature profiles. NMR has the important restriction of reduced sample size, due to the small space available inside NMR spectrometer bores. In the works discussed here, the sample can fit inside a cylinder with radius no larger than 9 mm and a couple of centimeters tall. Compaction studies measuring the static packing fraction between individual mechanical perturbations have used capacitive probes [57,58], γ -ray transmission [59,60] and X-ray microtomography [61]. However these techniques were not designed to probe the granular bed during the perturbation but to obtain an overall estimation of the static density after the perturbation. Invasive probes have been used to measure grain motion within the bed. In reference [62] the displacement of a porous disc immersed inside the bed was directly compared to the displacement of the 2

We will not address the extensive work performed on vibrated monolayers and one dimensional columns of beads.

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supporting vibrating plate3. Torsional probes immersed inside the bed have been used [63–65], as well as an immersed rotating vane [66]. A rotary type viscometer was used to check the relation of fluidity to packing structure, the latter obtained from electrical resistance measurements in a bed of conductive particles [67]. Heat probes have been used to study the heat transfer properties of vibrated beds [68–70]. As a final remark for this section, let us note that previous applications of capacitance probes to VGBs have focused on determining the overall density of the bed, bed’s dilation and the size of the gap at the bottom of the bed. This is a marked difference of the literature on VGBs with that of other applications like spouted beds or fluidized beds where more elaborated capacitance probes have been applied both to make single point measurements and to map the solids distribution with electrical capacitance tomography. 3. Previous work on density profiles of vibrated granular beds In the previous section, it was shown how several different observables can be determined on a VGB. Here we focus on reported determinations of the bulk solid fraction of a VGB, with variations both in time and throughout the column. Discrete element computer simulations, have access to the bulk density with fine temporal and spatial resolution, since the positions of all grains are known at each time step. Therefore, most of the previous work reporting the evolution of the bulk density in vibrated beds corresponds to computational studies. For example, spatiotemporal density profiles help to appreciate the upward motion of a density wave [71], to illustrate the compacting effect of the interstitial gas eVladBrecha in VGBs and to show the Leidenfrost effect in wet vibrated beds [73]. Theoretical models predicting the packing distribution are usually based on the Navier-Stokes granular hydrodynamics [74,75],see for example references [76–80]. For the experimental case, monitoring the bulk density is more difficult, since the position of all the grains is not easy to determine, except in simplified configurations. To our knowledge, three techniques have been used to generate spatiotemporal density profiles of a VGB, and we discuss their observations and resolution below. In quasi-two dimensional experiments with a single vertical layer of grains (typically spheres or discs), it is possible to spot every particle using image analysis techniques [32–34]. Early works focused on the steady state behavior of a vibrated bed, and, although time averaged spatial density profiles were shown in references [32,81], they pay little or no attention to the dynamical evolution of the packing fraction. Spatiotemporal density profiles with time resolution of the order of 1/100th of an oscillation period, have been used by [82] to show the propagation of a shock wave after the impact of a VGB with the container, and by [83] to illustrate the collapse of a granular gas after the vibration is turned off. Yang and Candela [84], using NMR measured the time averaged density (over a single vibration cycle) and its variation with height and with Γ, for a small bed with three layers of mustard seeds 4. Yan et al. [50] and Huan et al. [51] used a combination of pulsed field gradient NMR and magnetic resonance imaging to measure the time dependent density. In those two works, a spatiotemporal density profile is shown to illustrate that, for the experimental configuration used, the time variation of the bed was rather small. Huntley et al. using NMR, reported a spatiotemporal density profile [52], however due to the experimental restrictions the temporal resolution was 1/12th of a vibration cycle. Caprihan et al. [54] paid special attention to image the vibrating bed at different phases of oscillation, they show a series of 24 two dimensional NMR images of the vibrated bed spanning two cycles of oscillation, with spatial resolution of 16 slices 0.4 cm wide. 3 This particular work is mainly dedicated to vibrated fluidized beds, but they discuss the case of no aeration. 4 Seeds are used because their high oil content has good magnetic contrast.

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K. Asencio et al. / Powder Technology 270 (2015) 10–19

Cx

Amplifier C-V

Signal generator

P1

Output for processing

Σ BandPass filter

Amplifier

Detector

Adder

LowPass filter

Phase Shifter

P2

Fig. 1. Block diagram of the AC measuring system used as capacitive measuring devices. See Section 4.1 for details.

An optical technique based on the measurement of the intensity of transmitted laser light (related to the packing density of a granular bed of glass spheres) was used by Matuttis et al. [49], who reported space-time density measurements with a time resolution of 1 point per cycle of oscillation, and seven locations along the whole column in the direction of oscillation. A similar approach, but with diffuse light served Ehelrs et al. [48] to generate spatiotemporal density profiles spanning a thousand cycles of oscillation with a time resolution of the order of ten time the oscillation period, showing the evolution of a binary mixture (a single sensor was moved along the column to obtain information at different heights). After reviewing experimental approaches to determine the solids distribution in VGBs, we can see that video tracking has been applied on 2D experiments and only three techniques have been applied in 3D experiments: NMR, and two optical transmission techniques using laser and diffuse light. None of these studies have focused on discussing density variations within a single oscillation cycle. The lack of interest for rapid density fluctuations may rely on the previous interest on steady state regimes, in order to compare with hydrodynamic theoretical models, like the Garzó-Dufty theory [77] or to a special interest on long time effects like the segregation process studied by Ehlers et al. [48]. However in many other situations the granular bed shows a cyclic transition from a jammed to an agitated state during a single oscillation cycle. This jammed/agitated regime may have important implications for phenomena like segregation and granular transport since it is possible that density fluctuations may shed light on the onset of flow in these processes [20,85–89]. Additionally, predictions of non-stationary models regarding shock wave propagation [76] and air effects on VGBs [72] still lack experimental confirmation on three dimensional beds. Therefore any effort to determine solids distribution with enough time resolution is worth doing. In the interest of measuring the solids distribution of VGBs, in principle NMR will be an ideal first option because it can offer 3D imaging. Besides NMR has the added benefit that it can measure velocity profiles. Another option could be the recently developed electrical capacitance volume tomography (ECVT) with reported real time 3D imaging potential [90,91]. It is much more versatile than NMR due to the fact that custom arrays of sensor can be prepared for specific applications, however, we could not find any reported applications on VGBs. The downside of tomography techniques is their cost. However if a complete 3D reconstruction is not mandatory, a much less expensive and less complicated alternative is to determine the density distribution along a single direction. In VGBs the direction of vibration represents the most important to monitor since typically forcing and gravity impose a lack of symmetry along this direction. In fact all references cited in this section on applications to VGBs, report density variations along the direction of vibration, even works using NMR tomography. Optical techniques like those of

references [48,49] represent a low cost option to map density fluctuations along a single direction, but are restricted to partially transparent grains. In this work we developed a low cost option based on an array of capacitance sensors, that may also be used with opaque grains and has enough time resolution to probe a VGB within a single cycle of vibration. 4. Methodological aspects 4.1. Capacitance sensor electronics Capacitance sensor devices are based on three different types of circuit: (1) AC measuring method (2) impedance analyzer and (3) capacitor charge/discharge method [92]. In our particular case, we decided to use the first option, using a circuit whose diagram is shown in Fig 1. The core of the circuit is the capacitance-to-voltage converter which has a relation between the input (Vi) and output (Vo) voltages given by: Vo ¼

jϖC x R f V; jϖC f R f þ 1 i

ð1Þ

where Cx and Cf are the sensing and feedback capacitances respectively, ϖ is the angular frequency injected to the excitation electrode. Since we are interested in detecting the capacitance change measured by the electrodes due to changes of the density distribution of a dielectric material between them, the circuit includes a voltage reference shifter based on a phase shifter and voltage amplifier. Our measuring apparatus uses an array of identical circuits like the one depicted in Fig. 1, one per each sensing electrode5. Due to the fact that passive components used for the circuitry have factory tolerance it is expected that the output signal of the different circuits differ among themselves. To obtain the same response to the excitation signal for each pair of electrodes and its corresponding circuit, two gain control potentiometers (P1, P2) were included (see Fig. 1). During calibration, the response of each electrode is fine tuned using those two potentiometers, in order to obtain an even response throughout the whole electrode array. 4.2. Experimental setup The vibration system is composed of an electromagnetic shaker (VTS-150 from Dynamical Solutions) driven by an audio amplifier (Crown 5000). A function generator (Tektronix AFG-3051C) is used to generate sinusoidal signals. The trunnion base of the shaker rests on a 16 mm thick aluminum sheet with three aligning screws. A 5

An alternative approach is to use a single circuit with proper multiplexation.

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piezoelectric accelerometer is used to monitor the acceleration. The signal from the accelerometer is monitored with a PC via a USB data acquisition card (NI-6009). We used two types of granular materials: Delrin spheres with diameter d = 3 mm and single grain density ρg = 1400 kg/m 3 and soda-lime glass spheres with diameter d = 1 mm and single grain density ρg = 2500 kg/m3. We used two containers in order to have a quasi-two dimensional and a three dimensional granular bed. For the quasi-two dimensional case, the container was a rectangular prism, made of 3 mm thick acrylic sheets, with an inner horizontal rectangular cross section of 3 mm× 60 mm and 200 mm tall, able to hold a single vertical layer of Delrin spheres. The initial height (at rest) of the bed was almost 30 mm, equivalent to 11 layers of grains ordered in a triangular lattice. For the three dimensional case, the container was also an acrylic rectangular prism, with an horizontal rectangular cross section of 10.1 mm × 60 mm and 200 mm tall, able to hold a column of glass spheres, 10 diameters thick, with initial height at rest of 40 mm. Anti-static cleaner was used to avoid electrostatic adhesion of particles to the container wall, it was sprayed on the inner walls of the container before each experimental run. 4.3. Capacitive technique Capacitive measurements were done using a single excitation electrode on one side of the container, and an array of 15 sensing electrodes on the opposite side (see Fig. 2). We used 15 sensing electrodes since our available data acquisition system had 16 channels (we used a pair of NI-6009 USB data acquisition boards), and one of them was used for the accelerometer signal. The acquisition frequency used was 5 kHz. Each sensing electrode is a horizontal strip spanning the whole width of the container, 3 mm (quasi-two dimensional case) or 5 mm (three dimensional case) tall with 1 mm of separation between electrodes (see Fig. 3). The complete set of sensing electrodes is surrounded by a 1 cm grounded guard. The space between electrodes has 0.3 mm tall copper strips connected to this guard. This is of key importance to reduce overlapping between the signals of nearby electrodes. The area of the excitation electrode covers the sensing electrodes and their guard. A simple calibration is done to identify the voltage of signals corresponding to an empty container and a container filled with material (poured and tapped), in order to get a linear scale between those two values. The information obtained during measurement, is a set of 15 signals indicating the overall solids fraction of the volume between each sensing electrode and the excitation electrode. 4.4. Video analysis technique Video measurements of the bed density are performed using a similar setup as in reference [83]. For the latter it is necessary to record the motion of the bed, obtaining enough frames within a single vibration cycle. Nowadays, this is typically performed with high-speed video cameras. It is possible to have a reasonable spatial pixel resolution, with an outstanding time resolution of hundreds of images within a single vibration cycle. The only downside of high-speeds cameras is their cost, and the fact that in order to average over many cycles of oscillation, a considerable shooting and post-processing time may be necessary and the storage of image data can become an issue. A cheaper option is to record the motion of the bed with stroboscopic illumination, or triggering a digital camera with the same frequency of oscillation. In this way, a single frame is obtained in each cycle and controlling the delay between the strobe or the trigger with respect to the oscillation, it is possible to obtain information of all the phases within a cycle. The information obtained will be noisy, since each phase point will correspond to a different oscillation cycle. Video measurements were performed without the electrodes, filming the container with a CMOS digital camera (Pixelink BL-741), with backlight from a halogen reflector and a diffuser. The camera frame rate was adjusted to be slightly lower (a typical

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difference of the order of 0.3%) than the oscillation frequency in order to obtain a stroboscopic effect. In this way we could record images of different phases of the oscillation (a minimum of 150 images for a single oscillation cycle, chosen to have a comparable time resolution to that of the capacitive technique). To determine the solid fraction from the images, we define observation windows spanning the whole container in a direction perpendicular to the motion of the container, and with an arbitrary height. Within each observation window, a pixel counting algorithm is used after obtaining binary images with proper thresholding, and a set of solid fractions for different heights are obtained for each image.

5. Comparison of video and capacitive techniques in a 2D bed The capacitive system described in the last sections is able to generate spatiotemporal profiles of the density of a VGB. As a way to illustrate its validity we will show density profiles obtained with video analysis and with our capacitor array. Due to the dimensional limitations of the video technique, the comparison was done for a quasi-two dimensional bed. Spatiotemporal density profiles for the quasi-two dimensional bed are presented in Fig. 4, with comparable time resolution (a minimum of 150 points within a single oscillation cycle), and imposing the same spatial resolution (choosing the size of observation windows in the video analysis, equal to the size of the measuring electrodes used for the capacitive measurements). Below each profile, the motion of the container is included as a reference guide for the time scale. The grayscale indicating the volume fraction reaches a value of 0.93, this value is larger than the two dimensional limit of ≈ 0.91 because there is a small degree of overlapping between layers of grains. Some snapshots of the vibrating container are included in Fig. 5 to illustrate the actual configuration of the column at four different phases of oscillation. The upper row of Fig. 4 shows a case with low Γ. As is well known in the literature on VGBs, the critical value of Γ for a point mass on a sinusoidally vibrating plate is Γ = 1, however, for a granular bed, the lift off value of Γ is usually larger. The bed for the case of Γ = 1.4 in Fig. 4 does not lift off, even thou the container is moving, resulting in a flat profile for both video and capacitive cases. The upper boundary is not flat in neither case since the upper layer of grains occupies a portion of the corresponding measuring window (video case) or electrode (capacitive case). The boundary for the video case occupies two measuring windows, located at heights 7 and 8, while for the capacitive case it is located at heights 7, 8, 9 and 10. The boundary is more diffuse for the capacitive case due to the unavoidable boundary effect, minimized with the use of guards but always present, that results in a small overlap of the signals of consecutive electrodes within the array. This effect lowers the capacitance signal of electrodes at heights 7 an 8 and increases the signal of electrodes at vertical positions 9 and 10. It is important to note that the lower electrode (at height 1) is not affected by the lack of material below it (it does not show a lower density than its video counterpart, as is the case for the electrode at height 7), due to the fact the guard below the electrode is larger than the rest. The center row of Fig. 4 shows a case of bouncing bed in a coherent regime6. The bed moves as a whole, bouncing on the container within a single oscillation cycle. The white space at the lower part of the profile is associated with the gap formed between the bottom of the granular column and the container’s floor. The bottom row of Fig. 4 shows a case of bouncing bed in a subharmonic coherent regime. The bed performs two consecutive flights, one larger than the other. The short flight shows a gap lower than a single particle diameter, therefore it is harder to 6 Several researchers have proposed classifications for different states observed in a VGB [93–95]. In a coherent state, most grains move in a similar way, except for those on the top of the column. This coherence is not present in other states like gasified or Newtonian [94] states, or arching states.

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Fig. 2. Side view showing the configuration of the container and the electrodes, used for measurements with the capacitive technique. All the components indicated rest on the vibrating plate. Electrodes are fixed to the container with screws. Besides the support plate attached to the vibrating plate, additional fixed mechanical support is used to reduce the stress on the connectors.

appreciate in the profiles. In fact, in this case, the capacitive technique almost does not distinguish the second flight. In order to perform a fair comparison of video and capacitive techniques, we have to look at profiles obtained from the video technique using its full resolution, something that can be appreciated in Fig. 5. The profile at the left, was obtained using video analysis, with observation windows a single pixel tall. In that case the spatial resolution allows to observe the spatial arrangement of the grains in layers, corresponding to the dark and light strips observed when the bed is in contact with the container’s bottom (at time ≈0.3 T). At a time of ≈ 0.9 T the

bed reaches its maximum height and the ordering is partially lost, until it is recovered after the bed has landed (at a time ox 1.2 T). The profile at the center was obtained using vertical strips 1 diameter tall (approximately the same size of the electrodes used for the capacitance technique). The profile at the right is obtained from the capacitance data. The profiles of the center and right of Fig. 5 are extremely similar, the main difference being that the boundaries are more diffuse for the capacitance case, a reminiscence of the boundary effect of the electrodes, as we mentioned earlier. Although video profiles have a superior spatial resolution, the capacitive profiles have superior applicability.

Fig. 3. Left: Front view of the quasi-two dimensional container, without the excitation electrode used for capacitive measurements, showing the 11 layers of Delrin spheres used for the experiments of Section 5, and the sensing electrodes surrounded by a grounded guard. Right: Angled frontal view of the three dimensional container used for experiments of Section 6 filled with 1 mm glass spheres, without the excitation electrode, showing the sensing electrodes at the back.

K. Asencio et al. / Powder Technology 270 (2015) 10–19

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Fig. 4. Comparison of spatiotemporal density profiles of a quasi-two dimensional vibrated bed obtained with video and capacitive techniques, for three different values of the maximum dimensionless acceleration Γ. Here T is the period of oscillation.

Video profiles need about a couple of hours of postprocessing. Capacitive profiles are practically real time, since the signal processing takes of the order of tens of milliseconds. Storing a large amount of images can become an issue, for example the raw image data needed to generate the profile of Fig. 4 center left, occupied more than 1 Gb. The signals for the corresponding capacitive profile needed less than 500 Kb. Video profiles can not be obtained when the sample under study has more than a single layer of grains, unless the analysis is restricted to the grains located at the boundaries.

6. Density profiles for 3D experiments In order to illustrate the capabilities of our sensor array we show spatiotemporal density profiles for a 3D granular bed of 1 mm glass beads (Fig. 6). Fig. 6A shows a bed in a coherent bouncing state, with the same periodicity of the container, while Fig. 6B shows a bed in a coherent bouncing state with twice the period of the container. From studies of a single point mass on a vibrating surface, it is known that a period doubling scenario is observed when Γ is varied. This has also been

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Fig. 5. Spatiotemporal density profiles for the quasi-two dimensional bed, illustrating different spatial resolutions. Left: profile obtained with video technique at full resolution. Images at the top row correspond to four different oscillation phases indicated by letters at the top of the profile and corresponding vertical red lines. Center: profile obtained with video technique, with spatial resolution equal to that of the capacitive technique. Right: profile obtained with the capacitive technique, where the spatial resolution is fixed by the size of the sensing electrodes, in this case, the size of a single grain.

observed for a single column of beads [96] and for quasi two dimensional beds [97,98]. The change of periodicity from the same to twice that of the container occurs for a value of Γ slightly lower than 4 in the case of a completely inelastic point mass, but shifts to higher Γ when the dissipating effect of air is taken into account [85]. In our case, our bed of 1 mm grains shows the same periodicity of the container for Γ = 4.1 and double the period of oscillation of the container for Γ = 7.3, both values shifted to higher Γ than the inelastic point mass. Although the accepted reference size below which air effects are determinant on the behavior of a VGB is ≈ 700 μm [101], our bed with grain size 1 mm is affected by air in the sense that the periodicity changes are observed for values of Γ shifted in the direction indicated by the simple model of Kroll [85]. Another important observation derived from the profile in Fig. 6A, is that the core of the bouncing bed has a relatively steady density throughout the whole period, something that can be appreciated by looking at the electrodes in positions 3, 4, and 5, whose signals remain basically unchanged, unlike the rest. Electrodes in positions 1 and 2 show variations associated with the gap at the bottom of the column and electrodes in positions higher than 5 show variations associated with the top of the column. This observation confirms a result obtained by computer simulations including the effect of air [72], where a similar granular column was vibrated at Γ = 4.7 and a bouncing regime was appreciated, with a central bulk of basically steady density, crowned by a low density region. If the value of Γ is increased to 7.3 like the profile in Fig. 6B, the core of the column shows more temporal variation than the case of Fig. 6A. In fact it is possible to appreciate diagonal bands of lower density spanning the whole column, for example near times 2 and 4, these bands are associated with the dilation of the column during its flight. The bands appear inclined to the right, indicating that the lower part of the column starts descending before the upper part, and after landing a compacting front propagates to the top of the column. These shock waves resemble previous observation in computer simulations [72,71,76,99] and in two dimensional experiments [82,100].

The profile in Fig. 6C illustrates a spontaneous change of periodicity. We can identify three low density bands spanning the whole column, near times 1, 5 and 9, indicating that the column performs bouncing with a period equal to three times the period of oscillation of the container. Suddenly, the periodicity of bed motion changes to twice the period of the container, something signaled by the low density bands appearing around times 11, 13 and 15. This change of periodicity may seem a reminiscent of the chaotic behavior shown by a point mass on a vibrating plate [102–104]. A refined study (not the object of the present work) should be performed before stating that the profile shown corresponds to a bed bouncing in a chaotic way. Finally, the profile in Fig. 6D shows a case of granular Leidenfrost effect. The profile clearly shows a stable bulk at heights 4, 5 and 6 supported by a low density zone at heights 1, 2 and 3. Both the density of the bulk and the density of the supporting low density portion are quite stable throughout the whole time window shown, spanning a little bit more than 16 cycles of the container. The low density portion is associated to a highly agitated portion of the bed that is able to support a dense bulk on top of it, like water vapor is able to isolate a drop of liquid water on a hot surface. The solid fraction measured by electrodes at heights 1 and 2 are ≈ 0.22 and 0.35 respectively, both comparable to the densities observed in the lower “gaseous” portions of the bed in three dimensional computer simulations [105,73,106]. The quasi-two dimensional experiment of reference [107] shows densities of 0.5 for the gaseous part of the column during Leidenfrost-like behavior, however a direct comparison with our case is not straightforward due to the different dimensionality. 7. Closing remarks We have presented a capacitive system capable of performing real time measurements of density variations along a single direction in vibrated granular beds. The system was validated comparing spatiotemporal density profiles generated with video analysis techniques

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Fig. 6. Spatiotemporal density profiles for a 3D granular bed of 1 mm diameter soda-lime glass spheres. A: frequency 15 Hz, amplitude 4.51 mm, Γ = 4.1;B: frequency 20 Hz, amplitude 4.52 mm, Γ = 7.3;C: frequency 30 Hz, amplitude 3.2 mm, Γ = 11.5; frequency 30 Hz, amplitude 4.5 mm, Γ = 16.4.

and obtained with the capacitive technique in a quasi-two dimensional granular bed. Both techniques show equivalent results. The capacitive technique disadvantages were a lower spatial resolution and a lower definition of the boundaries than the video analysis technique. However it offers three fundamental advantages: the ability to perform real time measurements, the ability to probe three dimensional granular beds, and its relative low cost, not only compared to the cost of high-speed video systems, but also compared with the cost of nuclear magnetic resonance (the other technique used in the literature to obtain spatiotemporal density profiles of three dimensional VGB) or electric capacitance volume tomography. These tomographic techniques may be mandatory when density variations along more than one direction are important, for example when studying subharmonic arching [54] or bunkering [108]. Spatiotemporal profiles shown in Section 6 have (up to our knowledge) no precedent in the literature and reach a level of detail similar to that shown by computer simulations. They show that the capacitive

technique is able to observe structural differences for several regimes of a vibrated granular bed. The spatial resolution of the capacitive technique is limited by the number and size of electrodes used, something that depends on the application and the resources available 7. If one of the dimensions of the electrodes is of the order of inter-electrode spacing, boundary effects will become an issue, despite the use of guards, since these guards will also be small if they are to fit between a small inter-electrode spacing. Future work may explore strategies beyond the use of guards, to minimize the overlapping of the signal between nearby electrodes. For example a future improvement could be to use different frequencies for different electrode pairs (instead of using a single excitation electrode with a single frequency).

7 For example, a rather small capacitance tomograph has been developed to monitor spray coming out of an inhaler [109].

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K. Asencio et al. / Powder Technology 270 (2015) 10–19

With the configuration used here the capacitive technique is able to give information with enough temporal resolution as to appreciate density variations within a single cycle of oscillation. Besides it has enough spatial resolution as to appreciate phenomena previously appreciated only in computer simulations or single vertical layers, such as the observation of the granular Leidenfrost effect, and the presence of a dense bulk with rather stable density crowned by a low density region when the bed is bouncing. Future research may include the study of the effect of air on the vibrating column, in order to test previous results obtained by computer simulations [72] and the study of the motion of intruders immersed in the granular column.

Acknowledgments This work was supported by the following projects: IVIC-857, FONACIT-PEII-2011001085. We thank J.R. Darias and V. Idler for important discussions, P. Silva, J.R. Darias and C. Colonnello for logistical help and C. Mendoza for revising the manuscript.

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