Intermittent gravity-driven flow of grains through narrow pipes
aa r X i v : . [ c ond - m a t . s o f t ] F e b Intermittent gravity-driven flow of grains throughnarrow pipes ✩ , ✩✩ Carlos A. Alvarez b , Erick de Moraes Franklin a a Faculty of Mechanical Engineering - University of Campinas - UNICAMPe-mail: [email protected] Mendeleyev, 200 - Campinas - SP - CEP: 13083-970Brazil b Faculty of Mechanical Engineering - University of Campinas - UNICAMPe-mail: [email protected] Mendeleyev, 200 - Campinas - SP - CEP: 13083-970Brazil
Abstract
Grain flows through pipes are frequently found in various settings, suchas in pharmaceutical, chemical, petroleum, mining and food industries. Inthe case of size-constrained gravitational flows, density waves consisting ofalternating high- and low-compactness regions may appear. This study in-vestigates experimentally the dynamics of density waves that appear in grav-itational flows of fine grains through vertical and slightly inclined pipes. Theexperimental device consisted of a transparent glass pipe through which dif-ferent populations of glass spheres flowed driven by gravity. Our experimentswere performed under controlled ambient temperature and relative humidity,and the granular flow was filmed with a high-speed camera. Experimentalresults concerning the length scales and celerities of density waves are pre-sented, together with a one-dimensional model and a linear stability analysis.The analysis exhibits the presence of a long-wavelength instability, with themost unstable mode and a cut-off wavenumber whose values are in agreementwith the experimental results.
Keywords:
Fine grains, gravitational flow, narrow pipes, instability, ✩ c (cid:13) ✩✩ Accepted Manuscript for Physica A, v. 465, p. 725-741,DOI:10.1016/j.physa.2016.08.071
Preprint submitted to Physica A May 4, 2018 ensity waves 2 . Introduction
Granular materials play an important role in our daily lives, for instance,arid regions occupy about 20% of Earth’s surface, the global annual produc-tion of grains and aggregates is approximately ten billion metric tons, andthe processing of granular media consumes roughly 10% of all the energy pro-duced worldwide [1]. As a result, gravitational flows of these materials arefrequently observed in nature and industry. However, the behavior of gran-ular flows is not well understood, as granular matter is a discrete mediumwhose rheology is unknown. Given the importance of granular flows, con-siderable work has been done to understand their dynamics and instabilities[2, 3, 4, 5, 6].Gravitational grain flows in pipes are common in industry. Some ex-amples are the transport of grains in the food industry, the transport ofsand in civil constructions, and the transport of powders in the chemicaland pharmaceutical industries. When the grains and the tube diameter aresize-constrained, granular flow may give rise to instabilities. These insta-bilities consist of alternating high- and low-compactness regions (regions ofhigh and low grain concentration, respectively), and are characterized by in-termittency, oscillating patterns and even blockages [7, 8, 9]. Although thisinstability may appear under vacuum conditions [10, 11], in the case of finegrains these patterns are recognized as the result of the interaction betweensmall-size falling grains and trapped air.Lee [12] investigated the density waves in granular flows through verticaltubes and hoppers using analytical techniques and numerical simulations.The author used mass and momentum equations to describe density and ve-locity fields. In the equations, the law of friction proposed by Bagnold [13]was employed and the effects of both air pressure and drag caused on grainswere neglected. For the vertical tubes, the author found that kinetic wavesexist and partially obtained a dispersion relation for the dynamic waves,which he did not solve. The numerical simulations were performed usingmolecular dynamics (MD), and the author found indications that the den-sity waves are of kinetic nature. However, because air effects (pressure anddrag) were absent in both the stability analysis and the numerical simula-tion, the results are not suitable in the case of fine grains in narrow pipes.In addition, in the case of density waves, the grains in high-density regionsare in permanent contact; therefore, MD is not an adequate method since itassumes binary and instantaneous contacts.3aafat et al. [9] studied the formation of density waves in pipes ex-perimentally. The experiments were performed in a 1 . m long tube withan internal diameter D of 2 . mm using glass splinters and glass beads withmean grain diameter d of 0 . mm to 0 . mm and 0 . mm , respectively. Theyobserved density waves for moderate grain flow rate and when the ratio be-tween the pipe and the grain diameter is 6 ≤ D/d ≤
30. Furthermore, theyproposed that the friction between the grains and the forces between thetrapped air and the grains are responsible for the density waves.Aider et al. [7] presented an experimental study of the granular flowpatterns in vertical pipes. The experiments were performed in a tube similarto that of Raafat et al. [9] using glass beads with mean diameter of 125 µm .The density variations were measured using a linear CCD (charge coupleddevice) camera with frequencies of up to 2 kHz . Aider et al. [7] observedthat the density waves consisted of high-compactness plugs ( c ≈ c is the compactness) separated by low-density regions; furthermore, thedensity waves appeared when the grain flow rate ˙ m was 1 . g/s − . g/s (oscillating waves) or 2 . g/s − g/s (propagative waves). The authors alsonoted that humidity H must be within 35% and 75%, otherwise the grainsclogged the tube due to capillary forces ( H >
H < D = 3 mm , 1 . m long) and the glass beads ( d = 125 µm glass beads) were more or less the same as those of Aider et al. [7], and alinear CCD camera was used. In addition, capacitance sensors were used tomeasure the compactness of grains at two different locations, and the pres-sure distribution was also measured. The experimental data showed thatthe characteristic length of the high-compactness regions of the density waveregime is in the order of 10 mm .Recently, Franklin and Alvarez [14] presented a linear stability analysisand experimental results for the vertical chute of grains in a narrow pipe.They found a dimensional dispersion relation to be solved numerically, andthe analysis was limited to some small ranges of grains and pipes. The exper-iments were performed in a 1 m long glass tube of 3 mm internal diameteraligned vertically, and the grains consisted of glass beads of specific mass ρ s = 2500 kg/m divided in two different populations: grains with diameterwithin 212 µm ≤ d ≤ µm and within 106 µm ≤ d ≤ µm . Franklinand Alvarez [14] reported the existence of granular plugs with length in the4ange 3 < λ/D <
11, where λ is the plug length.Numerical studies on intermittent granular flows in pipes have been car-ried out in recent years. Ellingsen et al. [15] studied the gravitational flowof grains through a narrow pipe under vacuum conditions. They performednumerical simulations based on a one-dimensional model for the granularflow where the collisions were modeled using two coefficients of restitution,one among grains and the other between the grains and the pipe walls. Anarrow pipe was assumed and periodic boundary conditions were employed.The numerical results showed that granular waves could form in the absenceof air if the dissipation caused by the collisions among the grains was smallerthan that between the grains and the walls. However, the proposed modelcannot predict the wavelength of the density waves in the presence of inter-stitial gas. Verb¨ucheln et al. [16], using particle-based numerical simulations,found that density waves depend on the mass flow rate, particle distribution,and geometrical parameters of the pipe (pipe diameter and wall roughness).The authors reported plugs moving with constant velocity along the pipe.Moreover, they observed that the plugs do not break up with the impact ofthe smaller particle groups that fall onto it. In other words, the frictionalforces that yield the arches leading to plug formation are strong enough tosustain the downward pressure on the granular column. The numerical simu-lations were conducted with different pipe diameters, but the granular plugsusually appeared when the diameter was 3 mm .The density waves appearing in gravitational flows have similarities withthe waves appearing in the vertical pneumatic conveying in dense regimes[17, 18, 19, 20]. Konrad [17] presented a review on the pneumatic conveyingof grains in horizontal and vertical tubes. For dense upward flows in verticaltubes, the author explained the formation of plugs as the interaction betweenthe pressure differences between consecutive air bubbles (low compactnessregions) and the weight of grains within the granular plug. The former isdirected upwards, whereas the latter is subjected to the Janssen effect andresults in a drag force directed downwards.Borzone and Kinzing [18] studied the formation of plugs in a vertical pipeof 25 . mm internal diameter. Coal particles with diameters in the range16 µm ≤ d ≤ µm were pneumatically conveyed, and the pressure dropand the plug length were measured. The authors reported granular plugswith lengths in the range 2 < λ/D < m long hori-5ontal section and in a 3 m long vertical section, both of 57 mm internaldiameter, using polyamide chips measuring 3 mm × mm × mm . Theymeasured the density waves with a high-speed camera and an Electrical Ca-pacitance Tomography (ECT) device. In the vertical test section, Jaworskiand Dyakowski [20] reported the existence of series of granular plugs, thelength of each plug ranging between 2 D and 4 D .The objective of the present study is to determine the wavelengths andcelerities of density waves that appear when fine grains fall through verticaland slightly inclined pipes. An experimental investigation was undertaken,using high-frequency movies to measure wavelengths and celerities. Our ex-periments were performed under controlled ambient temperature and relativehumidity, and the granular flow was filmed with a high-speed camera. More-over, this paper presents a one-dimensional flow model based on the workof Bertho et al. [21] with the inclusion of closure equations for the frictionterms, and also a linear stability analysis. The flow model is made dimension-less and the stability analysis takes into consideration the main mechanismsinvolved, namely the Janssen effect, the interaction between the grains andthe air, and gravity, and the results are then compared to the experimentaldata.The next sections describe the experimental set-up and the experimentalresults. The following sections present the physics and the main equationsof the one-dimensional model, the stability analysis of the granular flow, andthe discussion of the main results. The conclusion section follows.
2. Experimental device
The experimental device consisted of a conical hopper with an openingangle of 60 o , a reservoir, a 1.0 m long vertical glass pipe with an internaldiameter D = 3 mm (test section), and an exit valve (Fig. 1).The grains consisted of glass beads of specific mass ρ s = 2500 kg/m di-vided in two different populations, namely: grain diameters within 106 µm ≤ d ≤ µm and within 212 µm ≤ d ≤ µm . The tube was verticallyaligned (within ± o ) and both the reservoir entrance and the exit valve wereat atmospheric pressure. The temperature and the humidity measurementswere within ± ◦ C and ± igure 1: Experimental set-up the contact forces in the wave regime, we performed experiments with bothsmooth surface grains (beads used in only a few experiments) and roughsurface grains (which passed many times through the glass tube and theirsurface was roughened by collisions with the walls and with other grains).The mass flow rate was determined from the time variation of the measuredmass, using a chronometer and a balance with ± d . The measured diameters were d = 110 ± µm for the batch of grains with diameters between 106 and 212 µm , and d =225 ± µm for the batch of grains with diameters between 212 and 300 µm . As an example, Fig. 2 illustrates some images acquired by SEM, whichshows some irregularly-shaped grains.The grains falling through the transparent pipe were filmed with a high-speed camera acquiring at a maximum frequency of 1000 Hz with a 1280 px × px spatial resolution. In order to provide the necessary light for lowexposure times while avoiding beat frequencies between the light source and7a) (b)(c) (d) Figure 2: Grain images taken by SEM. (a) grain samples of 212 − µm at 100 X. (b)individual grain at 2000 X, (c) grain samples of 106 − µm at 300 X. (d) grain samplesof 212 − µm at 300 X. the camera frequency, LED (Low Emission Diode) lamps were branched to acontinuous current source. For these experiments, the camera frequency wasset to 250 Hz and 500 Hz . The density waves were experimentally obtainedwith a moderate constriction at the bottom of the tube. Figure 3 shows aschematic view of these waves, where the high- and low-compactness regionsare designated as plugs and air bubbles, respectively, and λ is the meanlength of plugs.A large number of images were acquired during the tests (in the orderof thousands); therefore, we developed an image processing code in orderto treat sequentially all the images obtained. The image processing codeidentifies the patterns in the RGB (red, green and blue) images, tracks themalong images, and calculates the lengths and the celerities of the granular8 igure 3: Schematic view of the density waves, λ is the length of the granular plug. plugs. Figure 4 illustrates three frames of a test run for which the camerafrequency was set to 250 Hz , but the time between these frames is 0.06 s .It was observed that the celerity of density waves was negative (contrary togranular flow direction). Subsection 2.1 describes briefly the image processingcode developed in MatLab environment. An image processing code was developed during this work to automati-cally treat the images obtained. Therefore, MatLab scripts were written inorder to automatically identify the granular plugs in the RGB images, tofollow them along the images, and to compute the lengths and celerities ofgranular plugs. The following is a brief explanation of the code.Each image, which corresponds to a movie frame, is acquired in RGBformat during the tests and it is saved as a matrix. The first steps of thecode are to open sequentially each one of these matrices, and to load scalingand threshold information, related to the image and light calibrations (inorder to convert pixels to mm, a calibration image was acquired before each9 igure 4: Images acquired during the tests showing density waves with negative celerity.The time between frames is 0.06 s . test). Next, in each image the code detects the regions that contain thegranular flow, and that is limited by the tube walls; therefore, the code onlyconsiders information belonging to this region. In the following, each imageis converted to grayscale, and then to binary scale. Next, with the smallestgranular plug assumed to scale with the tube diameter, the code searches thegrain regions with lengths greater than 3 mm , and these are considered asthe granular plugs. Once the plugs have been identified, the code saves theirupper and bottom positions in a matrix. Finally, the mean celerity of plugs isdetermined by calculating the slope of their upper positions, the mean lengthof plugs is determined by calculating the difference between their upper andbottom positions, and spatiotemporal diagrams were constructed. Figure 5shows an example of plugs identified by the code for one image. From left10o right, Fig. 5 shows the image in RGB format, the image converted tograyscale, the image converted to binary format, and the identified upperand bottom positions of the plugs, marked with asterisks. The main steps ofthe code are shown below in a schematic form.1. Entry:(a) Image(b) Frequency(c) Scale(d) Threshold2. Loop (until the total number of images is achieved):(a) Read image(b) Select image area that contains the grain flows(c) Convert RGB image to grayscale image(d) Convert grayscale image to binary image(e) Search areas greater than the tube diameter(f) Show images while they are computing(g) Store granular plug positions at different times3. Post-processing:(a) Computation of the mean celerity of granular plugs v p (b) Computation of the mean length of granular plugs λ
4. Graphical display:(a) Plot the upper and bottom plug positions as a function of time(spatiotemporal diagram)(b) Plot the mean length of plugs λ as a function of mass flow rate ˙ m (c) Plot the mean celerity of plugs v p as a function of mass flow rate˙ m
3. Experimental results
Density waves were observed as soon as the granular flow began. Thedensity waves had positive or negative mean celerity, and they sometimespresented an oscillatory component. Aider et al. [7] proposed that differentcelerity behaviors are due to different granular flow rates and humidity.Two wave flow regimes were identified based on spatiotemporal diagrams.The first corresponding to density waves that propagated at a constant celer-ity (known as propagative wave regime), and the second corresponding to11 igure 5: Example of plugs identification by the code. From left to right: image in RGBformat, image converted to grayscale, image converted to binary format, and identifiedthe upper and bottom plug positions, marked with asterisks. oscillating waves, with low and large amplitude oscillations. Previous works[7, 8] have reported that density waves appeared when the mass flow rate ˙ m was within 1.5 − g/s , (with plug compactness typically within 55 − m values ten times smaller than those previously reported, andslightly smaller relative humidity values than those previously reported.In our opinion, the relatively small mass flow rates observed in our studyare due to two factors. The first is associated to the relative humidity in ourtests. We observed density waves at relative humidity values smaller thanthose reported in previous works. In addition, in our experiments the relativehumidity was continuously controlled, and this parameter was kept constantduring each test (Tabs. 1, 2). The second is associated to the geometry andsuperficial roughness of the grains used. The images taken by SEM allowedto observe that some grains were not perfect spheres (Figs. 2c, 2d), and thatsome grains contained little incrustations on their surface (Fig. 2b).12n order to determine the mean celerity v p and the mean length λ of gran-ular plugs, we constructed spatiotemporal diagrams. These diagrams showthe plug positions as a function of time; therefore, v p can be obtained fromthe mean slope of the plug positions, and λ can be obtained by subtractingthe upper and bottom positions of each plug. This flow regime appeared with smooth and rough surface grains, anddiameters within 106 and 212 µm . In all tests within this diameter range,the density waves propagated upward, i.e., contrary to flow direction, whichaccording to our coordinate reference system represented a negative celerity.Figure 6 displays two spatiotemporal diagrams in the case of rough surfacegrains. The diagrams in Fig. 6 are representative of all tests with grains inthis diameter range.The vertical axis corresponds to the distances measured along the pipe,increasing from top to bottom, and the horizontal axis corresponds to time,increasing from left to right. The height of the field of view is 237 mm .The top of this figure is located at 30 cm downstream of the tube’s upperend. The literature reports that between 7 and 20 cm is required for fullydeveloped plugs to build up [7, 22]. The variations as functions of time ofthe upper and bottom plug positions are shown by black squares and bluecircles, respectively. The mean celerity of plugs v p , can be determined by theslope obtained for the upper zone. The slope is constant for a plug and alsofrom one plug to another, then v p can be easily measured. In addition, v p varies as a function of the mass flow rate ˙ m , ˙ m = 0.55 g/s for Fig. 6.Figure 7 presents a spatiotemporal diagram for which the top is locatedat 45 cm downstream of the tube’s upper end, for the case of smooth surfacegrains. From Figs 6 and 7, we conclude that v p is somewhat independentof the vertical position of the analyzed field, i.e., all captured plugs insidethe diagrams have the same mean celerity v p ; therefore, we can assert that v p is constant along the tube. This is in agreement with Aider et al. [7],who also reported that v p is constant once a stationary propagation regimehas been reached. In order to verify if the observed plugs were moving ina stationary regime, we analyzed spatiotemporal diagrams constructed overdifferent time intervals during each test run. The two diagrams presentedin Fig. 6 belong to the same test. The test starts in Fig. 6a, the durationis almost one second, and after 2.4 seconds (not shown in Fig. 6), the test13 .3 1.4 1.5 1.6 1.7 1.8 1.9 200.050.10.150.2 Time (s) P o s i t i on ( m ) (a) Time (s) P o s i t i on ( m ) (b) Figure 6: Spatiotemporal diagrams corresponding to propagative wave regime for roughsurface grains. The upper and bottom plug positions are shown by black squares andblue circles, respectively. The mean celerity of plugs v p is shown in the dashed line, v p =-0.0538 m/s, ˙ m = 0.55 g/s. P o s i t i on ( m ) Figure 7: Spatiotemporal diagram corresponding to propagative wave regime for smoothsurface grains. The upper and bottom plug positions are shown by black squares andblue circles, respectively. The mean celerity of plugs v p is shown in the dashed line, v p =-0.0408 m/s, ˙ m = 0.40 g/s. Spatiotemporal diagrams using different time intervals for the same testruns were constructed for both rough and smooth surface grains. The resultswere always similar, with granular plugs propagating upward with a constantmean celerity v p , and the mean celerity was negative for all the tests, i.e., theplugs always propagated upward, contrary to the flow direction. In previousworks [7, 8, 9], the authors reported that the plugs propagated downward,in same direction of the flow. This difference may be related to the differentenvironmental conditions and the mass flow rates used in our tests. In thestudies mentioned in this paper, the mass flow rates were ten times greaterthan our rates, and the relative humidity was slightly larger.The spatiotemporal diagrams in Figs. 6 and 7 show that the bottom ofplugs, identified by blue circles in the diagrams, exhibit strong curvatures fol-lowed by stronger slopes in some regions. This suggests that in these regionssome grains are accelerated by gravity as they fall out of a plug, and into the15ubble immediately below with a free fall motion. The curvature then givesthe accelerating transient and the stronger slope the free fall velocity. Thegrains then leave the bubble, and are slowed down when they reach the upperboundary of the next plug. Figure 8 shows an enlarged view of a spatiotem-poral diagram for which v p = -0.0366 m/s and ˙ m = 0.37 g/s. In this figure,the strong curvatures followed by stronger constant slopes (continuous linesdrawn in Fig. 8) are highlighted. From the constant slope lines, the grains’velocity was computed to be in the order of 0.8 m/s , which corresponds tothe free fall velocity in the bubble region. For comparison, we estimatedthe final velocity by assuming free fall in the bubble and neglecting the airresistance. This gives approximately 0.9 m/s , which is in agreement withthe velocity obtained from the spatiotemporal diagram. Figure 8: Enlarged view of a spatiotemporal diagram. The mean celerity of plugs v p =-0.0366 m/s and the mass flow rate ˙ m = 0.37 g/s. Table 1 illustrates the main results obtained for the propagative waveregime. For each test run, Tab. 1 presents the mass flow rate ˙ m , the roomrelative humidity H , the surface state of grains, the mean celerity v p , andthe mean length of plugs λ . The results of Tab. 1 are presented in Figs. 9and 10 in order to investigate the relation between the mean celerity of plugsand the mass flow rate, and between the mean length of plugs and the massflow rate. Previous works have shown that the mean celerity of plugs has alinear dependence on the mass flow rate [7, 8, 9]. Therefore, we investigatedif there would be linear dependence in our experiments, despite the upward16otion of plugs. run ˙ m H surface v p λ · · · g/s % · · · m/s cm Table 1: ˙ m , H , surface state, v p , λ for each test run in the propagative wave regime Figure 9 presents the variation of v p (absolute values) as a function of ˙ m .The celerity of plugs increases linearly with ˙ m and ranges between -0.0366and -0.0963 m/s . The linear dependency of v p with ˙ m is in agreement withthat reported in the literature [7, 8, 9]. In addition, Fig. 9 shows that thecelerity of smooth-grained plugs is larger than the celerity of coarse-grainedplugs.Figure 10 presents the length of plugs λ as a function of the mass flowrate ˙ m , and it shows that λ is almost independent of ˙ m . The mean size ofplugs obtained from the spatiotemporal diagrams was λ ≈ D . In addition,the length of the plugs appears to be independent of the surface state of thegrains. 17 .4 0.6 0.8 10.030.040.050.060.070.080.090.1 | v p | ( m / s ) ˙ m (g/s) Figure 9: Variation of mean celerity of plugs v p as a function of mass flow rate ˙ m in thepropagative wave regime. The symbols (cid:13) and △ correspond to tests using rough grainsand smooth grains, respectively, and the continuous and dashed lines represent linearfittings. The oscillating regime appeared when the experiments were conductedusing a batch of rough-surface glass spheres and diameters ranging between212 and 300 µm . The mass flow rates were between 0.25 and 0.4 g/s . Inthis regime, it was observed that the density waves oscillated over a non-zerodrift celerity, which can be considered as the mean celerity of plugs v p . Theplugs always propagated downward, in the same direction as the granularflow, and v p was constant along the tube.Density waves oscillating at low amplitude appeared in some tests. Fig-ure 11 displays spatiotemporal diagrams with these characteristics. The fre-quency of the observed oscillations was 7 Hz . Figures 11a and 11b correspondto the diagrams obtained with values very close to the superior and inferiorlimits of ˙ m determined for this wave regime. These figures show that theoscillating frequency is independent of the mass flow rate.Table 2 illustrates the values of the mass flow rate ˙ m , the room relativehumidity H , the mean celerity v p , and the mean length of plugs λ , for each18 .3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1012345 ˙ m (g/s) λ ( c m ) Figure 10: Mean length of plugs λ as a function of mass flow rate ˙ m in the propagativewave regime. The symbols ♦ and ▽ correspond to the tests using smooth grains and roughgrains, respectively. The size of the plugs is approximately 30 mm ( λ ≈ D ). test run, measured for density waves oscillating at low amplitude.The results of Tab. 2 are shown in Figs. 12 and 13 in order to investigatethe relation between the mean celerity of plugs and the mass flow rate, andbetween the mean length of plugs and the mass flow rate. Figure 12 presentsthe mean celerity of plugs v p as a function of mass flow rate ˙ m for the lowamplitude oscillating waves, and shows that there is an approximate lineardependence of v p on ˙ m . The range of v p values is within 0.2 − m/s ,which is considerably lower than previously reported results; however, thelinear dependence remains. Figure 13 displays the mean length of plugs λ asa function of mass flow rate ˙ m in the oscillating wave regime, and shows that λ is independent of the mass flow rate, with its value of approximately 10 D .We note that the propagative regime presented identical results; therefore,we conclude that the mean length of plugs is independent of both the massflow rate and the wave regime.The granular flow surrounded by air is a two phase flow system, and someanalogies with other two phase systems can be explored, as, for example, air-water flows. These systems are characterized by high compressibility (due to19 P o s i t i on ( m ) (a) P o s i t i on ( m ) (b) Figure 11: Spatiotemporal diagram of low amplitude oscillating waves. The upper andbottom plug positions are shown by black squares and blue circles, respectively. Thedashed line corresponds to the mean drift celerity v p . (a) v p = 0.0250 m/s , ˙ m = 0.26 g/s .(b) v p = 0.0501 m/s , ˙ m = 0.37 g/s . The frequency of oscillations is in the order of 7 Hz . m H v p λ · · · g/s % m/s cm Table 2: ˙ m , H , v p , λ for each test run, in the case of density waves oscillating at lowamplitude air bubbles) and high density ratios (due to mean density of the plugs com-pared to air). The sound velocity computed for those systems is generallylow, in the order of a few m/s , resulting in low resonance frequencies. Usingsome approximations, it is possible to calculate a resonance frequency equiv-alent to 4 Hz in this case. This value is in accordance with the frequencycalculated only using the spatiotemporal diagrams (7 Hz ), indicating thatthe mechanism of acoustic resonance may be related to the plugs length.In the oscillating wave regime, oscillating flows with amplitudes slightlylarger than those observed in Fig. 11 were also detected. The frequency ofthe waves was in the order of 4 Hz . Figure 14 shows a typical spatiotemporaldiagram corresponding to this situation, which shows that the mean celerityof plugs v p , represented by the dashed line, is constant and positive (plugspropagate downward).The amplitude of the oscillations suggests that the granular plugs main-tain static contact with the tube walls for some time, and afterward theystart to move again, in a typical stick-slip movement. The friction forces21 .25 0.3 0.35 0.40.010.020.030.040.050.060.07 | v p | ( m / s ) ˙ m (g/s) Figure 12: Mean celerity of plugs v p as a function of mass flow rate ˙ m in the oscillatingwave regime. The continuous line corresponds to a linear fitting. between the plugs and the tube walls must be large enough to balance theweight of the grains and cause the plugs to stop for some time. Under theseconditions, there is a redirection of forces within the plug, and the Jansseneffect [23] is expected. Franklin and Alvarez [14], using the Janssen’s theory,calculated the mean size of a compact plug and reported the length of plugswas in the range 3 < λ/D <
11, which is in accordance with the results ofthe present study. 22 .25 0.3 0.35 0.4012345 ˙ m (g/s) λ ( c m ) Figure 13: Mean length of plugs λ as a function of mass flow rate ˙ m in the oscillatingwave regime. The size of the plugs is of approximately 30 mm ( λ ≈ D ). P o s i t i on ( m ) Figure 14: Spatiotemporal diagram of large amplitude oscillating waves. Black squaresand blue circles correspond to the upper and bottom plug positions, respectively. Thedashed line corresponds to the mean drift celerity v p , and the mean values are v p = 0.0190 m/s , ˙ m = 0.25 g/s . The frequency of oscillations is in the order of 4 Hz . . Gravity-driven flow of grains through pipes: a one-dimensionalmodel The analyzed problem consists of cohesionless fine grains falling from ahopper through a narrow tube. The ratio between the tube diameter andthe mean grain diameter is within 6 ≤ D/d ≤
30, the humidity is within35 < H < − o ≤ θ ≤ o , where θ is the pipe inclination with respectto the gravitational acceleration. Within this scope, density waves consistingof alternating high- and low- compactness regions are expected [7, 8]. Inthe high-concentration regions, which are plugs of granular material, thecompactness varies but is close to its maximum value, and grains in the plugperiphery are in contact with the tube wall. Therefore, there is a redirectionof forces within the plug and the Janssen effect is expected if the plugs arelong enough [1, 24]. In the low-concentration regions, which are air bubbleswith dispersed free-falling grains, the air pressure increases owing to thestresses caused by the neighboring plugs as well as the volume decrease causedby the free-falling grains. Figure 15 shows the layout of the gravitationalgranular flow.A one-dimensional model is proposed for this problem. The model con-sists of an equation of motion for the grains in a compact plug (or a compactregime), an air pressure equation, and a mass conservation equation of thegrains. These equations, displayed below, are used in the stability analysis inSection 5. The stability analysis is performed to determine the length scaleof the granular plugs. The mass transport equation of the grains is given by Eq. 1. ∂c∂t + c ∂v s ∂z + v s ∂c∂z = 0 (1)where c is the compactness of granular plugs, v s is the local grain velocity, z is the vertical coordinate, and t is the time. Normalizing Eq. 1 by thecharacteristic length L c = D , time t c = p D/g , and velocity v c = √ gD , weobtain Eq. 2: ∂c∂t ∗ + c ∂v ∗ s ∂z ∗ + v ∗ s ∂c∂z ∗ = 0 (2)25 igure 15: Layout of the gravitational granular flow through a narrow pipe. z is thevertical coordinate, λ is the length of the granular plugs, and θ is the pipe inclination withrespect to the gravitational acceleration ~g . where z ∗ = z/D , t ∗ = t/ p D/g , v s ∗ = v s / √ gD . The equation of motion for the grains in compact regime is given by abalance between the grains acceleration, the friction between the grains andthe tube wall, the forces due to air pressure and granular tension distribution,and the weight. This balance is given by Eq. 3, ρ s (cid:18) ∂cv s ∂t + ∂cv s ∂z (cid:19) = ρ s cg cos θ − ∂P∂z − ∂σ zz ∂z − D σ zr (3)where ρ s is the specific mass of each grain, g is the gravitational acceleration, P is the air pressure, σ zz is the vertical stress operating on the grains, and26 zr is the stress between the tube wall and the grains. We use here theclosure of σ zz and σ zr proposed by Franklin and Alvarez [14]. The first oneis to take into account the redirection of forces through a constant coefficient(dimensionless) [1] κ : σ zr = µ s κσ zz , where µ s ≈ tan(32 o ) is the frictioncoefficient between the grains, and the grains and the pipe walls. The secondis to model σ zr as a function of the square of the grains velocity: σ zr =1 / ρ s µ s v s . The third is to consider that capillary forces can be modeled asa multiplicative constant b ≥ b = 1 and did not change it.The resulting equation is ∂cv s ∂t + ∂cv s ∂z = c g cos θ − ρ s ∂P∂z − v s κ ∂v s ∂z − D µ s v s (4)Normalizing Eq. 4 by the characteristic length L c , time t c , velocity v c and pressure P c = ρ s gD , we obtain Eq. 5 c ∂v ∗ s ∂t ∗ + v ∗ s (cid:18) ∂c∂t ∗ + v ∗ s ∂c∂z ∗ + c ∂v ∗ s ∂z ∗ (cid:19) + cv ∗ s ∂v ∗ s ∂z ∗ = c cos θ − ∂P ∗ ∂z ∗ − v ∗ s κ ∂v ∗ s ∂z ∗ − µ s ( v ∗ s ) (5)In Eq. 5 we identify the normalized mass conservation equation (Eq. 2);therefore, with an additional simplification, we obtain Eq. 6. c ∂v ∗ s ∂t ∗ + cv ∗ s ∂v ∗ s ∂z ∗ − c cos θ + ∂P ∗ ∂z ∗ + v ∗ s κ ∂v ∗ s ∂z ∗ + 2 µ s ( v ∗ s ) = 0 (6)where P ∗ = P/ ( ρ s gD ) For the air pressure, an equation based on the work of Bertho et al. [21]is used. Bertho et al. [21] combined the mass conservation equations forthe air and grains, the isentropic relation for the air, and Darcy’s equationrelating the air flow through packed grains to the pressure gradient to obtainEq. 7 ∂P∂t + v s ∂P∂z + γP (1 − c ) ∂v s ∂z − B ∂ P∂z = 0 (7)where γ is the ratio of specific heats (1 . B is a coefficient givenby 27 = γP (1 − c ) d µ a c (8)where µ a is the dynamic viscosity of air. In Eq. 8, B was obtained byestimating the permeability of grains using the Carman–Kozeny equation.Normalizing Eq. 7 by the characteristic length L c , time t c , velocity v c andpressure P c , we obtain Eq. 9 ∂P ∗ ∂t ∗ = − v ∗ s ∂P ∗ ∂z ∗ − γP ∗ (1 − c ) ∂v ∗ s ∂z ∗ + B ∗ ∂ P ∗ ∂z ∗ (9)where B ∗ = Bg − / D − / is a dimensionless coefficient.
5. Stability analysis
A linear stability analysis is presented based on Eqs. 2, 6 and 9, which aresolved for P ∗ , c and v ∗ . The main objective is to find the typical length for thehigh-density regions of the granular flow. The initial state, considered as thebasic state, is a steady, dense uniform flow of grains known as compact regime[7]. This state is then perturbed and we investigate if a preferential modeexists, i.e., we investigate if an initial compact regime will be fractured ingranular plugs with a preferential wavelength. Thus, the analysis considers abasic state in which the pressure is equal to the characteristic pressure, P c , thegrain velocity is equal to the characteristic velocity v c , and the compactnessis equal to an average dense compactness c (where, in compact regime c ≈ . O (1), and the perturbation, of O ( ǫ ), ǫ ≪
1. In dimensionlessform: P ∗ = 1 + ˜ P , v ∗ s = 1 + ˜ v s , c = c + ˜ c (10)where ˜ P ≪
1, ˜ v ≪ c ≪ O ( ǫ ), we obtain ∂ ˜ c∂t ∗ + c ∂ ˜ v s ∂z ∗ + ∂ ˜ c∂z ∗ = 0 (11)28 ∂ ˜ v s ∂t ∗ + c ∂ ˜ v s ∂z ∗ − ˜ c cos θ + ∂ ˜ P∂z ∗ + 1 κ ∂ ˜ v s ∂z ∗ + 4 µ s ˜ v s = 0 (12) ∂ ˜ P∂t ∗ = − ∂ ˜ P∂z ∗ − γ (1 − c ) ∂ ˜ v s ∂z ∗ + B ∗ ∂ ˜ P∂z ∗ (13)where B ∗ is a constant obtained by replacing P by P c in B ∗ . Equations11, 12 and 13 form a linear system with constant coefficients; therefore, thesolutions can be found by considering the following normal modes:˜ c = ˆ c e i ( k ∗ z ∗ − ω ∗ t ∗ ) + c.c. ˜ v s = ˆ v s e i ( k ∗ z ∗ − ω ∗ t ∗ ) + c.c. ˜ P = ˆ P e i ( k ∗ z ∗ − ω ∗ t ∗ ) + c.c. (14)where k ∗ = kD = 2 πD/λ ∈ R is the dimensionless wavenumber in the z ∗ direction, λ is the wavelength in the z ∗ direction, ˆ c ∈ C ˆ v ∈ C and ˆ P ∈ C are the dimensionless amplitudes, and c.c. stands for the complex conjugate.Let ω ∗ ∈ C , ω ∗ = ω ∗ r + iω ∗ i , where ω ∗ r = ω r / ( kv c ) ∈ R is the dimensionlessangular frequency and ω ∗ i = ω i t c ∈ R is the dimensionless growth rate. Byinserting the normal modes in Eqs. 11, 12 and 13, we obtain Eq. 15. ik ∗ − iω ∗ ik ∗ c − cos θ − iω ∗ c + ik ∗ c + κ − ik ∗ + 4 µ s ik ∗ γ − c ik ∗ − iω ∗ + ik ∗ + B ∗ k ∗ ˆ c ˆ v s ˆ P == (15)The existence of non-trivial solutions for this system requires its deter-minant to be zero. This results in 29 c ω ∗ + ω ∗ [ k ∗ ( − ic − iκ − ) − k ∗ c B ∗ − µ s ] ++ ω ∗ [ k ∗ (2 c B ∗ + κ − B ∗ ) + ik ∗ (2 κ − − γ − c + 3 c − µ s B ∗ )+ k ∗ ( c cos θ + 8 µ s )]++ ik ∗ (cid:16) − c − κ − + 4 µ s B ∗ + c cos θ B ∗ + γ − c (cid:17) ++ k ∗ ( − µ s − c cos θ ) + B ∗ k ∗ ( − κ − − c ) = 0 (16)Equation 16 is solved to find ω ∗ ( k ∗ ). In order to solve it, constant b wasassumed to be equal to 1. Constant B ∗ was computed using the characteristicvalues of our experiments, i.e., D = 3 mm , d = d = 0 . mm (the lattervalue obtained by SEM), and we assumed that µ s = tan(32 o ), κ ≈ .
5, and c ≈ .
55 [1, 24]. Additionally, we assumed θ ≤ o . The imaginary partof ω ∗ ( k ∗ ), ω ∗ i ( k ∗ ), was investigated in order to obtain the typical length ofgranular plugs.Figure 16 shows the dimensionless growth rate ω ∗ i as a function of di-mensionless wavenumber k ∗ . The continuous line corresponds to one rootof Eq. 16, the dashed-dot line and the dashed line to the others. Figure16a, for the broad range of wavenumbers, shows that small wavelengths arestable. Figure 16b illustrates the 0 . kD . . k ∗ ≈ . k ∗ ≈ .
6. This corresponds to wavelengths in the order of10 D .
6. Discussion
These experiments and those of Franklin and Alvarez [14] showed thatthe plug sizes are 3 < λ/D <
11, which is in perfect agreement with theproposed model. However, as only one tube diameter was employed, we thencompare these results with the previously published results.In a series of papers, Raafat et al. [9], Aider et al. [7], and Bertho et al. [8]presented experiments of granular flows through a tube. In particular, withregard to the characteristics of density waves, Raafat et al. [9] reported the30 * ω i * ω i,1* ω i,2* ω i,3* (a) * ω i * (b) Figure 16: Dimensionless growth rate ω ∗ i as a function of dimensionless wavenumber k ∗ .The continuous line corresponds to one root of Eq. 16, the dashed-dot line and the dashedline to the others. λ/D ≈
10 and that it was approximately independent of theflow rate. Bertho et al. [8] also reported that the size of plugs was λ/D ≈ λ bubble /D ≈
10. Thesemeasurements are in agreement with the lengths predicted by the proposedmodel.Because the dense waves appearing in the vertical chute of grains in nar-row tubes share similarities with the waves appearing in the vertical pneu-matic conveying in dense regime, comparisons with some experiments onpneumatic conveying are presented next. Borzone and Kinzing [18] reportedplugs with 2 < λ/D <
15 while Jaworski and Dyakowski [20] reported theexistence of series of granular plugs, where each plug has a length between2 D and 4 D , in agreement with the proposed model.The results of the model are in agreement with the results reported byFranklin and Alvarez [14], even though in the present study the compactness c was assumed as a variable, different from that work, where the authorsfixed c as constant. In addition, through θ it is possible to consider smalldeviations of the tube with respect to the vertical alignment. This allows totake into consideration small angle variations that may have occurred in thecited works.Peng and Herrmann [25], using lattice-gas automata, studied densitywaves in vertical pipes. The authors reported that both the energy dissipa-tion and the roughness of the walls of the pipe are essential to the formationof density waves. However, they did not measure the typical length of granu-lar plugs. Poschell [22] investigated the formation of density waves using 2DMolecular Dynamics simulations. The author found that the kinetic energyof the falling grains increases up to a characteristic threshold, from whichdensity waves of no definite wavelength appear. Ellingsen et al. [15] studiedthe gravitational flow of grains through a narrow pipe under vacuum condi-tions. Their numerical results showed that granular waves could form in theabsence of air if the dissipation caused by the collisions among the grains wassmaller than that between the grains and the walls. However, the proposedmodel cannot predict the wavelength of the density waves in the presence ofinterstitial gas. Lately, Verb¨ucheln et al. [16], using particle-based numericalsimulations, studied a method to homogenize granular flows in order to avoidthe formation of density waves. They did not report the value of the ratiobetween the plug length and the tube’s diameter, λ/D ; however, in theirsimulations the granular plugs usually appeared when the diameter was 3mm. 32he present model is based on the work of Bertho et al. [21], with theinclusion of closure equations for the friction terms. Bertho et al. [21] solvednumerically their model equations and found that the formation of a decom-paction wave was captured by the model; however, the numerical simulationsdid not explain the length of plugs that may appear in narrow pipes. Dif-ferent from Bertho et al. [21], we included closure equations that allowed usto perform a linear stability analysis, which showed the existence of a mostunstable mode. This analysis explains the typical length of plugs appearingin narrow pipes.As far as we know, Lee [12] performed the only stability analysis previousto our work. In his analysis, Lee neglected air effects (pressure and drag);therefore, the analysis was not able to find the correct length scale of plugs.Different from Lee [12], we considered air effects, and the present stabilityanalysis predicts a wavelength that agrees with typical lengths observed indifferent experiments.The present analysis considers only one median diameter as the scale forthe grain size. Although dispersion around the median diameter is admitted,it is expected that the analysis may fail in the case of large dispersions aroundthe medium value. If this is the case, other scales for the grain size maybe necessary. In addition, cohesion among grains may be present. In thisproblem, cohesion has two sources: (i) capillarity, when air relative humidityis high, commonly above 70%, and (ii) electrostatic forces, when air relativehumidity is low, commonly below 30%. While the model considers capillarityvia a multiplicative factor (Subsection 4.2), it does not consider electrostaticforces. The present paper is concerned with air relative humidity between30% and 70%, so that these cohesion effects are not very strong; however,we understand that further investigations on cohesion forces, as well as onpolydisperse grains, are necessary in order to fully understand the problem.We note that this study is interested in the formation and dynamics ofgranular plugs. However, the formation of plugs must be avoided in many in-dustrial applications. In these cases, the formation of plugs can be preventedby controlling the air relative humidity or the flow rate of gains.The final observation concerns the lowest plug in the gravitational denseflow. Bertho et al. [8] reported that at the lower portion of the tube (tubeexit) a different plug is formed. The length of this plug varies with theflow rate. For ˙ m from 1 . g/s to 3 . g/s , they found that the length of thebottom plug varies from λ/D ≈
30 to λ/D ≈
7. Conclusions
This paper focused on the density waves that appear when fine grainsfall through a narrow tube. Its main objective was to experimentally deter-mine the wavelengths and celerities of density waves that appear when finegrains fall through vertical and slightly inclined pipes. In our experiments,different sized glass spheres flowed driven by gravity through a vertical glasspipe, and the granular flow was filmed with a high-speed camera. The am-bient temperature and relative humidity were controlled. Both the meancelerity and wavelengths were calculated using an image processing code de-veloped in MatLab environment. When varying the mass flow rate, two waveregimes occurred: the density waves either propagated at a constant celerityor oscillated over a mean drift celerity. The oscillations suggest a stick-slipdisplacement of the plugs. The density waves appeared with mass flow ratesbetween 0.1 and 0.95 g/s , ten times smaller than previously reported values.The paper also presented a stability analysis based on equations proposedby Bertho et al. [21], with small modifications and in dimensionless form.In our analysis, the basic state is a compact regime, and we investigated ifit would be fractured in granular plugs with a preferential wavelength. Thestability analysis predicts a wavelength in the order of 10 D for the high-density regions. This predicted length scale is in good agreement with ourexperimental results as well as with previously published results.
8. Acknowledgments
Carlos A. Alvarez is grateful to the Ecuadorian government foundationSENESCYT for the scholarship grant (no. 2013-AR2Q2850). Erick deMoraes Franklin is grateful to FAPESP (grant no. 2012/19562-6), to CNPq(grant no. 471391/2013-1) and to FAEPEX/UNICAMP (conv. 519.292,project AP0008/2013) for the financial support provided.341] J. Duran, Sands, powders and grains: an introduction to the physics ofgranular materials, 2nd Edition, Springer, 1999.[2] C. Campbell, Granular material flows - an overview, Powder Technology162 (3) (2006) 208 – 229.[3] H. Elbelrhiti, P. Claudin, B. Andreotti, Field evidence for surface-wave-induced instability of sand dunes, Nature 437 (04058).[4] E. M. Franklin, Linear and nonlinear instabilities of a granular bed:determination of the scales of ripples and dunes in rivers, Appl. Math.Model. 36 (2012) 1057–1067.[5] GDR-MiDi, On dense granular flows, The European Physical Journal E14 (4). doi:10.1140/epje/i2003-10153-0 .[6] H. M. Jaeger, S. R. Nagel, Physics of the granular state, Science255 (5051) (1992) 1523–1531.[7] J.-L. Aider, N. Sommier, T. Raafat, J.-P. Hulin, Experimental study ofa granular flow in a vertical pipe: A spatiotemporal analysis, Phys. Rev.E 59 (1999) 778–786.[8] Y. Bertho, F. Giorgiutti-Dauphin´e, T. Raafat, E. J. Hinch, H. J. Her-rmann, J. P. Hulin, Powder flow down a vertical pipe: the effect of airflow, J. Fluid Mech. 459 (2002) 317–345.[9] T. Raafat, J. P. Hulin, H. J. Herrmann, Density waves in dry granularmedia falling through a vertical pipe, Phys. Rev. E 53 (1996) 4345–4350.[10] S. B. Savage, Gravity flow of cohesionless granular materials in chutesand channels, J. Fluid Mech. 92 (1979) 53–96.[11] C. H. Wang, R. Jackson, S. Sundaresan, Instabilities of fully developedrapid flow of granular material in channel, J. Fluid Mech. 342 (1997)179–197.[12] J. Lee, Density waves in the flows of granular media, Phys. Rev. E 49(1994) 281–298. 3513] R. A. Bagnold, Experiments on a gravity-free dispersion of large solidspheres in a newtonian fluid under shear, Proc. R. Soc. London Ser. A225 (1160) (1954) 49–63.[14] E. M. Franklin, C. A. Zambrano, Length scale of density waves in thegravitational flow of fine grains in pipes, J. Braz. Soc. Mech. Sci. Eng.37 (5) (2015) 1507–1513. doi:10.1007/s40430-014-0291-3doi:10.1007/s40430-014-0291-3