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Experimental study on the dynamic characteristics of wall normal stresses during silo discharge
Powder Technology 363 (2020) 509–518
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Experimental study on the dynamic characteristics of wall normal stresses during silo discharge Xin Wang a,1, Cai Liang a,⁎,1, Xiuqi Guo a, Yangyang Chen a, Daoyin Liu a, Jiliang Ma a, Xiaoping Chen a, Haiquan An b,⁎ a b
Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing 210096, China National Institute of Clean-and-Low-Carbon Energy, Beijing 102211, China
a r t i c l e
i n f o
Article history: Received 13 August 2019 Received in revised form 9 December 2019 Accepted 7 January 2020 Available online 08 January 2020 Keywords: Bulk solid Silo Gravity discharge Wall normal stress Stress fluctuation
a b s t r a c t The characteristics and mechanisms of stress fluctuation are not fully understood, especially for the large fluctuation at the cylinder/hopper transition. In this paper, we performed wall normal stress tests in a laboratory-scale silo which consisted of the cylinder section and hopper section. The amplitude and period characteristics of wall normal stress fluctuation at the top of the hopper were investigated. The result shows that the peak of the stress fluctuation and the maximum of the fluctuation range at the hopper top can reach 1.60 times and 0.64 times of the theoretical value, respectively. The amplitude of the stress fluctuation at the hopper top is independent of the outlet diameter, while the period of the stress fluctuation decreases with the increase of the outlet diameter. Furthermore, the stress-state transition and the shear zones in bulk solids are reasonably responsible for the periodic stress fluctuation at the hopper top. © 2020 Elsevier B.V. All rights reserved.
1. Introduction Dynamic characteristics of wall normal stress often occur in silo during discharge [1–4]. It can cause the silo's structural failure and affect the stability of bulk solid handling [1,4]. Here, dynamic wall stresses were divided into two categories: stress pulsation and stress fluctuation. When the wall stress presents a periodic sharp rise and slow decline [5–7], it can be considered as stress pulsation. Its another feature was that the frequency was often far greater than 1 Hz. While the stress fluctuation showed a relatively slow rise, meanwhile the stress increased and decreased almost at the same speed. Moreover, its frequency was less than 1 Hz [8–10]. Most studies about the dynamic wall stresses were focused on the stress pulsation as it was always accompanied by the “silo music” and “silo shock” phenomenon. Kmita [11] used magneto-elastic sensors to measure the normal stresses on the silo wall. He found that the frequency of stress pulsation was regular, and the silo with the granular material can be identified as one of the self-induced vibration systems. Roberts [5] measured the wall normal stress and shear stress in a pilot scale mass-flow bin and gave the method to predict the amplitude and period of stress pulsation. He proposed the solid dilation during flow and the slip-stick motion between stored solids and silo walls ⁎ Corresponding authors. E-mail addresses: [email protected] (C. Liang), [email protected] (H. An). 1 These authors contributed equally: Xin Wang, Cai Liang.
https://doi.org/10.1016/j.powtec.2020.01.023 0032-5910/© 2020 Elsevier B.V. All rights reserved.
were responsible for the stress pulsation. In recent years, the accelerometers and numerical methods were widely used to investigate the pulsation. Wensrich [12] buried the uniaxial accelerometers in the bulk solid to get the acceleration of material during “silo shock”. The propagation of rarefaction and compression waves were observed. Wilde et al. [13] used the triaxial piezoelectric accelerometers fixed on the silo structure to measure the acceleration, and proposed that the pulsation was mainly produced by the change of the shearing direction at the hopper. Muite et al. [14] measured the accelerations both in the granular material and on the silo structure. They found that the internal slip-stick of bulk solid was a source of the pulsation. Kobyłka et al. [3] used the discrete element method (DEM) to simulate the discharge initiation of a grain silo and studied the propagation of a rarefaction–compaction wave. The results indicated that the rarefaction–compaction waves had a contribution to the stress pulsation. Wang et al. [15] predicted the dynamic pressure in hopper discharge by a finite element (FE) method, gave the conclusion that the compaction wave propagation was responsible for the stress pulsation. The stress fluctuation was often observed near the cylinder/hopper transition as the dense bulk solid discharged from the silo. The detailed investigation about its characteristic and source is relatively less than the stress pulsation. Blair-Fish and Bransby [8] investigated the flow patterns and wall stresses during the dense sand discharged from two-dimensional bin-hoppers. In their study, the shear zones (or called rupture surfaces, the zone suffering intense shear deformation) in material were observed by X-ray radiograph, and the fluctuation of wall stress with time near the bulker/hopper transition were traced. They
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Fig. 1. Size distribution of glass particles.
bin. When the bulk density was large and the cone angle of the hopper was between 65° and 75°, there were significant stress fluctuations with a long period for glass beads near the transition. Tüzün and Nedderman [9] recorded the wall stresses during the materials discharged through the bottom orifice in a tall bunker of rectangular cross-section. At the top of the stagnant zone, they observed continuous wall stresses fluctuations with a period of about 60–80 s. Moreover, at greater heights, there were only the stress pulsations caused by the slip-stick motion. Böhrnsen et al. [17] simultaneously measured the wall normal stress at the symmetrically opposite position near the bunker/hopper transition. It could be found that the periodic stresses fluctuated alternately with time, which was in agreement with the alternate formation of shear zone from the right and left [8]. Wang et al. [10] simulated the wall normal stress in a cylindrical silo with conical hopper by FE method. They found that the stress fluctuations at the verticalconverging transition had much larger amplitudes, and proposed that it may be caused by the orientation charge of the major principal stress from vertical to the converging direction. From the review of literature above, it can be found that the research on stress fluctuation is less than that of stress pulsation. Meanwhile, the study on the sources of stress fluctuation is less thorough than that of stress pulsation. As the stress fluctuations always have relatively larger amplitudes, it could cause the silo structure failure. Clearly, there is a need to research this subject more intensively. In this paper, we carried out an experimental study on the wall stress fluctuation in a laboratory-scale silo during discharge. The wall normal stresses were measured by the pressure sensors arranged in the wall at different heights. The wall stress fluctuation near the cylinder/hopper transition was investigated in particular. The amplitude and period characteristics of the stress fluctuation at the top of the hopper were present. Moreover, the effect of the outlet diameter on the amplitude and period were discussed and analyzed. Furthermore, we proposed a reasonable hypothesis for the cause of wall stress fluctuation at the hopper top by theoretical analysis. 2. Materials and experimental systems 2.1. Material properties
Fig. 2. Scanning electron micrograph of glass particles.
found that the formation and re-formation of shear zones were responsible for the fluctuation of wall stress. Moriyama and Jimbo [16] studied the effect of filling methods on the wall pressure near the transition in a
The bulk solid in the present study was composed of spherical glass particles. Fig. 1 shows the particle size distribution. The particle size ranged from 0.07 mm to 1.41 mm, and most of them were distributed around 0.84 mm. The scanning electron micrograph of the surface morphology is shown in Fig. 2. It could be found that the glass particles had a high degree of sphericity. The surfaces were smooth, no cracks
Fig. 3. Scheme of the experimental apparatus. ((1) plexiglass silo, (2) load cells, (3) steel frame, (4) ball valve, (5) pressure sensors, (6) data acquisition module, (7) strain test analyzer, (8) computer)
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and pore structure. The effective internal angle of friction, ϕ, was 26° tested by a direct shear tester (Geocomp ShearTrac II, USA). The angle of wall friction, ϕw, was 6.5° tested by the powder flow tester (Brookfield PFT, USA).
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cylinder section. In each test, the silo was first emptied completely and then filled by the rain method. The initial filling level was 50 cm apart from the cylinder/hopper transition. The silo weight and wall normal stress were beginning to be measured when the ball valve was opened.
2.2. Experiment setup 3. Results and discussion The experimental apparatus is shown schematically in Fig. 3. It mainly consists of three parts: a) silos with different hopper, b) load cells (Chaoyuan YBY-0.8, China) and pressure sensors (Chaoyuan BWM-0.02, China), c) data acquisition system. The silo was made of transparent plexiglass, and it contained the cylinder section and hopper section. The height and inside diameter of the cylinder section were 50 cm and 39 cm, respectively. Three hoppers with different outlet sizes were used in the present study. The outlet diameters were 20 mm, 32 mm and 50 mm, respectively. The apex half-angle of the hoppers was 22.5°. The silo was held upright on a steel frame through three lug-supports which fixed on the upper of the cylinder section. Three load cells were placed between the lug supports and the steel frame to measure the weight of silo during discharge. The diameter and thickness of the load cells were 30 mm and 8 mm, respectively. The full scale for each cell was 80 kg. Furthermore, twelve holes were made in the wall of the silo to arrange the pressure sensors. The detail locations of sensors for all cases are shown in Fig. 4. The diameter of the holes was 25 mm, which was consistent with that of the pressure sensors. The pressure sensors were load-cell-type soil pressure gauges. The thickness and measurement range of them were 7 mm and 0–20 kPa, respectively. The sensors' pressure-sensitive surfaces were leveled with the silo's inner wall as much as possible to reduce its impact on the flow of bulk solids. The data of load cells and pressure sensors were collected respectively by the data acquisition module (Advantech ADAM-6017, China) and the strain test analyzer (Chaoyuan YBY2001, China), finally saved in the computer. To get the initially dense bulk solids, the so-called “rain method” was used to fill the silo. The glass particles were loaded into the silo through a sieve which was fixed at the top of the
3.1. Stress distribution of powder during silo discharge The wall normal stresses of the whole discharge process were measured in the experiment. Fig. 5 shows the distribution characteristics of wall stress in the initial stage of discharge. It can be found that the peak of wall stress moves from the outlet to the cylinder/ hopper transition. This phenomenon follows the ‘stress switch’ theory put forward by Walters [18,19]. After the silo has been freshly filled from completely empty, the stress field gets into the active state. The major principal stress is approximately vertical [20], as shown in Fig. 6(a). At the beginning of discharge, the bulk solid dilates locally in the vertical direction. Hence, the vertical supporting stresses decrease. Meanwhile, the conical hopper makes the flow channel converged and leads to the compression of bulk solid in the horizontal direction. The major principal stresses at the axis turn to horizontal and form an ‘arched’ shape to support the upper bulk solid. The stress field gets into the passive state. As a result, the wall normal stresses increase and form a peak of wall stress at the front of dilation, as shown in Fig. 6(b). Moreover, the peak moves upward with the dilation front and rests at the cylinder/hopper transition [17,21]; the passive stress field is fully developed as shown in Fig. 6(c). Fig. 7 shows the wall normal stresses in the whole discharge process. It could be found that there are small-amplitude oscillations with high frequency in the stresses at all sensor locations. The slip-stick motion and compression wave propagation are responsible for the highfrequency oscillations [9,15]. It is noteworthy that there are obvious large-periodic stress fluctuations at the top of the hopper (sensor 7) in Fig. 8. The coefficient of variation was adopted to confirm the periodicity of the fluctuation. It can be calculated by: C:V: ¼
SD 100% MN
ð1Þ
where SD denotes the standard deviation of the period, MN is the mean of the period.
Fig. 4. Location of pressure sensors in the wall for all cases (mm).
Fig. 5. Wall normal stresses at the beginning of discharge (Do = 20 mm).
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Fig. 6. Development of active and passive states in silos. (Modified after Ref. [17], the solid lines represent the directions of principal stress. a) Filling conditions prevailing during initial filling of an empty silo, b) emptying conditions in the lower part of the hopper, c) emptying conditions in the entire hopper [20].)
The results show that the C.V. is less than 15% for all cases. It indicates that the stress fluctuations at the hopper top are regular. The following part of this paper will be focused on the characteristics of periodic stress fluctuations at the hopper top.
3.2. Fluctuation amplitude of the wall normal stress during discharge 3.2.1. Data selection and preprocessing The stresses in the relatively steady state were chosen to be investigated. The time ranges of extracted data for three cases were 99.6–281 s, 46.7–110.6 s and 12.5–34.7 s, respectively. All of those contained four complete fluctuation periods. The frequency characteristics of these stresses were analyzed using Fourier transformation and presented in Fig. 9. As this paper focuses on the large amplitude fluctuation of stress at the top of the hopper, the high-frequency fluctuations with small amplitude were filtered out. To retain as much fluctuation information as possible, the cutoff frequencies were set as 0.1 Hz, 0.25 Hz and 0.625 Hz, respectively. The results are shown in Fig. 10.
3.2.2. Theoretical stress prediction method Most silo design standards in the world are based on the Janssen equation [15], which derived from the equilibrium of forces on a slice element of bulk solid. Walker [22] and Walters [18,19] extended and improved the Janssen's analysis to make the theoretical predictions more accurate. As the stresses investigated were located at the hopper top and the lowering velocities of bulk solids' upper surface were constant in the relatively steady state of our experiment, the unsteady flow in the present experiment could be regarded as a quasi-static flow with varying height of the bulk solids [23]. Therefore, the theoretical method based on Walker and Walters' work which is valid for quasi-static flow was adopted in this paper, as follows [24]: The surcharge on the top of hopper can be calculated by
σ zz;Z ¼
ρb gD 4μ K Dc Z 1− exp − w wA 4μ w K wA Dc D
Fig. 7. Wall normal stress traces during discharge (Do = 20 mm).
ð2Þ
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Fig. 8. Wall normal stress traces at the top of the hopper (Do = 20 mm).
in which: K wA ¼
Dc ¼
1− sinϕ cosðω−ϕw Þ 1 þ sinϕ cosðω−ϕw Þ
pffiffiffiffiffiffiffiffiffiffi 2 1 þ sin ϕ þ 2 sinϕ 1−c h i 1 þ sin2 ϕ þ ð4=3=cÞ sinϕ 1−ð1−cÞ3=2
c ¼ tan2 ϕw = tan2 ϕ
ð4Þ
ð5Þ
sinϕw sinϕ
ω ¼ arcsin
ð3Þ
ð6Þ
where Z is the distance between cylinder/hopper transition and the equivalent material-level, ρb is the bulk density, g is the gravity acceleration, D is the diameter of the cylinder, μw is the wall friction coefficient, ϕ is the effective angle of internal friction, and ϕw is the angle of wall friction. The wall normal stress in the hopper for the passive state is given by σ w ¼ ðσ hh Þw
1 þ sinϕ cosðω þ ϕw Þ 1 − sinϕ cosðω þ ϕw þ 2α Þ
ð7Þ
in which: ðσ hh Þw ¼ Dh σ hh σ hh ¼ σ zz;Z
Dh ¼
h h0
ð8Þ
m þ
" m −1 # ρb gh h 1− m −1 h0
pffiffiffiffiffiffiffiffiffiffiffi 1 þ sin2 ϕ−2 sinϕ 1−c0 h i 1 þ sin2 ϕ−ð4=3=c0 Þ sinϕ 1−ð1−c0 Þ3=2
tan2 η c0 ¼ tan2 ϕ tanη ¼ m ¼ B¼
sinϕ sinðω þ ϕw þ 2α Þ 1 þ sinϕ cosðω þ ϕw þ 2α Þ
2B D tanα h
sinϕ sinðω þ ϕw þ 2α Þ 1− sinϕ cosðω þ ϕw þ 2α Þ
ð9Þ
3.2.3. Amplitude characteristics of the wall normal stress As the differences in the stress between the flowing and static cases are small except the region near the outlet [24]. The static theoretical value of stresses calculated from Eq. (7) can be used for the comparative analysis with the dynamic measured stresses [25]. Fig. 11 shows the characteristics of stress fluctuation on the hopper top. It can be found that the experimentally measured curve fluctuates around the theoretically predicted curve. Moreover, the linear fitting curve of stress peak values shows the declined trend with the discharging, and so does the theoretically predicted curve. Chiefly because there is an arched stress field at the top of the hopper during discharge as shown in Fig. 6(c). It carries part of the upper bulk solids weight. The weight decreases with the discharging. Therefore, the wall normal stress on the hopper top, which needed to maintain the equilibrium of forces, decreases gradually. Consequently, the linear fitting value of stress peaks decreases with the discharging. Furthermore, the comparison between the measured values and the theoretically predicted values was made, as shown in Table 1. Under three different conditions, the maximum difference between the peak stress and the theoretical value, namely ΔPmax, can reach 3.86 kPa. It is about 60% of the theoretical value. The average peak stress is about 1.3 times of the theoretical value. Fig. 12 shows the ratios of ΔPp-v to Pt,m, where the ΔPp-v denotes the difference between adjacent peaks and valleys, Pt,m denotes the mean theoretically predicted values between adjacent measured peak and valley. It could be seen that the ΔPp-v amounts to 64% of the theoretical value at most, and the average ratios under the three conditions are 43%, 40% and 47%, respectively. Furthermore, from Table 1 and Fig. 12, it can be found that the difference of fluctuation amplitude characteristics under different conditions is small within an order of magnitude. The outlet diameter has little effect on the amplitude characteristics of the stress fluctuation.
ð10Þ
ð11Þ
ð12Þ
ð13Þ ð14Þ
where h is the height of material measured vertically from the apex of hopper, h0 is the hopper height measured from the apex, and α is the hopper apex half-angle.
3.2.4. Sources of the large stress fluctuation Previous studies indicated that shear zones (or rupture surfaces) might occur in the bulk solid during discharging from silo [8,15,26–33]. It can be observed by the X-radiographic method [8] and the PIV method [33]. The schematic distribution of shear zones is present in Fig. 13. The bulk solids in the shear zones are suffered intense shear deformation and consequently dilated to a less dense. It is widely accepted that the shear zones develop from the cylinder/hopper transition [20,34]. Then, the shear zones move downwards with the bulk solid while discharge. The regions between the shear zones slide down the hopper wall as almost rigid blocks [8]. The periodic formation of shear zones is responsible for the wall stress fluctuation [8,15,17,26,33–35]. Here, Mohr's circle was also used to analyze the stresses in bulk solid. Fig. 14 shows Mohr's circles for the stresses adjacent to the wall. It can be seen that there are two intersection points of Mohr's circle with the wall yield locus (WYL). Point W and W′ correspond to active
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Fig. 10. Preprocessing of measured stress. Fig. 9. Frequency spectrum of the measured wall normal stress.
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Table 1 Comparison between the measured stress and the theoretical prediction. Do (mm)
ΔPmax (kPa)
ΔPmean (kPa)
ΔPmax/Ptheory
ΔPmean/ Ptheory
20 32 50
3.86 3.22 3.76
2.28 1.96 2.85
60% 47% 60%
30% 25% 37%
Fig. 12. Ratios of ΔPp-v to Ptheory.
Fig. 11. Amplitude characteristics of stress fluctuation at the hopper top.
Fig. 13. Distribution of shear zones. (Modified after Ref. [31]).
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Fig. 16. Period of stress fluctuation at the hopper top.
case in Fig. 14(a), the following equation can be driven, σ w0 sinω= sinϕw þ cosðω þ ϕw Þ ¼ sinω= sinϕw − cosðω−ϕw Þ σw
ð15Þ
Fig. 15 compares the ratios of the measured peak to valley and the value calculated from Eq. (15). It can be found that the ratios are smaller than the calculated value. It conforms to the state in Fig. 14(b), where the bulk solid is in a static state inside. Hence, it is consistent with the shear zones theory as shown in Fig. 13. Considering the results shown in Fig. 15 and the top of the hopper is in the transition zone of stress state shown in Fig. 6(c), the transition of stress state is a reasonable cause of the stress fluctuation. Therefore, the transition of stress state and the shear zones in bulk solids are reasonably responsible for the stress fluctuation with large amplitudes at the top of the hopper. Fig. 14. Mohr's circle for the stresses adjacent to the wall.
3.3. Fluctuation period of the wall normal stress during discharge failure and passive failure, respectively. Moreover, the Mohr's circle is tangent to the effective yield locus (EYL) in Fig. 14(a). Therefore, the bulk solid adjacent to the wall is in an incipient failure state. For the
Fig. 16 demonstrates the period of stress fluctuation in different cases. It can be found that the average period is respectively 45.35 s, 15.98 s and 5.55 s when the outlet diameter is 20 mm, 32 mm and
Fig. 15. Comparison between the ratio of peak to valley and value calculated by Eq.(15).
Fig. 17. Period versus velocity.
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2. The normal stresses at all positions along the wall fluctuate during the discharging. In particular, there is a periodic fluctuation with low frequency and high amplitude at the top of the hopper. 3. The peak of periodic stress fluctuation can reach 1.60 times of the theoretically predicted stress, and the maximum of fluctuation range is about 64% of the theoretically predicted stress. The amplitude characteristics of the stress fluctuation do not depend on the outlet diameter. 4. The period of stress fluctuation at the top of the hopper decreases with the increase of outlet diameter, and it is inversely proportional to the average velocity of bulk solid in the cylinder section during discharge. 5. The ratios of the measured stress peak to valley at the hopper top are smaller than the stress ratios of active failure to passive failure in an incipient failure state, which indicates the bulk solid adjacent to the wall is in a static state inside. The transition of stress state and the shear zones in bulk solids are reasonably responsible for the periodic stress fluctuation. Declaration of Competing Interest Fig. 18. Volume per unit period for different Do.
50 mm. The average period decreases with the increase of outlet diameter. Moreover, the average velocity of bulk solid in the cylinder during discharge, namely v, was adopted to investigate the periodic characteristics. It can be calculated by [6], Q v¼ ρB A
ð16Þ
where Q is the discharge rate, and A is the cross-sectional area of the cylindrical section. Fig. 17 illustrates the plot of the stress fluctuation period versus average velocity in the cylindrical section. The fitting curve indicates that there is a significant inverse proportional relationship between the fluctuation period and average velocity. The average downward volumes in an average period for different cases are present in Fig. 18. It can be found that the average downward volumes are almost constant for different outlet diameters. As reported by Michalowski [30], the shape of shear zones does not depend on the outlet diameter. Therefore, the slip-down volume is constant when a new shear zone forms from the cylinder/hopper transition for different outlet diameter. It is consistent with the results in Fig. 18. When the shear zones move downwards, the upper material will enter the space that has been moved, which can be considered as a refilling. It may charge the stress state and consequently cause the stress fluctuation. As the periodic formation of shear zones is responsible for the fluctuation of wall stress, the period of stress fluctuation has an inverse proportional relationship with the average velocity in the cylinder. 4. Conclusions In the present study, we performed the wall normal stress testes in a laboratory-scale silo which consisted of the cylinder section and hopper sections. The distribution of wall normal stresses and the stress fluctuation characteristics at the top of the hopper were investigated. The theoretical stress predicted by Walker and Walters's method was used to analyze the experimental data. The formation of shear zones and the Mohr's circle were adopted to discuss the source of stress fluctuation. The conclusions are as follow: 1. There is a peak of wall stress moving upward from the outlet to the cylinder/hopper transition at the beginning of discharge. Moreover, the major principal stress in the hopper switches to being nearly horizontal, and correspondingly the stress field changes from the active state to the passive state.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The present study was funded by the National Natural Science Foundation of China (51676048). References [1] J. Tejchman, G. Gudehus, Silo-music and silo-quake experiments and a numerical Cosserat approach, Powder Technol. 76 (1993) 201–212. [2] K. Grudzien, A. Romanowski, Z. Chaniecki, M. Niedostatkiewicz, D. Sankowski, Description of the silo flow and bulk solid pulsation detection using ECT, Flow Meas. Instrum. 21 (2010) 198–206. [3] R. Kobyłka, J. Horabik, M. Molenda, Numerical simulation of the dynamic response due to discharge initiation of the grain silo, Int. J. Solids Struct. 106-107 (2017) 27–37. [4] K. Wilde, J. Tejchman, M. Rucka, M. Niedostatkiewicz, Experimental and theoretical investigations of silo music, Powder Technol. 198 (2010) 38–48. [5] A.W. Roberts, Shock loads in silos -the 'Silo Quaking' problem, Bulk Solids Handl. 16 (1996) 59–70. [6] A.W. Roberts, C.M. Wensrich, Flow dynamics or ‘quaking’ in gravity discharge from silos, Chem. Eng. Sci. 57 (2002) 295–305. [7] H. B, Spannungsschwankungen im Schüttgut beim Entleeren in einem Silo, Technische Universität Braunschweig, Germany, 1998. [8] P.M. Blair-Fish, P.L. Bransby, Flow patterns and wall stresses in a mass-flow bunker, J. Eng. Ind. Trans. ASME 95 (1) (1973) 17–26. [9] U. Tüzün, R.M. Nedderman, Gravity flow of granular materials round obstacles—II: investigation of the stress profiles at the wall of a silo with inserts, Chem. Eng. Sci. 40 (1985) 337–351. [10] Y. Wang, Y. Lu, J.Y. Ooi, Finite element modelling of wall pressures in a cylindrical silo with conical hopper using an arbitrary Lagrangian–Eulerian formulation, Powder Technol. 257 (2014) 181–190. [11] J. Kmita, Silo as a system of self-induced vibration, J. Struct. Eng. 111 (1985) 190–204. [12] C. Wensrich, Experimental behaviour of quaking in tall silos, Powder Technol. 127 (2002) 87–94. [13] K. Wilde, M. Rucka, J. Tejchman, Silo music — mechanism of dynamic flow and structure interaction, Powder Technol. 186 (2008) 113–129. [14] B.K. Muite, S.F. Quinn, S. Sundaresan, K.K. Rao, Silo music and silo quake: granular flow-induced vibration, Powder Technol. 145 (2004) 190–202. [15] Y. Wang, Y. Lu, J.Y. Ooi, Numerical modelling of dynamic pressure and flow in hopper discharge using the arbitrary Lagrangian–Eulerian formulation, Eng. Struct. 56 (2013) 1308–1320. [16] R. Moriyama, G. Jimbo, The effect of filling methods on the wall pressure near the transition in a bin, Bulk Solids Handl. 5 (1985) 603–609. [17] J.U. Böhrnsen, H. Antes, M. Ostendorf, J. Schwedes, Silo discharge: measurement and simulation of dynamic behavior in bulk solids, Chem. Eng. Technol. 27 (2004) 71–76. [18] J.K. Walters, A theoretical analysis of stresses in silos with vertical walls, Chem. Eng. Sci. 28 (1973) 13–21. [19] J.K. Walters, A theoretical analysis of stresses in axially-symmetric hoppers and bunkers, Chem. Eng. Sci. 28 (1973) 779–789. [20] D. Schulze, Powders and Bulk Solids, Springer, Heidelberg, 2008.
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Nomenclature A: cross-sectional area of the cylindrical section, m2 B: parameter defined by Eq. (12) for the passive state, dimensionless c: parameter defined by Eq. (5), dimensionless
c′: parameter defined by Eq. (11), dimensionless D: diameter of cylindrical section, m Do: diameter of outlet, m Dc: distribution factor defined by Eq. (4), dimensionless Dh: distribution factor defined by Eq. (10), dimensionless f: frequency, Hz g: acceleration of gravity, m/s2 h: distance measured vertically from the apex of hopper, m h0: hopper height measured from the apex, m KwA: Janssen constant for the active state at the wall, dimensionless m⁎: parameter defined by Eq. (13), dimensionless Ptheory: theoretically predicted stress, Pa Pt,m: mean theoretically predicted stress between adjacent measured peak and valley, Pa Q: discharge rate, kg/s Tp: period, s v: average velocity of bulk solid in the cylinder, m/s Z: distance between cylinder/hopper transition and the equivalent material-level, m Greek symbols
α: hopper apex half-angle, degree μw: wall friction coefficient, dimensionless ρb: bulk density, kg/m3 σhh: normal stress at the at height-h surface in hopper, Pa (σhh)w: σhh acting on a vertical surface adjacent to the wall, Pa σw: normal stress at the hopper wall, Pa σ zz;Z : surcharge normal stress on the top of hopper, Pa ω: angle defined by Eq.(6), degree ϕ: effective angle of internal friction, degree ϕw: angle of wall friction, degree ΔPmax: maximum difference between the measured peak stress and the theoretically predicted stress, Pa ΔPmean: average difference between the measured peak stress and the theoretically predicted stress, Pa ΔPp-v: difference between adjacent peaks and valleys, Pa
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Experimental study on the dynamic characteristics of wall normal stresses during silo discharge Xin Wang a,1, Cai Liang a,⁎,1, Xiuqi Guo a, Yangyang Chen a, Daoyin Liu a, Jiliang Ma a, Xiaoping Chen a, Haiquan An b,⁎ a b
Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing 210096, China National Institute of Clean-and-Low-Carbon Energy, Beijing 102211, China
a r t i c l e
i n f o
Article history: Received 13 August 2019 Received in revised form 9 December 2019 Accepted 7 January 2020 Available online 08 January 2020 Keywords: Bulk solid Silo Gravity discharge Wall normal stress Stress fluctuation
a b s t r a c t The characteristics and mechanisms of stress fluctuation are not fully understood, especially for the large fluctuation at the cylinder/hopper transition. In this paper, we performed wall normal stress tests in a laboratory-scale silo which consisted of the cylinder section and hopper section. The amplitude and period characteristics of wall normal stress fluctuation at the top of the hopper were investigated. The result shows that the peak of the stress fluctuation and the maximum of the fluctuation range at the hopper top can reach 1.60 times and 0.64 times of the theoretical value, respectively. The amplitude of the stress fluctuation at the hopper top is independent of the outlet diameter, while the period of the stress fluctuation decreases with the increase of the outlet diameter. Furthermore, the stress-state transition and the shear zones in bulk solids are reasonably responsible for the periodic stress fluctuation at the hopper top. © 2020 Elsevier B.V. All rights reserved.
1. Introduction Dynamic characteristics of wall normal stress often occur in silo during discharge [1–4]. It can cause the silo's structural failure and affect the stability of bulk solid handling [1,4]. Here, dynamic wall stresses were divided into two categories: stress pulsation and stress fluctuation. When the wall stress presents a periodic sharp rise and slow decline [5–7], it can be considered as stress pulsation. Its another feature was that the frequency was often far greater than 1 Hz. While the stress fluctuation showed a relatively slow rise, meanwhile the stress increased and decreased almost at the same speed. Moreover, its frequency was less than 1 Hz [8–10]. Most studies about the dynamic wall stresses were focused on the stress pulsation as it was always accompanied by the “silo music” and “silo shock” phenomenon. Kmita [11] used magneto-elastic sensors to measure the normal stresses on the silo wall. He found that the frequency of stress pulsation was regular, and the silo with the granular material can be identified as one of the self-induced vibration systems. Roberts [5] measured the wall normal stress and shear stress in a pilot scale mass-flow bin and gave the method to predict the amplitude and period of stress pulsation. He proposed the solid dilation during flow and the slip-stick motion between stored solids and silo walls ⁎ Corresponding authors. E-mail addresses: [email protected] (C. Liang), [email protected] (H. An). 1 These authors contributed equally: Xin Wang, Cai Liang.
https://doi.org/10.1016/j.powtec.2020.01.023 0032-5910/© 2020 Elsevier B.V. All rights reserved.
were responsible for the stress pulsation. In recent years, the accelerometers and numerical methods were widely used to investigate the pulsation. Wensrich [12] buried the uniaxial accelerometers in the bulk solid to get the acceleration of material during “silo shock”. The propagation of rarefaction and compression waves were observed. Wilde et al. [13] used the triaxial piezoelectric accelerometers fixed on the silo structure to measure the acceleration, and proposed that the pulsation was mainly produced by the change of the shearing direction at the hopper. Muite et al. [14] measured the accelerations both in the granular material and on the silo structure. They found that the internal slip-stick of bulk solid was a source of the pulsation. Kobyłka et al. [3] used the discrete element method (DEM) to simulate the discharge initiation of a grain silo and studied the propagation of a rarefaction–compaction wave. The results indicated that the rarefaction–compaction waves had a contribution to the stress pulsation. Wang et al. [15] predicted the dynamic pressure in hopper discharge by a finite element (FE) method, gave the conclusion that the compaction wave propagation was responsible for the stress pulsation. The stress fluctuation was often observed near the cylinder/hopper transition as the dense bulk solid discharged from the silo. The detailed investigation about its characteristic and source is relatively less than the stress pulsation. Blair-Fish and Bransby [8] investigated the flow patterns and wall stresses during the dense sand discharged from two-dimensional bin-hoppers. In their study, the shear zones (or called rupture surfaces, the zone suffering intense shear deformation) in material were observed by X-ray radiograph, and the fluctuation of wall stress with time near the bulker/hopper transition were traced. They
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Fig. 1. Size distribution of glass particles.
bin. When the bulk density was large and the cone angle of the hopper was between 65° and 75°, there were significant stress fluctuations with a long period for glass beads near the transition. Tüzün and Nedderman [9] recorded the wall stresses during the materials discharged through the bottom orifice in a tall bunker of rectangular cross-section. At the top of the stagnant zone, they observed continuous wall stresses fluctuations with a period of about 60–80 s. Moreover, at greater heights, there were only the stress pulsations caused by the slip-stick motion. Böhrnsen et al. [17] simultaneously measured the wall normal stress at the symmetrically opposite position near the bunker/hopper transition. It could be found that the periodic stresses fluctuated alternately with time, which was in agreement with the alternate formation of shear zone from the right and left [8]. Wang et al. [10] simulated the wall normal stress in a cylindrical silo with conical hopper by FE method. They found that the stress fluctuations at the verticalconverging transition had much larger amplitudes, and proposed that it may be caused by the orientation charge of the major principal stress from vertical to the converging direction. From the review of literature above, it can be found that the research on stress fluctuation is less than that of stress pulsation. Meanwhile, the study on the sources of stress fluctuation is less thorough than that of stress pulsation. As the stress fluctuations always have relatively larger amplitudes, it could cause the silo structure failure. Clearly, there is a need to research this subject more intensively. In this paper, we carried out an experimental study on the wall stress fluctuation in a laboratory-scale silo during discharge. The wall normal stresses were measured by the pressure sensors arranged in the wall at different heights. The wall stress fluctuation near the cylinder/hopper transition was investigated in particular. The amplitude and period characteristics of the stress fluctuation at the top of the hopper were present. Moreover, the effect of the outlet diameter on the amplitude and period were discussed and analyzed. Furthermore, we proposed a reasonable hypothesis for the cause of wall stress fluctuation at the hopper top by theoretical analysis. 2. Materials and experimental systems 2.1. Material properties
Fig. 2. Scanning electron micrograph of glass particles.
found that the formation and re-formation of shear zones were responsible for the fluctuation of wall stress. Moriyama and Jimbo [16] studied the effect of filling methods on the wall pressure near the transition in a
The bulk solid in the present study was composed of spherical glass particles. Fig. 1 shows the particle size distribution. The particle size ranged from 0.07 mm to 1.41 mm, and most of them were distributed around 0.84 mm. The scanning electron micrograph of the surface morphology is shown in Fig. 2. It could be found that the glass particles had a high degree of sphericity. The surfaces were smooth, no cracks
Fig. 3. Scheme of the experimental apparatus. ((1) plexiglass silo, (2) load cells, (3) steel frame, (4) ball valve, (5) pressure sensors, (6) data acquisition module, (7) strain test analyzer, (8) computer)
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and pore structure. The effective internal angle of friction, ϕ, was 26° tested by a direct shear tester (Geocomp ShearTrac II, USA). The angle of wall friction, ϕw, was 6.5° tested by the powder flow tester (Brookfield PFT, USA).
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cylinder section. In each test, the silo was first emptied completely and then filled by the rain method. The initial filling level was 50 cm apart from the cylinder/hopper transition. The silo weight and wall normal stress were beginning to be measured when the ball valve was opened.
2.2. Experiment setup 3. Results and discussion The experimental apparatus is shown schematically in Fig. 3. It mainly consists of three parts: a) silos with different hopper, b) load cells (Chaoyuan YBY-0.8, China) and pressure sensors (Chaoyuan BWM-0.02, China), c) data acquisition system. The silo was made of transparent plexiglass, and it contained the cylinder section and hopper section. The height and inside diameter of the cylinder section were 50 cm and 39 cm, respectively. Three hoppers with different outlet sizes were used in the present study. The outlet diameters were 20 mm, 32 mm and 50 mm, respectively. The apex half-angle of the hoppers was 22.5°. The silo was held upright on a steel frame through three lug-supports which fixed on the upper of the cylinder section. Three load cells were placed between the lug supports and the steel frame to measure the weight of silo during discharge. The diameter and thickness of the load cells were 30 mm and 8 mm, respectively. The full scale for each cell was 80 kg. Furthermore, twelve holes were made in the wall of the silo to arrange the pressure sensors. The detail locations of sensors for all cases are shown in Fig. 4. The diameter of the holes was 25 mm, which was consistent with that of the pressure sensors. The pressure sensors were load-cell-type soil pressure gauges. The thickness and measurement range of them were 7 mm and 0–20 kPa, respectively. The sensors' pressure-sensitive surfaces were leveled with the silo's inner wall as much as possible to reduce its impact on the flow of bulk solids. The data of load cells and pressure sensors were collected respectively by the data acquisition module (Advantech ADAM-6017, China) and the strain test analyzer (Chaoyuan YBY2001, China), finally saved in the computer. To get the initially dense bulk solids, the so-called “rain method” was used to fill the silo. The glass particles were loaded into the silo through a sieve which was fixed at the top of the
3.1. Stress distribution of powder during silo discharge The wall normal stresses of the whole discharge process were measured in the experiment. Fig. 5 shows the distribution characteristics of wall stress in the initial stage of discharge. It can be found that the peak of wall stress moves from the outlet to the cylinder/ hopper transition. This phenomenon follows the ‘stress switch’ theory put forward by Walters [18,19]. After the silo has been freshly filled from completely empty, the stress field gets into the active state. The major principal stress is approximately vertical [20], as shown in Fig. 6(a). At the beginning of discharge, the bulk solid dilates locally in the vertical direction. Hence, the vertical supporting stresses decrease. Meanwhile, the conical hopper makes the flow channel converged and leads to the compression of bulk solid in the horizontal direction. The major principal stresses at the axis turn to horizontal and form an ‘arched’ shape to support the upper bulk solid. The stress field gets into the passive state. As a result, the wall normal stresses increase and form a peak of wall stress at the front of dilation, as shown in Fig. 6(b). Moreover, the peak moves upward with the dilation front and rests at the cylinder/hopper transition [17,21]; the passive stress field is fully developed as shown in Fig. 6(c). Fig. 7 shows the wall normal stresses in the whole discharge process. It could be found that there are small-amplitude oscillations with high frequency in the stresses at all sensor locations. The slip-stick motion and compression wave propagation are responsible for the highfrequency oscillations [9,15]. It is noteworthy that there are obvious large-periodic stress fluctuations at the top of the hopper (sensor 7) in Fig. 8. The coefficient of variation was adopted to confirm the periodicity of the fluctuation. It can be calculated by: C:V: ¼
SD 100% MN
ð1Þ
where SD denotes the standard deviation of the period, MN is the mean of the period.
Fig. 4. Location of pressure sensors in the wall for all cases (mm).
Fig. 5. Wall normal stresses at the beginning of discharge (Do = 20 mm).
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Fig. 6. Development of active and passive states in silos. (Modified after Ref. [17], the solid lines represent the directions of principal stress. a) Filling conditions prevailing during initial filling of an empty silo, b) emptying conditions in the lower part of the hopper, c) emptying conditions in the entire hopper [20].)
The results show that the C.V. is less than 15% for all cases. It indicates that the stress fluctuations at the hopper top are regular. The following part of this paper will be focused on the characteristics of periodic stress fluctuations at the hopper top.
3.2. Fluctuation amplitude of the wall normal stress during discharge 3.2.1. Data selection and preprocessing The stresses in the relatively steady state were chosen to be investigated. The time ranges of extracted data for three cases were 99.6–281 s, 46.7–110.6 s and 12.5–34.7 s, respectively. All of those contained four complete fluctuation periods. The frequency characteristics of these stresses were analyzed using Fourier transformation and presented in Fig. 9. As this paper focuses on the large amplitude fluctuation of stress at the top of the hopper, the high-frequency fluctuations with small amplitude were filtered out. To retain as much fluctuation information as possible, the cutoff frequencies were set as 0.1 Hz, 0.25 Hz and 0.625 Hz, respectively. The results are shown in Fig. 10.
3.2.2. Theoretical stress prediction method Most silo design standards in the world are based on the Janssen equation [15], which derived from the equilibrium of forces on a slice element of bulk solid. Walker [22] and Walters [18,19] extended and improved the Janssen's analysis to make the theoretical predictions more accurate. As the stresses investigated were located at the hopper top and the lowering velocities of bulk solids' upper surface were constant in the relatively steady state of our experiment, the unsteady flow in the present experiment could be regarded as a quasi-static flow with varying height of the bulk solids [23]. Therefore, the theoretical method based on Walker and Walters' work which is valid for quasi-static flow was adopted in this paper, as follows [24]: The surcharge on the top of hopper can be calculated by
σ zz;Z ¼
ρb gD 4μ K Dc Z 1− exp − w wA 4μ w K wA Dc D
Fig. 7. Wall normal stress traces during discharge (Do = 20 mm).
ð2Þ
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513
Fig. 8. Wall normal stress traces at the top of the hopper (Do = 20 mm).
in which: K wA ¼
Dc ¼
1− sinϕ cosðω−ϕw Þ 1 þ sinϕ cosðω−ϕw Þ
pffiffiffiffiffiffiffiffiffiffi 2 1 þ sin ϕ þ 2 sinϕ 1−c h i 1 þ sin2 ϕ þ ð4=3=cÞ sinϕ 1−ð1−cÞ3=2
c ¼ tan2 ϕw = tan2 ϕ
ð4Þ
ð5Þ
sinϕw sinϕ
ω ¼ arcsin
ð3Þ
ð6Þ
where Z is the distance between cylinder/hopper transition and the equivalent material-level, ρb is the bulk density, g is the gravity acceleration, D is the diameter of the cylinder, μw is the wall friction coefficient, ϕ is the effective angle of internal friction, and ϕw is the angle of wall friction. The wall normal stress in the hopper for the passive state is given by σ w ¼ ðσ hh Þw
1 þ sinϕ cosðω þ ϕw Þ 1 − sinϕ cosðω þ ϕw þ 2α Þ
ð7Þ
in which: ðσ hh Þw ¼ Dh σ hh σ hh ¼ σ zz;Z
Dh ¼
h h0
ð8Þ
m þ
" m −1 # ρb gh h 1− m −1 h0
pffiffiffiffiffiffiffiffiffiffiffi 1 þ sin2 ϕ−2 sinϕ 1−c0 h i 1 þ sin2 ϕ−ð4=3=c0 Þ sinϕ 1−ð1−c0 Þ3=2
tan2 η c0 ¼ tan2 ϕ tanη ¼ m ¼ B¼
sinϕ sinðω þ ϕw þ 2α Þ 1 þ sinϕ cosðω þ ϕw þ 2α Þ
2B D tanα h
sinϕ sinðω þ ϕw þ 2α Þ 1− sinϕ cosðω þ ϕw þ 2α Þ
ð9Þ
3.2.3. Amplitude characteristics of the wall normal stress As the differences in the stress between the flowing and static cases are small except the region near the outlet [24]. The static theoretical value of stresses calculated from Eq. (7) can be used for the comparative analysis with the dynamic measured stresses [25]. Fig. 11 shows the characteristics of stress fluctuation on the hopper top. It can be found that the experimentally measured curve fluctuates around the theoretically predicted curve. Moreover, the linear fitting curve of stress peak values shows the declined trend with the discharging, and so does the theoretically predicted curve. Chiefly because there is an arched stress field at the top of the hopper during discharge as shown in Fig. 6(c). It carries part of the upper bulk solids weight. The weight decreases with the discharging. Therefore, the wall normal stress on the hopper top, which needed to maintain the equilibrium of forces, decreases gradually. Consequently, the linear fitting value of stress peaks decreases with the discharging. Furthermore, the comparison between the measured values and the theoretically predicted values was made, as shown in Table 1. Under three different conditions, the maximum difference between the peak stress and the theoretical value, namely ΔPmax, can reach 3.86 kPa. It is about 60% of the theoretical value. The average peak stress is about 1.3 times of the theoretical value. Fig. 12 shows the ratios of ΔPp-v to Pt,m, where the ΔPp-v denotes the difference between adjacent peaks and valleys, Pt,m denotes the mean theoretically predicted values between adjacent measured peak and valley. It could be seen that the ΔPp-v amounts to 64% of the theoretical value at most, and the average ratios under the three conditions are 43%, 40% and 47%, respectively. Furthermore, from Table 1 and Fig. 12, it can be found that the difference of fluctuation amplitude characteristics under different conditions is small within an order of magnitude. The outlet diameter has little effect on the amplitude characteristics of the stress fluctuation.
ð10Þ
ð11Þ
ð12Þ
ð13Þ ð14Þ
where h is the height of material measured vertically from the apex of hopper, h0 is the hopper height measured from the apex, and α is the hopper apex half-angle.
3.2.4. Sources of the large stress fluctuation Previous studies indicated that shear zones (or rupture surfaces) might occur in the bulk solid during discharging from silo [8,15,26–33]. It can be observed by the X-radiographic method [8] and the PIV method [33]. The schematic distribution of shear zones is present in Fig. 13. The bulk solids in the shear zones are suffered intense shear deformation and consequently dilated to a less dense. It is widely accepted that the shear zones develop from the cylinder/hopper transition [20,34]. Then, the shear zones move downwards with the bulk solid while discharge. The regions between the shear zones slide down the hopper wall as almost rigid blocks [8]. The periodic formation of shear zones is responsible for the wall stress fluctuation [8,15,17,26,33–35]. Here, Mohr's circle was also used to analyze the stresses in bulk solid. Fig. 14 shows Mohr's circles for the stresses adjacent to the wall. It can be seen that there are two intersection points of Mohr's circle with the wall yield locus (WYL). Point W and W′ correspond to active
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Fig. 10. Preprocessing of measured stress. Fig. 9. Frequency spectrum of the measured wall normal stress.
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Table 1 Comparison between the measured stress and the theoretical prediction. Do (mm)
ΔPmax (kPa)
ΔPmean (kPa)
ΔPmax/Ptheory
ΔPmean/ Ptheory
20 32 50
3.86 3.22 3.76
2.28 1.96 2.85
60% 47% 60%
30% 25% 37%
Fig. 12. Ratios of ΔPp-v to Ptheory.
Fig. 11. Amplitude characteristics of stress fluctuation at the hopper top.
Fig. 13. Distribution of shear zones. (Modified after Ref. [31]).
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Fig. 16. Period of stress fluctuation at the hopper top.
case in Fig. 14(a), the following equation can be driven, σ w0 sinω= sinϕw þ cosðω þ ϕw Þ ¼ sinω= sinϕw − cosðω−ϕw Þ σw
ð15Þ
Fig. 15 compares the ratios of the measured peak to valley and the value calculated from Eq. (15). It can be found that the ratios are smaller than the calculated value. It conforms to the state in Fig. 14(b), where the bulk solid is in a static state inside. Hence, it is consistent with the shear zones theory as shown in Fig. 13. Considering the results shown in Fig. 15 and the top of the hopper is in the transition zone of stress state shown in Fig. 6(c), the transition of stress state is a reasonable cause of the stress fluctuation. Therefore, the transition of stress state and the shear zones in bulk solids are reasonably responsible for the stress fluctuation with large amplitudes at the top of the hopper. Fig. 14. Mohr's circle for the stresses adjacent to the wall.
3.3. Fluctuation period of the wall normal stress during discharge failure and passive failure, respectively. Moreover, the Mohr's circle is tangent to the effective yield locus (EYL) in Fig. 14(a). Therefore, the bulk solid adjacent to the wall is in an incipient failure state. For the
Fig. 16 demonstrates the period of stress fluctuation in different cases. It can be found that the average period is respectively 45.35 s, 15.98 s and 5.55 s when the outlet diameter is 20 mm, 32 mm and
Fig. 15. Comparison between the ratio of peak to valley and value calculated by Eq.(15).
Fig. 17. Period versus velocity.
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2. The normal stresses at all positions along the wall fluctuate during the discharging. In particular, there is a periodic fluctuation with low frequency and high amplitude at the top of the hopper. 3. The peak of periodic stress fluctuation can reach 1.60 times of the theoretically predicted stress, and the maximum of fluctuation range is about 64% of the theoretically predicted stress. The amplitude characteristics of the stress fluctuation do not depend on the outlet diameter. 4. The period of stress fluctuation at the top of the hopper decreases with the increase of outlet diameter, and it is inversely proportional to the average velocity of bulk solid in the cylinder section during discharge. 5. The ratios of the measured stress peak to valley at the hopper top are smaller than the stress ratios of active failure to passive failure in an incipient failure state, which indicates the bulk solid adjacent to the wall is in a static state inside. The transition of stress state and the shear zones in bulk solids are reasonably responsible for the periodic stress fluctuation. Declaration of Competing Interest Fig. 18. Volume per unit period for different Do.
50 mm. The average period decreases with the increase of outlet diameter. Moreover, the average velocity of bulk solid in the cylinder during discharge, namely v, was adopted to investigate the periodic characteristics. It can be calculated by [6], Q v¼ ρB A
ð16Þ
where Q is the discharge rate, and A is the cross-sectional area of the cylindrical section. Fig. 17 illustrates the plot of the stress fluctuation period versus average velocity in the cylindrical section. The fitting curve indicates that there is a significant inverse proportional relationship between the fluctuation period and average velocity. The average downward volumes in an average period for different cases are present in Fig. 18. It can be found that the average downward volumes are almost constant for different outlet diameters. As reported by Michalowski [30], the shape of shear zones does not depend on the outlet diameter. Therefore, the slip-down volume is constant when a new shear zone forms from the cylinder/hopper transition for different outlet diameter. It is consistent with the results in Fig. 18. When the shear zones move downwards, the upper material will enter the space that has been moved, which can be considered as a refilling. It may charge the stress state and consequently cause the stress fluctuation. As the periodic formation of shear zones is responsible for the fluctuation of wall stress, the period of stress fluctuation has an inverse proportional relationship with the average velocity in the cylinder. 4. Conclusions In the present study, we performed the wall normal stress testes in a laboratory-scale silo which consisted of the cylinder section and hopper sections. The distribution of wall normal stresses and the stress fluctuation characteristics at the top of the hopper were investigated. The theoretical stress predicted by Walker and Walters's method was used to analyze the experimental data. The formation of shear zones and the Mohr's circle were adopted to discuss the source of stress fluctuation. The conclusions are as follow: 1. There is a peak of wall stress moving upward from the outlet to the cylinder/hopper transition at the beginning of discharge. Moreover, the major principal stress in the hopper switches to being nearly horizontal, and correspondingly the stress field changes from the active state to the passive state.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The present study was funded by the National Natural Science Foundation of China (51676048). References [1] J. Tejchman, G. Gudehus, Silo-music and silo-quake experiments and a numerical Cosserat approach, Powder Technol. 76 (1993) 201–212. [2] K. Grudzien, A. Romanowski, Z. Chaniecki, M. Niedostatkiewicz, D. Sankowski, Description of the silo flow and bulk solid pulsation detection using ECT, Flow Meas. Instrum. 21 (2010) 198–206. [3] R. Kobyłka, J. Horabik, M. Molenda, Numerical simulation of the dynamic response due to discharge initiation of the grain silo, Int. J. Solids Struct. 106-107 (2017) 27–37. [4] K. Wilde, J. Tejchman, M. Rucka, M. Niedostatkiewicz, Experimental and theoretical investigations of silo music, Powder Technol. 198 (2010) 38–48. [5] A.W. Roberts, Shock loads in silos -the 'Silo Quaking' problem, Bulk Solids Handl. 16 (1996) 59–70. [6] A.W. Roberts, C.M. Wensrich, Flow dynamics or ‘quaking’ in gravity discharge from silos, Chem. Eng. Sci. 57 (2002) 295–305. [7] H. B, Spannungsschwankungen im Schüttgut beim Entleeren in einem Silo, Technische Universität Braunschweig, Germany, 1998. [8] P.M. Blair-Fish, P.L. Bransby, Flow patterns and wall stresses in a mass-flow bunker, J. Eng. Ind. Trans. ASME 95 (1) (1973) 17–26. [9] U. Tüzün, R.M. Nedderman, Gravity flow of granular materials round obstacles—II: investigation of the stress profiles at the wall of a silo with inserts, Chem. Eng. Sci. 40 (1985) 337–351. [10] Y. Wang, Y. Lu, J.Y. Ooi, Finite element modelling of wall pressures in a cylindrical silo with conical hopper using an arbitrary Lagrangian–Eulerian formulation, Powder Technol. 257 (2014) 181–190. [11] J. Kmita, Silo as a system of self-induced vibration, J. Struct. Eng. 111 (1985) 190–204. [12] C. Wensrich, Experimental behaviour of quaking in tall silos, Powder Technol. 127 (2002) 87–94. [13] K. Wilde, M. Rucka, J. Tejchman, Silo music — mechanism of dynamic flow and structure interaction, Powder Technol. 186 (2008) 113–129. [14] B.K. Muite, S.F. Quinn, S. Sundaresan, K.K. Rao, Silo music and silo quake: granular flow-induced vibration, Powder Technol. 145 (2004) 190–202. [15] Y. Wang, Y. Lu, J.Y. Ooi, Numerical modelling of dynamic pressure and flow in hopper discharge using the arbitrary Lagrangian–Eulerian formulation, Eng. Struct. 56 (2013) 1308–1320. [16] R. Moriyama, G. Jimbo, The effect of filling methods on the wall pressure near the transition in a bin, Bulk Solids Handl. 5 (1985) 603–609. [17] J.U. Böhrnsen, H. Antes, M. Ostendorf, J. Schwedes, Silo discharge: measurement and simulation of dynamic behavior in bulk solids, Chem. Eng. Technol. 27 (2004) 71–76. [18] J.K. Walters, A theoretical analysis of stresses in silos with vertical walls, Chem. Eng. Sci. 28 (1973) 13–21. [19] J.K. Walters, A theoretical analysis of stresses in axially-symmetric hoppers and bunkers, Chem. Eng. Sci. 28 (1973) 779–789. [20] D. Schulze, Powders and Bulk Solids, Springer, Heidelberg, 2008.
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Nomenclature A: cross-sectional area of the cylindrical section, m2 B: parameter defined by Eq. (12) for the passive state, dimensionless c: parameter defined by Eq. (5), dimensionless
c′: parameter defined by Eq. (11), dimensionless D: diameter of cylindrical section, m Do: diameter of outlet, m Dc: distribution factor defined by Eq. (4), dimensionless Dh: distribution factor defined by Eq. (10), dimensionless f: frequency, Hz g: acceleration of gravity, m/s2 h: distance measured vertically from the apex of hopper, m h0: hopper height measured from the apex, m KwA: Janssen constant for the active state at the wall, dimensionless m⁎: parameter defined by Eq. (13), dimensionless Ptheory: theoretically predicted stress, Pa Pt,m: mean theoretically predicted stress between adjacent measured peak and valley, Pa Q: discharge rate, kg/s Tp: period, s v: average velocity of bulk solid in the cylinder, m/s Z: distance between cylinder/hopper transition and the equivalent material-level, m Greek symbols
α: hopper apex half-angle, degree μw: wall friction coefficient, dimensionless ρb: bulk density, kg/m3 σhh: normal stress at the at height-h surface in hopper, Pa (σhh)w: σhh acting on a vertical surface adjacent to the wall, Pa σw: normal stress at the hopper wall, Pa σ zz;Z : surcharge normal stress on the top of hopper, Pa ω: angle defined by Eq.(6), degree ϕ: effective angle of internal friction, degree ϕw: angle of wall friction, degree ΔPmax: maximum difference between the measured peak stress and the theoretically predicted stress, Pa ΔPmean: average difference between the measured peak stress and the theoretically predicted stress, Pa ΔPp-v: difference between adjacent peaks and valleys, Pa