- Email: [email protected]
The influence of flow rate on the decrease in pressure beneath a conical pile
Powder Technology 212 (2011) 296–298
Contents lists available at ScienceDirect
Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Short Communication
The influence of flow rate on the decrease in pressure beneath a conical pile Jianguo Liu, Qicheng Sun ⁎, Feng Jin State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 8 February 2011 Received in revised form 29 April 2011 Accepted 12 May 2011 Available online 18 May 2011 Keywords: Granular materials Sandpile Stress depression Tactile pressure sensing
a b s t r a c t The decrease in the stress at the bottom of a sandpile exhibits the preparation dependence known to occur for granular materials. By using a grid-based tactile pressure sensor, we carefully measured the progressive development of the pressure profile at the bottom of conical piles formed by funneling rice grains onto the sensor. A significant decrease in the stress was observed at small flow rates, but it gradually disappeared at greater flow rates. This result is helpful in understanding the stress distribution within sandpiles. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The study of stress distributions in granular materials has been a subject of research in the engineering community for many years [1]. In the past decades, the dependence of granular stresses on the preparation of granular materials has inspired an upsurge of interest [2]. One typical example is that the pressure distribution below sandpiles, instead of always displaying a single central peak, may show a dip when the pile is formed by funneling the grains onto the peak. The reason for the dip in the stress profile of a sandpile remains unknown. It might be interpreted that strong force chains are oriented preferentially in the direction of the slope such that arches are formed, shielding the center from some of the weight [3]. Recent numerical studies further revealed the links between the pressure dip and the contact orientation and contact force orientation distributions [4]. The analysis using the theory of granular elasticity indicated that the dip could be produced if the center was less compact [5]. Many experimental methods have been proposed for measuring the normal forces on the surface of the base plate, such as capacitive normal stress sensors, carbon marks, and highly precise electronic balances. See Refs. [4–9] for more details. A careful analysis requires quantitative information about the progressive evolution of the pressure distribution beneath sandpiles; however, it is difficult to obtain such information experimentally at this stage of development. By using the grid-based tactile pressure-sensing technology developed by TEKSCAN Company, this difficulty could be overcome.
⁎ Corresponding author. Tel.: + 86 10 62796574; fax: + 86 10 62773576. E-mail address: [email protected] (Q. Sun). 0032-5910/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2011.05.009
A sensor is made with a thin (~ 0.1 mm thick) matrix-based sensor consisting of two flexible polyester sheets with silver conductive electrodes printed on them. Such are able to measure stresses at a large number of points in close proximity, thus allowing for a realistic normal stress distribution at a sampling frequency up to 200 Hz. The use of the sensors with granular materials was first reported in Ref. [10], which included the procedure for calibration analysis and measured the stress distribution that develops on the boundary between an interfacial shear device and the sensor. The method was also used for measuring the soil stress distribution in dynamic loadings and for studying the effect of grain size on the stress distribution measurements along a boundary with a solid surface [11,12]. The present experiments were carried out to explore the effect of the flow rate used to create the conical pile, as shown in Fig. 1 (left). A funnel containing 3 kg of rice grains was placed at a fixed height above the sensor. The grains discharged at different flow rates through different sizes of a round opening. The pressure distribution was acquired, and the effect of the flow rate on the formation of a pressure dip was analyzed. 2. Experimental setup As shown in Fig. 1 (middle), the sensing region dimensions of the sensor (model 5315) were 488 × 427 mm. The sensor contains 48 columns and 42 rows, resulting in 2,016 sensels spaced every 10 mm in each direction. The standard pressure range is 34.47 kPa for this sensor. By using the default sensitivity (64/255 of the standard), the real pressure range would be 8.65 kPa. The resolution of the sensor is 1/255. A 1.5 cm rigid float glass plate was placed under the sensor. At the bottom of the funnel, the diameter D of a small opening was 1, 1.5, 1.7, 2, 2.2 or 2.5 cm. The distance from the funnel bottom to the sensor
J. Liu et al. / Powder Technology 212 (2011) 296–298
297
Fig. 1. Formation of a pile as grains flow from a funnel onto the sensor (left). Top view of a conical rice pile (middle). Distribution of the measured weight (right).
was kept at 11 cm. A total of 3 kg of rice was discharged from the funnel. The weight distribution was acquired at a sampling frequency of 5 Hz as a function of time and quantified across all or part of the sensor, as shown in Fig. 1 (right). Before the experiments, an equilibration procedure was conducted to confirm that each sensel had a uniform output from a uniform load and was correct for uneven output. We used a large silicon bag filled with approximately 100 kg of water to apply a uniform pressure to the entire area of the sensor. The following calibration determined the relationship between the rice weight (2.0 kg, 3.1 kg) and the impedance of the loaded sensels. The dimension of a rice grain is approximately 2.5 mm × 1.8 mm × 0.8 mm. The friction coefficient between rice grains and the sensor was approximately 0.33. 3. Result analysis and discussion Fig. 2 depicts the measured weight W of rice grains discharged on the sensor. For D = 1.5 cm and time t b 145 s, the grains were continuously falling and W increased approximately linearly with a flow rate Q of approximately 20.1 g/s. During the progressive formation of the pile, successive small avalanches occurred, resulting in fluctuations of W. We also noticed that the time difference between avalanches tended to increase when the rice pile became larger, for example, when increasing from approximately 5 s at t b 100 s to approximately 15 s at 100 s b t b 145 s. At t = 145 s, all grains had been discharged, and the instant value of W was measured to be approximately 2.9 kg ( ~ 3% different from the real weight of 3 kg), but it was able to increase to 3.0 kg by t = 155 s because of the relaxation of the sensels. We acquired the distribution of W at this time. It should be pointed out that the sensor exhibits drift with an applied constant load. At the end of grains falling (t N 155 s), the measured weight kept increasing, and the increase is an approximate range between 0% and 3% per log time (in a worst-case scenario). For the discharging process at t b 145 s, Q could be well fit as a linear function of D 2, Q = 14.79D 2 − 14.48, as shown in the inset of Fig. 2. For D b 1 cm, the flow was blocked by a few grains arched around the opening.
Fig. 2. The measure weight of rice grains on the sensor during pile formation.
Fig. 3. Evolution of the pressure distribution during a test. The scattered points are the experimental data, and solid curves represent the averaged value.
Fig. 3 shows the evolution of the pressure profile as the pile is progressively formed. The scattered points are the experimental data, and the solid curves represent the averaged value. The distance from the center of the pile is r. For t = 10–11 s, the pile is relatively thin, and the impact force of falling grains on the sensor causes a larger pressure near r = 0. The dip in pressure becomes clear until t = 40–41 s. As the time increases, the pressure at approximately r = 4 cm increases faster in comparison with the pressure at r = 0, so a dip gradually formed. These results are consistent with the experimental observations from the large-scale conical stockpile tests and recent simulations [4]. Fig. 4 shows the pressure distributions as the measured weight increased to the real weight of 3 kg, and it is fitted with a third-order polynomial function, shown as a solid line. r is scaled by R, which is the final distance from the center axis for the conical piles. The pressure P at the distance r is scaled by the averaged hydrostatic pressure at the base, Mg/(πR 2), where M = 3 kg. The error bars represent the deviation of several runs, typically 5. A clear pressure minimum can be seen at r/R = 0. The pressure maximum occurs at a position of r/R ≈ 0.3, which agrees reasonably well with previous conical pile data. For D = 1 cm, Q = 20.1 g/s, P = 1.12 at r/R = 0.3 and P = 0.73 at r/R = 0, that is, the value is approximately 35% lower. These values are within a reasonable range [4,6]. One important observation would be that the pressure dips varies at different Q. For D = 2.5 cm and Q = 81.3 g/s, we find P = 1.08 at r/R = 0.3 and P = 1.02 at r/R = 0, that is, a larger flow rate may produce a central peak of pressure. We imagine that the rain-like method of preparing a pile could be treated as a funneling method with a very large opening.
Fig. 4. Fitted normalized pressures at the bottom of a pile versus the horizontal distance to the center. In the inset, the pressure distribution for D = 1.5 com is shown as circles with error bars, and it is fitted with a third-order polynomial function that is shown as a solid line.
298
J. Liu et al. / Powder Technology 212 (2011) 296–298
In summary, many research studies have focused on the effect of pile size and funnel height on the decrease in the stress. Our work specifically explored the effect of flow rate for a fixed funnel height above the conical base. A pressure dip at the center of the pile was observed as the flow rate became smaller, and the dip gradually disappeared as the flow rate increased. To better understand the phenomenon of pressure dip, the properties of both the contact structure and the force network should be carefully studied at the particle scale. Supplementary materials related to this article can be found online at doi:10.1016/j.powtec.2011.05.009. Acknowledgements The authors acknowledge the support of the National Key Basic Research Program of China (No. 2010CB731504) and the research funding from the State Key Laboratory of Hydroscience and Engineering, Tsinghua University (Nos. 2008-ZY-6; 2010-TC-1). References [1] L.D. Landau, E.M. Lifshitz, Theory of elasticity, 3rd ed. Pergamon, New York, 1986.
[2] A. Watson, Physics: searching for the sand pile pressure dip, Science 273 (1996) 579–580. [3] J.P. Wittmer, M.E. Cates, P. Claudin, J.-P. Bouchaud, An explanation for the central stress minimum in sandpiles, Nature 382 (1996) 336–338. [4] J. Ai, J.F. Chen, J.M. Rotter, J.Y. Ooi, Numerical and experimental studies of the base pressures beneath stockpiles, Granular Matter 12 (2010) 1-9-9. [5] D.O. Krimer, M. Pfitzner, K. Brauer, Y. Jiang, M. Liu, Granular elasticity: general considerations and the stress dip in sand piles, Phys. Rev. E 74 (2006) 061310. [6] L. Vanel, D. Howell, D. Clark, R.P. Behringer, E. Clement, Memories in sand: experimental tests of construction history on stress distributions under sandpiles, Phys. Rev. E 60 (1999) R05040. [7] G. Løvoll, K.J. Måløy, E.G. Flekkøy, Force measurements on static granular materials, Phys. Rev. E 60 (1999) 5872–5878. [8] D.L. Blair, N.W. Mueggenburg, A.H. Marshall, H.M. Jaeger, S.R. Nagel, Force distributions in three-dimensional granular assemblies: effects of packing order and interparticle friction, Phys. Rev. E 63 (2004) 041304. [9] A.P.F. Atman, P. Brunet, J. Geng, G. Reydellet, P. Claudin, R.P. Behringer, E. Clement, From the stress response function (back) to the sand pile ‘dip’, Eur. Phys. J. E. 17 (2005) 93–100. [10] S.G. Paikowsky, E.L. Hajduk, Calibration and use of grid-based tactile pressure sensors in granular material, A.S.T.M. Geotechnical Testing J. 20 (1997) 218–241. [11] S.G. Paikowslcy, C.J. Palmer, L.E. Rolwes. The use of tactile sensor technology for measuring soil stress distribution. GeoCongress 2006: Geotechnical Engineering in the Information Technology Age. 1–6. [12] J.C. Stith, Railroad track pressure measurements at the rail/tie plate interface using Tekscan sensors, Master's thesis, Department of Civil Engineering, University of Kentucky. (2004).
Contents lists available at ScienceDirect
Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Short Communication
The influence of flow rate on the decrease in pressure beneath a conical pile Jianguo Liu, Qicheng Sun ⁎, Feng Jin State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 8 February 2011 Received in revised form 29 April 2011 Accepted 12 May 2011 Available online 18 May 2011 Keywords: Granular materials Sandpile Stress depression Tactile pressure sensing
a b s t r a c t The decrease in the stress at the bottom of a sandpile exhibits the preparation dependence known to occur for granular materials. By using a grid-based tactile pressure sensor, we carefully measured the progressive development of the pressure profile at the bottom of conical piles formed by funneling rice grains onto the sensor. A significant decrease in the stress was observed at small flow rates, but it gradually disappeared at greater flow rates. This result is helpful in understanding the stress distribution within sandpiles. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The study of stress distributions in granular materials has been a subject of research in the engineering community for many years [1]. In the past decades, the dependence of granular stresses on the preparation of granular materials has inspired an upsurge of interest [2]. One typical example is that the pressure distribution below sandpiles, instead of always displaying a single central peak, may show a dip when the pile is formed by funneling the grains onto the peak. The reason for the dip in the stress profile of a sandpile remains unknown. It might be interpreted that strong force chains are oriented preferentially in the direction of the slope such that arches are formed, shielding the center from some of the weight [3]. Recent numerical studies further revealed the links between the pressure dip and the contact orientation and contact force orientation distributions [4]. The analysis using the theory of granular elasticity indicated that the dip could be produced if the center was less compact [5]. Many experimental methods have been proposed for measuring the normal forces on the surface of the base plate, such as capacitive normal stress sensors, carbon marks, and highly precise electronic balances. See Refs. [4–9] for more details. A careful analysis requires quantitative information about the progressive evolution of the pressure distribution beneath sandpiles; however, it is difficult to obtain such information experimentally at this stage of development. By using the grid-based tactile pressure-sensing technology developed by TEKSCAN Company, this difficulty could be overcome.
⁎ Corresponding author. Tel.: + 86 10 62796574; fax: + 86 10 62773576. E-mail address: [email protected] (Q. Sun). 0032-5910/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2011.05.009
A sensor is made with a thin (~ 0.1 mm thick) matrix-based sensor consisting of two flexible polyester sheets with silver conductive electrodes printed on them. Such are able to measure stresses at a large number of points in close proximity, thus allowing for a realistic normal stress distribution at a sampling frequency up to 200 Hz. The use of the sensors with granular materials was first reported in Ref. [10], which included the procedure for calibration analysis and measured the stress distribution that develops on the boundary between an interfacial shear device and the sensor. The method was also used for measuring the soil stress distribution in dynamic loadings and for studying the effect of grain size on the stress distribution measurements along a boundary with a solid surface [11,12]. The present experiments were carried out to explore the effect of the flow rate used to create the conical pile, as shown in Fig. 1 (left). A funnel containing 3 kg of rice grains was placed at a fixed height above the sensor. The grains discharged at different flow rates through different sizes of a round opening. The pressure distribution was acquired, and the effect of the flow rate on the formation of a pressure dip was analyzed. 2. Experimental setup As shown in Fig. 1 (middle), the sensing region dimensions of the sensor (model 5315) were 488 × 427 mm. The sensor contains 48 columns and 42 rows, resulting in 2,016 sensels spaced every 10 mm in each direction. The standard pressure range is 34.47 kPa for this sensor. By using the default sensitivity (64/255 of the standard), the real pressure range would be 8.65 kPa. The resolution of the sensor is 1/255. A 1.5 cm rigid float glass plate was placed under the sensor. At the bottom of the funnel, the diameter D of a small opening was 1, 1.5, 1.7, 2, 2.2 or 2.5 cm. The distance from the funnel bottom to the sensor
J. Liu et al. / Powder Technology 212 (2011) 296–298
297
Fig. 1. Formation of a pile as grains flow from a funnel onto the sensor (left). Top view of a conical rice pile (middle). Distribution of the measured weight (right).
was kept at 11 cm. A total of 3 kg of rice was discharged from the funnel. The weight distribution was acquired at a sampling frequency of 5 Hz as a function of time and quantified across all or part of the sensor, as shown in Fig. 1 (right). Before the experiments, an equilibration procedure was conducted to confirm that each sensel had a uniform output from a uniform load and was correct for uneven output. We used a large silicon bag filled with approximately 100 kg of water to apply a uniform pressure to the entire area of the sensor. The following calibration determined the relationship between the rice weight (2.0 kg, 3.1 kg) and the impedance of the loaded sensels. The dimension of a rice grain is approximately 2.5 mm × 1.8 mm × 0.8 mm. The friction coefficient between rice grains and the sensor was approximately 0.33. 3. Result analysis and discussion Fig. 2 depicts the measured weight W of rice grains discharged on the sensor. For D = 1.5 cm and time t b 145 s, the grains were continuously falling and W increased approximately linearly with a flow rate Q of approximately 20.1 g/s. During the progressive formation of the pile, successive small avalanches occurred, resulting in fluctuations of W. We also noticed that the time difference between avalanches tended to increase when the rice pile became larger, for example, when increasing from approximately 5 s at t b 100 s to approximately 15 s at 100 s b t b 145 s. At t = 145 s, all grains had been discharged, and the instant value of W was measured to be approximately 2.9 kg ( ~ 3% different from the real weight of 3 kg), but it was able to increase to 3.0 kg by t = 155 s because of the relaxation of the sensels. We acquired the distribution of W at this time. It should be pointed out that the sensor exhibits drift with an applied constant load. At the end of grains falling (t N 155 s), the measured weight kept increasing, and the increase is an approximate range between 0% and 3% per log time (in a worst-case scenario). For the discharging process at t b 145 s, Q could be well fit as a linear function of D 2, Q = 14.79D 2 − 14.48, as shown in the inset of Fig. 2. For D b 1 cm, the flow was blocked by a few grains arched around the opening.
Fig. 2. The measure weight of rice grains on the sensor during pile formation.
Fig. 3. Evolution of the pressure distribution during a test. The scattered points are the experimental data, and solid curves represent the averaged value.
Fig. 3 shows the evolution of the pressure profile as the pile is progressively formed. The scattered points are the experimental data, and the solid curves represent the averaged value. The distance from the center of the pile is r. For t = 10–11 s, the pile is relatively thin, and the impact force of falling grains on the sensor causes a larger pressure near r = 0. The dip in pressure becomes clear until t = 40–41 s. As the time increases, the pressure at approximately r = 4 cm increases faster in comparison with the pressure at r = 0, so a dip gradually formed. These results are consistent with the experimental observations from the large-scale conical stockpile tests and recent simulations [4]. Fig. 4 shows the pressure distributions as the measured weight increased to the real weight of 3 kg, and it is fitted with a third-order polynomial function, shown as a solid line. r is scaled by R, which is the final distance from the center axis for the conical piles. The pressure P at the distance r is scaled by the averaged hydrostatic pressure at the base, Mg/(πR 2), where M = 3 kg. The error bars represent the deviation of several runs, typically 5. A clear pressure minimum can be seen at r/R = 0. The pressure maximum occurs at a position of r/R ≈ 0.3, which agrees reasonably well with previous conical pile data. For D = 1 cm, Q = 20.1 g/s, P = 1.12 at r/R = 0.3 and P = 0.73 at r/R = 0, that is, the value is approximately 35% lower. These values are within a reasonable range [4,6]. One important observation would be that the pressure dips varies at different Q. For D = 2.5 cm and Q = 81.3 g/s, we find P = 1.08 at r/R = 0.3 and P = 1.02 at r/R = 0, that is, a larger flow rate may produce a central peak of pressure. We imagine that the rain-like method of preparing a pile could be treated as a funneling method with a very large opening.
Fig. 4. Fitted normalized pressures at the bottom of a pile versus the horizontal distance to the center. In the inset, the pressure distribution for D = 1.5 com is shown as circles with error bars, and it is fitted with a third-order polynomial function that is shown as a solid line.
298
J. Liu et al. / Powder Technology 212 (2011) 296–298
In summary, many research studies have focused on the effect of pile size and funnel height on the decrease in the stress. Our work specifically explored the effect of flow rate for a fixed funnel height above the conical base. A pressure dip at the center of the pile was observed as the flow rate became smaller, and the dip gradually disappeared as the flow rate increased. To better understand the phenomenon of pressure dip, the properties of both the contact structure and the force network should be carefully studied at the particle scale. Supplementary materials related to this article can be found online at doi:10.1016/j.powtec.2011.05.009. Acknowledgements The authors acknowledge the support of the National Key Basic Research Program of China (No. 2010CB731504) and the research funding from the State Key Laboratory of Hydroscience and Engineering, Tsinghua University (Nos. 2008-ZY-6; 2010-TC-1). References [1] L.D. Landau, E.M. Lifshitz, Theory of elasticity, 3rd ed. Pergamon, New York, 1986.
[2] A. Watson, Physics: searching for the sand pile pressure dip, Science 273 (1996) 579–580. [3] J.P. Wittmer, M.E. Cates, P. Claudin, J.-P. Bouchaud, An explanation for the central stress minimum in sandpiles, Nature 382 (1996) 336–338. [4] J. Ai, J.F. Chen, J.M. Rotter, J.Y. Ooi, Numerical and experimental studies of the base pressures beneath stockpiles, Granular Matter 12 (2010) 1-9-9. [5] D.O. Krimer, M. Pfitzner, K. Brauer, Y. Jiang, M. Liu, Granular elasticity: general considerations and the stress dip in sand piles, Phys. Rev. E 74 (2006) 061310. [6] L. Vanel, D. Howell, D. Clark, R.P. Behringer, E. Clement, Memories in sand: experimental tests of construction history on stress distributions under sandpiles, Phys. Rev. E 60 (1999) R05040. [7] G. Løvoll, K.J. Måløy, E.G. Flekkøy, Force measurements on static granular materials, Phys. Rev. E 60 (1999) 5872–5878. [8] D.L. Blair, N.W. Mueggenburg, A.H. Marshall, H.M. Jaeger, S.R. Nagel, Force distributions in three-dimensional granular assemblies: effects of packing order and interparticle friction, Phys. Rev. E 63 (2004) 041304. [9] A.P.F. Atman, P. Brunet, J. Geng, G. Reydellet, P. Claudin, R.P. Behringer, E. Clement, From the stress response function (back) to the sand pile ‘dip’, Eur. Phys. J. E. 17 (2005) 93–100. [10] S.G. Paikowsky, E.L. Hajduk, Calibration and use of grid-based tactile pressure sensors in granular material, A.S.T.M. Geotechnical Testing J. 20 (1997) 218–241. [11] S.G. Paikowslcy, C.J. Palmer, L.E. Rolwes. The use of tactile sensor technology for measuring soil stress distribution. GeoCongress 2006: Geotechnical Engineering in the Information Technology Age. 1–6. [12] J.C. Stith, Railroad track pressure measurements at the rail/tie plate interface using Tekscan sensors, Master's thesis, Department of Civil Engineering, University of Kentucky. (2004).