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Measurement of solids velocity in a conical Hopper by mass tracer particles
Chemical Engineering Science 54 (1999) 455—459
Measurement of solids velocity in a conical Hopper by mass tracer particles H.-Y. Xie *, K. Shinohara Department of Chemical Process & Control Engineering, College of Chemical Engineering, Dalian University of Technology, Dalian 116012, China Particulate Laboratory, Division of Materials Sciences Engineering, Graduate School of Hokkaido University, Hokkaido University, North 13 West 8, Sapporo 001 Japan Received 21 October 1996
Abstract Nedderman’s method is used and developed to measure particle velocity inside a conical hopper by placing a mass of host particles, which dyed a different color, into an initially static bed. The velocities of glass beads and sand particles of different sizes were measured in a steel hopper and a Perspex hopper. The effect of wall friction on the solids flow was also investigated by increasing wall roughness with sandpaper stuck on the inside hopper wall. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Particle velocity; Mass tracer particles; Conical hopper
1. Introduction Much research on the flow of granular materials inside conical hoppers has been carried out both theoretically (e.g., Jenike, 1961 and 1987; Brown and Richards, 1966) and experimentally (e.g., Bradsby and Blair-Fish, 1975; Polderman et al., 1987; Nedderman, 1988, Cleaver and Nedderman, 1993a). In the measurement of solids velocity inside the conical hopper, X-ray technique has been used to detect the trace of marker particles (e.g. Bradsby and Blair-Fish, 1975). Nedderman (1988) developed the method of measuring passout time of marker particles, from which the velocity can be deduced for radial flow. By releasing marker particles through a guide-tube into the established solids flow, this method avoids the problems associated with the dilation that takes place on initiation of flow. Cleaver and Nedderman (1993a) showed that the burrowing of marker particles into the solids material after free-fall down the guide-tube affects the passout time for heavier marker particles, but this effect could be avoided by proper selection of marker particles. Moreover, Nedderman’s method was further
developed by introducing marker particles in the established flow, then stopping the flow, removing the guidetube in order to prevent the hold-up effect of the guide-tube on marker particles, and then restarting the flow. They reported that the passout time of markers was not affected by stopping and restarting the flow. Shinohara and Nakamura (1995) have developed an online method to measure solids velocity by a bore scope technique. This is an invasive method—the bore scope may well disturb the flow field and affect the velocity profile. In the present work, Nedderman’s method was used to measure the passout time of tracer particles by placing a mass of tracer particles into an initially static bed on a batch scale at a selected height level and inclination angle. The effect of wall friction on the solids flow inside a conical hopper was also investigated by increasing the wall roughness with sandpaper stuck on the inside hopper wall. Glass beads, spherical and angular sand particles of different sizes were used in the experiments.
2. Experimental method
* Corresponding author.
Experiments were carried out in a steel conical hopper approximately 0.6 m in height, and a Perspex conical
0009-2509/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 2 3 2 - 2
456
H.-Y. Xie, K. Shinohara/Chemical Engineering Science 54 (1999) 455—459
hopper approximately 0.8 m in height, as shown in Fig. 1. Both the hopper half angles are 15° each and the base of the hopper can be fitted with a matching conical insert of the same angle to change the opening size, D. In the present experiments, four opening sizes of 1, 2, 3 and 4 cm were used to obtain different discharge rates. Tracer particles, which were dyed in different colors from the host particles, were used to measure passout time by placing a mass of them at a selected position, in terms of height h from the apex and angle h of inclination R from the central axis, in an initially static bed to operate on a batch scale. In the experiments, five height levels between 0.3 m and 0.5 m with intervals of 5 cm, and five inclination angles of 0, 6.8, 9.6, 12.3, and 15° in the steel hopper and seven inclination angles of 0, 2.6, 5.1, 7.7, 10.2, 12.7 and 15° in the Perspex hopper were selected. A large tray was used to collect the particles discharged from the hopper, and an inverted cone was placed below the hopper outlet to disperse the particles in order to reveal the tracer particles clearly. The experi-
mental procedure was as follows: the bottom of the cone was plugged and the hopper was filled to a selected level h , about 10 g of tracer particles was introduced on the R top of this level at a selected inclination angle h through an adjustable guide-tube (the radial distance from hopper apex r can thus be obtained from the selected h and h), R then the guide-tube was withdrawn and the hopper was filled up to a desired bed height h . A stopwatch was G started when the discharge of particles was initiated, and it was stopped when the tracer particles appeared in the heap of discharging particles. The velocity in the radial direction at the selected inclination could be then obtained from the measured passout time. Glass beads and sand particles of different sizes were used in the experiments. Their particle size d and density . o , bulk density o , shape and the angle of repose u are . given in Table 1. Two grades (A400 and A240 in the order of increasing surface roughness) of aluminum oxide sandpaper (Nippon Coated Abrasive) were stuck on the inside hopper wall to examine the effects of wall friction.
3. Experimental results Fig. 2 shows the passout time t of tracer particles placed at different tracer levels h , both along the central R axis and on the wall, vs initial bed height h , for G 250—600 km spherical sand particles at a discharge rate of 0.7 kg/s. The figure shows that the initial bed height has little effect on the measured passout time when the initial bed height is 5 cm or more above the level of tracer particles, corroborating what was reported by Cleaver and Nedderman (1993a). For radial and incompressible flow in a hopper, the radial particle velocity can be given as (e.g. Nedderman, 1988): dr C l , "! P dt r
Fig. 1. Illustration of experimental apparatus.
(1)
where C is a velocity constant which could be a function of f (h, , , D, d , o ). Integration of Eq. (1) gives U . . 3Ct"r. (2)
Table 1 Physical properties of particles used in the experiments Particles
d N (km)
o N (g/cm)
o (g/cm)
Shape (!)
u (deg)
Glass beads (G1) Glass beads (G2) Glass beads (G3) Sand (S1) Sand (S2) Sand (S3) Sand (S4)
500 1000 2000 250—600 600—800 800—1000 1000—2500
2.6 2.6 2.6 2.9 2.9 2.9 —
1.50 1.50 1.57 1.83 1.71 1.73 1.30
Spherical Spherical Spherical Spherical Spherical Spherical Angular
23 23 23 28 28 28 35
H.-Y. Xie, K. Shinohara/Chemical Engineering Science 54 (1999) 455—459
457
Fig. 2. Passout time t of tracer particles at different tracer levels h vs initial bed height h on the central axis and on the wall (spherical sand 250—600 km, D"3 cm, bare steel wall, discharge rate 0.7 kg/s).
Fig. 5. Velocity constants C obtained by Eq. (2) at different tracer levels vs inclination angle, h (spherical sand particles 250—600 km, D"2 cm, discharge rate 0.26 kg/s).
Fig. 3. Passout time t versus the cube of radial distance from apex, r.
Fig. 6. Velocity constants C at different tracer levels vs C for all experimental results. Fig. 4. 3Ct vs the cube of radial distance from apex, r.
An example of the measured passout time vs the cube of the radial distance from the apex r is shown in Fig. 3 for spherical sand particles of 250—600 km at a discharge rate of 0.26 kg/s with sandpaper A240 stuck on the inside hopper wall. The figure shows that the passout time t is proportional to r at different inclination angles h. Thus,
the velocity constants C can be obtained from linear regression of the plot t!r with precision and the velocity constants C obtained from the measured passout times t by C"r/3t should be independent of r. Fig. 4 plots 3Ct vs r for all experimental results, showing that linear regression of the plot t!r could give satisfactory accuracy (within $10%) for the velocity constant C. Fig. 5 shows how C varies with
458
H.-Y. Xie, K. Shinohara/Chemical Engineering Science 54 (1999) 455—459
Fig. 7. Velocity constants C compared with velocity constants C ob tained by the linear regression of t!r plot.
Fig. 8. Ratio of velocity constants C
/C
vs inclination angle, h.
Table 2 Particle discharge rates (kg/s) at different hopper opening sizes and different grades of sandpapers Particles (km)
G500 G1000 G2000 S250—600 S600—800 S800—1000 G500 G1000 S250—600 S600—800 S1000—2500 S1000—2500
Hopper and hopper opening size (cm)
Steel, 2 Steel, 2 Steel, 2 Steel, 2 Steel, 2 Steel, 2 Steel, 3 Steel, 3 Steel, 3 Steel, 3 Perspex, 1 Perspex, 4
Sandpaper none
A400
A240
0.253 0.228 0.229 0.262 0.236 0.212 0.654 0.664 0.703 0.644 0.058 2.678
0.248 0.228 — 0.262 0.234 0.215 0.665 0.652 0.737 0.653
0.248 0.231 0.222 0.262 0.234 0.216 0.685 0.668 0.734 0.652
Fig. 9. Effect of wall friction on velocity profiles.
opening size D of 2 cm. The figure shows that the velocity ratio profile becomes steeper as wall friction increases.
Acknowledgements inclination angle h, but remains practically constant with tracer levels, for 250—600 lm spherical sand particles at hopper opening of 2 cm and with different grades of sandpaper stuck on the inside hopper wall. For all other experimental results, Fig. 6 shows that, at any inclination angle h, the velocity constant C obtained from Eq. (2) at different tracer levels varies within $10% deviation from the average value C taken over the different tracer levels. Fig. 7 compares the velocity constant C obtained by linear regression of the plot t!r and the velocity constant obtained from Eq. (2) and averaged over different tracer levels C for all experimental re sults, showing that the two methods give closely similar results. Fig. 8 shows the variation of the ratio of velocity constant C /C (C being C at the central axis) with inclination angle h for glass beads, spherical and angular sand particles with bare hopper wall, while Fig. 9 shows the effect of wall friction on the same velocity constant ratio for 500 lm glass beads at the hopper
This research was sponsored by a internal research fund of Dalian University of Technology. Most of the experiments were performed at Hokkaido University under the sponsorship of the International Research and Exchange Program of Hokkaido University and Kajima Foundation.
Notation C C
C D d N h h G h R
velocity constant, m/s velocity constant C obtained from Eq. (2) and averaged over different tracer levels, m/s velocity constant C at the center axis, m/s diameter of hopper outlet, m, or cm particle size, km bed height, m initial bed height from apex, m height level of tracer particles from apex, m
H.-Y. Xie, K. Shinohara/Chemical Engineering Science 54 (1999) 455—459
r t v P
radial distance from hopper apex, m passout time of particles, s radial velocity, m/s
Greek letters h u o N o
inclination angle, deg angle of repose, deg particle density, g/cm bulk density, g/cm
References Bransby, P.L., & Blair-Fish, P.M. (1975). Initial deformations during mass flow from a bunker: observation and idealizations. Powder ¹echnol., 11, 273. Brown, R.L., & Richards, J.C. (1966). Principles of powder mechanics. Oxford: Pergamon Press. Cleaver, J.A.S., & Nedderman, R.M. (1993a). Measurement of velocity profiles in conical hoppers. Chem. Engng Sci., 48, 3703. Cleaver, J.A.S., & Nedderman, R.M. (1993b). Theoretical prediction of stress and velocity Profiles in conical hoppers. Chem. Engng Sci., 48, 3693.
459
Huang, J.J. (1997). Experimental and theoretical studies on solids flow inside a conical hopper. B.Eng. Dissertation (in Chinese), Department of Chemical Process & Control Engineering, Dalian University of Technology, Dalian, China. Jenike, A.W. (1961). Gravity flow of bulk solids, Bull. 108, University of Utah, USA. Jenike, A.W. (1987). A theory of flow of particulate solids in converging and diverging channels based on a conical yield function. Powder ¹echnol., 50, 229. Nedderman, R.M. (1988). The measurement of the velocity profile in a granular material discharging from a conical hopper. Chem Engng Sci., 43, 1507. Nedderman, R.M. (1992). Statics and kinematics of granular materials. Cambridge UK: Cambridge University Press. Polderman, H.G., Boom, J., de Hilster, E., & Scott, A.M. (1987). Solids flow velocity profiles in mass flow hoppers. Powder ¹echnol., 42, 737. Rose, H.F., & Tanaka, T. (1959). ¹he Engineer (London) 208, 465. Shinohara, K., & Nakamura, H. (1995). Measurement of particle velocity in a cone. Advan. Powder ¹echnol., 6, 249. Verghese, T.M., & Nedderman, R.M. (1993). A theoretical prediction of the discharge rate from conical hoppers in both the mass and core flow regime. ¹rans Instn Chem Engng, 71, 637. Tuzun, U., Houlsby, G.T., Nedderman, R.M., & Savage, S.B. (1982). The flow of granular materials II: Velocity distributions in slow flow. Chem Engng Sci., 37, 1969.
Measurement of solids velocity in a conical Hopper by mass tracer particles H.-Y. Xie *, K. Shinohara Department of Chemical Process & Control Engineering, College of Chemical Engineering, Dalian University of Technology, Dalian 116012, China Particulate Laboratory, Division of Materials Sciences Engineering, Graduate School of Hokkaido University, Hokkaido University, North 13 West 8, Sapporo 001 Japan Received 21 October 1996
Abstract Nedderman’s method is used and developed to measure particle velocity inside a conical hopper by placing a mass of host particles, which dyed a different color, into an initially static bed. The velocities of glass beads and sand particles of different sizes were measured in a steel hopper and a Perspex hopper. The effect of wall friction on the solids flow was also investigated by increasing wall roughness with sandpaper stuck on the inside hopper wall. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Particle velocity; Mass tracer particles; Conical hopper
1. Introduction Much research on the flow of granular materials inside conical hoppers has been carried out both theoretically (e.g., Jenike, 1961 and 1987; Brown and Richards, 1966) and experimentally (e.g., Bradsby and Blair-Fish, 1975; Polderman et al., 1987; Nedderman, 1988, Cleaver and Nedderman, 1993a). In the measurement of solids velocity inside the conical hopper, X-ray technique has been used to detect the trace of marker particles (e.g. Bradsby and Blair-Fish, 1975). Nedderman (1988) developed the method of measuring passout time of marker particles, from which the velocity can be deduced for radial flow. By releasing marker particles through a guide-tube into the established solids flow, this method avoids the problems associated with the dilation that takes place on initiation of flow. Cleaver and Nedderman (1993a) showed that the burrowing of marker particles into the solids material after free-fall down the guide-tube affects the passout time for heavier marker particles, but this effect could be avoided by proper selection of marker particles. Moreover, Nedderman’s method was further
developed by introducing marker particles in the established flow, then stopping the flow, removing the guidetube in order to prevent the hold-up effect of the guide-tube on marker particles, and then restarting the flow. They reported that the passout time of markers was not affected by stopping and restarting the flow. Shinohara and Nakamura (1995) have developed an online method to measure solids velocity by a bore scope technique. This is an invasive method—the bore scope may well disturb the flow field and affect the velocity profile. In the present work, Nedderman’s method was used to measure the passout time of tracer particles by placing a mass of tracer particles into an initially static bed on a batch scale at a selected height level and inclination angle. The effect of wall friction on the solids flow inside a conical hopper was also investigated by increasing the wall roughness with sandpaper stuck on the inside hopper wall. Glass beads, spherical and angular sand particles of different sizes were used in the experiments.
2. Experimental method
* Corresponding author.
Experiments were carried out in a steel conical hopper approximately 0.6 m in height, and a Perspex conical
0009-2509/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 2 3 2 - 2
456
H.-Y. Xie, K. Shinohara/Chemical Engineering Science 54 (1999) 455—459
hopper approximately 0.8 m in height, as shown in Fig. 1. Both the hopper half angles are 15° each and the base of the hopper can be fitted with a matching conical insert of the same angle to change the opening size, D. In the present experiments, four opening sizes of 1, 2, 3 and 4 cm were used to obtain different discharge rates. Tracer particles, which were dyed in different colors from the host particles, were used to measure passout time by placing a mass of them at a selected position, in terms of height h from the apex and angle h of inclination R from the central axis, in an initially static bed to operate on a batch scale. In the experiments, five height levels between 0.3 m and 0.5 m with intervals of 5 cm, and five inclination angles of 0, 6.8, 9.6, 12.3, and 15° in the steel hopper and seven inclination angles of 0, 2.6, 5.1, 7.7, 10.2, 12.7 and 15° in the Perspex hopper were selected. A large tray was used to collect the particles discharged from the hopper, and an inverted cone was placed below the hopper outlet to disperse the particles in order to reveal the tracer particles clearly. The experi-
mental procedure was as follows: the bottom of the cone was plugged and the hopper was filled to a selected level h , about 10 g of tracer particles was introduced on the R top of this level at a selected inclination angle h through an adjustable guide-tube (the radial distance from hopper apex r can thus be obtained from the selected h and h), R then the guide-tube was withdrawn and the hopper was filled up to a desired bed height h . A stopwatch was G started when the discharge of particles was initiated, and it was stopped when the tracer particles appeared in the heap of discharging particles. The velocity in the radial direction at the selected inclination could be then obtained from the measured passout time. Glass beads and sand particles of different sizes were used in the experiments. Their particle size d and density . o , bulk density o , shape and the angle of repose u are . given in Table 1. Two grades (A400 and A240 in the order of increasing surface roughness) of aluminum oxide sandpaper (Nippon Coated Abrasive) were stuck on the inside hopper wall to examine the effects of wall friction.
3. Experimental results Fig. 2 shows the passout time t of tracer particles placed at different tracer levels h , both along the central R axis and on the wall, vs initial bed height h , for G 250—600 km spherical sand particles at a discharge rate of 0.7 kg/s. The figure shows that the initial bed height has little effect on the measured passout time when the initial bed height is 5 cm or more above the level of tracer particles, corroborating what was reported by Cleaver and Nedderman (1993a). For radial and incompressible flow in a hopper, the radial particle velocity can be given as (e.g. Nedderman, 1988): dr C l , "! P dt r
Fig. 1. Illustration of experimental apparatus.
(1)
where C is a velocity constant which could be a function of f (h, , , D, d , o ). Integration of Eq. (1) gives U . . 3Ct"r. (2)
Table 1 Physical properties of particles used in the experiments Particles
d N (km)
o N (g/cm)
o (g/cm)
Shape (!)
u (deg)
Glass beads (G1) Glass beads (G2) Glass beads (G3) Sand (S1) Sand (S2) Sand (S3) Sand (S4)
500 1000 2000 250—600 600—800 800—1000 1000—2500
2.6 2.6 2.6 2.9 2.9 2.9 —
1.50 1.50 1.57 1.83 1.71 1.73 1.30
Spherical Spherical Spherical Spherical Spherical Spherical Angular
23 23 23 28 28 28 35
H.-Y. Xie, K. Shinohara/Chemical Engineering Science 54 (1999) 455—459
457
Fig. 2. Passout time t of tracer particles at different tracer levels h vs initial bed height h on the central axis and on the wall (spherical sand 250—600 km, D"3 cm, bare steel wall, discharge rate 0.7 kg/s).
Fig. 5. Velocity constants C obtained by Eq. (2) at different tracer levels vs inclination angle, h (spherical sand particles 250—600 km, D"2 cm, discharge rate 0.26 kg/s).
Fig. 3. Passout time t versus the cube of radial distance from apex, r.
Fig. 6. Velocity constants C at different tracer levels vs C for all experimental results. Fig. 4. 3Ct vs the cube of radial distance from apex, r.
An example of the measured passout time vs the cube of the radial distance from the apex r is shown in Fig. 3 for spherical sand particles of 250—600 km at a discharge rate of 0.26 kg/s with sandpaper A240 stuck on the inside hopper wall. The figure shows that the passout time t is proportional to r at different inclination angles h. Thus,
the velocity constants C can be obtained from linear regression of the plot t!r with precision and the velocity constants C obtained from the measured passout times t by C"r/3t should be independent of r. Fig. 4 plots 3Ct vs r for all experimental results, showing that linear regression of the plot t!r could give satisfactory accuracy (within $10%) for the velocity constant C. Fig. 5 shows how C varies with
458
H.-Y. Xie, K. Shinohara/Chemical Engineering Science 54 (1999) 455—459
Fig. 7. Velocity constants C compared with velocity constants C ob tained by the linear regression of t!r plot.
Fig. 8. Ratio of velocity constants C
/C
vs inclination angle, h.
Table 2 Particle discharge rates (kg/s) at different hopper opening sizes and different grades of sandpapers Particles (km)
G500 G1000 G2000 S250—600 S600—800 S800—1000 G500 G1000 S250—600 S600—800 S1000—2500 S1000—2500
Hopper and hopper opening size (cm)
Steel, 2 Steel, 2 Steel, 2 Steel, 2 Steel, 2 Steel, 2 Steel, 3 Steel, 3 Steel, 3 Steel, 3 Perspex, 1 Perspex, 4
Sandpaper none
A400
A240
0.253 0.228 0.229 0.262 0.236 0.212 0.654 0.664 0.703 0.644 0.058 2.678
0.248 0.228 — 0.262 0.234 0.215 0.665 0.652 0.737 0.653
0.248 0.231 0.222 0.262 0.234 0.216 0.685 0.668 0.734 0.652
Fig. 9. Effect of wall friction on velocity profiles.
opening size D of 2 cm. The figure shows that the velocity ratio profile becomes steeper as wall friction increases.
Acknowledgements inclination angle h, but remains practically constant with tracer levels, for 250—600 lm spherical sand particles at hopper opening of 2 cm and with different grades of sandpaper stuck on the inside hopper wall. For all other experimental results, Fig. 6 shows that, at any inclination angle h, the velocity constant C obtained from Eq. (2) at different tracer levels varies within $10% deviation from the average value C taken over the different tracer levels. Fig. 7 compares the velocity constant C obtained by linear regression of the plot t!r and the velocity constant obtained from Eq. (2) and averaged over different tracer levels C for all experimental re sults, showing that the two methods give closely similar results. Fig. 8 shows the variation of the ratio of velocity constant C /C (C being C at the central axis) with inclination angle h for glass beads, spherical and angular sand particles with bare hopper wall, while Fig. 9 shows the effect of wall friction on the same velocity constant ratio for 500 lm glass beads at the hopper
This research was sponsored by a internal research fund of Dalian University of Technology. Most of the experiments were performed at Hokkaido University under the sponsorship of the International Research and Exchange Program of Hokkaido University and Kajima Foundation.
Notation C C
C D d N h h G h R
velocity constant, m/s velocity constant C obtained from Eq. (2) and averaged over different tracer levels, m/s velocity constant C at the center axis, m/s diameter of hopper outlet, m, or cm particle size, km bed height, m initial bed height from apex, m height level of tracer particles from apex, m
H.-Y. Xie, K. Shinohara/Chemical Engineering Science 54 (1999) 455—459
r t v P
radial distance from hopper apex, m passout time of particles, s radial velocity, m/s
Greek letters h u o N o
inclination angle, deg angle of repose, deg particle density, g/cm bulk density, g/cm
References Bransby, P.L., & Blair-Fish, P.M. (1975). Initial deformations during mass flow from a bunker: observation and idealizations. Powder ¹echnol., 11, 273. Brown, R.L., & Richards, J.C. (1966). Principles of powder mechanics. Oxford: Pergamon Press. Cleaver, J.A.S., & Nedderman, R.M. (1993a). Measurement of velocity profiles in conical hoppers. Chem. Engng Sci., 48, 3703. Cleaver, J.A.S., & Nedderman, R.M. (1993b). Theoretical prediction of stress and velocity Profiles in conical hoppers. Chem. Engng Sci., 48, 3693.
459
Huang, J.J. (1997). Experimental and theoretical studies on solids flow inside a conical hopper. B.Eng. Dissertation (in Chinese), Department of Chemical Process & Control Engineering, Dalian University of Technology, Dalian, China. Jenike, A.W. (1961). Gravity flow of bulk solids, Bull. 108, University of Utah, USA. Jenike, A.W. (1987). A theory of flow of particulate solids in converging and diverging channels based on a conical yield function. Powder ¹echnol., 50, 229. Nedderman, R.M. (1988). The measurement of the velocity profile in a granular material discharging from a conical hopper. Chem Engng Sci., 43, 1507. Nedderman, R.M. (1992). Statics and kinematics of granular materials. Cambridge UK: Cambridge University Press. Polderman, H.G., Boom, J., de Hilster, E., & Scott, A.M. (1987). Solids flow velocity profiles in mass flow hoppers. Powder ¹echnol., 42, 737. Rose, H.F., & Tanaka, T. (1959). ¹he Engineer (London) 208, 465. Shinohara, K., & Nakamura, H. (1995). Measurement of particle velocity in a cone. Advan. Powder ¹echnol., 6, 249. Verghese, T.M., & Nedderman, R.M. (1993). A theoretical prediction of the discharge rate from conical hoppers in both the mass and core flow regime. ¹rans Instn Chem Engng, 71, 637. Tuzun, U., Houlsby, G.T., Nedderman, R.M., & Savage, S.B. (1982). The flow of granular materials II: Velocity distributions in slow flow. Chem Engng Sci., 37, 1969.