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Flushing phenomenon of fine powder
Powder Technology 126 (2002) 275 – 282 www.elsevier.com/locate/powtec
Flushing phenomenon of fine powder Shigeru Kuchii a,*, Yuji Tomita b,1 a
Department of Control and Information Systems Engineering, Kitakyushu National College of Technology, 5-20-1, Shii, Kokura-minami, Kitakyushu 802-0985, Japan b Department of Mechanical and Control Engineering, Kyushu Institute of Technology, 1-1, Sensuicho, Tobata, Kitakyushu 804-8550, Japan Received 22 October 2000; received in revised form 20 February 2002; accepted 6 March 2002
Abstract When a load is applied to a free surface of cohesive fine powder in a vessel, the powder can spurt from a small orifice, which we call the flushing. A gas – solid two-phase flow model using Darcy’s law examines this phenomenon. The numerical simulation by this model is qualitatively verified by the measurement for a case when the powder in a rectangular vessel spurts from a slit by a constant piston load. It is found that near the slit, there appears a high-speed region of particle of which velocity is considerably larger than that of free-flowing due to the gravity. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Two-phase flow; Darcy’s law; Fine powder; Flow visualization; Flushing region; Velocity profile; Numerical analysis
1. Introduction In handling of fine powder, an uncontrollable flow that is sometimes called flushing can occur and becomes a source of operating difficulties. For example, this often happens in rotary feeders when aerated powders are handled. It becomes difficult to control the mass flow rate of powder since the powder leaks out from the clearance where the flow is not expected. An unexpected large quantity of mass flow is also observed in a funnel flow hopper. Rathbone and Nedderman [1] call this phenomenon flooding and note that this occurs when the upper part of a rathole becomes unstable and falls down the rathole to strike the bunker bottom. In this case, aeration occurs due to airflow caused by increased pressure difference between the powder and the outside. Rathbone et al. [2] experimentally showed that for fully aerated materials, the flow rate is comparable to that of an inviscid fluid. In most cases, the flushing occurs when fine powders that contain a great deal of air are subjected to some external disturbances. The flow is transient and does not last for a long time. It is practically important to know in advance whether a powder in question is flushing. Carr [3– 5] proposed a method to evaluate the *
Corresponding author. Tel.: +81-93-964-7259; fax: +81-93-964-7259. E-mail addresses: [email protected] (S. Kuchii), [email protected]. (Y. Tomita). 1 Tel.: + 81-93-884-3157; fax: + 81-93-871-8591.
flowability and floodability of powder by the physical properties. Bruff [6] developed a simple test as an indicator of flushing. Geldart and Wong [7] suggested that powders in group A and some in group C are likely to cause the flushing according to their classification [8]. We [9] assume that the flushing occurs due to air stream generated by increased interstitial gas pressure in the powder when a load is applied to the powder and showed that the voidage is one of key factors for the flushing. The powder that gives rise to the flushing is usually fine and has large voidage. This is because the particles can build a weak structure of relatively large voidage by the cohesive force between particles. If a load is applied to such powder in a vessel to reduce the volume, the interstitial gas firstly supports the load since the particle structure is too weak to share the load. Thus, the interstitial gas pressure is increased. If the vessel opens to a low-pressure region through a small opening, gas flow occurs and the particles are forced to spurt out from the opening. At the same time, consolidation of powder proceeds and the load shared by the particles soon increases, which in turn reduces the gas pressure and the flushing finally ceases. In gravity flow of fine powders from an orifice of vessel bottom, the powders do not always flow out even if the size of orifice is large enough as compared with the particle size. This is because a powder arch is formed over the orifice. If we apply a suitable load on the top surface to break the arch structure, the particles will spurt out of the orifice. It is difficult for cohesionless particles,
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however, to produce the flushing by the same method since the voidage of such particles is usually low and the load is almost always supported by the particle skeleton or by the effective stress. In this paper, we will examine the phenomenon based on a gas – solid two-phase flow model using the Darcy’s law. An experiment was done in a rectangular vessel to observe the particle flow pattern during the flushing and to test our flushing model. However, computation based on the linearized equations, which omit the effective stress, is carried out to see the particle velocity distributions in the vessel at the beginning of the flushing when the effective stress is small.
where m is the kinematic viscosity of gas, gi is the gravitational acceleration and k is the permeability of Darcy’s law, that is,
2. Theory
p ¼ qf RT
If p is the interstitial pressure, rijs is the stress in the particle that supports the particle skeleton and e is the voidage, the total stress of powder rij is given by
where R represents the gas constant and T the absolute temperature. We assume the process is isothermal, that is,
rij ¼ ð1 eÞrsij epdij
rij ¼ rijV pdij
ð2Þ
Substituting Eq. (2) for Eq. (1), the stress of solid phase is written by rijV as ð1 eÞrsij ¼ rijV ð1 eÞpdij
ð3Þ
If we use the Cartesian coordinate, the equations of continuity for solid and gas phases are given by A A fð1 eÞqs g þ fð1 eÞqs ui g ¼ 0 At Axi
ð4Þ
and A A ðeqf Þ þ ðeqf vi Þ ¼ 0 At Axi
ð5Þ
where qs represents the solid density, qf the gas density, ui the solid velocity and vi the gas velocity. According to Zienkiewics and Shiomi [11], the equations of motion for both phases become, by using the effective stress, as ð1 eÞqs
Aui Aui þ uj At Axj
eqf
k¼
Avi Avi þ vj At Axj
¼ e
Ap e2 qf m ðvi ui Þ eqf gi ð7Þ Axi k
1 e3 d2 180 ð1 eÞ2
ð8Þ
where d is the particle size. The equation of state of gas is given by
T ¼ const:
ð9Þ
ð10Þ
ð1Þ
where rij is Kronecker’s delta and we follow the usual convention that a compressive stress is negative. Terzaghi and Peck [10] formulated rij by p and the effective stress rijV as follows,
and
¼ ð1 eÞ þ
Ap e2 qf m ðvi ui Þ þ Axi k
ArijV ð1 eÞqs gi Axj
ð6Þ
3. Experiment We produce the flushing by applying a piston load on the top surface of powder in a rectangular vessel that has a slit at the bottom. The dimension of the vessel is 60 mm in length, 13 mm in width and 80 mm in height. The slit is 3 13 mm and is set up in the center of the bottom. The vessel was made of transparent acrylic plates to observe the particle flow. The images of particle motion were taken in a field size of 60 80 mm by a high-speed video camera whose shutter speed is 1000 frames per second. To examine the particle motion in the vessel, we used tracer particles of 0.7-mm diameter. The 361 particles were placed in the nodes of a checkered pattern of 3 mm spacing 19 in length by 19 in height. To set the tracer particles in their initial positions, we at first put the particles in regularly positioned pits of a plate, cover with this the front side of the vessel that is then filled with the powder and replace the plate with the transparent acrylic plate. Before using this particle, we had observed the powder flow by using almost similar particles as the test powder but different color, which were filled in checkered pattern in advance. It was confirmed by comparing both flows obtained by different particles that the present particles practically trace the powder motion. The initial voidage e0 was determined by measuring the mass of powder that was poured into the vessel by using a funnel gently and as even as possible. To obtain large voidage, we used a sieve through which the powder was filled into the vessel. The powder is filled in the vessel 60 mm in height. The piston pressure DP was 5.03 kPa. A laser displacement sensor monitored the piston motion. There is a clearance between the piston and the vessel, and we insert
S. Kuchii, Y. Tomita / Powder Technology 126 (2002) 275–282 Table 1 Physical properties of tested particles Particle
Soft wheat flour
FCC catalyst
Particle size (Am) Material density (kg/m3) Angle of repose (degrees) Carr’s flowability index Carr’s floodability index
51.6 1480 49.0 40.5 56.0
59.0 2565 28.2 94.0 71.0
a sheet of cardboard between the top surface of the powder and the piston to avoid the leakage of powder and gas through the clearance. It was confirmed by observing the particle flow during experiment that the leakage was prevented. Powders used in this work are soft wheat flour and FCC catalyst. Table 1 shows the physical properties of the powders including the flowability and floodability indices of Carr, which were measured by Hosokawa Powder Characteristics Tester. By this experiment, it was found that the soft wheat flour is cohesive and gives rise to the flushing, while the FCC catalyst is cohesionless free-flowing and does not give rise to the flushing. Both powders belong to group A of Geldart’s map.
4. Numerical procedure It is supposed that the effective stress in the powder is minimum before the piston load is applied. The effective stress acts to decelerate the powder flow and finally stops the flushing. Thus, it is possible to omit the effective stress in order to examine the flow near the onset of flushing. Furthermore, we will linearize the governing equations since the flow in the vessel is slow. We also omit the gravity term since it is not critical for the flushing. Indeed, the flushing occurs through a horizontal nozzle [9]. We assume that the flow is two-dimensional, where x coordinate is horizontally set along the vessel bottom and y coordinate is taken vertically upward from the slit center. We put pressure and voidage as p ¼ p0 þ pV
ð11Þ
and e ¼ e0 þ eV
ð12Þ
where p0 and e0 are the initial pressure and voidage, respectively, and we assume that pVbp0
ð13Þ
contain the products of small quantities, the equation of continuity for solid phase becomes AeV Aux Auy ¼ ð1 e0 Þ þ ð15Þ At Ax Ay where qs is assumed constant. If we substitute Eqs. (9) and (10) for Eq. (5), omit the small terms and use Eq. (15), the continuity equation for gas phase becomes ApV 1 e0 Aux Auy ¼ p0 þ At e0 Ax Ay Avx Avy þ p0 Ax Ay
ð14Þ
Futhermore, we assume that the generated velocities of gas and solid phases are also small quantities like pVand eV whose products are neglected. If we omit the terms that
ð16Þ
The equations of motion for both phases become e20 qf 0 m Aux 1 ApV þ ¼ ðvx ux Þ k0 qs ð1 e0 Þ qs Ax At
ð17Þ
e20 qf 0 m Auy 1 ApV þ ¼ ðvy uy Þ qs Ay At k0 qs ð1 e0 Þ
ð18Þ
Avx 1 ApV e0 m ¼ ðvx ux Þ qf 0 Ax k0 At
ð19Þ
and Avy 1 ApV e0 m ¼ ðvy uy Þ qf 0 Ay k0 At
ð20Þ
where we assume that the permeability is constant as k0 that is evaluated by substituting e0 for e in Eq. (8). The powder in the vessel flows out from the bottom slit by the piston load. The piston motion decreases a region in the vessel occupied by the powder. We calculated the flow of solid and gas in this region, and apply the method of SOLA-VOF [12], which can solve the velocity and pressure fields in time marching for a region having moving boundaries. Table 2 shows calculating conditions, where p0 is the atmospheric pressure and the mesh points for calculation consist of 22 in x-direction by 52 in y-direction. Table 2 Conditions of numerical calculation Computing range Initial conditions
and eVbe0
277
Boundary conditions
Numbers of mesh
0 V x V 0.030 (m) 0 V y V 0.075 (m) Pressure: p = p0 (0 V x V 0.030, 0 V y V yp) Velocity: ux = uy = vx = vy = 0 (0 V x V 0.030, 0 V y V yp) Voidage: e = e0 (0 V x V 0.030, 0 V y V yp) Pressure: p = p0 + DP (0 V x V 0.030, y = yp); p = p0 (0 V x V 0.0015, y = 0) Velocity: ux = vx = 0 (x = 0, 0.03, 0 V y V yp); uy = vy = 0 (0.0015 V x V 0.030, y = 0) x y = 22 52
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5. Results and discussion 5.1. Flow pattern during the flushing Fig. 1 shows the photographs of flow of FCC catalyst in the vessel for different times from the beginning of measurement, where Vp is the measured piston velocity and B is the slit width, that is, B = 3 mm. Fig. 1(a) shows
Fig. 1. Flow pattern of FCC catalyst (e0 = 0.618).
that the particles already discharge at the slit before the piston load is applied. It is well known that when the particles is discharged freely by the gravity alone from the slit, the discharge of particles depends on the slit width B and the gravitational acceleration g, and pffiffiffiffiffiffithe velocity of particles at the slit is in proportion to gB [13], which is 0.1715 m/s in this case. Therefore, in p the figure, we scale ffiffiffiffiffiffiffiffi pffiffiffiffiffiffi the velocity and time by gB and B=g , respectively. Indeed, it is found that Vp is the same order of magnitude as the mean particle velocity in the vessel cross-section that is predicted by the discharge without the piston load if the bulk density case pffiffiffiffiffiis ffi constant, that is, in the p ffiffiffiffiffiffi of Fig. 1(b), Vp ¼ 0:03 gBc0:005 m=s and ðB=2aÞ gBc0:008 m=s where a is the half length of the vessel and is 30 mm. As the piston goes down, there appears a funnel type of free surface of powder as shown in Fig. 1(c). If the apex of the funnel reaches the slit, the discharge stops. This flow pattern is typical of free-flowing materials in the gravity discharge without surcharge on the top surface of powder and is different from that in the flushing phenomenon. This suggests that the piston load does not affect the particle discharge from the vessel for the free-flowing materials. Fig. 2 shows the photographs of flow of soft wheat flour. The flow pattern is distinctly different from that in Fig. 1 and seems to be typical of flushing phenomenon. As shown in Fig. 2(a), the particles are at rest unless the piston load is applied, which is different from the case for free-flowing materials. When the load is applied, the particle flow firstly occurs near the slit and below the piston as shown in Fig. 2(b). It is found that the piston velocity is much higher than that in Fig. 1, and then the duration of particle motion becomes shorter than that in Fig. 1. It is noted that the particle free surface does not appear and the flow stops on the way. This is because the pressure difference necessary to produce the particle flow disappears by the increased effective stress. In the following, we exclusively focus on the flow of soft wheat flour since the motion of FCC catalyst is not caused by flushing but gravity. The particle velocity in the gravity discharge is much smaller than that of the flushing particles as shown above, and the particles in the gravity discharge are assumed to move in the contact with each other in the vessel. Then, the effective stress is an important governing factor of the flow. However, it is still difficult for the present model to predict the onset of flushing since the calculation is based on a fluid model. Even if the effective stress is small, it is necessary to consider it, of which constitutive equation should be a plastic one having a yield stress, in order to explain the onset. Thus, we cannot give a criterion on the applicability of the present calculation model in terms of the particle properties, and the calculation for the FCC catalyst by this model will be insignificant while giving similar results as those for soft wheat flour since the particle size and the density are almost the same with each other as shown in Table 1.
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279
region near the slit, which we call the flushing region, and apslow ffiffiffiffiffiffi flow region in both corners near the bottom. Since pffiffiffiffiffiffi gB is 0.1715 m/sm the particle velocity is larger than gB in the pffiffiffiffiffi ffi flushing region and is the same order of magnitude as gB in the plug-like flow region, being slightly smaller than Vp. Fig. 5 shows the numerical result on the same condition as the measurement in Fig. 3. It is confirmed that there are the flushing region near the slit and the plug-like flow region below the piston as in the measurement. The numerical result reproduces the particle flow qualitatively. The
Fig. 2. Flow pattern of soft wheat flour (e0 = 0.726).
Fig. 3 shows the photographs of flow of soft wheat flour when e0 = 0.819 and ffiFig. 4 is the corresponding velocity qffiffiffiffiffiffiffiffiffiffiffiffiffiffi contours of u2x þ u2y in different times, where the piston velocity Vp is 0.205 m/s in (a), 0.223 m/s in (b) and 0.223 m/s in (c), respectively. The dark circular region in the photograph is an afterimage of lens. It is noted that the particle velocity is increased with an increasing e0. There are a plug-like flow region below the piston, a high-speed
Fig. 3. Flow pattern of soft wheat flour (e0 = 0.819).
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Fig. 4. Velocity contours of soft wheat flour for e0 = 0.819 (experiment).
piston velocity of the calculation is larger than that of measurement. This is because the effective stress that will act to resist the particle motion is omitted in the calculation. It is found that the particle motion firstly occurs near the slit as well as just p below ffiffiffiffiffiffi the piston, and that the particle velocity is larger than gB in a region near the slit, which is called the flushing region. The calculation confirms these results for the same conditions as measurement.
Fig. 5. Velocity contours of soft wheat flour for e0 = 0.819 (numerical calculation).
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281
5.2. Velocity distribution Fig. 6 shows the relationship pffiffiffiffiffiffiffiffi between umax/Vp and y/a for soft wheat flour at t= B=g ¼ 3:1 for two different initial voidages where umax is the downward particle velocity on y axis and yp is the current piston position. In the measurement, umax/Vp increases with e0 near the slit while the calculation fails to explain the effect of e0 on the profile near the slit. This indicates that the resistance to the particles motion is reduced when the voidage is large. Similar experimental results are obtained for other piston positions. However, the effect of voidage disappears toward the piston, which shows that the particles near the piston are moved en masse by the piston. Near the piston region where umax/Vp is almost constant, we can assume that the distribution for a given time will take the following. x uy ¼f a Vp
ð21Þ Fig. 7. uy/Vp vs. x/a for soft wheat flour for e0 = 0.819.
Fig. 7 shows the distribution by Eq. (21) with z/a as a parameter, where z is a distance from the current piston position yp, that is, z = yp y. Since the number of tracer particles is limited, we plotted the experimental particle velocity collecting into each small range of z. Although the scatter is considerable in the measurement, the profile is pluglike and is weakly dependent on x/a. The numerical result also shows a similar profile, where it is noted that no-slip condition at the wall is not satisfied since the governing equations contain the first-order derivatives as to the velocity. Near the slit region, the effect of piston motion disappears and the motion of particles is in local equilibrium like
Fig. 6. umax/Vp vs. y/a for soft wheat flour.
a turbulent jet. Thus, we can expect the following velocity distribution for given e0 and yp, uy x ¼f y umax
ð22Þ
Fig. 8 shows the distribution by Eq. (22), which shows that uy/umax is a function of x/y and weakly dependent on z as well. The numerical result is also dependent on z. We can expect a kind of self-preserving flow in the region far from a piston. Such a flow is also found in a blow tank solid conveyor for free-flowing powders [14]. Generally, in such
Fig. 8. uy/umax vs. x/y for soft wheat flour for e0 = 0.819.
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S. Kuchii, Y. Tomita / Powder Technology 126 (2002) 275–282
a region, particle velocity distribution is expressed by Eq. (22). In this region of z/a, the effect of piston motion almost disappears.
6. Conclusion We have defined the flushing phenomenon as spurting of powder due to air stream generated by an increase in interstitial gas pressure in the powder bed when an external load is applied to the powder. In order to test this mechanism, we carried out experiments and formulated the phenomenon based on gas – solid two-phase flow modeling to perform numerical analysis for given experimental condition. It is found that when a piston load is applied, the particle flow at first begins near the slit, which is remote from the piston, and the velocity of particles is considerably larger than that of free-flowing due to the gravitational acceleration. The numerical analysis explains the phenomenon qualitatively. List of symbols a half length of the vessel B slit width d particle diameter g gravitational acceleration k permeability of Darcy’s law k0 permeability of Darcy’s law of initial voidage p fluid pressure p0 atmospheric pressure pV pressure disturbance R gas constant t time T temperature umax maximum velocity of uy yp current piston velocity ux x-component of solid velocity uy y-component of solid velocity vx x-component of gas velocity
vy Vp x y z DP
y-component of gas velocity piston velocity Cartesian coordinate in horizontal direction Cartesian coordinate in vertical direction distance from the current piston position piston pressure
Greek letters dij Kronecker’s delta e voidage e0 initial voidage eV voidage disturbance rij total stress of powder rijs interparticle stress rijV effective stress m kinematic viscosity of fluid qf fluid density qf0 fluid density of initial voidage qs particle density
References [1] T. Rathbone, R.M. Nedderman, Powder Technol. 51 (1987) 115. [2] T. Rathbone, R.M. Nedderman, J.F. Davidson, Chem. Eng. Sci. 42 (4) (1987) 725. [3] R.L. Carr Jr., Chem. Eng. 1 (Feb. 1965) 69. [4] R.L. Carr Jr., Chem. Eng. 18 (Jan. 1965) 163. [5] R.L. Carr Jr., Chem. Eng. 13 (Oct. 1969) 7. [6] W. Bruff, J. Eng. Ind. (ASME) 91 (1969) 323. [7] D. Geldart, A.C.Y. Wong, Chem. Eng. Sci. 40 (1985) 653. [8] D. Geldart, Powder Technol. 7 (1973) 285. [9] Y. Tomita, H. Ikeuchi, S. Kuchii, K. Funatsu, J. Rheol. 38 (1994) 231. [10] K. Terzaghi, R.B. Peck, Theoretical Soil Mechanics in Engineering Particles, vol. 51, Wiley, New York, 1948. [11] O.C. Zienkiewics, T. Shiomi, Int. J. Numer. Anal. Methods Geomech. 8 (1984) 71 – 99. [12] B.D. Nichols, C.W. Hirt, R.S. Hotchkiss, 1980, LA-8355. [13] R.L. Brown, J.C. Richards, Trans. Inst. Chem. Eng. 37 (2) (1959) 108. [14] Y. Tomita, T. Jotaki, S. Makimoto, K. Fukushima, Powder Technol. 32 (1982) 1.
Flushing phenomenon of fine powder Shigeru Kuchii a,*, Yuji Tomita b,1 a
Department of Control and Information Systems Engineering, Kitakyushu National College of Technology, 5-20-1, Shii, Kokura-minami, Kitakyushu 802-0985, Japan b Department of Mechanical and Control Engineering, Kyushu Institute of Technology, 1-1, Sensuicho, Tobata, Kitakyushu 804-8550, Japan Received 22 October 2000; received in revised form 20 February 2002; accepted 6 March 2002
Abstract When a load is applied to a free surface of cohesive fine powder in a vessel, the powder can spurt from a small orifice, which we call the flushing. A gas – solid two-phase flow model using Darcy’s law examines this phenomenon. The numerical simulation by this model is qualitatively verified by the measurement for a case when the powder in a rectangular vessel spurts from a slit by a constant piston load. It is found that near the slit, there appears a high-speed region of particle of which velocity is considerably larger than that of free-flowing due to the gravity. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Two-phase flow; Darcy’s law; Fine powder; Flow visualization; Flushing region; Velocity profile; Numerical analysis
1. Introduction In handling of fine powder, an uncontrollable flow that is sometimes called flushing can occur and becomes a source of operating difficulties. For example, this often happens in rotary feeders when aerated powders are handled. It becomes difficult to control the mass flow rate of powder since the powder leaks out from the clearance where the flow is not expected. An unexpected large quantity of mass flow is also observed in a funnel flow hopper. Rathbone and Nedderman [1] call this phenomenon flooding and note that this occurs when the upper part of a rathole becomes unstable and falls down the rathole to strike the bunker bottom. In this case, aeration occurs due to airflow caused by increased pressure difference between the powder and the outside. Rathbone et al. [2] experimentally showed that for fully aerated materials, the flow rate is comparable to that of an inviscid fluid. In most cases, the flushing occurs when fine powders that contain a great deal of air are subjected to some external disturbances. The flow is transient and does not last for a long time. It is practically important to know in advance whether a powder in question is flushing. Carr [3– 5] proposed a method to evaluate the *
Corresponding author. Tel.: +81-93-964-7259; fax: +81-93-964-7259. E-mail addresses: [email protected] (S. Kuchii), [email protected]. (Y. Tomita). 1 Tel.: + 81-93-884-3157; fax: + 81-93-871-8591.
flowability and floodability of powder by the physical properties. Bruff [6] developed a simple test as an indicator of flushing. Geldart and Wong [7] suggested that powders in group A and some in group C are likely to cause the flushing according to their classification [8]. We [9] assume that the flushing occurs due to air stream generated by increased interstitial gas pressure in the powder when a load is applied to the powder and showed that the voidage is one of key factors for the flushing. The powder that gives rise to the flushing is usually fine and has large voidage. This is because the particles can build a weak structure of relatively large voidage by the cohesive force between particles. If a load is applied to such powder in a vessel to reduce the volume, the interstitial gas firstly supports the load since the particle structure is too weak to share the load. Thus, the interstitial gas pressure is increased. If the vessel opens to a low-pressure region through a small opening, gas flow occurs and the particles are forced to spurt out from the opening. At the same time, consolidation of powder proceeds and the load shared by the particles soon increases, which in turn reduces the gas pressure and the flushing finally ceases. In gravity flow of fine powders from an orifice of vessel bottom, the powders do not always flow out even if the size of orifice is large enough as compared with the particle size. This is because a powder arch is formed over the orifice. If we apply a suitable load on the top surface to break the arch structure, the particles will spurt out of the orifice. It is difficult for cohesionless particles,
0032-5910/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 ( 0 2 ) 0 0 0 7 6 - 1
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S. Kuchii, Y. Tomita / Powder Technology 126 (2002) 275–282
however, to produce the flushing by the same method since the voidage of such particles is usually low and the load is almost always supported by the particle skeleton or by the effective stress. In this paper, we will examine the phenomenon based on a gas – solid two-phase flow model using the Darcy’s law. An experiment was done in a rectangular vessel to observe the particle flow pattern during the flushing and to test our flushing model. However, computation based on the linearized equations, which omit the effective stress, is carried out to see the particle velocity distributions in the vessel at the beginning of the flushing when the effective stress is small.
where m is the kinematic viscosity of gas, gi is the gravitational acceleration and k is the permeability of Darcy’s law, that is,
2. Theory
p ¼ qf RT
If p is the interstitial pressure, rijs is the stress in the particle that supports the particle skeleton and e is the voidage, the total stress of powder rij is given by
where R represents the gas constant and T the absolute temperature. We assume the process is isothermal, that is,
rij ¼ ð1 eÞrsij epdij
rij ¼ rijV pdij
ð2Þ
Substituting Eq. (2) for Eq. (1), the stress of solid phase is written by rijV as ð1 eÞrsij ¼ rijV ð1 eÞpdij
ð3Þ
If we use the Cartesian coordinate, the equations of continuity for solid and gas phases are given by A A fð1 eÞqs g þ fð1 eÞqs ui g ¼ 0 At Axi
ð4Þ
and A A ðeqf Þ þ ðeqf vi Þ ¼ 0 At Axi
ð5Þ
where qs represents the solid density, qf the gas density, ui the solid velocity and vi the gas velocity. According to Zienkiewics and Shiomi [11], the equations of motion for both phases become, by using the effective stress, as ð1 eÞqs
Aui Aui þ uj At Axj
eqf
k¼
Avi Avi þ vj At Axj
¼ e
Ap e2 qf m ðvi ui Þ eqf gi ð7Þ Axi k
1 e3 d2 180 ð1 eÞ2
ð8Þ
where d is the particle size. The equation of state of gas is given by
T ¼ const:
ð9Þ
ð10Þ
ð1Þ
where rij is Kronecker’s delta and we follow the usual convention that a compressive stress is negative. Terzaghi and Peck [10] formulated rij by p and the effective stress rijV as follows,
and
¼ ð1 eÞ þ
Ap e2 qf m ðvi ui Þ þ Axi k
ArijV ð1 eÞqs gi Axj
ð6Þ
3. Experiment We produce the flushing by applying a piston load on the top surface of powder in a rectangular vessel that has a slit at the bottom. The dimension of the vessel is 60 mm in length, 13 mm in width and 80 mm in height. The slit is 3 13 mm and is set up in the center of the bottom. The vessel was made of transparent acrylic plates to observe the particle flow. The images of particle motion were taken in a field size of 60 80 mm by a high-speed video camera whose shutter speed is 1000 frames per second. To examine the particle motion in the vessel, we used tracer particles of 0.7-mm diameter. The 361 particles were placed in the nodes of a checkered pattern of 3 mm spacing 19 in length by 19 in height. To set the tracer particles in their initial positions, we at first put the particles in regularly positioned pits of a plate, cover with this the front side of the vessel that is then filled with the powder and replace the plate with the transparent acrylic plate. Before using this particle, we had observed the powder flow by using almost similar particles as the test powder but different color, which were filled in checkered pattern in advance. It was confirmed by comparing both flows obtained by different particles that the present particles practically trace the powder motion. The initial voidage e0 was determined by measuring the mass of powder that was poured into the vessel by using a funnel gently and as even as possible. To obtain large voidage, we used a sieve through which the powder was filled into the vessel. The powder is filled in the vessel 60 mm in height. The piston pressure DP was 5.03 kPa. A laser displacement sensor monitored the piston motion. There is a clearance between the piston and the vessel, and we insert
S. Kuchii, Y. Tomita / Powder Technology 126 (2002) 275–282 Table 1 Physical properties of tested particles Particle
Soft wheat flour
FCC catalyst
Particle size (Am) Material density (kg/m3) Angle of repose (degrees) Carr’s flowability index Carr’s floodability index
51.6 1480 49.0 40.5 56.0
59.0 2565 28.2 94.0 71.0
a sheet of cardboard between the top surface of the powder and the piston to avoid the leakage of powder and gas through the clearance. It was confirmed by observing the particle flow during experiment that the leakage was prevented. Powders used in this work are soft wheat flour and FCC catalyst. Table 1 shows the physical properties of the powders including the flowability and floodability indices of Carr, which were measured by Hosokawa Powder Characteristics Tester. By this experiment, it was found that the soft wheat flour is cohesive and gives rise to the flushing, while the FCC catalyst is cohesionless free-flowing and does not give rise to the flushing. Both powders belong to group A of Geldart’s map.
4. Numerical procedure It is supposed that the effective stress in the powder is minimum before the piston load is applied. The effective stress acts to decelerate the powder flow and finally stops the flushing. Thus, it is possible to omit the effective stress in order to examine the flow near the onset of flushing. Furthermore, we will linearize the governing equations since the flow in the vessel is slow. We also omit the gravity term since it is not critical for the flushing. Indeed, the flushing occurs through a horizontal nozzle [9]. We assume that the flow is two-dimensional, where x coordinate is horizontally set along the vessel bottom and y coordinate is taken vertically upward from the slit center. We put pressure and voidage as p ¼ p0 þ pV
ð11Þ
and e ¼ e0 þ eV
ð12Þ
where p0 and e0 are the initial pressure and voidage, respectively, and we assume that pVbp0
ð13Þ
contain the products of small quantities, the equation of continuity for solid phase becomes AeV Aux Auy ¼ ð1 e0 Þ þ ð15Þ At Ax Ay where qs is assumed constant. If we substitute Eqs. (9) and (10) for Eq. (5), omit the small terms and use Eq. (15), the continuity equation for gas phase becomes ApV 1 e0 Aux Auy ¼ p0 þ At e0 Ax Ay Avx Avy þ p0 Ax Ay
ð14Þ
Futhermore, we assume that the generated velocities of gas and solid phases are also small quantities like pVand eV whose products are neglected. If we omit the terms that
ð16Þ
The equations of motion for both phases become e20 qf 0 m Aux 1 ApV þ ¼ ðvx ux Þ k0 qs ð1 e0 Þ qs Ax At
ð17Þ
e20 qf 0 m Auy 1 ApV þ ¼ ðvy uy Þ qs Ay At k0 qs ð1 e0 Þ
ð18Þ
Avx 1 ApV e0 m ¼ ðvx ux Þ qf 0 Ax k0 At
ð19Þ
and Avy 1 ApV e0 m ¼ ðvy uy Þ qf 0 Ay k0 At
ð20Þ
where we assume that the permeability is constant as k0 that is evaluated by substituting e0 for e in Eq. (8). The powder in the vessel flows out from the bottom slit by the piston load. The piston motion decreases a region in the vessel occupied by the powder. We calculated the flow of solid and gas in this region, and apply the method of SOLA-VOF [12], which can solve the velocity and pressure fields in time marching for a region having moving boundaries. Table 2 shows calculating conditions, where p0 is the atmospheric pressure and the mesh points for calculation consist of 22 in x-direction by 52 in y-direction. Table 2 Conditions of numerical calculation Computing range Initial conditions
and eVbe0
277
Boundary conditions
Numbers of mesh
0 V x V 0.030 (m) 0 V y V 0.075 (m) Pressure: p = p0 (0 V x V 0.030, 0 V y V yp) Velocity: ux = uy = vx = vy = 0 (0 V x V 0.030, 0 V y V yp) Voidage: e = e0 (0 V x V 0.030, 0 V y V yp) Pressure: p = p0 + DP (0 V x V 0.030, y = yp); p = p0 (0 V x V 0.0015, y = 0) Velocity: ux = vx = 0 (x = 0, 0.03, 0 V y V yp); uy = vy = 0 (0.0015 V x V 0.030, y = 0) x y = 22 52
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5. Results and discussion 5.1. Flow pattern during the flushing Fig. 1 shows the photographs of flow of FCC catalyst in the vessel for different times from the beginning of measurement, where Vp is the measured piston velocity and B is the slit width, that is, B = 3 mm. Fig. 1(a) shows
Fig. 1. Flow pattern of FCC catalyst (e0 = 0.618).
that the particles already discharge at the slit before the piston load is applied. It is well known that when the particles is discharged freely by the gravity alone from the slit, the discharge of particles depends on the slit width B and the gravitational acceleration g, and pffiffiffiffiffiffithe velocity of particles at the slit is in proportion to gB [13], which is 0.1715 m/s in this case. Therefore, in p the figure, we scale ffiffiffiffiffiffiffiffi pffiffiffiffiffiffi the velocity and time by gB and B=g , respectively. Indeed, it is found that Vp is the same order of magnitude as the mean particle velocity in the vessel cross-section that is predicted by the discharge without the piston load if the bulk density case pffiffiffiffiffiis ffi constant, that is, in the p ffiffiffiffiffiffi of Fig. 1(b), Vp ¼ 0:03 gBc0:005 m=s and ðB=2aÞ gBc0:008 m=s where a is the half length of the vessel and is 30 mm. As the piston goes down, there appears a funnel type of free surface of powder as shown in Fig. 1(c). If the apex of the funnel reaches the slit, the discharge stops. This flow pattern is typical of free-flowing materials in the gravity discharge without surcharge on the top surface of powder and is different from that in the flushing phenomenon. This suggests that the piston load does not affect the particle discharge from the vessel for the free-flowing materials. Fig. 2 shows the photographs of flow of soft wheat flour. The flow pattern is distinctly different from that in Fig. 1 and seems to be typical of flushing phenomenon. As shown in Fig. 2(a), the particles are at rest unless the piston load is applied, which is different from the case for free-flowing materials. When the load is applied, the particle flow firstly occurs near the slit and below the piston as shown in Fig. 2(b). It is found that the piston velocity is much higher than that in Fig. 1, and then the duration of particle motion becomes shorter than that in Fig. 1. It is noted that the particle free surface does not appear and the flow stops on the way. This is because the pressure difference necessary to produce the particle flow disappears by the increased effective stress. In the following, we exclusively focus on the flow of soft wheat flour since the motion of FCC catalyst is not caused by flushing but gravity. The particle velocity in the gravity discharge is much smaller than that of the flushing particles as shown above, and the particles in the gravity discharge are assumed to move in the contact with each other in the vessel. Then, the effective stress is an important governing factor of the flow. However, it is still difficult for the present model to predict the onset of flushing since the calculation is based on a fluid model. Even if the effective stress is small, it is necessary to consider it, of which constitutive equation should be a plastic one having a yield stress, in order to explain the onset. Thus, we cannot give a criterion on the applicability of the present calculation model in terms of the particle properties, and the calculation for the FCC catalyst by this model will be insignificant while giving similar results as those for soft wheat flour since the particle size and the density are almost the same with each other as shown in Table 1.
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region near the slit, which we call the flushing region, and apslow ffiffiffiffiffiffi flow region in both corners near the bottom. Since pffiffiffiffiffiffi gB is 0.1715 m/sm the particle velocity is larger than gB in the pffiffiffiffiffi ffi flushing region and is the same order of magnitude as gB in the plug-like flow region, being slightly smaller than Vp. Fig. 5 shows the numerical result on the same condition as the measurement in Fig. 3. It is confirmed that there are the flushing region near the slit and the plug-like flow region below the piston as in the measurement. The numerical result reproduces the particle flow qualitatively. The
Fig. 2. Flow pattern of soft wheat flour (e0 = 0.726).
Fig. 3 shows the photographs of flow of soft wheat flour when e0 = 0.819 and ffiFig. 4 is the corresponding velocity qffiffiffiffiffiffiffiffiffiffiffiffiffiffi contours of u2x þ u2y in different times, where the piston velocity Vp is 0.205 m/s in (a), 0.223 m/s in (b) and 0.223 m/s in (c), respectively. The dark circular region in the photograph is an afterimage of lens. It is noted that the particle velocity is increased with an increasing e0. There are a plug-like flow region below the piston, a high-speed
Fig. 3. Flow pattern of soft wheat flour (e0 = 0.819).
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Fig. 4. Velocity contours of soft wheat flour for e0 = 0.819 (experiment).
piston velocity of the calculation is larger than that of measurement. This is because the effective stress that will act to resist the particle motion is omitted in the calculation. It is found that the particle motion firstly occurs near the slit as well as just p below ffiffiffiffiffiffi the piston, and that the particle velocity is larger than gB in a region near the slit, which is called the flushing region. The calculation confirms these results for the same conditions as measurement.
Fig. 5. Velocity contours of soft wheat flour for e0 = 0.819 (numerical calculation).
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5.2. Velocity distribution Fig. 6 shows the relationship pffiffiffiffiffiffiffiffi between umax/Vp and y/a for soft wheat flour at t= B=g ¼ 3:1 for two different initial voidages where umax is the downward particle velocity on y axis and yp is the current piston position. In the measurement, umax/Vp increases with e0 near the slit while the calculation fails to explain the effect of e0 on the profile near the slit. This indicates that the resistance to the particles motion is reduced when the voidage is large. Similar experimental results are obtained for other piston positions. However, the effect of voidage disappears toward the piston, which shows that the particles near the piston are moved en masse by the piston. Near the piston region where umax/Vp is almost constant, we can assume that the distribution for a given time will take the following. x uy ¼f a Vp
ð21Þ Fig. 7. uy/Vp vs. x/a for soft wheat flour for e0 = 0.819.
Fig. 7 shows the distribution by Eq. (21) with z/a as a parameter, where z is a distance from the current piston position yp, that is, z = yp y. Since the number of tracer particles is limited, we plotted the experimental particle velocity collecting into each small range of z. Although the scatter is considerable in the measurement, the profile is pluglike and is weakly dependent on x/a. The numerical result also shows a similar profile, where it is noted that no-slip condition at the wall is not satisfied since the governing equations contain the first-order derivatives as to the velocity. Near the slit region, the effect of piston motion disappears and the motion of particles is in local equilibrium like
Fig. 6. umax/Vp vs. y/a for soft wheat flour.
a turbulent jet. Thus, we can expect the following velocity distribution for given e0 and yp, uy x ¼f y umax
ð22Þ
Fig. 8 shows the distribution by Eq. (22), which shows that uy/umax is a function of x/y and weakly dependent on z as well. The numerical result is also dependent on z. We can expect a kind of self-preserving flow in the region far from a piston. Such a flow is also found in a blow tank solid conveyor for free-flowing powders [14]. Generally, in such
Fig. 8. uy/umax vs. x/y for soft wheat flour for e0 = 0.819.
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a region, particle velocity distribution is expressed by Eq. (22). In this region of z/a, the effect of piston motion almost disappears.
6. Conclusion We have defined the flushing phenomenon as spurting of powder due to air stream generated by an increase in interstitial gas pressure in the powder bed when an external load is applied to the powder. In order to test this mechanism, we carried out experiments and formulated the phenomenon based on gas – solid two-phase flow modeling to perform numerical analysis for given experimental condition. It is found that when a piston load is applied, the particle flow at first begins near the slit, which is remote from the piston, and the velocity of particles is considerably larger than that of free-flowing due to the gravitational acceleration. The numerical analysis explains the phenomenon qualitatively. List of symbols a half length of the vessel B slit width d particle diameter g gravitational acceleration k permeability of Darcy’s law k0 permeability of Darcy’s law of initial voidage p fluid pressure p0 atmospheric pressure pV pressure disturbance R gas constant t time T temperature umax maximum velocity of uy yp current piston velocity ux x-component of solid velocity uy y-component of solid velocity vx x-component of gas velocity
vy Vp x y z DP
y-component of gas velocity piston velocity Cartesian coordinate in horizontal direction Cartesian coordinate in vertical direction distance from the current piston position piston pressure
Greek letters dij Kronecker’s delta e voidage e0 initial voidage eV voidage disturbance rij total stress of powder rijs interparticle stress rijV effective stress m kinematic viscosity of fluid qf fluid density qf0 fluid density of initial voidage qs particle density
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