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Incipient motion of solid particles in horizontal pneumatic conveying
51
Powder Technology, 72 (1992) 51-61
Incipient motion of solid particles in horizontal Francisco Deparhent
pneumatic
conveying
J. Cabrejos of Mechanical Engineering, University of Pittsburgh, Pittsburgh, PA 15261 (USA)
and George
E. Klinzing
Department of Chemical and Petroleum Engineering Univemiry of Pittsburgh, Pittsburgh PA 15261 (USA) (Received
July 22, 1991; in revised form January
30, 1992)
Abstract A technique for finding the minimum pickup velocity of solid particles in horizontal pneumatic conveying has been developed. A general semi-empirical correlation based on the Archimedes number which is valid over a range of particle size from 10 to 1000 pm is presented for the prediction of the minimum gas velocity required to pick up particles. Experiments have been carried out with fine particles (glass beads, alumina and coal) in a 52 mm id. transparent pipeline. Noteworthy is the existence of a minimum point in the curves of minimum pickup velocity as a function of particle diameter for particle diameters of approximately 100 Frn. Maximum conveyability is expected to occur at this particle size. Above and below this minimum point, higher gas velocities are required to pick up particles lying on the bottom of a pipe. Experiments in an actual pneumatic conveying system clarify the practical significance and usefulness of the minimum pickup velocity. Thus, it is suggested that the proposed test procedure be used as a unified technique in determining the minimum transport velocity of solid particles in horizontal pneumatic conveying systems.
Introduction
Conveying velocity is considered one of the most important parameters in the pneumatic transport of solids. Successful design and operation of pneumatic conveying systems depends upon the prediction or determination of the minimum conveying velocity at which the solids may be conveyed steadily through a pipeline. An unnecessarily high conveying velocity will result in higher energy costs due to an increased pressure drop in the system, increased erosion, and increased solids degradation. On the other hand, a low conveying velocity can result in the deposition of solids on the bottom of the pipe or can even block the pipeline. Thus, a general correlation to predict the minimum conveying velocity of solids in dilute-phase systems may be very useful for engineers and designers. There are many terms used to refer to minimum conveying velocity: saltation velocity, pickup velocity, suspension velocity, deposition velocity, rolling or sliding velocity, critical velocity, etc. Definitions of these terms are based on visual observations and pressure drop measurements, and they are often applied to indicate some transition in the way in which a particle is moving or begins to move. In fact, the terms pickup velocity and saltation velocity are often used interchangeably in dilute-phase conveying.
0032-5910/92/$5.00
Bagnold [l] was one of the first investigators in this field. With his experience in the Sahara desert and his study on sand movement in a wind tunnel, he developed a theory explaining the formation of desert dunes. He used the terms ‘fluid threshold’ and ‘impact threshold’, which correspond to pickup and saltation, respectively. Since then extensive work has been carried out on the prediction of minimum conveying velocity and several published correlations have been reviewed in detail by Plasynski et al. [2] and Jones and Leung [3]. The ‘pickup velocity’ concept is closely related to saltation velocity. The air velocity in a horizontal line at which the particles start to drop out of suspension and settle on the bottom of the line is usually called saltation velocity. Pickup velocity may be defined as the fluid velocity required to resuspend a particle initially at rest on the bottom of a line [4]. Pickup differs from saltation in relation to the initial state of the particle. Keeping gas velocity above minimum pickup velocity in all horizontal sections of a conveying line ensures no deposition of solids in the system. In general, this term can be defined as the safe fluid velocity for the horizontal conveyance of solids. If this velocity is set at the beginning of a piping system, the velocity will increase along the pipeline due to pressure drop and density decrease so the rest of the pipeline should be well above this lower pickup velocity bound.
0 1992 - Elsevier Sequoia. All rights reserved
52
The mechanism of pickup and saltation is complex. It was not possible to find a unique and general expression that correlates with all the parameters affecting the minimum conveying velocity. The main reason for the diversity of data on the pneumatic conveying of solids is the variety of materials handled. Only two studies aiming to predict the pickup velocity of particles were found: the graphic method proposed by Halow [5] (fluid velocity required to initiate sliding, rolling and suspension of particles) and the correlation done by Azizov and Toshov [6] (blowing-away critical velocity). Zenz [7] experimentally determined single-particle saltation velocities (minimum fluid velocity required to carry solids at a specified rate without allowing them to settle out in any horizontal pipe) for different spherical and angular materials and proposed a graphic representation based on two dimensionless parameters, $!&&$ and ?m. He showed that the minimum velocity required to pick up a particle from a layer and transport it in suspension can be 2 to 2.5 times greater than the minimum velocity required to transport an injected particle without saltation. The models proposed by these investigators will be used later for comparison with the present work. Matsumoto et al. [8] experimentally studied the effect of particle size on the minimum transport velocity for horizontal pneumatic conveying of solids over a wide range of particle sizes from 20 to 1600 pm. They found that for a given material there exists a particle size which provides the maximum conveyability of solids and which establishes a distinction between fine- and coarse-particles. Matsumoto et al.3 data of minimum
transport velocity (saltation) of glass and copper beads as a function of particle size show a minimum at diameters of 500 and 100 pm, respectively. Their correlations for the minimum transport velocity in horizontal pneumatic conveying are based on the solids loading ratio, pipe diameter, and material characteristics.
Experimental A technique for finding the pickup velocity of solid particles in horizontal pneumatic conveying has been developed. This technique is based on visual observation of the behavior of particles lying on the bottom of a transparent pipe when the air streamvelocity is gradually increased. Simple and inexpensive experiments can be performed to measure the minimum pickup velocity for a given material. The setup consists of a transparent, 7 m long, 52 mm i.d. pipeline with a removable section, a compressed regulated air supply, and a solids collector. The details of the experimental setup and procedure can be found elsewhere [9]. Figure 1 shows schematically the setup designed to carry out pickup velocity experiments. In order to carry out the measurements, a 1 m long layer of particles with approximately half of the crosssection area of the pipe left free is created using a two gate arrangement and a removable section. The procedure to create the initial layer of particles on the bottom of the pipeline is graphically shown in Fig. 2. A constant air flow rate is set through the pipeline Paper Filter Bags
t
b-3m-+lm+-3m_-_Y
I,
I /
r
Transparent Exelon 52 mm I.D. Pipeline
Removable Section
Initial Layer of Solids
r
I
1
hA YY
Control Valve
Pressure Regulator
Main Valve
I-
Solids Collector 8 House Air Supply
Fig. 1. Schematic
representation
of the setup used for pickup velocity experiments.
53
1
Solids
(1)
(2)
(4)
Fig. 2. Solids filling procedure to create the initial layer of particles inside the bottom of the removable section.
0
minimum pickup velocity “gp”*--__-----_-_---_-_-
Fig. 3. Determination of the minimum pickup velocity, system reaches automatically the equilibrium condition.
as the
and the layer erodes slowly as the air picks up the particles on top of it. As the free cross-section area increases, the air velocity over the layer decreases and so does the capacity of the stream to pick up more TABLE
1. Characteristics
Material
of the materials
particles. Since the described phenomenon takes place continuously, a final equilibrium is automatically reached when no more erosion occurs. This condition corresponds to the minimum pickup velocity, as shown in Fig, 3. The velocity is then increased and a similar test is performed until a similar non pickup condition is again achieved. It is important to point out that the rate at which equilibrium is reached depends on the air flow rate but the final minimum pickup velocity is independent of the initial conditions. Pickup velocity is defined as the air stream velocity required to pick up or entrain particles from a layer of solids on the bottom of the pipe. By measuring the air flow rate and the free cross-section area remaining over the particles, the minimum pickup velocity can easily be calculated using the following equation:
A wide variety of solids varying in size, shape, and density was tested. Table 1 summarizes the characteristics of all the materials used. Glass beads, crushed glass, and alumina particles were sieved with Tyler standard screens to give several fractions, each having a fairly uniform particle size. The particle size of the ESP dust (alumina dust from electrostatic precipitators) and the coals was determined with a Microtrac particle size analyzer, where the mean particle diameter corresponds to d,,.
used for pickup velocity experiments
Density
Particle
(kg mm3)
(pm)
(pm)
glass beads
2480
crushed
2480
d,<45 45 cd,<53 53 -cd,<75 75 -cd, < 150 3OO
22 49 64 112 450 49 112 225 450 800 22 60 112 225 450 7.76 21.5 14.8 6.75
glass
alumina
3750
ESP dust Wyodak coal Elkhom coal Illinois #6 coal
3000 1300 1300 1300
size
Mean d,
Shape
0.03 0.15 0.24 0.61 3.37 0.12 0.42 1.08 2.19 3.15 0.05 0.27 0.69 1.71 3.48 0 0.02 0.01 0
spherical spherical spherical spherical spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical
54
Results
20
ESP dust
Figures 4-7 show the results obtained for the minimum pickup velocity of each material tested. The vertical bars in the graphs represent the range - 2a< U,, < + 2u, which corresponds statistically to 95% of the values. Several experiments were performed to check accuracy and reproducibility.
F
15
5 2 m 3
Illinois 10
wal
#6
it
t U. Elkhom
coal Wyodak
lb
coal
2’0 dp
(w)
’
Fig. 7. Minimum pickup velocity for ESP dust and coals (nonspherical particles, 3000 and 1300 kg rne3, respectively).
2-
0 0
200
I 400
i 600
600
dp
1000
(w)
Fig. 4. Minimum pickupvelocity for glass beads (spherical particles, 2480 kg m-‘).
I
Z_( i -0 I
200
400
600
coo
dp
1000 '
(w)
Fig. 5. Minimum pickup velocity for crushed glass (non-spherical particles, 2480 kg me3).
20 3 g
16-
2
12-
3
* 84-
0 0
200
400
600
so0
1
dp W’O Fig. 6. Minimum pickup particles, 3750 kg me3).
velocity
for alumina
(non-spherical
One of the most important results obtained is the existence of a minimum point in the curves of minimum pickup velocity as a function of particle diameter, which corroborates Matsumoto et al.‘s findings [8]. These minimum points appear at particle diameters of 70, 100 and 150 pm for glass beads, crushed glass and alumina, respectively. Noteworthy also in the test for the finer materials such as ESP dust and fine coals is the manner in which the particles left the surface of the layer. For the ESP dust, the particles were lifted from the top of the layer, while in the case of the coals, the particles disappeared from the leading edge of the layer rather than from the top. Interparticle cohesiveness is drawn up in order to explain these two behaviors, ESP dust being less cohesive than the fine coals tested. Another interesting finding is that spherical glass beads have a lower pickup velocity than crushed glass solids. This can be attributed to the higher degree of interlocking of the crushed particles that results in a higher resistance to pickup in comparison with spherical material. Also of note is the fact that the denser materials required higher pickup velocities indicating the controlling effect of the inertial forces for these dense materials. The Geldart fluidization diagram is widely used for classifying powders into four groups based on the mean particle size and the density difference between the particles and the fluidizing gas [lo]. The distinction between powders A (aeratable) and powders B (sandlike) is located at 90 and 60 pm for glass and alumina, respectively. Although Geldart’s diagram is related to the fluidization characteristics of powders and not the pneumatic conveyance, it is interesting to note that these values are close to those found for the minimum point in the curves of minimum pickup velocity as a function of particle diameter. Further work is required to study the similarity, if any, between Geldart’s diagram and the minimum points of the pickup velocity plots.
5.5
Particle dynamics
Y
A model was developed for the incipient motion of a single sphere initially at rest on the bottom of a horizontal pipe and subjected to a steady fully developed turbulent flow of air. The behavior of the particle is determined mainly by the interaction between the forces acting on it, the physical properties of the particle and the velocity of the air stream. The most important forces involved in the pickup mechanism of a single particle are shown in Fig. 8. Gravitational, buoyant, and adhesive forces depend only on the physical properties of the particle and gas density, and are independent of the gas stream velocity. The drag and lift forces depend on the gas stream velocity, while the frictional force is proportional to the coefficient of sliding friction. Forces due to electrostatics and lift due to particle rotation were not taken into account in the model. The fact that the gas stream adheres to the wall (no slip condition) means that frictional forces retard the motion of the gas in a thin layer near the wall, the so-called boundary layer. Based on the thickness of the laminar sublayer (a,), a distinction between ‘large’ and ‘small’ particles is made. A small particle (d, < 6,) lies deep in the boundary layer, mainly inside the laminar sublayer, and is exposed to smaller velocities and to a linear velocity profile [ll]. Larger particles (~,zs- SJ face larger velocities and flatter velocity profiles, as shown in Fig. 9. Although the distinction between large and small particles is not precise, it helps to determine the significant forces acting on the particle and the velocity profile facing the particle. The adhesive force, mainly van der Waals attraction between the sphere and the wall, is negligible for large particles. Also in this case, the almost uniform velocity profile of the gas stream exerts no lift over the sphere. A single particle initially at rest on the bottom of a horizontal pipe begins to move when the forces in
4 4 Air stream
Pipewall
x _i
FW&(
///jj/jj7jjjjjjjjjjjjjjjjjjjjjjjjj/ ,.
Lifl Buoyancy
)$
\/ Gravity + +
Adhesion
Fig. 8. Forces acting on a single sphere initially at rest on the bottom of a pipe, facing a steady fully developed turbulent flow of air.
4
VelocitvProfile
Small Particle
’
Large Particle
Fig. 9. Schematic representation of the velocity profile and two particles of different sizes lying on the bottom of a pipe, showing the thickness of the laminar sublayer (6,).
the horizontal direction are zero (Newton’s second law). If the gas velocity is gradually increased, the drag exerted on the particle will gradually increase until it finally equals the frictional force. Combining the expressions developed for each of the forces involved and solving for the gas velocity, it is possible to find the gas velocity required to initiate motion or to pickup a single particle lying on the bottom of a pipe (refer to the Appendix for more details). l For a large single sphere, the following equation was developed:
(2) l
For a small single sphere, the following equation was found:
1.54
lo-4[1-(~)“s]-2cDpfd~(~)1 -6.35
( )I
Vluo 1O-3 pfdP3 -&
l/8
(3)
To assess the relative significance of the two models, it is helpful to display graphically the solution of eqns. (2) and (3). F’g 1 ure 10 compares the minimum pickup velocity for the following case: a sphere of different densities (1000,2000,3000 kg m-‘) lying on the bottom of a 52 mm i.d. horizontal pipe (employed in the experiments) subject to an air stream passing over it. The coefficient of sliding friction between the particle and the wall is assumed to be constant and equal to 0.45. Single-particle experiments were carried out with glass beads and ceramic spheres to verify the model developed for the incipient motion of a single particle. Figure .ll compares the experimental results (mean values) with the minimum pickup velocity predicted by the model. It can be seen that calculated and measured values are in good agreement.
56
by other investigators, although neither Halow nor Azizov and Toshov predict any minimum point in the curves of gas velocity as a function of particle diameter. Zenz’s method predicts such a minimum point at d, equal to 50 pm for the given example. Also, none of these methods takes into account the coefficient of sliding friction between the particle and the pipe wall. As expected, pickup velocity is higher than saltation velocity. 100 2ObO
dp
@m)
30bo dp
200 4000
OLm)
General correlation for multiple-particle systems
Fig. 10. Comparison of the minimum pickup velocity for a sphere using both the small- and large-particle models, with an enlargement of the low range of particle diameters (52 mm i.d. pipe).
6
l
3‘
o
Glass&ad
Combining the single-particle model with the experimental results for the minimum pickup velocity of a layer of particles, a general correlation with practical applicability for prediction of the minimum gas velocity required to pick up particles lying on the bottom of a horizontal pipe was developed. This semi-empirical correlation is valid over the range of particle sizes from 10 to 1000 pm and has the following expression:
0
U,, = [1.27Ar-“3+ 0.036Ar’”
Ceramic
00
lm. - . 0
+0.45][0.70Ar-“5 where:
2
3
4
5 dp
6
7
(mm)
10
!
d
(4)
I
1
Fig. 11. Comparison between the experimental results and the predicted values of the minimum pickup velocity for single spheres.
0
+ l]U,,,,
I 200
400
600 dp
660
lOi0
bm)
Fig. 12. Comparison between the model developed spheres and the work done by other investigators.
for single
In order to compare the proposed model with the work done by Halow [5], Azizov and Toshov [6], and Zenz [7] Fig. 12 shows the results for the case of a single sphere of density 2480 kg mm3 lying on the bottom of a 52 mm i.d. horizontal pipe facing an air stream. It is important to point out that the proposed model is in quite good agreement with the methods proposed
Ar= J_ -ps-pr dp3, Archimedes number. v* Pf
(5)
The first term on the right hand side of eqn. (4) takes into account the effects of particle interactions with other particles while the second term accounts for the particle shape (equal to one for spherical particles). The Archimedes number, defined by eqn. (5), was used to correlate these effects. It characterizes the basic particle/gas properties and is related with the Froude number, the Reynolds number and the ratio of gravity to buoyant force. The model for prediction of the minimum pickup velocity of a single spherical particle ( UgPUO) already takes into account pipe diameter, air properties, particle characteristics, coefficient of sliding friction, drag coefficient, and the presence of surrounding walls. In order to evaluate the accuracy of the proposed correlation, the minimum pickup velocity was calculated (eqn. (4)) and plotted against the experimental results for all the materials tested, as shown in Fig. 13. The agreement between measured and predicted values for the minimum pickup velocity, shown in Fig. 13, was found to be generally within the accuracy of the tests, considering the large number of variables involved in the pickup mechanism and the dispersion
57
0
4
6
12
Ugpu
Fig. 13. Comparison between the predicted for minimum pickup velocity.
16
predlcted
(m/s)
and measured
values
of the experimental measurements. It can be seen that except for a few cases, the general correlation developed for the minimum gas velocity required to pick up particles lying on the bottom of a pipe predicts slightly higher values as compared with the experimental results. Two indicators widely used for comparison, the average absolute deviation (AAD) and the square root of the mean of the square of the deviations (E), were found to be equal to 2.1 m s-l and 31%, respectively, for all the materials tested in the particle size range between 10 to 1000 pm. The indicators are defined as follows for a data set of N points:
E=
J
1 N-l
.j~ (&,u “F-
u,~
y-d)’
1oo%
gpu measured
(7) Thus, eqn. (4) can be used satisfactorily to predict the minimum pickup velocity of solid particles in horizontal pneumatic conveying in the particle size range between 10 to 1000 pm. It is important to point out that for spherical particles all the points lie within 15%, as shown in Fig. 13. For smaller particles, the agreement between predicted and measured minimum pickup velocity is less reliable.
Pilot-plant
tests
In order to compare the minimum pickup velocity (Up,“) obtained by using the new technique proposed and the minimum transport velocity that would be required to convey solids without allowing them to settle out in a horizontal line, experiments in an actual pilotplant have been carried out. This will clarify the practical
significance and usefulness of the concept of minimum pickup velocity. The closed-loop pneumatic conveying system used, shown schematically in Fig. 14, consists of a 14.5 m long horizontal test pipeline with a 2.2 m vertical section, a return line, T-bends and several transparent sections placed along the pipeline; a compressed regulated air supply that provides the air velocity and pressure necessary to convey the solids; a hopper, placed on a scale and connected through flexible connections to the collector and feeder in order to weigh the solids inventory continuously; a solids collector with a paper filter bag, placed on top of the solids hopper; and the solids feeder, a butterfly valve that ensures a constant gravity downflow of solids at different flow rates depending on the position of the valve. The pipeline was grounded by wrapping a conducting wire so that electrostatic effects could be minimized. Experiments were carried out by reducing the gas velocity at a constant flow rate of solids until the particles began to drop out of suspension and formed a dune on the bottom of the transparent section in the horizontal line (saltation velocity). The particles were 450 pm spherical glass beads with a density of 2480 kg mP3, one of the materials used in the pickup velocity experiments. Figure 15 shows the results obtained for the saltation velocity at different solids loading ratios and the minimum pickup velocity. It is important to point out that the minimum pickup velocity, as defined in the present work, is independent of the solids loading ratio. Two of the correlations widely used for prediction of saltation velocity are also included in Fig. 15. Meyers et al. [12] defined saltation velocity for coarse particles as the optimal operating condition of a horizontal pneumatic conveying system (pressure minimum point), whereas Matsumoto et al. [8] defined it as the minimum superficial air velocity required for the transport of solids without the formation of a stationary bed on the bottom of a horizontal pipe. It can be seen that both correlations predict slightly higher saltation velocities as compared with the experimental results, for solids loading ratios over 5. Zenz [7] predicted minimum pickup velocities 2 to 2.5 times greater than the saltation velocity. Figure 15 shows that this looks to be the case only in the very dilute-phase conveyance, i.e. for solids loading ratios less than 0.5. As seen in Fig. 15, keeping gas velocity above the minimum pickup velocity in all horizontal sections of a conveying line ensures no deposition of solids in the system. So, the practical significance of the minimum pickup velocity is that it can be used as the safe gas velocity for the horizontal conveyance of solids. The
58
Filter bag
Flexible connection
to equalize the pressure
Hopper
Transparent section
I
I
A/(
I
3
I
-
YY
Control Valve Rotameter
Fig. 14. Schematic
I
..,_L
Transparent section
Transparent section
Manometer
oQ4
representation
P
Pressure Regulator
~c
Main Valve House Air Supply
of the setup used for saltation
velocity experiments.
Conclusions
2
r-Gzizq
1
1
Fig. 15. Comparison between minimum pickup velocity (CT,,) and saltation velocity (U,) as a function of the solids loading ratio.
usefulness of the concept of minimum pickup velocity is that one of the most important parameters for the design of pneumatic conveying systems, i.e. the minimum transport velocity, can be predicted by performing relatively simple and inexpensive experiments with the technique developed.
The concept of pickup velocity of solids was studied from an experimental and theoretical viewpoint. The new technique proposed for prediction of minimum pickup velocity of solids was successfully tested. In addition, a general semi-empirical correlation was developed, based on a model for the incipient motion of a single particle, the boundary layer thickness, and the Archimedes number. The pickup velocity for different solids cannot be easily predicted because this parameter is influenced by many diverse variables. Among these, the characteristics of the material itself, such as particle size, density and shape, the coefficient of sliding friction, and the particle interaction with other particles are the most important variables that affect pickup velocity. Experiments in an actual pneumatic conveying system have ratified the usefulness of the concept of minimum pickup velocity. Both technique and correlation provide new alternatives for engineers and designers to predict the minimum transport velocity of solids. Finally, it is suggested that the proposed test procedure be used as
59
a unified technique in determining the minimum transport velocity of solid particles in horizontal pneumatic conveying systems.
List of symbols A free AH A, Ar Cl3 D dP F, Fb Fd F* Fg F, P g
Qair R% s u, u SP u u
gPU ECPUO
free cross-section area remaining over the layer of solids, m2 constant in eqn. (AlO), Nm particle projected area normal to the flow, m2 Archimedes number, drag coefficient, pipe diameter, m mean particle diameter, m London-van der Waals adhesive force, N buoyant force, N drag force, N frictional force, N gravitational force, N lift force, N normal force, N coefficient of sliding friction, acceleration due to gravity, m sm2 volumetric flow rate of air, m3 s-l particle Reynolds number, separation between the sphere and the wall in eqn. (AlO), m mean gas velocity, m s-l gas velocity at the center of the sphere in eqn. (A13), m s-* minimum pickup velocity, m s- ’ minimum pickup velocity for single particle, m S
UF.S U,
-1
saltation velocity, m s-l terminal settling velocity, m s-l
Greek letters thickness of the laminar sublayer, m 6 V fluid kinematic viscosity, m2 s- ’ solids loading ratio, IJ solid particle density, kg mm3 PS fluid density, kg mm3 Pf shear stress at the wall, N m-’ 70
References R. A. Bagnold, The Physics of Blown Sand and Desert Dunes, Methuen, London, 1941. S. Plasynski, S. Dhodapkar, G. Klinzing and F. Cabrejos, AZChE Symp. Ser., 87 (1991) 78. P. J. Jones and L. S. Leung, Z’neumotransport 4, Paper Cl (June, 1978), pp. 1-12.
4 S. J. Rossetti, in N. P. Cheremisinoff and R. Gupta (eds.), Handbook of Fluti in Motion, Ann Arbor Science Butterworths, Boston, 1983, pp. 635-652. 5 J. Halow, Chem. Erg. Sci., 28 (1973) 1. 6 A. Azizov and V. R. Toshov, J. Appl. Mech. Tech. Phys., 27 (1987) 855. 7 F. Zenz, Znd. Erg Chem., Fundam., 3 (1964) 65. 8 S. Matsumoto, M. Kikuta and S. Maeda, J. Chem. Erg. Jpn., 10 (1977) 273. 9 F. Cabrejos, it4.S. Thesis, School of Engineering, University of Pittsburgh, 1991. 10 E. Geldart, Gas Fluidization Technology, Wiley, New York, 1986. 11 H. Schlichting, Boundary-Layer Theory, (4th edn.) McGrawHill, New York, 1979. 12 S. Meyers, R. Marcus and F. Rizk, Bulk Solids Handling, 5 (1985) 779. 13 R. D. Marcus, L. S. Leung, G. E. Klinzing and F. Rizk, Pneumatic Conveying of Solids, Chapman & Hall (London, 1990). 14 A. H. Kom, Chem. Erg., 57 (1950) 108. 15 H. C. Hamaker, Physicu, 4 (1937) 1058. 16 W. B. Pietsch, J. Eng. Znd. ASME (1969) 435. 17 P. G. Saffian, J. Fluid Mech., 22 (1965) 385. 18 P. G. Satfman, J. Fluid Mech., 31 (1968) 624.
Appendix: sphere
Modet for the incipient
motion of a single
Lalge-particle model For ‘large spheres’, the size of the particle is considered much larger than the thickness of the laminar sublayer (d,s SJ. The significant forces acting on a large-single particle lying on the bottom of a horizontal pipe are: drag force (acting in the same direction as the flow), friction force (acting in the opposite direction as the drag force), buoyant force and gravitational force. The velocity distribution of the steady fully developed turbulent flow of air past the sphere is assumed to be uniform in this case. The occurrence and effects of gravitational force (F,), which is always present, are well known and for a spherical particle the equation is: F, =mpg=
: psdp3g
(Al)
where mp is the mass of a particle of density pS and diameter d, Provided that the particle is completely surrounded by the fluid, the buoyant force (F,,) or upthrust can be expressed as:
w The drag force (FJ caused by a uniform flow on a single spherical particle at rest is given as [13]:
wo where C,, denotes the drag coefficient for a single particle in an undisturbed and unbounded fluid. The presence of the pipe walls interferes with the flow, mainly in the region adjacent to the wall, and its effects cannot be neglected. Marcus et al. [13] recommend Munroe’s equation to account for the wall effect on the drag coefficient of a single particle in a pipe of diameter D. Substitution in eqn. (A3) yields the expression:
(A4) Friction is the resistance that is encountered when a solid body slides or tends to slide over a solid surface. The standard form for the frictional force (Ff) is: Ff =fsFn
(A3
where fs denotes the coefficient of sliding friction (here defined as the tangent of the angle at which the pipe must be raised above the horizontal to cause the particle, initially at rest, to begin to slide). In this case, the normal force (F,) acting on the particle is equal to the difference between the gravitational and the buoyant forces: F,=F,--Fb
(A6)
The adhesive forces between the particle and the wall are negligible for large particles. Also in this case, the uniform velocity distribution of the flow exerts no lifting over a spherical particle [14]. Applying Newton’s second law, a large-single particle initially at rest on the bottom of a pipe will begin to move when the forces in the horizontal direction are zero. In other words, if the fluid velocity is gradually increased, the drag exerted on the particle will gradually increase until it finally equals the frictional force. Thus, motion will take place when:
&=F,=fJE;,-F,I
Gw
Substituting eqns. (Al), (42) and (A4) (for each of the forces involved in eqn. (A7) leads to:
Solving for the gas velocity U,, the velocity of the uniform flow required to initiate motion or to pickup a single particle lying on the bottom of a horizontal pipe can be written as:
(A9
It is important to point out that eqn. (A9) is an implicit equation because the drag coefficient is a function of the particle Reynolds number, and thus the gas velocity. Small-particle model
In the case of a ‘small sphere’ lying on the bottom of a pipe inside the laminar sublayer (d, < S,), the most important forces acting on the particle are shown in Fig. 8. The particle is subjected to a steady fully developed turbulent flow of air with a linear velocity distribution as shown in Fig. 9. The expressions for gravitational and buoyant forces remain the same as those developed in the large-particle model, and are given by eqns. (Al) and (A2), respectively. Hamaker [15] suggested the following expression for the adhesive force (F,), mainly London-van der Waals attraction between a spherical particle close to a flat wall, that acts downward. Considering a separation of 0.08 pm between the particle and the wall, suggested by Pietsch [16], and a value of lo-” erg for the constant A, given by Hamaker, the adhesive force then becomes: F,=
44, 24s2
=
lo-l9 d, 12 (8 lo-s)2
= 1.302 10-6d,
(AlO)
The drag force acting on the small-single particle in the same direction as the flow differs from the expression developed before for the large-particle model. In this case, the velocity distribution of the flow is not uniform. Thus, the gas velocity U, in eqn. (A4) is considered at the center of the particle. Using Blasius’ correlation for the friction factor inside a smooth pipe, the linear velocity profile of the flow in the laminar sublayer can be written as [ll]:
(All) Evaluating eqn. (All) at the center of the particle 6~= d,/2), one can find an expression for the drag force (FJ on a small particle in terms of the mean gas velocity. Considering the wall effects yields:
The particle Reynolds number used to calculate the drag coefficient is also based on the gas velocity at the center of the particle. The lift force (F,) acting upwards on a small-single sphere due to shear flow can be computed using the expression developed by Satfman [17, 181. Using eqn. (All) for the velocity distribution:
61
F,=6.46p&U,,, aUd%2 J ?Y4 (Al3)
where U,,, denotes the gas velocity at the center of the sphere. The frictional force holds the same expression as in the large-particle model, given by eqn. (A5). In this case the normal force (F,)acting on the particle, as shown in Fig. 8, is:
F,,=F,+F,-F,--Fb
(AW
Similarly, a small-single particle initially at rest on the bottom of a pipe will begin to move when the forces acting in the horizontal direction are zero. Applying Newton’s second law, motion will take place when:
Fc,=Ff=fs[Fg+Fa-F,-FJ
(AW
When the gas velocity is gradually increased, the drag and lift exerted on the particle will gradually increase until the drag force equals the frictional force, and the particle will slide or roll on the bottom of the pipe. These criteria are used to determine the minimum gas velocity to pick up a small-single particle lying on the bottom of a pipe. A similar procedure of substitution of the expressions for each force involved into eqn. (A15) then yields: 1.54 10-4[ l-($ICoptd+&‘)ln
=fs ;gd,3(p,-p,)+1.302 -6.35
10-3pfd,3
lo-“dp
6416)
Obviously, this is an implicit equation and a numerical method should be used in order to solve for the minimum pickup velocity of a single particle, Ugpuo.
Powder Technology, 72 (1992) 51-61
Incipient motion of solid particles in horizontal Francisco Deparhent
pneumatic
conveying
J. Cabrejos of Mechanical Engineering, University of Pittsburgh, Pittsburgh, PA 15261 (USA)
and George
E. Klinzing
Department of Chemical and Petroleum Engineering Univemiry of Pittsburgh, Pittsburgh PA 15261 (USA) (Received
July 22, 1991; in revised form January
30, 1992)
Abstract A technique for finding the minimum pickup velocity of solid particles in horizontal pneumatic conveying has been developed. A general semi-empirical correlation based on the Archimedes number which is valid over a range of particle size from 10 to 1000 pm is presented for the prediction of the minimum gas velocity required to pick up particles. Experiments have been carried out with fine particles (glass beads, alumina and coal) in a 52 mm id. transparent pipeline. Noteworthy is the existence of a minimum point in the curves of minimum pickup velocity as a function of particle diameter for particle diameters of approximately 100 Frn. Maximum conveyability is expected to occur at this particle size. Above and below this minimum point, higher gas velocities are required to pick up particles lying on the bottom of a pipe. Experiments in an actual pneumatic conveying system clarify the practical significance and usefulness of the minimum pickup velocity. Thus, it is suggested that the proposed test procedure be used as a unified technique in determining the minimum transport velocity of solid particles in horizontal pneumatic conveying systems.
Introduction
Conveying velocity is considered one of the most important parameters in the pneumatic transport of solids. Successful design and operation of pneumatic conveying systems depends upon the prediction or determination of the minimum conveying velocity at which the solids may be conveyed steadily through a pipeline. An unnecessarily high conveying velocity will result in higher energy costs due to an increased pressure drop in the system, increased erosion, and increased solids degradation. On the other hand, a low conveying velocity can result in the deposition of solids on the bottom of the pipe or can even block the pipeline. Thus, a general correlation to predict the minimum conveying velocity of solids in dilute-phase systems may be very useful for engineers and designers. There are many terms used to refer to minimum conveying velocity: saltation velocity, pickup velocity, suspension velocity, deposition velocity, rolling or sliding velocity, critical velocity, etc. Definitions of these terms are based on visual observations and pressure drop measurements, and they are often applied to indicate some transition in the way in which a particle is moving or begins to move. In fact, the terms pickup velocity and saltation velocity are often used interchangeably in dilute-phase conveying.
0032-5910/92/$5.00
Bagnold [l] was one of the first investigators in this field. With his experience in the Sahara desert and his study on sand movement in a wind tunnel, he developed a theory explaining the formation of desert dunes. He used the terms ‘fluid threshold’ and ‘impact threshold’, which correspond to pickup and saltation, respectively. Since then extensive work has been carried out on the prediction of minimum conveying velocity and several published correlations have been reviewed in detail by Plasynski et al. [2] and Jones and Leung [3]. The ‘pickup velocity’ concept is closely related to saltation velocity. The air velocity in a horizontal line at which the particles start to drop out of suspension and settle on the bottom of the line is usually called saltation velocity. Pickup velocity may be defined as the fluid velocity required to resuspend a particle initially at rest on the bottom of a line [4]. Pickup differs from saltation in relation to the initial state of the particle. Keeping gas velocity above minimum pickup velocity in all horizontal sections of a conveying line ensures no deposition of solids in the system. In general, this term can be defined as the safe fluid velocity for the horizontal conveyance of solids. If this velocity is set at the beginning of a piping system, the velocity will increase along the pipeline due to pressure drop and density decrease so the rest of the pipeline should be well above this lower pickup velocity bound.
0 1992 - Elsevier Sequoia. All rights reserved
52
The mechanism of pickup and saltation is complex. It was not possible to find a unique and general expression that correlates with all the parameters affecting the minimum conveying velocity. The main reason for the diversity of data on the pneumatic conveying of solids is the variety of materials handled. Only two studies aiming to predict the pickup velocity of particles were found: the graphic method proposed by Halow [5] (fluid velocity required to initiate sliding, rolling and suspension of particles) and the correlation done by Azizov and Toshov [6] (blowing-away critical velocity). Zenz [7] experimentally determined single-particle saltation velocities (minimum fluid velocity required to carry solids at a specified rate without allowing them to settle out in any horizontal pipe) for different spherical and angular materials and proposed a graphic representation based on two dimensionless parameters, $!&&$ and ?m. He showed that the minimum velocity required to pick up a particle from a layer and transport it in suspension can be 2 to 2.5 times greater than the minimum velocity required to transport an injected particle without saltation. The models proposed by these investigators will be used later for comparison with the present work. Matsumoto et al. [8] experimentally studied the effect of particle size on the minimum transport velocity for horizontal pneumatic conveying of solids over a wide range of particle sizes from 20 to 1600 pm. They found that for a given material there exists a particle size which provides the maximum conveyability of solids and which establishes a distinction between fine- and coarse-particles. Matsumoto et al.3 data of minimum
transport velocity (saltation) of glass and copper beads as a function of particle size show a minimum at diameters of 500 and 100 pm, respectively. Their correlations for the minimum transport velocity in horizontal pneumatic conveying are based on the solids loading ratio, pipe diameter, and material characteristics.
Experimental A technique for finding the pickup velocity of solid particles in horizontal pneumatic conveying has been developed. This technique is based on visual observation of the behavior of particles lying on the bottom of a transparent pipe when the air streamvelocity is gradually increased. Simple and inexpensive experiments can be performed to measure the minimum pickup velocity for a given material. The setup consists of a transparent, 7 m long, 52 mm i.d. pipeline with a removable section, a compressed regulated air supply, and a solids collector. The details of the experimental setup and procedure can be found elsewhere [9]. Figure 1 shows schematically the setup designed to carry out pickup velocity experiments. In order to carry out the measurements, a 1 m long layer of particles with approximately half of the crosssection area of the pipe left free is created using a two gate arrangement and a removable section. The procedure to create the initial layer of particles on the bottom of the pipeline is graphically shown in Fig. 2. A constant air flow rate is set through the pipeline Paper Filter Bags
t
b-3m-+lm+-3m_-_Y
I,
I /
r
Transparent Exelon 52 mm I.D. Pipeline
Removable Section
Initial Layer of Solids
r
I
1
hA YY
Control Valve
Pressure Regulator
Main Valve
I-
Solids Collector 8 House Air Supply
Fig. 1. Schematic
representation
of the setup used for pickup velocity experiments.
53
1
Solids
(1)
(2)
(4)
Fig. 2. Solids filling procedure to create the initial layer of particles inside the bottom of the removable section.
0
minimum pickup velocity “gp”*--__-----_-_---_-_-
Fig. 3. Determination of the minimum pickup velocity, system reaches automatically the equilibrium condition.
as the
and the layer erodes slowly as the air picks up the particles on top of it. As the free cross-section area increases, the air velocity over the layer decreases and so does the capacity of the stream to pick up more TABLE
1. Characteristics
Material
of the materials
particles. Since the described phenomenon takes place continuously, a final equilibrium is automatically reached when no more erosion occurs. This condition corresponds to the minimum pickup velocity, as shown in Fig, 3. The velocity is then increased and a similar test is performed until a similar non pickup condition is again achieved. It is important to point out that the rate at which equilibrium is reached depends on the air flow rate but the final minimum pickup velocity is independent of the initial conditions. Pickup velocity is defined as the air stream velocity required to pick up or entrain particles from a layer of solids on the bottom of the pipe. By measuring the air flow rate and the free cross-section area remaining over the particles, the minimum pickup velocity can easily be calculated using the following equation:
A wide variety of solids varying in size, shape, and density was tested. Table 1 summarizes the characteristics of all the materials used. Glass beads, crushed glass, and alumina particles were sieved with Tyler standard screens to give several fractions, each having a fairly uniform particle size. The particle size of the ESP dust (alumina dust from electrostatic precipitators) and the coals was determined with a Microtrac particle size analyzer, where the mean particle diameter corresponds to d,,.
used for pickup velocity experiments
Density
Particle
(kg mm3)
(pm)
(pm)
glass beads
2480
crushed
2480
d,<45 45 cd,<53 53 -cd,<75 75 -cd, < 150 3OO
22 49 64 112 450 49 112 225 450 800 22 60 112 225 450 7.76 21.5 14.8 6.75
glass
alumina
3750
ESP dust Wyodak coal Elkhom coal Illinois #6 coal
3000 1300 1300 1300
size
Mean d,
Shape
0.03 0.15 0.24 0.61 3.37 0.12 0.42 1.08 2.19 3.15 0.05 0.27 0.69 1.71 3.48 0 0.02 0.01 0
spherical spherical spherical spherical spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical non-spherical
54
Results
20
ESP dust
Figures 4-7 show the results obtained for the minimum pickup velocity of each material tested. The vertical bars in the graphs represent the range - 2a< U,, < + 2u, which corresponds statistically to 95% of the values. Several experiments were performed to check accuracy and reproducibility.
F
15
5 2 m 3
Illinois 10
wal
#6
it
t U. Elkhom
coal Wyodak
lb
coal
2’0 dp
(w)
’
Fig. 7. Minimum pickup velocity for ESP dust and coals (nonspherical particles, 3000 and 1300 kg rne3, respectively).
2-
0 0
200
I 400
i 600
600
dp
1000
(w)
Fig. 4. Minimum pickupvelocity for glass beads (spherical particles, 2480 kg m-‘).
I
Z_( i -0 I
200
400
600
coo
dp
1000 '
(w)
Fig. 5. Minimum pickup velocity for crushed glass (non-spherical particles, 2480 kg me3).
20 3 g
16-
2
12-
3
* 84-
0 0
200
400
600
so0
1
dp W’O Fig. 6. Minimum pickup particles, 3750 kg me3).
velocity
for alumina
(non-spherical
One of the most important results obtained is the existence of a minimum point in the curves of minimum pickup velocity as a function of particle diameter, which corroborates Matsumoto et al.‘s findings [8]. These minimum points appear at particle diameters of 70, 100 and 150 pm for glass beads, crushed glass and alumina, respectively. Noteworthy also in the test for the finer materials such as ESP dust and fine coals is the manner in which the particles left the surface of the layer. For the ESP dust, the particles were lifted from the top of the layer, while in the case of the coals, the particles disappeared from the leading edge of the layer rather than from the top. Interparticle cohesiveness is drawn up in order to explain these two behaviors, ESP dust being less cohesive than the fine coals tested. Another interesting finding is that spherical glass beads have a lower pickup velocity than crushed glass solids. This can be attributed to the higher degree of interlocking of the crushed particles that results in a higher resistance to pickup in comparison with spherical material. Also of note is the fact that the denser materials required higher pickup velocities indicating the controlling effect of the inertial forces for these dense materials. The Geldart fluidization diagram is widely used for classifying powders into four groups based on the mean particle size and the density difference between the particles and the fluidizing gas [lo]. The distinction between powders A (aeratable) and powders B (sandlike) is located at 90 and 60 pm for glass and alumina, respectively. Although Geldart’s diagram is related to the fluidization characteristics of powders and not the pneumatic conveyance, it is interesting to note that these values are close to those found for the minimum point in the curves of minimum pickup velocity as a function of particle diameter. Further work is required to study the similarity, if any, between Geldart’s diagram and the minimum points of the pickup velocity plots.
5.5
Particle dynamics
Y
A model was developed for the incipient motion of a single sphere initially at rest on the bottom of a horizontal pipe and subjected to a steady fully developed turbulent flow of air. The behavior of the particle is determined mainly by the interaction between the forces acting on it, the physical properties of the particle and the velocity of the air stream. The most important forces involved in the pickup mechanism of a single particle are shown in Fig. 8. Gravitational, buoyant, and adhesive forces depend only on the physical properties of the particle and gas density, and are independent of the gas stream velocity. The drag and lift forces depend on the gas stream velocity, while the frictional force is proportional to the coefficient of sliding friction. Forces due to electrostatics and lift due to particle rotation were not taken into account in the model. The fact that the gas stream adheres to the wall (no slip condition) means that frictional forces retard the motion of the gas in a thin layer near the wall, the so-called boundary layer. Based on the thickness of the laminar sublayer (a,), a distinction between ‘large’ and ‘small’ particles is made. A small particle (d, < 6,) lies deep in the boundary layer, mainly inside the laminar sublayer, and is exposed to smaller velocities and to a linear velocity profile [ll]. Larger particles (~,zs- SJ face larger velocities and flatter velocity profiles, as shown in Fig. 9. Although the distinction between large and small particles is not precise, it helps to determine the significant forces acting on the particle and the velocity profile facing the particle. The adhesive force, mainly van der Waals attraction between the sphere and the wall, is negligible for large particles. Also in this case, the almost uniform velocity profile of the gas stream exerts no lift over the sphere. A single particle initially at rest on the bottom of a horizontal pipe begins to move when the forces in
4 4 Air stream
Pipewall
x _i
FW&(
///jj/jj7jjjjjjjjjjjjjjjjjjjjjjjjj/ ,.
Lifl Buoyancy
)$
\/ Gravity + +
Adhesion
Fig. 8. Forces acting on a single sphere initially at rest on the bottom of a pipe, facing a steady fully developed turbulent flow of air.
4
VelocitvProfile
Small Particle
’
Large Particle
Fig. 9. Schematic representation of the velocity profile and two particles of different sizes lying on the bottom of a pipe, showing the thickness of the laminar sublayer (6,).
the horizontal direction are zero (Newton’s second law). If the gas velocity is gradually increased, the drag exerted on the particle will gradually increase until it finally equals the frictional force. Combining the expressions developed for each of the forces involved and solving for the gas velocity, it is possible to find the gas velocity required to initiate motion or to pickup a single particle lying on the bottom of a pipe (refer to the Appendix for more details). l For a large single sphere, the following equation was developed:
(2) l
For a small single sphere, the following equation was found:
1.54
lo-4[1-(~)“s]-2cDpfd~(~)1 -6.35
( )I
Vluo 1O-3 pfdP3 -&
l/8
(3)
To assess the relative significance of the two models, it is helpful to display graphically the solution of eqns. (2) and (3). F’g 1 ure 10 compares the minimum pickup velocity for the following case: a sphere of different densities (1000,2000,3000 kg m-‘) lying on the bottom of a 52 mm i.d. horizontal pipe (employed in the experiments) subject to an air stream passing over it. The coefficient of sliding friction between the particle and the wall is assumed to be constant and equal to 0.45. Single-particle experiments were carried out with glass beads and ceramic spheres to verify the model developed for the incipient motion of a single particle. Figure .ll compares the experimental results (mean values) with the minimum pickup velocity predicted by the model. It can be seen that calculated and measured values are in good agreement.
56
by other investigators, although neither Halow nor Azizov and Toshov predict any minimum point in the curves of gas velocity as a function of particle diameter. Zenz’s method predicts such a minimum point at d, equal to 50 pm for the given example. Also, none of these methods takes into account the coefficient of sliding friction between the particle and the pipe wall. As expected, pickup velocity is higher than saltation velocity. 100 2ObO
dp
@m)
30bo dp
200 4000
OLm)
General correlation for multiple-particle systems
Fig. 10. Comparison of the minimum pickup velocity for a sphere using both the small- and large-particle models, with an enlargement of the low range of particle diameters (52 mm i.d. pipe).
6
l
3‘
o
Glass&ad
Combining the single-particle model with the experimental results for the minimum pickup velocity of a layer of particles, a general correlation with practical applicability for prediction of the minimum gas velocity required to pick up particles lying on the bottom of a horizontal pipe was developed. This semi-empirical correlation is valid over the range of particle sizes from 10 to 1000 pm and has the following expression:
0
U,, = [1.27Ar-“3+ 0.036Ar’”
Ceramic
00
lm. - . 0
+0.45][0.70Ar-“5 where:
2
3
4
5 dp
6
7
(mm)
10
!
d
(4)
I
1
Fig. 11. Comparison between the experimental results and the predicted values of the minimum pickup velocity for single spheres.
0
+ l]U,,,,
I 200
400
600 dp
660
lOi0
bm)
Fig. 12. Comparison between the model developed spheres and the work done by other investigators.
for single
In order to compare the proposed model with the work done by Halow [5], Azizov and Toshov [6], and Zenz [7] Fig. 12 shows the results for the case of a single sphere of density 2480 kg mm3 lying on the bottom of a 52 mm i.d. horizontal pipe facing an air stream. It is important to point out that the proposed model is in quite good agreement with the methods proposed
Ar= J_ -ps-pr dp3, Archimedes number. v* Pf
(5)
The first term on the right hand side of eqn. (4) takes into account the effects of particle interactions with other particles while the second term accounts for the particle shape (equal to one for spherical particles). The Archimedes number, defined by eqn. (5), was used to correlate these effects. It characterizes the basic particle/gas properties and is related with the Froude number, the Reynolds number and the ratio of gravity to buoyant force. The model for prediction of the minimum pickup velocity of a single spherical particle ( UgPUO) already takes into account pipe diameter, air properties, particle characteristics, coefficient of sliding friction, drag coefficient, and the presence of surrounding walls. In order to evaluate the accuracy of the proposed correlation, the minimum pickup velocity was calculated (eqn. (4)) and plotted against the experimental results for all the materials tested, as shown in Fig. 13. The agreement between measured and predicted values for the minimum pickup velocity, shown in Fig. 13, was found to be generally within the accuracy of the tests, considering the large number of variables involved in the pickup mechanism and the dispersion
57
0
4
6
12
Ugpu
Fig. 13. Comparison between the predicted for minimum pickup velocity.
16
predlcted
(m/s)
and measured
values
of the experimental measurements. It can be seen that except for a few cases, the general correlation developed for the minimum gas velocity required to pick up particles lying on the bottom of a pipe predicts slightly higher values as compared with the experimental results. Two indicators widely used for comparison, the average absolute deviation (AAD) and the square root of the mean of the square of the deviations (E), were found to be equal to 2.1 m s-l and 31%, respectively, for all the materials tested in the particle size range between 10 to 1000 pm. The indicators are defined as follows for a data set of N points:
E=
J
1 N-l
.j~ (&,u “F-
u,~
y-d)’
1oo%
gpu measured
(7) Thus, eqn. (4) can be used satisfactorily to predict the minimum pickup velocity of solid particles in horizontal pneumatic conveying in the particle size range between 10 to 1000 pm. It is important to point out that for spherical particles all the points lie within 15%, as shown in Fig. 13. For smaller particles, the agreement between predicted and measured minimum pickup velocity is less reliable.
Pilot-plant
tests
In order to compare the minimum pickup velocity (Up,“) obtained by using the new technique proposed and the minimum transport velocity that would be required to convey solids without allowing them to settle out in a horizontal line, experiments in an actual pilotplant have been carried out. This will clarify the practical
significance and usefulness of the concept of minimum pickup velocity. The closed-loop pneumatic conveying system used, shown schematically in Fig. 14, consists of a 14.5 m long horizontal test pipeline with a 2.2 m vertical section, a return line, T-bends and several transparent sections placed along the pipeline; a compressed regulated air supply that provides the air velocity and pressure necessary to convey the solids; a hopper, placed on a scale and connected through flexible connections to the collector and feeder in order to weigh the solids inventory continuously; a solids collector with a paper filter bag, placed on top of the solids hopper; and the solids feeder, a butterfly valve that ensures a constant gravity downflow of solids at different flow rates depending on the position of the valve. The pipeline was grounded by wrapping a conducting wire so that electrostatic effects could be minimized. Experiments were carried out by reducing the gas velocity at a constant flow rate of solids until the particles began to drop out of suspension and formed a dune on the bottom of the transparent section in the horizontal line (saltation velocity). The particles were 450 pm spherical glass beads with a density of 2480 kg mP3, one of the materials used in the pickup velocity experiments. Figure 15 shows the results obtained for the saltation velocity at different solids loading ratios and the minimum pickup velocity. It is important to point out that the minimum pickup velocity, as defined in the present work, is independent of the solids loading ratio. Two of the correlations widely used for prediction of saltation velocity are also included in Fig. 15. Meyers et al. [12] defined saltation velocity for coarse particles as the optimal operating condition of a horizontal pneumatic conveying system (pressure minimum point), whereas Matsumoto et al. [8] defined it as the minimum superficial air velocity required for the transport of solids without the formation of a stationary bed on the bottom of a horizontal pipe. It can be seen that both correlations predict slightly higher saltation velocities as compared with the experimental results, for solids loading ratios over 5. Zenz [7] predicted minimum pickup velocities 2 to 2.5 times greater than the saltation velocity. Figure 15 shows that this looks to be the case only in the very dilute-phase conveyance, i.e. for solids loading ratios less than 0.5. As seen in Fig. 15, keeping gas velocity above the minimum pickup velocity in all horizontal sections of a conveying line ensures no deposition of solids in the system. So, the practical significance of the minimum pickup velocity is that it can be used as the safe gas velocity for the horizontal conveyance of solids. The
58
Filter bag
Flexible connection
to equalize the pressure
Hopper
Transparent section
I
I
A/(
I
3
I
-
YY
Control Valve Rotameter
Fig. 14. Schematic
I
..,_L
Transparent section
Transparent section
Manometer
oQ4
representation
P
Pressure Regulator
~c
Main Valve House Air Supply
of the setup used for saltation
velocity experiments.
Conclusions
2
r-Gzizq
1
1
Fig. 15. Comparison between minimum pickup velocity (CT,,) and saltation velocity (U,) as a function of the solids loading ratio.
usefulness of the concept of minimum pickup velocity is that one of the most important parameters for the design of pneumatic conveying systems, i.e. the minimum transport velocity, can be predicted by performing relatively simple and inexpensive experiments with the technique developed.
The concept of pickup velocity of solids was studied from an experimental and theoretical viewpoint. The new technique proposed for prediction of minimum pickup velocity of solids was successfully tested. In addition, a general semi-empirical correlation was developed, based on a model for the incipient motion of a single particle, the boundary layer thickness, and the Archimedes number. The pickup velocity for different solids cannot be easily predicted because this parameter is influenced by many diverse variables. Among these, the characteristics of the material itself, such as particle size, density and shape, the coefficient of sliding friction, and the particle interaction with other particles are the most important variables that affect pickup velocity. Experiments in an actual pneumatic conveying system have ratified the usefulness of the concept of minimum pickup velocity. Both technique and correlation provide new alternatives for engineers and designers to predict the minimum transport velocity of solids. Finally, it is suggested that the proposed test procedure be used as
59
a unified technique in determining the minimum transport velocity of solid particles in horizontal pneumatic conveying systems.
List of symbols A free AH A, Ar Cl3 D dP F, Fb Fd F* Fg F, P g
Qair R% s u, u SP u u
gPU ECPUO
free cross-section area remaining over the layer of solids, m2 constant in eqn. (AlO), Nm particle projected area normal to the flow, m2 Archimedes number, drag coefficient, pipe diameter, m mean particle diameter, m London-van der Waals adhesive force, N buoyant force, N drag force, N frictional force, N gravitational force, N lift force, N normal force, N coefficient of sliding friction, acceleration due to gravity, m sm2 volumetric flow rate of air, m3 s-l particle Reynolds number, separation between the sphere and the wall in eqn. (AlO), m mean gas velocity, m s-l gas velocity at the center of the sphere in eqn. (A13), m s-* minimum pickup velocity, m s- ’ minimum pickup velocity for single particle, m S
UF.S U,
-1
saltation velocity, m s-l terminal settling velocity, m s-l
Greek letters thickness of the laminar sublayer, m 6 V fluid kinematic viscosity, m2 s- ’ solids loading ratio, IJ solid particle density, kg mm3 PS fluid density, kg mm3 Pf shear stress at the wall, N m-’ 70
References R. A. Bagnold, The Physics of Blown Sand and Desert Dunes, Methuen, London, 1941. S. Plasynski, S. Dhodapkar, G. Klinzing and F. Cabrejos, AZChE Symp. Ser., 87 (1991) 78. P. J. Jones and L. S. Leung, Z’neumotransport 4, Paper Cl (June, 1978), pp. 1-12.
4 S. J. Rossetti, in N. P. Cheremisinoff and R. Gupta (eds.), Handbook of Fluti in Motion, Ann Arbor Science Butterworths, Boston, 1983, pp. 635-652. 5 J. Halow, Chem. Erg. Sci., 28 (1973) 1. 6 A. Azizov and V. R. Toshov, J. Appl. Mech. Tech. Phys., 27 (1987) 855. 7 F. Zenz, Znd. Erg Chem., Fundam., 3 (1964) 65. 8 S. Matsumoto, M. Kikuta and S. Maeda, J. Chem. Erg. Jpn., 10 (1977) 273. 9 F. Cabrejos, it4.S. Thesis, School of Engineering, University of Pittsburgh, 1991. 10 E. Geldart, Gas Fluidization Technology, Wiley, New York, 1986. 11 H. Schlichting, Boundary-Layer Theory, (4th edn.) McGrawHill, New York, 1979. 12 S. Meyers, R. Marcus and F. Rizk, Bulk Solids Handling, 5 (1985) 779. 13 R. D. Marcus, L. S. Leung, G. E. Klinzing and F. Rizk, Pneumatic Conveying of Solids, Chapman & Hall (London, 1990). 14 A. H. Kom, Chem. Erg., 57 (1950) 108. 15 H. C. Hamaker, Physicu, 4 (1937) 1058. 16 W. B. Pietsch, J. Eng. Znd. ASME (1969) 435. 17 P. G. Saffian, J. Fluid Mech., 22 (1965) 385. 18 P. G. Satfman, J. Fluid Mech., 31 (1968) 624.
Appendix: sphere
Modet for the incipient
motion of a single
Lalge-particle model For ‘large spheres’, the size of the particle is considered much larger than the thickness of the laminar sublayer (d,s SJ. The significant forces acting on a large-single particle lying on the bottom of a horizontal pipe are: drag force (acting in the same direction as the flow), friction force (acting in the opposite direction as the drag force), buoyant force and gravitational force. The velocity distribution of the steady fully developed turbulent flow of air past the sphere is assumed to be uniform in this case. The occurrence and effects of gravitational force (F,), which is always present, are well known and for a spherical particle the equation is: F, =mpg=
: psdp3g
(Al)
where mp is the mass of a particle of density pS and diameter d, Provided that the particle is completely surrounded by the fluid, the buoyant force (F,,) or upthrust can be expressed as:
w The drag force (FJ caused by a uniform flow on a single spherical particle at rest is given as [13]:
wo where C,, denotes the drag coefficient for a single particle in an undisturbed and unbounded fluid. The presence of the pipe walls interferes with the flow, mainly in the region adjacent to the wall, and its effects cannot be neglected. Marcus et al. [13] recommend Munroe’s equation to account for the wall effect on the drag coefficient of a single particle in a pipe of diameter D. Substitution in eqn. (A3) yields the expression:
(A4) Friction is the resistance that is encountered when a solid body slides or tends to slide over a solid surface. The standard form for the frictional force (Ff) is: Ff =fsFn
(A3
where fs denotes the coefficient of sliding friction (here defined as the tangent of the angle at which the pipe must be raised above the horizontal to cause the particle, initially at rest, to begin to slide). In this case, the normal force (F,) acting on the particle is equal to the difference between the gravitational and the buoyant forces: F,=F,--Fb
(A6)
The adhesive forces between the particle and the wall are negligible for large particles. Also in this case, the uniform velocity distribution of the flow exerts no lifting over a spherical particle [14]. Applying Newton’s second law, a large-single particle initially at rest on the bottom of a pipe will begin to move when the forces in the horizontal direction are zero. In other words, if the fluid velocity is gradually increased, the drag exerted on the particle will gradually increase until it finally equals the frictional force. Thus, motion will take place when:
&=F,=fJE;,-F,I
Gw
Substituting eqns. (Al), (42) and (A4) (for each of the forces involved in eqn. (A7) leads to:
Solving for the gas velocity U,, the velocity of the uniform flow required to initiate motion or to pickup a single particle lying on the bottom of a horizontal pipe can be written as:
(A9
It is important to point out that eqn. (A9) is an implicit equation because the drag coefficient is a function of the particle Reynolds number, and thus the gas velocity. Small-particle model
In the case of a ‘small sphere’ lying on the bottom of a pipe inside the laminar sublayer (d, < S,), the most important forces acting on the particle are shown in Fig. 8. The particle is subjected to a steady fully developed turbulent flow of air with a linear velocity distribution as shown in Fig. 9. The expressions for gravitational and buoyant forces remain the same as those developed in the large-particle model, and are given by eqns. (Al) and (A2), respectively. Hamaker [15] suggested the following expression for the adhesive force (F,), mainly London-van der Waals attraction between a spherical particle close to a flat wall, that acts downward. Considering a separation of 0.08 pm between the particle and the wall, suggested by Pietsch [16], and a value of lo-” erg for the constant A, given by Hamaker, the adhesive force then becomes: F,=
44, 24s2
=
lo-l9 d, 12 (8 lo-s)2
= 1.302 10-6d,
(AlO)
The drag force acting on the small-single particle in the same direction as the flow differs from the expression developed before for the large-particle model. In this case, the velocity distribution of the flow is not uniform. Thus, the gas velocity U, in eqn. (A4) is considered at the center of the particle. Using Blasius’ correlation for the friction factor inside a smooth pipe, the linear velocity profile of the flow in the laminar sublayer can be written as [ll]:
(All) Evaluating eqn. (All) at the center of the particle 6~= d,/2), one can find an expression for the drag force (FJ on a small particle in terms of the mean gas velocity. Considering the wall effects yields:
The particle Reynolds number used to calculate the drag coefficient is also based on the gas velocity at the center of the particle. The lift force (F,) acting upwards on a small-single sphere due to shear flow can be computed using the expression developed by Satfman [17, 181. Using eqn. (All) for the velocity distribution:
61
F,=6.46p&U,,, aUd%2 J ?Y4 (Al3)
where U,,, denotes the gas velocity at the center of the sphere. The frictional force holds the same expression as in the large-particle model, given by eqn. (A5). In this case the normal force (F,)acting on the particle, as shown in Fig. 8, is:
F,,=F,+F,-F,--Fb
(AW
Similarly, a small-single particle initially at rest on the bottom of a pipe will begin to move when the forces acting in the horizontal direction are zero. Applying Newton’s second law, motion will take place when:
Fc,=Ff=fs[Fg+Fa-F,-FJ
(AW
When the gas velocity is gradually increased, the drag and lift exerted on the particle will gradually increase until the drag force equals the frictional force, and the particle will slide or roll on the bottom of the pipe. These criteria are used to determine the minimum gas velocity to pick up a small-single particle lying on the bottom of a pipe. A similar procedure of substitution of the expressions for each force involved into eqn. (A15) then yields: 1.54 10-4[ l-($ICoptd+&‘)ln
=fs ;gd,3(p,-p,)+1.302 -6.35
10-3pfd,3
lo-“dp
6416)
Obviously, this is an implicit equation and a numerical method should be used in order to solve for the minimum pickup velocity of a single particle, Ugpuo.