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The inclined plane particle classifier
Powder Technology. 28 (1981) 129 - 134 0 Elsevier Sequoia S-A, Lauunn e-Printed
129 in
The
Netherlands
Plane Particle Clasitier
The hdined J M. BEECKMANS Faculty
of
(Received
EngineeringSrrence. The Uniuersity of Western Ontario. London, May 6.1980;
SUMMARY
A
novel
pneumatic
particle
classifier,
classifier (IPC), is described The performance of the IPC was determined for a variety of operating condrtions, using limestone particies in the sire mnge 0.1 - 1 .O mm. Reasonably sharp separations were obtained, and it was found posstble to correlate the observed cut s&zes by numerica!ly solvmg the particle trajectory equations for the flow field and boundary conditions a;lpropnate to each experimentThe IPC appears to be a useful alternative to screening for dry classification, which avoids some of the difficulties encountered in industrial dry screening of particles below I mm in size. termed
the
inclined
Ont. (Canada N6A
589)
in revised form June 24.19BO)
ing coarser particles. It was to fill this perceived need that the IPC was conceived and developed_
plane
INTRODUCTION
In this paper some results are reported on classification of limestone particles in the size range O-1 - 1.0 mm, using a novel aerodynamic classifying device which has been termed the Inclined Plane Classifier (IPC)_ The IPC was designed vvlth the following desiderata in mind: simplicity, low energy consumption, low maintenance, high throughput capabllity, flexibility, and reasonable sharpness of separation. A need for such a machine was perceived, because of the problems and costs associated with dry screening of granular solids in the indicated size range. It is also frequently true that extreme precision m classifkation is not required, in which case it may be possible to achieve the desired result without using screening. Several aerodynamic particle classifiers exist designed primarily for classification of particles below approximately 0.1 mm, such as the Hukki classifiers Cl], but none was designed expressly for process-
DESCRJFI’ION
OF THE
INCLINED
PLANE
CLASSIFIER If sharp classification is to be achieved it is essential that all particles of a given size entering the classifier be subjected to the same aerodynamic and geometric conditions. These conditions are satisfied in the IPC by feeding the material to be classified through a thin chute, from which it emerges into classification space m the form of a planar sheet_ Figure 1 shows a schematic sectional elevation of the device; the dimension of the machine
normal to the plane of the figure is arbim, and it may be made as wide as necessary to achieve the required throughput capacity.
Fig. 1. Vertical section through the IPC
Material to be classified was fed continuously from a vibratory feeder onto the feedmg tray T, from which it slid into feeding chute C, which was made as thk as possible consistent with permitting unimpeded flow of
the large& particles in the nurture. The feeding chute was inclmed at an angle 0 to the honzontal, and smooth flow of the ma+kal was facilitated by vibrations induced by rotary vikator V located near the top of the chute, as illustrared. At the outlet to the chute the particles were essentially coplanar and had parallel velocity vectors. They then entered the classification space where they were subjected to transverse aerodynamic forces induced by a flow of air directed at right angles to the direction of particle motion. Air entered through inlet slot I, and -as. drawn out through outler 0 The air flow produced perturbations in the trajectories of the particles, the perturbations being largest with the smallest pzticles. A planar divider P with sharp leading edge K was located in or cIose to the plane of the chute, and served to partition the stream of particles into fine and coarse fractions. The particles then settled out of the upper (U) and lower (L) settling chambers, and were collected through their respective outlets S and S. In practice it may be desirable to recirculate part or all of the air, since a proportion of the tie particles may not settle out in the upper chamber. In this case, if only partial recirculation of air is used, the balance may be exhausted through an appropriate collection device such as a mter or cyclone_ The air withdrawn may then be replaced by a-7 equal flow through the feeding chute. It is evident that the cut sire of the device may be conizolled by selection of the operating conclitions. These Include the air flowrate. the angle B and the size of the classiiication gap between the divider -tie edge and the chute outlet.
EXPERIMENTAL
was variable. The pressure drop between the upper and lower settling chambers was measured by means of a Magnehelic gauge, with a fulI scale deflection of 6.5 mm of _ water, and 0.15 mm subdivisions. The experimental procedure was as follows. The air flow was turned on and adjusted, using a damper in the duct connecting the outlet 0 and the air fan. GranuIar limestone was then fed onto tray T at a controlled rate from a vibratory feeder (not shown in Fig. 1) untrl an adequate quantity had been processed through the classifier_ The air flow was then turned off, and the material which had accumulated in the settling chambers was removed through outlets S and S’ and weighed_ The size distributions of the product fractions were then Tound by sieving analysis. The mean velmity of particles exiting horn +‘-e chute at various angles of inclination of the chute to the horizontal was estimaw by placing an open compartmented box horizontally in the lower settling chamber (Fig. 2). The_compartments in the box were created by means of vertical baffles, wluch served to classify the collected material according to horizontal distance from the exit of the chute. The mean x and y coordinates of the collected material, together with the mean size of the particles, permitted an estimate to be made of mean particle velocity at the outlet of the chute, as outlined below.
PROCEDURES
The following were the dimensions of the device which was used in this study: depth of upper chamber 610 mm; width of upper chamber 28Lmm, length of upper chamber 914 mm; width of feeding tray and chute 229 mm; thickness of feeding chute 3.2 mm; length of chute 203 mm; depth of lower chamber 305 mm; length of lower chamber 451 m.m; width of lower chamber 281 nun. The width of the classi5cation gap between the outlet of the chute and the knife edge K
Fig 2. Arrangement particle velocdy at
TRAJECTORY
for determination of the mean chute exit (schematic).
the
EQUATIONS
In this section the particle trajectory differential equations are derived. These equations are valid for both the chute exit velocity
131
experiments (V = 0) and the classification experiments. Reference is made to Fig. 3, which ilhrstrates the geometrical relationships, and the nomenclature and coordinate system used in connection with the trajectory calculations_ The origin of the coordinate system is taken at point A, which corresponds to the outlet of the chute, and the x-axis is parallel to the floor of the chute, with the y-axis normal to the x-ax= and in the same vertical plane. Note that the knife edge B is assumed to be located at a distance 6 from the projection of the plane of the chute through point A. Figure 4 gives the velocity diagram for the air and particle motions and the force diagram for forces acting on a particle. By resolving the velocity vectors into their components in the x and y duections it is possible to derive an expression for U’,, the velocity of the zir relahve to that of the particle_
The drag force Fn on the particle is pamlIe to U,, with magnitude Fn = (s/8)Cr,p,V:d2
The net force F on the particle is the vector sum of the drag and gravitational forces; the components of this force in the x and y directions are F,
= F,U,,/U,
tgm
F,
= F,U,,/U,
-gm
=n
dV,
clm!znAH
(‘3)
-
Vy)/8
-
mg cos 8 j (7)
(dtlm) dV,
= [~C,d2p,Ur(U~
-
V,)/8
+ mg sin f?]
(dtlm)
(8)
(1)
Schiller and Nauman’s equation [Z] used to compute the drag coefficient:
(2)
Cn = (24/Re)(l+
(3)
relation-
-01-
Fig_ 4. Velocity and force diagmxas used for deriving particle trajectory equations.
0 15Re0--)
wz;1s Pa) (9b)
= ~f dU,l~
Equations (1) and (2) were integrated numerically by computer for assumed values of 8. m, V (at the exit to the chute) and U_ U (see Fig. 3) are given by the relations u,
= U6 /( wz + 62)1’2
(10)
u,
= UWI(W2
(11)
dy = V,
YEULtlTI
(5)
cos 8
= [rrC,d2prUr(Uy
Finally, computed
Fig. 3. Coordinate system and geometrical ships for trajectory calculations
sin 8
Equating eqns. (5) and (6) to products of particle mass times component of acceleration in the r and y directions, respectively, we obtain after simplification
Re u, V,)l
(4)
+ &2)1’2
the particle trajectories were by integrating chc: equations
dt
(12)
dt
(13)
and dx=
V,
Equations (7) (8), (12) and (13) were integrated simultaneously to yield the desired relationship between x and y, i e. the particle trajectory_ The initiaI boundary conditions = 0, and V, = V,, the initial werex=y=V, velocity of the particle as it left the chute. The arrangement used in experiments made for the purpose of determining V. is shcwn schematically in Fig. 2. In this case the final to the coordinates x,, ym corresponding most probable particle kajectory were measured experimentally for each run. Equations (7), (8), (12) and (13) were then intcg-
132 rated using a value of 0 for U and an assumed value for V,_ A number of trials w-e made to find the value of V0 which caused the particle trajectory to pass through the point x,, ym, and this value was assumed to represent the chute exit velocity.
EXPERIMENTAL
RESULTS
(a) Chute exit velocity calculations Chute exit velocity was found to be a func-
tion of both the angle 19and the particle feed rat-e F, but it did not appear to depend significantly on particle size. Two limestone samples were used in the experrents reported in this paper; lot A had a geometric mean diameter, by weight, of 0.214 mm, with geometric standard deviation 3.9, and lot B had corresponding values of 0.325 mm and 3_5_ The results of the experiments are given in Table l_ The data were analyzed by mnltrple regression, resulting in the following equation: V0 = -0.0528
+ 0.01788
+ 3.21 X lo-’
8F (14)
TABLE
1
Resultsofchuteexitvelocityexperiments
e (de& 25 35
45
55 60
70 25 35 45 55 65
F (kg/m h)
a
Ym
xm
(mm)
(ml
(ml
(m/s)
3.3 95.3 7-4 75-3 219 5.7 13.7 44.9 86-S 178 3-7 218 15 3 86.8 165 2.0 228 12.7 124 12.3 322 ll_l 20.0 255 13.0 328
0325
O-178 O-178 0 246 0.246 O-343 0.343 0.310 0.343 0.343 0310 O-103 0.103 0.129 0.129 0.129 O-138 0.138 0.187 0.187 0 224 0.224 0.232 0.108 O-108 0.129 0.129
0.062 0.073 0.084 0.120 0.120 O-102 0.080 0.119 0120 0.116 0.048 0.052 0.048 0.061 0053 0.034 0.038 0.039 0.033 0.065 0.090 CL068 0.039 0.058 0.038 0.049
o-445 0.54 O-72 0.61 0.97 080 0.63 0.95 0.96
O-214
_
vo
0 equals the angle of inclination of the chute to the horizontal, in dew, F equals the perikle feed rate, in kg/m h, and V, equals the chute exit velocity in metres/second. The standard deviation from the regression equation was 0.123 m/s. A total of 26 experimental results were used for calculating eqn. (14). Regression terms involving particle size, F, F2 and 8* were not significant and were not retained. (b) Particle
classifwation
experiments
Experiments were made using the granular limestone particles used for the chute velocity determination experiments. Screening analyses were performed on both products from the classfier, from which the proportion of particles within a even size range reporting to the fines could be determined_ It was found that plots of proportion of particles within a given size range appearing in the fines (i.e. elevated above the knife edge) uersus midinterval size were generally linear when plotted on logarithmic probability paper, and Fig_ 5 shows a typical example. From such plots it was a simple matter to determine the cut size dx, corresponding to the 50% ordinate, and (I~, taken a the ratio of d, to the diamekr corresponding to the 84% ordinate, or alternatively as the ratio of the diameter corresponding to the 16% ordinate to dm_ Results of a series of runs at various values of 0, F and air flowrates are given in Table 2. Most of these runs were performed with W = 76 mm, but in a few runs this value
0.99 l-09
1.30 110 1.22 135 l-16 1.47 0.31 0.26 0 57 0.83 0.72 0.80 1.60 1.10 190
Fig. 5. Typical classificationplotrpercentofparticles mthln agivensizeintemal elevatedabove knife edge, us_ mid-interval size.
133 TABLB
2
ResuIts ofclassificationexperiments Runno.
1 2 3 4 5 6 7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
was
8 (d-1 27 25 27 25 25 25 25 25 36 34 35 30 30 29 25 24 20 20 20 20 21 25 25 25 E 34 34
AP (mm water)
F
(“mm) 76 76 76 76 102 102 102 102 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76
2.28 2-79 2.3 28 23 23 2-3 Z-3 1.3 07 05 0.4 0.2 0.15 o-2 0.08 O-3 0.18 O-35 0 50 0.43 0.28 0.18 O-16 031 0 23 03 0.23
315 318 10 10 121 261 335 302 16 41 32 45 17 12 61 33 61 57 49 44 41 74 32 58 77 88 109 69
increased to102
mm.Thedistance6
(kg/m h)
be-
tween the plane of the chute floor and the plane containing the lmife edge was equal to 8 mm in all runs. In calculating the theoretical cut size an estimate of U was needed, in order to be able to integrate the equations of motion of the particles. This was obtained by using the simplified orifice equation in conjunction with the observed pressure drop: (15) The value of the discharge coefficient C,, was adjusted to reduce the average difference between the theoretical and experimental cut sizes to as close to zero as possible; the final value used was 0.67. This value appeared reasonable in view of the elongated, rectangular shape of the opening, which would be expected to result in a somewhat higher value for Ce than would be obtained with circular orifices of equivalent area_
ue
1.20 1.26 1.37 1.40 1.30 1.26 l-19 1.22 l-20 121 l-21 1.21 1.20 1.13 1.15 1.24 1.22 1.35 119 1.28 1.21 1.32 1.15 1.19 1.19 1.20 116 1.18
(mm)
Theoretical (-1
0.627 0.627 0.571 0.683 0.627 0.661 0.582 0 588 0.403 0.402 0319 0 314 0235 0.202 0.213 0.129 0.246 0.180 0258 0.279 0.269 0.234 0.185 0.218 0.259 0.202 0 255 O-215
0.680 0.754 0.638 0697 0.606 0.619 0626 0.622 0.542 0.388 0335 0.268 0.202 0.179 0.205 0.132 0.214 0.173 0.228 0.272 0.253 O-219 O-182 0.182 0.278 O-235 O-270 0233
Experimental
cut size
cut size
Regressions were performed on the data in Table 2. The parameter (TV was found not to be signiScantIy related to any of the indepen%nt variables, whereas the ratio of experimental to calculated values of cut size was found to be dependent on both 8 and AP: Experimental = 1344
-
ds,,/Calculated 0.0551AP
-
d,
=
0.01068
In eqn. (16), AP is in millimetres 0 is in degrees.
(16)
of water and
CONCLUSIONS
The IPC was found to give reasonably sharp separations in the size range O-l- 1.0 mm, and its range of operation can probably be extended beyond these limits. There was acceptable agreement between theoretical and experimental cut sizes, especially in view of the fact that the air flowrate was not
134
measured directly_ It should also be pointed out that the drag coefficient equation used in the trajectory calculation is valid only for spherical particles. whereas the particles used had irregti Aa_pes. Finally, the experimental cut size determinations were based on sieving analyses of the fine and coarse dictions obtained from the classEl%r, which implied a particular deCn.ition of particle diameter_ In view of these facts, one would not expect highly precise correspondence between the experimental cut size and its theoretical prediction_ The device is not subject to blinding and is vutuahy we=-*, and hence it should prove useful as an alternative to screening where extreme sharpness of separation is not required. The maximum throughput rate per unit width which can be achieved before cg begins to increase significantly is not yet k-lOW-Il_
LIST
OF SYMBOLS
co
discharge coefficient (dimensionless) drag coefficient (dimensionless) particle diameter (m) medum particle diameter in chute exit velocity experiments (m) cut size (m) particle feed rate (kg/m h); also net force on particle (N) drag force on particle (N) gravitational force on particle (N) gravitational acceleration (m/s2) particle mass (kg) pressure differential across separation space (Pa) particle Reynolds number (dimensionIess)
cD d
a
dm F FD
F, g
p” Re
time
(s) air velocity (m/s) air velocity relative to particle velocity (m/s) projection U, on the x-axis (m/s) projection of U, on the x-axis (m/s) projection of U on the x-axis (m/s) projectron of U on the y-axis (m/s) particle velocity (m/s) chute exit velocity (m/s) projection of V on the x-axis (m/s) projection of V on the y-axis (m/s) width of projection of distance between the chute exit and the lmife edge onto the x-axis (m) distance between chute exrt and knife edge (m) coordinate parallel to particle velocity at chute exit (m) x coordinate of top of partitioned box, corresponding to location of median particles (m) coordinate normal to plane of chmute floor (m) y coordinate of top of partitioned box, corresponding to location of median particles (m) angle of inclination of chute (degrees) air viscosAy (kg/m s) fluid density (kg/m’) sharpness of separation parameter (dimensionless)
REFERENCES 1 R. T_ Hukki, Proc_ Xth Int_ Mineral Processing Congx. 1973. Inst. Min. MetaLl_, pp_ 195 - 212. 2 L. Schiller and A_ Nauman, 2. Ver. Dtsch Ing., 77 (1973) 3aa.
129 in
The
Netherlands
Plane Particle Clasitier
The hdined J M. BEECKMANS Faculty
of
(Received
EngineeringSrrence. The Uniuersity of Western Ontario. London, May 6.1980;
SUMMARY
A
novel
pneumatic
particle
classifier,
classifier (IPC), is described The performance of the IPC was determined for a variety of operating condrtions, using limestone particies in the sire mnge 0.1 - 1 .O mm. Reasonably sharp separations were obtained, and it was found posstble to correlate the observed cut s&zes by numerica!ly solvmg the particle trajectory equations for the flow field and boundary conditions a;lpropnate to each experimentThe IPC appears to be a useful alternative to screening for dry classification, which avoids some of the difficulties encountered in industrial dry screening of particles below I mm in size. termed
the
inclined
Ont. (Canada N6A
589)
in revised form June 24.19BO)
ing coarser particles. It was to fill this perceived need that the IPC was conceived and developed_
plane
INTRODUCTION
In this paper some results are reported on classification of limestone particles in the size range O-1 - 1.0 mm, using a novel aerodynamic classifying device which has been termed the Inclined Plane Classifier (IPC)_ The IPC was designed vvlth the following desiderata in mind: simplicity, low energy consumption, low maintenance, high throughput capabllity, flexibility, and reasonable sharpness of separation. A need for such a machine was perceived, because of the problems and costs associated with dry screening of granular solids in the indicated size range. It is also frequently true that extreme precision m classifkation is not required, in which case it may be possible to achieve the desired result without using screening. Several aerodynamic particle classifiers exist designed primarily for classification of particles below approximately 0.1 mm, such as the Hukki classifiers Cl], but none was designed expressly for process-
DESCRJFI’ION
OF THE
INCLINED
PLANE
CLASSIFIER If sharp classification is to be achieved it is essential that all particles of a given size entering the classifier be subjected to the same aerodynamic and geometric conditions. These conditions are satisfied in the IPC by feeding the material to be classified through a thin chute, from which it emerges into classification space m the form of a planar sheet_ Figure 1 shows a schematic sectional elevation of the device; the dimension of the machine
normal to the plane of the figure is arbim, and it may be made as wide as necessary to achieve the required throughput capacity.
Fig. 1. Vertical section through the IPC
Material to be classified was fed continuously from a vibratory feeder onto the feedmg tray T, from which it slid into feeding chute C, which was made as thk as possible consistent with permitting unimpeded flow of
the large& particles in the nurture. The feeding chute was inclmed at an angle 0 to the honzontal, and smooth flow of the ma+kal was facilitated by vibrations induced by rotary vikator V located near the top of the chute, as illustrared. At the outlet to the chute the particles were essentially coplanar and had parallel velocity vectors. They then entered the classification space where they were subjected to transverse aerodynamic forces induced by a flow of air directed at right angles to the direction of particle motion. Air entered through inlet slot I, and -as. drawn out through outler 0 The air flow produced perturbations in the trajectories of the particles, the perturbations being largest with the smallest pzticles. A planar divider P with sharp leading edge K was located in or cIose to the plane of the chute, and served to partition the stream of particles into fine and coarse fractions. The particles then settled out of the upper (U) and lower (L) settling chambers, and were collected through their respective outlets S and S. In practice it may be desirable to recirculate part or all of the air, since a proportion of the tie particles may not settle out in the upper chamber. In this case, if only partial recirculation of air is used, the balance may be exhausted through an appropriate collection device such as a mter or cyclone_ The air withdrawn may then be replaced by a-7 equal flow through the feeding chute. It is evident that the cut sire of the device may be conizolled by selection of the operating conclitions. These Include the air flowrate. the angle B and the size of the classiiication gap between the divider -tie edge and the chute outlet.
EXPERIMENTAL
was variable. The pressure drop between the upper and lower settling chambers was measured by means of a Magnehelic gauge, with a fulI scale deflection of 6.5 mm of _ water, and 0.15 mm subdivisions. The experimental procedure was as follows. The air flow was turned on and adjusted, using a damper in the duct connecting the outlet 0 and the air fan. GranuIar limestone was then fed onto tray T at a controlled rate from a vibratory feeder (not shown in Fig. 1) untrl an adequate quantity had been processed through the classifier_ The air flow was then turned off, and the material which had accumulated in the settling chambers was removed through outlets S and S’ and weighed_ The size distributions of the product fractions were then Tound by sieving analysis. The mean velmity of particles exiting horn +‘-e chute at various angles of inclination of the chute to the horizontal was estimaw by placing an open compartmented box horizontally in the lower settling chamber (Fig. 2). The_compartments in the box were created by means of vertical baffles, wluch served to classify the collected material according to horizontal distance from the exit of the chute. The mean x and y coordinates of the collected material, together with the mean size of the particles, permitted an estimate to be made of mean particle velocity at the outlet of the chute, as outlined below.
PROCEDURES
The following were the dimensions of the device which was used in this study: depth of upper chamber 610 mm; width of upper chamber 28Lmm, length of upper chamber 914 mm; width of feeding tray and chute 229 mm; thickness of feeding chute 3.2 mm; length of chute 203 mm; depth of lower chamber 305 mm; length of lower chamber 451 m.m; width of lower chamber 281 nun. The width of the classi5cation gap between the outlet of the chute and the knife edge K
Fig 2. Arrangement particle velocdy at
TRAJECTORY
for determination of the mean chute exit (schematic).
the
EQUATIONS
In this section the particle trajectory differential equations are derived. These equations are valid for both the chute exit velocity
131
experiments (V = 0) and the classification experiments. Reference is made to Fig. 3, which ilhrstrates the geometrical relationships, and the nomenclature and coordinate system used in connection with the trajectory calculations_ The origin of the coordinate system is taken at point A, which corresponds to the outlet of the chute, and the x-axis is parallel to the floor of the chute, with the y-axis normal to the x-ax= and in the same vertical plane. Note that the knife edge B is assumed to be located at a distance 6 from the projection of the plane of the chute through point A. Figure 4 gives the velocity diagram for the air and particle motions and the force diagram for forces acting on a particle. By resolving the velocity vectors into their components in the x and y duections it is possible to derive an expression for U’,, the velocity of the zir relahve to that of the particle_
The drag force Fn on the particle is pamlIe to U,, with magnitude Fn = (s/8)Cr,p,V:d2
The net force F on the particle is the vector sum of the drag and gravitational forces; the components of this force in the x and y directions are F,
= F,U,,/U,
tgm
F,
= F,U,,/U,
-gm
=n
dV,
clm!znAH
(‘3)
-
Vy)/8
-
mg cos 8 j (7)
(dtlm) dV,
= [~C,d2p,Ur(U~
-
V,)/8
+ mg sin f?]
(dtlm)
(8)
(1)
Schiller and Nauman’s equation [Z] used to compute the drag coefficient:
(2)
Cn = (24/Re)(l+
(3)
relation-
-01-
Fig_ 4. Velocity and force diagmxas used for deriving particle trajectory equations.
0 15Re0--)
wz;1s Pa) (9b)
= ~f dU,l~
Equations (1) and (2) were integrated numerically by computer for assumed values of 8. m, V (at the exit to the chute) and U_ U (see Fig. 3) are given by the relations u,
= U6 /( wz + 62)1’2
(10)
u,
= UWI(W2
(11)
dy = V,
YEULtlTI
(5)
cos 8
= [rrC,d2prUr(Uy
Finally, computed
Fig. 3. Coordinate system and geometrical ships for trajectory calculations
sin 8
Equating eqns. (5) and (6) to products of particle mass times component of acceleration in the r and y directions, respectively, we obtain after simplification
Re u, V,)l
(4)
+ &2)1’2
the particle trajectories were by integrating chc: equations
dt
(12)
dt
(13)
and dx=
V,
Equations (7) (8), (12) and (13) were integrated simultaneously to yield the desired relationship between x and y, i e. the particle trajectory_ The initiaI boundary conditions = 0, and V, = V,, the initial werex=y=V, velocity of the particle as it left the chute. The arrangement used in experiments made for the purpose of determining V. is shcwn schematically in Fig. 2. In this case the final to the coordinates x,, ym corresponding most probable particle kajectory were measured experimentally for each run. Equations (7), (8), (12) and (13) were then intcg-
132 rated using a value of 0 for U and an assumed value for V,_ A number of trials w-e made to find the value of V0 which caused the particle trajectory to pass through the point x,, ym, and this value was assumed to represent the chute exit velocity.
EXPERIMENTAL
RESULTS
(a) Chute exit velocity calculations Chute exit velocity was found to be a func-
tion of both the angle 19and the particle feed rat-e F, but it did not appear to depend significantly on particle size. Two limestone samples were used in the experrents reported in this paper; lot A had a geometric mean diameter, by weight, of 0.214 mm, with geometric standard deviation 3.9, and lot B had corresponding values of 0.325 mm and 3_5_ The results of the experiments are given in Table l_ The data were analyzed by mnltrple regression, resulting in the following equation: V0 = -0.0528
+ 0.01788
+ 3.21 X lo-’
8F (14)
TABLE
1
Resultsofchuteexitvelocityexperiments
e (de& 25 35
45
55 60
70 25 35 45 55 65
F (kg/m h)
a
Ym
xm
(mm)
(ml
(ml
(m/s)
3.3 95.3 7-4 75-3 219 5.7 13.7 44.9 86-S 178 3-7 218 15 3 86.8 165 2.0 228 12.7 124 12.3 322 ll_l 20.0 255 13.0 328
0325
O-178 O-178 0 246 0.246 O-343 0.343 0.310 0.343 0.343 0310 O-103 0.103 0.129 0.129 0.129 O-138 0.138 0.187 0.187 0 224 0.224 0.232 0.108 O-108 0.129 0.129
0.062 0.073 0.084 0.120 0.120 O-102 0.080 0.119 0120 0.116 0.048 0.052 0.048 0.061 0053 0.034 0.038 0.039 0.033 0.065 0.090 CL068 0.039 0.058 0.038 0.049
o-445 0.54 O-72 0.61 0.97 080 0.63 0.95 0.96
O-214
_
vo
0 equals the angle of inclination of the chute to the horizontal, in dew, F equals the perikle feed rate, in kg/m h, and V, equals the chute exit velocity in metres/second. The standard deviation from the regression equation was 0.123 m/s. A total of 26 experimental results were used for calculating eqn. (14). Regression terms involving particle size, F, F2 and 8* were not significant and were not retained. (b) Particle
classifwation
experiments
Experiments were made using the granular limestone particles used for the chute velocity determination experiments. Screening analyses were performed on both products from the classfier, from which the proportion of particles within a even size range reporting to the fines could be determined_ It was found that plots of proportion of particles within a given size range appearing in the fines (i.e. elevated above the knife edge) uersus midinterval size were generally linear when plotted on logarithmic probability paper, and Fig_ 5 shows a typical example. From such plots it was a simple matter to determine the cut size dx, corresponding to the 50% ordinate, and (I~, taken a the ratio of d, to the diamekr corresponding to the 84% ordinate, or alternatively as the ratio of the diameter corresponding to the 16% ordinate to dm_ Results of a series of runs at various values of 0, F and air flowrates are given in Table 2. Most of these runs were performed with W = 76 mm, but in a few runs this value
0.99 l-09
1.30 110 1.22 135 l-16 1.47 0.31 0.26 0 57 0.83 0.72 0.80 1.60 1.10 190
Fig. 5. Typical classificationplotrpercentofparticles mthln agivensizeintemal elevatedabove knife edge, us_ mid-interval size.
133 TABLB
2
ResuIts ofclassificationexperiments Runno.
1 2 3 4 5 6 7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
was
8 (d-1 27 25 27 25 25 25 25 25 36 34 35 30 30 29 25 24 20 20 20 20 21 25 25 25 E 34 34
AP (mm water)
F
(“mm) 76 76 76 76 102 102 102 102 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76
2.28 2-79 2.3 28 23 23 2-3 Z-3 1.3 07 05 0.4 0.2 0.15 o-2 0.08 O-3 0.18 O-35 0 50 0.43 0.28 0.18 O-16 031 0 23 03 0.23
315 318 10 10 121 261 335 302 16 41 32 45 17 12 61 33 61 57 49 44 41 74 32 58 77 88 109 69
increased to102
mm.Thedistance6
(kg/m h)
be-
tween the plane of the chute floor and the plane containing the lmife edge was equal to 8 mm in all runs. In calculating the theoretical cut size an estimate of U was needed, in order to be able to integrate the equations of motion of the particles. This was obtained by using the simplified orifice equation in conjunction with the observed pressure drop: (15) The value of the discharge coefficient C,, was adjusted to reduce the average difference between the theoretical and experimental cut sizes to as close to zero as possible; the final value used was 0.67. This value appeared reasonable in view of the elongated, rectangular shape of the opening, which would be expected to result in a somewhat higher value for Ce than would be obtained with circular orifices of equivalent area_
ue
1.20 1.26 1.37 1.40 1.30 1.26 l-19 1.22 l-20 121 l-21 1.21 1.20 1.13 1.15 1.24 1.22 1.35 119 1.28 1.21 1.32 1.15 1.19 1.19 1.20 116 1.18
(mm)
Theoretical (-1
0.627 0.627 0.571 0.683 0.627 0.661 0.582 0 588 0.403 0.402 0319 0 314 0235 0.202 0.213 0.129 0.246 0.180 0258 0.279 0.269 0.234 0.185 0.218 0.259 0.202 0 255 O-215
0.680 0.754 0.638 0697 0.606 0.619 0626 0.622 0.542 0.388 0335 0.268 0.202 0.179 0.205 0.132 0.214 0.173 0.228 0.272 0.253 O-219 O-182 0.182 0.278 O-235 O-270 0233
Experimental
cut size
cut size
Regressions were performed on the data in Table 2. The parameter (TV was found not to be signiScantIy related to any of the indepen%nt variables, whereas the ratio of experimental to calculated values of cut size was found to be dependent on both 8 and AP: Experimental = 1344
-
ds,,/Calculated 0.0551AP
-
d,
=
0.01068
In eqn. (16), AP is in millimetres 0 is in degrees.
(16)
of water and
CONCLUSIONS
The IPC was found to give reasonably sharp separations in the size range O-l- 1.0 mm, and its range of operation can probably be extended beyond these limits. There was acceptable agreement between theoretical and experimental cut sizes, especially in view of the fact that the air flowrate was not
134
measured directly_ It should also be pointed out that the drag coefficient equation used in the trajectory calculation is valid only for spherical particles. whereas the particles used had irregti Aa_pes. Finally, the experimental cut size determinations were based on sieving analyses of the fine and coarse dictions obtained from the classEl%r, which implied a particular deCn.ition of particle diameter_ In view of these facts, one would not expect highly precise correspondence between the experimental cut size and its theoretical prediction_ The device is not subject to blinding and is vutuahy we=-*, and hence it should prove useful as an alternative to screening where extreme sharpness of separation is not required. The maximum throughput rate per unit width which can be achieved before cg begins to increase significantly is not yet k-lOW-Il_
LIST
OF SYMBOLS
co
discharge coefficient (dimensionless) drag coefficient (dimensionless) particle diameter (m) medum particle diameter in chute exit velocity experiments (m) cut size (m) particle feed rate (kg/m h); also net force on particle (N) drag force on particle (N) gravitational force on particle (N) gravitational acceleration (m/s2) particle mass (kg) pressure differential across separation space (Pa) particle Reynolds number (dimensionIess)
cD d
a
dm F FD
F, g
p” Re
time
(s) air velocity (m/s) air velocity relative to particle velocity (m/s) projection U, on the x-axis (m/s) projection of U, on the x-axis (m/s) projection of U on the x-axis (m/s) projectron of U on the y-axis (m/s) particle velocity (m/s) chute exit velocity (m/s) projection of V on the x-axis (m/s) projection of V on the y-axis (m/s) width of projection of distance between the chute exit and the lmife edge onto the x-axis (m) distance between chute exrt and knife edge (m) coordinate parallel to particle velocity at chute exit (m) x coordinate of top of partitioned box, corresponding to location of median particles (m) coordinate normal to plane of chmute floor (m) y coordinate of top of partitioned box, corresponding to location of median particles (m) angle of inclination of chute (degrees) air viscosAy (kg/m s) fluid density (kg/m’) sharpness of separation parameter (dimensionless)
REFERENCES 1 R. T_ Hukki, Proc_ Xth Int_ Mineral Processing Congx. 1973. Inst. Min. MetaLl_, pp_ 195 - 212. 2 L. Schiller and A_ Nauman, 2. Ver. Dtsch Ing., 77 (1973) 3aa.