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State of equilibrium between dust pile and jet flow in slit nozzle impactor
Powder 0
Technology,
11 (1975)
173-181
Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
State of Equilibrium
between
Dust Pile and Jet Flow in Slit Nozzle
SHINICHI YUU, TATUO MURATAKA, Department
of Chemical
Engineering,
I%DEKAZU
Kyoto
University
Impactor
ISHIMURA and KOICHI IINOYA (Japan)
(Received May 20, 1974; in revised form September 20,1974)
SUMMARY The shape of the dust pile equilibrated on a collection plate with a two-dimensional impinging jet flow is studied experimentally and analytically in relation to re-entrainment in a cascade im-
pactor. Consequently, it becomes clear that the shape of the equilibrated dust pile may be expressed as a function of the dimensionless number B, which represents the ratio of the weight of piled dust per characteristic volume of the nozzle to the momentum of fluid jet per unit time, and of the effective angle of internal friction of the dust. A maximum weight and volume of the dust pile for an impactor can be estimated by use of the value of B, talc-ulated from the operating conditions. When an impactor is used for the separation of small quantity of a precious dust, the maximum weight and volume of the dust pile are the critical value.
INTRODUCTION
In relation not only to the mechanism of dust pile formation against air jet flow Dut also to the re-entrainment of collected particles in an impactor, it is interesting to examine how much dust is equilibrated with the impinging jet flow on a collection plate. However, no work has been published, presumably because the relationship between the properties of piled dust and the air flow around it are too complicated. On the assumptions that the angle between the maximum principal stress on the free surface of the piled dust and the vertical axis is T/Z radians and that the air flow around the dust pile is a potential flow [l] , the equilibrium state between the dust pile and a two-dimensional impinging jet flow is studied herein. Consequently the mechanism of this phenomenon is clarified to some extent.
1. THEORY 1.1 Fluid velocity
distribufion
The schematic view of the model is shown in Fig. 1. It is anticipated that the equilibrium dust pile may be wedge-shaped. On the assumption that the fluid flow is a potential one, the approximate fluid velocity distribution around the dust pile is described by the following equation obtained using the Schwarz and Christoffel transformation [2] :
fl is the angle between the free surface of the dust pile and the vertical axis at an arbitrary point, and the fluid velocity changes along the free surface of the dust pile.
X-lg. 1. Schematic
view of equilibrium
dust File.
Fig. 2. Schematic
view of equilibrium
stress.
1.2 Derivation of equilibrium equation The fundamental equation is based on the following idea: (Resultant surface force of piled dust) = (surface force by fluid) + (weight of piled dust)
(2)
Figure 2 shows a schematic view of the equilibrium state. Equation (2) is developed over a stationary volume element (AxAyZ). By dividing this developed equation by (&Ayl) and taking the limit as Ax and Ay approach zero, the equation of equilibrium is expressed as follows:
aa, arxr
3% : ah
ax+-ay
ay
are respectively the x and y components of resultant surface force
ax
of piled dust per unit volume,
wf
-
ay
ah,
+ -
ax
are respectively the x and y components of surface force per
unit volume by fluid, (pbg) is the y component of weight of piled dust per unit vohnne, and T__,= T~, under the equilibrium condition for an elementary volume. Therefore,
aa, aii,
aaxf
ax*ay=ax+ay
aTxrf
aa aT aa,, a7,,, __?+Z ay ax =~tX4T+Pbg
(3)
According to Jenike [3], the component stresses of the dust are as follows: 0,
= a (1 -
sin 8 cos 2w)
OY
=a(I+sinf?cos2w)
TX,
=asmesin2w
(4)
175
where 6 is the effective angle of internal friction of the dust, which is defined by the following equation 13 - 53 : 01-02
(5)
=sintI Dl+-
(32
w in eqn. (4) is the angle between the direction of the major principal stress u1 and the vertical axis in any arbitrary plane. Near the horizontal part of the free surface o is 7r/2; however, at other parts it is not always 7r/2. In this paper o is assumed to be n/2 for simplicity [4,5]. Therefore, eqn. (4) reduces to 0,
= u (1 +
0,
=o(l-SirlB)
TXY = Tyx
=
.sjne) (6) 0
On the free surface of the dust at equilibrium do _s zero. Hence
a0
dy -= dx
-- ax
(7)
ao ay
Substituting eqns. (3) and (6) into eqn. (7), the following equation is obtained:
(8)
where the shearing stress of the fluid is neglected. oxf and (s,,f are obtained by using the Bernoulli equation: Ozcf = (Tyf
=-p+2_po
(9)
Substituting eqns. (1) and (9) into eqn. (8), the final equilibrium ordinary differential equation is obtained:
-
(10)
where dimensionless quantities are introduced as defined by eqn. (II):
(11)
176
As the result, a dimensionless equilibrium equation is derived from eqn. (10) by using B and 8 as follows:
l-sine
(
d2u
=f(B,@)=
(rZ+Y2)(~w+4)*
x-the
cu12
where B is a dimensionless number defined by eqn. (13):
B=
%bga = weight PG
of dust pile per characteristic volume (0: I) of nozzle momentum of fluid jet per unit tune
(13)
Substituting the values of B and 6 into eqn. (12) and carrying out the calculation, the shape of the equilibrium dust pile is obtained. This is a two point boundary value problem, and the two boundary values are as follows:
7 = 0,
F=O
(15)
Equation (14) means that the equilibrium shape of the dust pile is almost a horizontal plane at a point which is seven times D, away in the direction of the x-axis from the central axis of the piled dust. This agrees with the experimental results. On the other hand, eqn. (15) means that the origin is located at the top of the piled dus:.
2. EXPERIMENTAL
APPARAT’JS
AND
PROCE;)URE
The experimental apparatus is shown in Fig. 3. Test dust, which is dried at about 100°C for not less than 24 hours, is fed to the impactor through a mixer type disperser. The experimental procedure used is as follows. The vacuum pump is started and the aerosol jet allowed to impinge on a
&%
mixer
diagram of experimental
apparatus.
177 TABLE
1
Experimental 1. Ploperties
conditions of dust fly ash
mass median diameter D,, particle density
, pm
pp. g/cm3
effective angle of internal friction of dust 0, deg 2. 3. 4. 5. 6. 7. 8. 9. 10.
CaC03
10.6
4.8
2.26
2.80 37.5
39.6
Air velocity of nozzle outlet ug = 5 - 25 m/set Reynolds number based on nozzle width R,j = 280 - 1820 Inlet air flow rate Q,-, = 11 - 43 I/min Clearance ratio R, = 5 Material of nozzle: brass Nozzle width D,: 0.0910, 0.0668 cm Nozzle length I: 4.550, 4.385 cm Material of collection plate: glass Dimensions of collection plate: 1.5 X 4.8 cm
tared collection plate. A large quantity of dust is piled up on the collection plate until it is equilibrated with the air jet, then the plate is removed and reweighed, and a photograph of the crosssection of the dust pile is taken. The nozzle is directed against the collection plate at a distance of five times the nozzle width. The state of dispersion of the test dusts has been checked by the following method. The particle size distributions, which are measured by a cascade impactor, of the test dusts generated by a mixer type disperser, have been compared with the distributions by a sedimentation balance [ 63, and they are in fairly good agreement with each other. Therefore the test dusts seem to be well dispersed by the mixer type disperser. The effective angle of internal friction of each dust used as feed is measured by the shear test apparatus [4]. The experimental conditions and details of the nozzles and the collection target plate are shown ill Table 1.
3. RESULTS
AND
DISCUSSION
3.1. Calculated and experimental results The calcu’dted shapes of the equilibrium dust piles for various values of i3 are shown in Fig. 4 (0 = 39.63 fly ash) and Fig. 5 (0 = 37.53 CaC03) respectively. As the cross-sectional area of the dust pile is bisymmetrical, only one side of each dust pile is shown in these figures. The origin on each figure indicates the top of the dust pile. The cross-sectional area and the volume of the dust pile are obtained from the shapes indicated in these figures. The calculated results for the dimensionless cross-sectional area % are shown in Fig. 6 (0 = 39.6”, fly ash) and Fig. 7 (0 = 37.54 CaCO, ) respectively. Similarly the calculated dimensionless weights M are shown in Figs. 8 and 9.3 and M are defined by the following equations:
cross-sectional area of dust pile s area of square with side equal to nozzle width D, = 0,’
g=
M=
weight of dust pile momentum
of air jet per unit time
=p9PbgS
(16)
(17)
P &Dc
In order to analyse these experimental results, the air jet velocity, equilibrium shape, and bull;
178
Fig. 4. Comparison
of calculated
shapes of equilibrium
dust pile with experimental
values (fly ash).
Fig. 5. Comparison
of calculated
shapes of equilibrium
dust pile with experimental
values (CaCOa).
density of the dust pile have to be measured. The air flow rate is measured by a gas meter to obtain the air jet velocity. The equilibrium shape is obtained from the photograph. The weight and the cross-sectional area of the dust pile for obtaining the bulk density are measured by use of a microbalance and a planimeter respectively, and the length of the dust pile is measured by micrometer. The bulk density pb is obtained by the following equation: the weight of the dust pile Pb = (the cross-sectional area of the dust pile) (the length of the dust pile)
(18)
The measured bulk density of pb is plotted against u. in Fig. 10. Experimental results of the shape, 3 and Iii are showzn in Figs. 4 - 9 with the calculated results. One of the photographs of an equilibrium dust pile is given as an example in Fig. 11.
20
-
.
2 -To L 0.1
0.05
F_
0.15
0.2
0.1
t-1 I,
1
0
lDbgDC
0.02
0.04 a=
111
0.1
0.08
2E!?%0
0
,
,
0.06
0.12
0.14
I-1
“;
Fig. 6. Comparison values (fly a&l>.
of calculated
dimensionless
crosssectional
areas of equilibrium
dust pile with experimental
Fig. 7. timparison
of calculated
dimensionless
crosssectional
areas of equilibrium
dust pile with experimental
(CaC03).
values
179
l-
0.8. 0.6
-
cl.4 0.3.
0.2
-
::
u
“3 p_ O-l-
0.08.
I
0.06
-
0.04
-
0.06 0.04
o.oj-
c
N
0.03
o.o2-
O_Oll 0.01
1
0.02
.
5=
.
0.05
0.03 ~
l/c Ja
0.1 0.08
.
0
.
I
0_080.1
2Db9DC
l-l
L) u;
Fig. 8. Comparison of calculated mental values (fly ash). Fig. 9. Comparison values (CaCO,).
ratio Al of weight of equilibrium
of calculated ratio of weight of equilibrium
dust pile to momentum
dust pile to momentum
dust : fly ash B = 0.178 t-1 DC= 0.091 Icml
of air jet with experi-
of air jet with experimental
magnificati.cn=<
:
0
0
5
t 10
I
15
I
20
25
up b/secl
Fig. 10. Experimental
values of bulk density.
Fig. 11. Photograph
of equilibrium
piled dust.
180 3.2 Discussion 3.2.1 Equilibrium shape of dust pile The results of Figs. 4 and 5 indicate that the shape of the equilibrium dust pile is expressed as a function of the dimensionless number B when 0 is constant. Decreasing the value of B means that the weight of piled dust per unit volume or the bulk density becomes smaller, or that the momentum of the air jet, in other words the jet velocity, becomes large-r. As shown in Fig. 10, it is evident that the bulk density becomes larger at higher air jet velocities, because the dust is more closely piled. Therefore, the smaller value of B means that the increment of the momentum of the air jet becomes larger than that of the bulk density. The results shown in Figs. 4 and 5 also indicate that in proportion to the value of B, for small B the quantity of piled dust at equilibrium is small and the apex angle of the piled dust is large, that is, the piled dust is depressed by a higher air jet velocity. In general, experimental results are in good agreement with calculated ones, as shown in Figs. 4 and 5. Accordingly the above theoretical explanation of the equilibrium state seems to be reasonable. 3.2.2 Dimensionless cross-sectional area of dust pile s The results of Figs. 6 and 7 indicate that the dimensionless cross-sectional area may be expressed as a function of B, similarly to the above equilibrium shape. Sfor fly ash (0 = 39.69 in Fig. 6 is larger than 3 for CaCOs (0 = 37.5“) in Fig. 7 at the same value of B, therefore the equilibrium S is proportional to the effective internal friction angle 8. It is also shown that s decreases rapidly when B becomes smaller than about 0.03. The trends of the experimental results are in good agreement with those of the calculated ones, but there are some differences between the experimental and calculated values. The discrepancies seem to be caused by the following three facts: (i) only the effective internal friction angle is taken into account as a dust property to derive the equilibrium equation (lo), (ii) w is assumed equal to 7r/2, (iii) the viscosity of the air is neglected. 3.2.3 Dimensionless weight of dust pile M For the weighing method in particle size measurement by an impactor 163, it is an important mztter to estimate the weight limit of the dust pile equilibrated with an air jet. From the experimentai results of Figs. 8 and 9 it is found that 111increases about 20% when 8 becomes about 2’ larger. AM uersus B plot gives a straight line on logarithmic paper. As the value of B is calculated from the operating conditions, the maximmn weight of dust pile without re-entrainment can be estimated by using these figures.
CONCLUSION
The following results have been obtained for the equilibrium dust pile: (1) The approximate theoretical explanation for the dust pile equilibrated with a two-dimensional air jet is given by a simple model. (2) Shape, cross-sectional area and weight of an equilibrium dust pile are expressed as a function of a dimensionless number B and the effective angle of internal friction of the dust. (3) The maximum weight and the volume of the dust pile on an impactor plate can be estimated by the use of B calculated for the operating conditions.
LIST
B D,
OF SYMBOLS
dimensionless number defined by eqn. (13) = 2p,.,gD,/pu~ nozzle width [cm]
[ - J
181
D Pm g
L
I M Qo
Rc Rej
s
s u UO --
x,
Y
z
P
a P9 PP Pb I-1 w 01 (TP D, ox,
ax,, Txr
*Y
Oyf 9 7Y.T
TXYf,
TYXf
mass median diameter of dust particles [pm] acceleration due to gravity [cm/see21 distance from nozzle outlet to collection target plate [cm] nozzle length [cm] ratio of weight of dust pile to momentum of air jet per unit time = 2pbgs;p uw, 1 - ] inlet air flow rate [cm3/sec] clearance ratio = (L/D,) [ - ] Reynolds number based on nozzle width = DC p uo/p [ - 1 cross-sectional area of dust pile [cm21 dimensionless cross-sectional area of dust pile = S/DC2 1. - ] complex velocity of fluid = (u,, - iuX) [m/set] air velocity at nozzle outlet [m/set] dimensionless coordinates = x/D,, yfD, [ - ] dimensionless complex number = X + i 7 [ - ] angle between free surface of dust pile and vertical axis at arbitrary point [radian], 13 effective angle of internal friction of dust [radian], [“I air and particle densities [g/cm31 bulk density of dust pile [g/cm31 air viscosity [g/cm set ] angle between direction of major principal stress and vertical axis [radian], [“I major principal stress of dust pile [g/set’ cm] minor principal stress of dust pile [g/sec2 cm] normal stresses of dust pile [g/sec2 cm] normal stresses by air flow [g/sec2 cm] shearing stresses of dust pile [g/see2 cm] shearing stresses by air flow [g/sec2 cm]
REFERENCES B. Fagela-Alabastro Estrella and J.D. Hellurns, Laminar gas jet impinging on an infinite liquid surface, Ind. Eng. Chem. Fundam., 6 (196’i) 580. L.M. Milne and C.B. Thomson, Theoretical Hydrodynamics, Macmillan, London, 4th edn., 1960, p. 289. A.W. Jenike, Steady gravity flow of frictional-cohesive solids in converging channels, Trans. ASME, J. Appl. Mech., 31 (1964) 5. A.W. Jenike, Gravity flow of bulk solids, Bull. 108, Eng. Expt. Sta., Utah State Univ., 1961. J.R. Johanson and A.W. Jenike, Stress and velocity fields in gravity flow of bulk solids, Bull. i16, Eng. Expt. Sta., Utah State Univ., 1962. K. Iinoya, S. Yuu, K. Makino and K. Nakano, On measurement of particle size distribution by cascade impactor - In case of setting the clearance ratio three for round nozzle -, Kagaku Kogaku, 33 (1969) 689.
Technology,
11 (1975)
173-181
Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
State of Equilibrium
between
Dust Pile and Jet Flow in Slit Nozzle
SHINICHI YUU, TATUO MURATAKA, Department
of Chemical
Engineering,
I%DEKAZU
Kyoto
University
Impactor
ISHIMURA and KOICHI IINOYA (Japan)
(Received May 20, 1974; in revised form September 20,1974)
SUMMARY The shape of the dust pile equilibrated on a collection plate with a two-dimensional impinging jet flow is studied experimentally and analytically in relation to re-entrainment in a cascade im-
pactor. Consequently, it becomes clear that the shape of the equilibrated dust pile may be expressed as a function of the dimensionless number B, which represents the ratio of the weight of piled dust per characteristic volume of the nozzle to the momentum of fluid jet per unit time, and of the effective angle of internal friction of the dust. A maximum weight and volume of the dust pile for an impactor can be estimated by use of the value of B, talc-ulated from the operating conditions. When an impactor is used for the separation of small quantity of a precious dust, the maximum weight and volume of the dust pile are the critical value.
INTRODUCTION
In relation not only to the mechanism of dust pile formation against air jet flow Dut also to the re-entrainment of collected particles in an impactor, it is interesting to examine how much dust is equilibrated with the impinging jet flow on a collection plate. However, no work has been published, presumably because the relationship between the properties of piled dust and the air flow around it are too complicated. On the assumptions that the angle between the maximum principal stress on the free surface of the piled dust and the vertical axis is T/Z radians and that the air flow around the dust pile is a potential flow [l] , the equilibrium state between the dust pile and a two-dimensional impinging jet flow is studied herein. Consequently the mechanism of this phenomenon is clarified to some extent.
1. THEORY 1.1 Fluid velocity
distribufion
The schematic view of the model is shown in Fig. 1. It is anticipated that the equilibrium dust pile may be wedge-shaped. On the assumption that the fluid flow is a potential one, the approximate fluid velocity distribution around the dust pile is described by the following equation obtained using the Schwarz and Christoffel transformation [2] :
fl is the angle between the free surface of the dust pile and the vertical axis at an arbitrary point, and the fluid velocity changes along the free surface of the dust pile.
X-lg. 1. Schematic
view of equilibrium
dust File.
Fig. 2. Schematic
view of equilibrium
stress.
1.2 Derivation of equilibrium equation The fundamental equation is based on the following idea: (Resultant surface force of piled dust) = (surface force by fluid) + (weight of piled dust)
(2)
Figure 2 shows a schematic view of the equilibrium state. Equation (2) is developed over a stationary volume element (AxAyZ). By dividing this developed equation by (&Ayl) and taking the limit as Ax and Ay approach zero, the equation of equilibrium is expressed as follows:
aa, arxr
3% : ah
ax+-ay
ay
are respectively the x and y components of resultant surface force
ax
of piled dust per unit volume,
wf
-
ay
ah,
+ -
ax
are respectively the x and y components of surface force per
unit volume by fluid, (pbg) is the y component of weight of piled dust per unit vohnne, and T__,= T~, under the equilibrium condition for an elementary volume. Therefore,
aa, aii,
aaxf
ax*ay=ax+ay
aTxrf
aa aT aa,, a7,,, __?+Z ay ax =~tX4T+Pbg
(3)
According to Jenike [3], the component stresses of the dust are as follows: 0,
= a (1 -
sin 8 cos 2w)
OY
=a(I+sinf?cos2w)
TX,
=asmesin2w
(4)
175
where 6 is the effective angle of internal friction of the dust, which is defined by the following equation 13 - 53 : 01-02
(5)
=sintI Dl+-
(32
w in eqn. (4) is the angle between the direction of the major principal stress u1 and the vertical axis in any arbitrary plane. Near the horizontal part of the free surface o is 7r/2; however, at other parts it is not always 7r/2. In this paper o is assumed to be n/2 for simplicity [4,5]. Therefore, eqn. (4) reduces to 0,
= u (1 +
0,
=o(l-SirlB)
TXY = Tyx
=
.sjne) (6) 0
On the free surface of the dust at equilibrium do _s zero. Hence
a0
dy -= dx
-- ax
(7)
ao ay
Substituting eqns. (3) and (6) into eqn. (7), the following equation is obtained:
(8)
where the shearing stress of the fluid is neglected. oxf and (s,,f are obtained by using the Bernoulli equation: Ozcf = (Tyf
=-p+2_po
(9)
Substituting eqns. (1) and (9) into eqn. (8), the final equilibrium ordinary differential equation is obtained:
-
(10)
where dimensionless quantities are introduced as defined by eqn. (II):
(11)
176
As the result, a dimensionless equilibrium equation is derived from eqn. (10) by using B and 8 as follows:
l-sine
(
d2u
=f(B,@)=
(rZ+Y2)(~w+4)*
x-the
cu12
where B is a dimensionless number defined by eqn. (13):
B=
%bga = weight PG
of dust pile per characteristic volume (0: I) of nozzle momentum of fluid jet per unit tune
(13)
Substituting the values of B and 6 into eqn. (12) and carrying out the calculation, the shape of the equilibrium dust pile is obtained. This is a two point boundary value problem, and the two boundary values are as follows:
7 = 0,
F=O
(15)
Equation (14) means that the equilibrium shape of the dust pile is almost a horizontal plane at a point which is seven times D, away in the direction of the x-axis from the central axis of the piled dust. This agrees with the experimental results. On the other hand, eqn. (15) means that the origin is located at the top of the piled dus:.
2. EXPERIMENTAL
APPARAT’JS
AND
PROCE;)URE
The experimental apparatus is shown in Fig. 3. Test dust, which is dried at about 100°C for not less than 24 hours, is fed to the impactor through a mixer type disperser. The experimental procedure used is as follows. The vacuum pump is started and the aerosol jet allowed to impinge on a
&%
mixer
diagram of experimental
apparatus.
177 TABLE
1
Experimental 1. Ploperties
conditions of dust fly ash
mass median diameter D,, particle density
, pm
pp. g/cm3
effective angle of internal friction of dust 0, deg 2. 3. 4. 5. 6. 7. 8. 9. 10.
CaC03
10.6
4.8
2.26
2.80 37.5
39.6
Air velocity of nozzle outlet ug = 5 - 25 m/set Reynolds number based on nozzle width R,j = 280 - 1820 Inlet air flow rate Q,-, = 11 - 43 I/min Clearance ratio R, = 5 Material of nozzle: brass Nozzle width D,: 0.0910, 0.0668 cm Nozzle length I: 4.550, 4.385 cm Material of collection plate: glass Dimensions of collection plate: 1.5 X 4.8 cm
tared collection plate. A large quantity of dust is piled up on the collection plate until it is equilibrated with the air jet, then the plate is removed and reweighed, and a photograph of the crosssection of the dust pile is taken. The nozzle is directed against the collection plate at a distance of five times the nozzle width. The state of dispersion of the test dusts has been checked by the following method. The particle size distributions, which are measured by a cascade impactor, of the test dusts generated by a mixer type disperser, have been compared with the distributions by a sedimentation balance [ 63, and they are in fairly good agreement with each other. Therefore the test dusts seem to be well dispersed by the mixer type disperser. The effective angle of internal friction of each dust used as feed is measured by the shear test apparatus [4]. The experimental conditions and details of the nozzles and the collection target plate are shown ill Table 1.
3. RESULTS
AND
DISCUSSION
3.1. Calculated and experimental results The calcu’dted shapes of the equilibrium dust piles for various values of i3 are shown in Fig. 4 (0 = 39.63 fly ash) and Fig. 5 (0 = 37.53 CaC03) respectively. As the cross-sectional area of the dust pile is bisymmetrical, only one side of each dust pile is shown in these figures. The origin on each figure indicates the top of the dust pile. The cross-sectional area and the volume of the dust pile are obtained from the shapes indicated in these figures. The calculated results for the dimensionless cross-sectional area % are shown in Fig. 6 (0 = 39.6”, fly ash) and Fig. 7 (0 = 37.54 CaCO, ) respectively. Similarly the calculated dimensionless weights M are shown in Figs. 8 and 9.3 and M are defined by the following equations:
cross-sectional area of dust pile s area of square with side equal to nozzle width D, = 0,’
g=
M=
weight of dust pile momentum
of air jet per unit time
=p9PbgS
(16)
(17)
P &Dc
In order to analyse these experimental results, the air jet velocity, equilibrium shape, and bull;
178
Fig. 4. Comparison
of calculated
shapes of equilibrium
dust pile with experimental
values (fly ash).
Fig. 5. Comparison
of calculated
shapes of equilibrium
dust pile with experimental
values (CaCOa).
density of the dust pile have to be measured. The air flow rate is measured by a gas meter to obtain the air jet velocity. The equilibrium shape is obtained from the photograph. The weight and the cross-sectional area of the dust pile for obtaining the bulk density are measured by use of a microbalance and a planimeter respectively, and the length of the dust pile is measured by micrometer. The bulk density pb is obtained by the following equation: the weight of the dust pile Pb = (the cross-sectional area of the dust pile) (the length of the dust pile)
(18)
The measured bulk density of pb is plotted against u. in Fig. 10. Experimental results of the shape, 3 and Iii are showzn in Figs. 4 - 9 with the calculated results. One of the photographs of an equilibrium dust pile is given as an example in Fig. 11.
20
-
.
2 -To L 0.1
0.05
F_
0.15
0.2
0.1
t-1 I,
1
0
lDbgDC
0.02
0.04 a=
111
0.1
0.08
2E!?%0
0
,
,
0.06
0.12
0.14
I-1
“;
Fig. 6. Comparison values (fly a&l>.
of calculated
dimensionless
crosssectional
areas of equilibrium
dust pile with experimental
Fig. 7. timparison
of calculated
dimensionless
crosssectional
areas of equilibrium
dust pile with experimental
(CaC03).
values
179
l-
0.8. 0.6
-
cl.4 0.3.
0.2
-
::
u
“3 p_ O-l-
0.08.
I
0.06
-
0.04
-
0.06 0.04
o.oj-
c
N
0.03
o.o2-
O_Oll 0.01
1
0.02
.
5=
.
0.05
0.03 ~
l/c Ja
0.1 0.08
.
0
.
I
0_080.1
2Db9DC
l-l
L) u;
Fig. 8. Comparison of calculated mental values (fly ash). Fig. 9. Comparison values (CaCO,).
ratio Al of weight of equilibrium
of calculated ratio of weight of equilibrium
dust pile to momentum
dust pile to momentum
dust : fly ash B = 0.178 t-1 DC= 0.091 Icml
of air jet with experi-
of air jet with experimental
magnificati.cn=<
:
0
0
5
t 10
I
15
I
20
25
up b/secl
Fig. 10. Experimental
values of bulk density.
Fig. 11. Photograph
of equilibrium
piled dust.
180 3.2 Discussion 3.2.1 Equilibrium shape of dust pile The results of Figs. 4 and 5 indicate that the shape of the equilibrium dust pile is expressed as a function of the dimensionless number B when 0 is constant. Decreasing the value of B means that the weight of piled dust per unit volume or the bulk density becomes smaller, or that the momentum of the air jet, in other words the jet velocity, becomes large-r. As shown in Fig. 10, it is evident that the bulk density becomes larger at higher air jet velocities, because the dust is more closely piled. Therefore, the smaller value of B means that the increment of the momentum of the air jet becomes larger than that of the bulk density. The results shown in Figs. 4 and 5 also indicate that in proportion to the value of B, for small B the quantity of piled dust at equilibrium is small and the apex angle of the piled dust is large, that is, the piled dust is depressed by a higher air jet velocity. In general, experimental results are in good agreement with calculated ones, as shown in Figs. 4 and 5. Accordingly the above theoretical explanation of the equilibrium state seems to be reasonable. 3.2.2 Dimensionless cross-sectional area of dust pile s The results of Figs. 6 and 7 indicate that the dimensionless cross-sectional area may be expressed as a function of B, similarly to the above equilibrium shape. Sfor fly ash (0 = 39.69 in Fig. 6 is larger than 3 for CaCOs (0 = 37.5“) in Fig. 7 at the same value of B, therefore the equilibrium S is proportional to the effective internal friction angle 8. It is also shown that s decreases rapidly when B becomes smaller than about 0.03. The trends of the experimental results are in good agreement with those of the calculated ones, but there are some differences between the experimental and calculated values. The discrepancies seem to be caused by the following three facts: (i) only the effective internal friction angle is taken into account as a dust property to derive the equilibrium equation (lo), (ii) w is assumed equal to 7r/2, (iii) the viscosity of the air is neglected. 3.2.3 Dimensionless weight of dust pile M For the weighing method in particle size measurement by an impactor 163, it is an important mztter to estimate the weight limit of the dust pile equilibrated with an air jet. From the experimentai results of Figs. 8 and 9 it is found that 111increases about 20% when 8 becomes about 2’ larger. AM uersus B plot gives a straight line on logarithmic paper. As the value of B is calculated from the operating conditions, the maximmn weight of dust pile without re-entrainment can be estimated by using these figures.
CONCLUSION
The following results have been obtained for the equilibrium dust pile: (1) The approximate theoretical explanation for the dust pile equilibrated with a two-dimensional air jet is given by a simple model. (2) Shape, cross-sectional area and weight of an equilibrium dust pile are expressed as a function of a dimensionless number B and the effective angle of internal friction of the dust. (3) The maximum weight and the volume of the dust pile on an impactor plate can be estimated by the use of B calculated for the operating conditions.
LIST
B D,
OF SYMBOLS
dimensionless number defined by eqn. (13) = 2p,.,gD,/pu~ nozzle width [cm]
[ - J
181
D Pm g
L
I M Qo
Rc Rej
s
s u UO --
x,
Y
z
P
a P9 PP Pb I-1 w 01 (TP D, ox,
ax,, Txr
*Y
Oyf 9 7Y.T
TXYf,
TYXf
mass median diameter of dust particles [pm] acceleration due to gravity [cm/see21 distance from nozzle outlet to collection target plate [cm] nozzle length [cm] ratio of weight of dust pile to momentum of air jet per unit time = 2pbgs;p uw, 1 - ] inlet air flow rate [cm3/sec] clearance ratio = (L/D,) [ - ] Reynolds number based on nozzle width = DC p uo/p [ - 1 cross-sectional area of dust pile [cm21 dimensionless cross-sectional area of dust pile = S/DC2 1. - ] complex velocity of fluid = (u,, - iuX) [m/set] air velocity at nozzle outlet [m/set] dimensionless coordinates = x/D,, yfD, [ - ] dimensionless complex number = X + i 7 [ - ] angle between free surface of dust pile and vertical axis at arbitrary point [radian], 13 effective angle of internal friction of dust [radian], [“I air and particle densities [g/cm31 bulk density of dust pile [g/cm31 air viscosity [g/cm set ] angle between direction of major principal stress and vertical axis [radian], [“I major principal stress of dust pile [g/set’ cm] minor principal stress of dust pile [g/sec2 cm] normal stresses of dust pile [g/sec2 cm] normal stresses by air flow [g/sec2 cm] shearing stresses of dust pile [g/see2 cm] shearing stresses by air flow [g/sec2 cm]
REFERENCES B. Fagela-Alabastro Estrella and J.D. Hellurns, Laminar gas jet impinging on an infinite liquid surface, Ind. Eng. Chem. Fundam., 6 (196’i) 580. L.M. Milne and C.B. Thomson, Theoretical Hydrodynamics, Macmillan, London, 4th edn., 1960, p. 289. A.W. Jenike, Steady gravity flow of frictional-cohesive solids in converging channels, Trans. ASME, J. Appl. Mech., 31 (1964) 5. A.W. Jenike, Gravity flow of bulk solids, Bull. 108, Eng. Expt. Sta., Utah State Univ., 1961. J.R. Johanson and A.W. Jenike, Stress and velocity fields in gravity flow of bulk solids, Bull. i16, Eng. Expt. Sta., Utah State Univ., 1962. K. Iinoya, S. Yuu, K. Makino and K. Nakano, On measurement of particle size distribution by cascade impactor - In case of setting the clearance ratio three for round nozzle -, Kagaku Kogaku, 33 (1969) 689.