Effect of Tapping on Flow of Particles from Storage Vessels Kunio Shinohara* and Tatsuo Tanaka Department of Chemical Process Engineering, Hokkaido University, Sapporo. Japan
This paper presents an approximate theory on the mechanism of flow of particles through an aperture under tapping. A simplified block-flow model is assumed that particles more or less move as masses or blocks rather than separate single particles within a vessel due to friction and cohesion among particles and a wa!l of the vessel. Thus, blocks directly above the discharge opening flow down predominantly. Based on the model, the effect of tapping on the flow criterion and the flow rate of particles is estimated by solids properties, geometrical dimensions of a storage vessel, and operational conditions. The solids pressure distributions within a cylindrical and a conical vessel are also investigated in relation to the flow mechanism.
Introduction An operation of tapping is in common use especially in such powder handling processes as compaction or filling, storage and supply, and sieving. The effect of tapping has qualitatively been investigated mainly in the field of compaction, where various empirical relations are found (Kawakita, 1964; Kuno, 1958; Taneya and Sone, 1962). Since the mechanism of tapping has not been analyzed on a theoretical basis, it is hard to say whether the tapping operation is useful in practice and what intensity of tapping is suitable for the operation requiring tapping. It is said that tapping is sometimes effective to release blockage of nonfree-flowing powders in a hopper, but sometimes it causes a much more stable arch to form above the discharge opening. This paper intends to deal approximately but analytically with the mec’lanism of flow of particulate solids through an aperture of a vessel, the whole of which is subjected to vertical tapping by rotating a cam ring, and to evaluate the effect of tapping by material properties, geometry of the vessel, and operational conditions. The flow criterion and the flow rate of particles irrespective of the degree of cohesion are theoretically derived on the basis of a simplified block-flow model. The apparent constancy of the flow rate under tapping is also discussed as the result of fluctuation of the solids stream. The present consideration would directly be the basis of the rate of sieving under tapping (Shinohara and Tanaka, 1975).
where I is defined as the intensity of tapping by ( A u / A t ) / g. When C, = 0 and I = 0, corresponding to noncohesive powder under gravity alone, eq 2 is reduced to the socalled Janssen equation. In the case of a conical hopper the force balance on a disk is expressed by
(P,
nD(kiJ.,Pt i- C,)dy (1) where Pt is the vertical solids pressure under tapping at the height, y , above the level of the bottom and A u / A t is the acceleration due to tapping impact. Solving eq 1 under the bound-ry condition of Pt = P,I at y = y 1
% .! (g g,
dP,)ay2 tan2 Q
+
$>TJJ’
tan2 Qdy =
or in the rearranged form
where Ptpis the pressure exerted on the wall perpendicularly, and (Y and @ are coefficients defined as P t , = (sin‘ e + k cos2 O)P, cy
= 2pw cot Q(sin2Q
13
= -2C,
t
(5)
k cos2 Q)
(6)
cot Q
(7)
Integrating eq 4 by using the boundary condition of Pt = Ptl a t y = y l gives cy f
Solids Pressure within d Vessel The distribution of solids pressure within a vessel is needed in consideration of the flow criterion and the flow rate. It could approximately be predicted after Janssen’s (1895) derivation even for cohesive powders under tapping. (See Figure 1 for the notations used in the solids pressure and block-flow model.) In the case of a cylindrical vessel with a flat bottom, the force balance on a solids disk of differential thickness, dy, during impact time of tapping, At, is represented by
+
1: P, = P t 1 ( k ) @ + f{l-
+
In the case of the conical vessel on which a cylindrical vessel is placed, Ptlin eq 8 equals the bottom pressure in the cylinder given by substituting y = 0 in eq 2 . Figure 2 depicts the distribution of solids pressure in both cases.
Flow Criterion under Tapping Fine powders often cause stoppage of flow within a hopper due to cohesion of particles, and even free-flowing coarse solids cannot pass through a smaller opening than that of a size several times the particle diameter. The latter fact would be mainly due to a frictional effect among particles, and it is quite often observed on a screen in a sieving process. Particulate solids are always accompanied more or less by cohesion and friction. A simplified blockInd. Eng. Chem.. Process Des. Develop., Vol. 14, No. 1 , 1975 1
200
100
I /-----0
01
0.2
03
0.4
05
0.6
0.7
0.8
0.9
d1.0 1.0
y l H (-)
Figure 2. Solids pressure distribution within a vessel under tapping.
Figure 1. Notation of solids pressure and block-flow model.
Table I , Material Properties and Operational Conditions Used Commonly for Calculation In the Static State, A s Shown in Figures 2, 3, and 4 Cylinder 1 2 3 4 D = 5.0 cm 8 e = 300 Cone 5 6 7 I 0.0 2.25 4.52 ~0 H =50.0cm D,=l.Ocm k,=0.01 p,=1.134g/cm3 C, = 0.25 G/cm2 C, = 0.20 G/cm2 p l = 0.50 G, = 0.30 In the Nearly Static and Dynamic States, A s Shown in Figures 5 to 9 A t = 0.03 sec pl = 0.075 p w = 0.10 C, = 0 . 0 G/cm' C, = 0.0 G/cm2 p, = 1.386 g/cm3 k , = 2 . 4 6 k,' = e.001 z ' ~= 20 cm/sec
___
flow model is, therefore, assumed in which particles move as agglomerated blocks rather than separate single particles. It is assumed that a hypothetical yield plane will develop vertically in the same shape and size as that of a discharge opening above the outlet, as was adopted in a hopper design in our previous paper (Shinohara, et al., 1968). Thus the force balance on a cylindrical bl-ock of particles above a circular opening leads to the flow criterion under tapping in static equilibrium as
71
ZD02cT
(9)
where HO is the height of the aperture above the bottom or the apex of the vessel. Integrating and transposing it on the left-hand side
where Fa indicates the degree of flowability, the positive sign of which suggests sure discharge under tapping. In the case of the cylindrical vessel, putting eq 2 into eq 10 gives the flowability as 2
Ind. Eng. Chern., Process Des. Develop., Vol. 14, No. 1, 1975
In the case of the conical vessel, putting eq 8 into eq 10
(%*)as+1}]
(12)
2H
According to the computed results of Fa with respect to y, one block of thickness, H - Ho, corresponding to y = H and CT = 0 in eq 11 and 12, gives the maximum value of Fa, that is, the largest probability of discharge under tapping, as shown in Figure 3. Hence, Fa,,,, = 0 presents the flow criterion under tapping, which gives the critical relationship among the minimum intensity of tapping, I, the aperture size, DO, and the maximum height of powder bed, H . In the case of the cylinder, from eq 11
In the case of the cone, from eq 12
7---/
,
loo/
L-0
02
04
I
06
08
10
y IH ( - )
Figure 3. Flowability of solids block under tapping.
100
200
300
LOO
500
603
700
803
H (crn)
Figure 4. Flow criterion of particles under tapping
(14)
where YOis defined as a function of H by
y - A =H- - - - - -Do 0 -
H
2Htan 0
(15)
In eq 13 and 14, 1 = 0 gives the flow criterion under gravit y alone, and I = infinity gives the blockage line under tapping. Figure 4 illustrates the interrelationship among DO,H, and I.
Acceleration of Particles Block When the flow criterion is satisfied and solids begin to discharge, the acceleration of particles is derived from the dynamic force balance on the cylindrical block as
large enough for particles to start with acceleration due to tapping impact and to decelerate to a stop due to frictional and cohesive resistance over the vertical yield plane of the block above the opening. Among such bulk properties as C,, pnr p , , and k , the pressure ratio, k , is the representative property which varies largely and sensitively with the state of packing and stress to exhibit fluctuation of the solids stream. k will change with discharge time from the minimum value of k,' in the nearly static state to the maximum, k
