Ind. Eng. Chem. Process Des. Dev. 1004, 2 3 , 423-428
of relief vents (Nomura and Tanaka 1980b),for example. Nomenclature ap = absorptivity of particle c d = dust concentration, g/cm3 C, = oxygen concentration, mol/cm3 C,, = specific heat of solid, cal/(g K) Cpi = specific heat of i gas at constant pressure, cal/(mol K) Do = diffusion coefficient of oxygen, cm2/s Dp,Dpo= particle size, cm F = view factor AHo%* = standard heat of formation, cal/mol KD = burning constant, s/cm2 k = thermal conductivity of gas, cal/(cm s K) k* = material constant, cm L = distance between particles, cm Mp = molecular weight of particle, g/mol ml,mo,m(0) = mass of a particle, g Nl = mole number of oxygen required for the combustion of one mole of particle NO, = mole number of oxygen in the space shared by a single particle, mol Np= mole number of particle burnt by exhausting NO,mole of oxygen, mol ~i = mole number of i gas, mol po = partial pressure of oxygen, atm R b = flame front radius, cm r = distance, cm Td, Tf, T = temperature of particle, flame and gas, respectively, ’K Ti, T, = initial temperature and surface temperature of particle, K Ti, = initial temperature of gas, K V = volume occupied by one particle, cm3 q, cp = emissivity of flame and particle l =~ K D ~ P ~ D ~ ~ TK~ ~ / P ~ C , ,
q
423
= e/?
0, O1 = time, s K = thermal diffusivity, cmz/s
5 = 12KDk/psCp pa, po = density of particle and air, g/cm3 u = Stefan-Boltzmann constant, cal/(cm2K4s) 7 = time needed for complete combustion of a single particle, 8
Registry No. Oxygen, 7782-44-7; aluminum, 7429-90-5; terephthalic acid, 100-21-0.
Literature Cited Craven, A. D.; Foster, M. 0. Combust. flame 1987, 1 1 , 408. Essenhigh, R. H.; Fells, I.Discuss. faraday Soc. 1980, 30, 208. Godsave, 0. A. E. “Fourth Symposlum on Combustion”; The Combustion Institute: Cambridge, MA, 1953; p 818. Hartmann. I.;Jacobson, M.; Willlama, R. P. U . S . Bur. Mines Rept. Invest. No. 5052 1954. Ishlhama, W.; Enomoto, H. “15th Symposlum on Combustion”; The Combustion Institute: Tokyo, Japan, 1974; p 479. Kumagal, S.; I d a , H. “Sixth Symposium on Combustlon”; The Combustion Institute: New Haven, CT, 1957; p 726. Maezawa, M. “Galsetsu AnrenXogaku”; Kyorltsu Shuppan (Japan), 1975; p 78. Mlesse, C. “Slxth Symposium on Combustion”; The Combustion Institute: New Haven, CT, 1957; p 732. Miteul, R.; Tanaka, T. Ind. €4. Chem. Process Des. D e v . 1973, 12, 384. Nomwa, S.; Tanaka. T. Kagaku Kogeku Ronbunshu (Jpn) 1978. 6 , 634. Noumra, S.; Tanaka, T. Ind. Eng. Chem. Process Des. Dev. 198Oa, 19, 451. Noumra, S.; Tanaka, T. J . Chem. Eng. Jpn. 1980b, 13, 309. Wise, H.; Lorel, J.; Wood, E. J. “Fifth Symposium on Combustlon”; The Com bustlon Instttute: Pittsburgh, PA, 1055; p 1321. Yagi, S.; Kunli, D. J . Chem. Soc.(Jpn). Ind. Chem. Sect. 1953, 5 6 , 131.
.
Received for review June 26,1980 Accepted August 29, 1983
Mechanism of Density Segregation of Particles in Filling Vessels Kunlo Shlnohara’ and Shln-lchl Mlyata D e p a m n t of Chemic&!/Process Engineering, Hokkakb Un/verslty, Sapporo, Japan 060
Our segregation model on particles of different size and shape is generally modified to describe the segregation mechanism of differently dense particles flowing down on the inclined heap surface. The analysis and experiments show that the heavy particles are collected near the central feed point and form a V-shaped narrow zone In a two-dimensional hopper. The segregation zone expands with increases of Initial mixing ratio of heavy particles, the feed rate, and the total flow length along the heap line. The denser component behaves like a smaller one in size segregation and like an angular one in shape Segregation in filling containers.
In industrial processes the mixtures of solids differing in particle density are often handled as raw materials and/or final products. Filling vessels with particulate matters is also essential to such operations as storage, mixing, grinding, contacting, and so on. Since most of unit operations in powder technology are associated with particle movement, segregation of solids mixture due to density difference will often take place and affect the quality of the mixture or the achievement of operations to a considerable extent. Only a few references are, however, available on overall observation of the density segregation in filling the conQ196-43Q5f84f 1 123-0423$01.5O/O
tainer (Fowler, 1961; Harris and Hildon, 1970). No analytical investigation except a push-away model (Tanaka, 1971) has been made in relation to the density-segregation mechanism. Our screening model on the size and shape segregation of particles flowing down on the inclined heap surface (Shinohara et al., 1970,1972,1979) is generally modified to interpret the mechanism of density segregation. As a result, the segregation patterns formed in a two-dimensional vessel are analytically described by the mixing ratio based on the net volume fraction of the denser component in the binary mixture. 0 1984 American Chemical Society
424
Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 3, 1984
increment of bulk volume of particles in each sublayer during differential time, At, is expressed by eq 1-3. For the remaining layer
Mixed F e e d
01
.i-
Block Number
[hR(m,n) - hR(m - 1 , n ) l h L W = hR(m,n - l)UR(m,n 1)Atw - hR(m - l,n)UR(m - 1,n)Atw VR (1)
+
For the segregating layer [hm(m,n) - hm(m - 1,n)JhLw = hm(m,n - l ) ~ m ( m , n - 1)Atw - hm(m - l , n ) ~ m ( m- 1,n)AtW - ( V R + Vs) (2)
For the separated layer
1
I j
\
[hs(m,n) - hs(m - l,n)]hLw = hs(m,n - l)us(m,n 1)Atw - hs(m - l , n ) ~ s ( m - 1,n)Atw Vs (3)
S t a t i o n a r y Layer
I
..
,
+
Zone N u m b e r
\
Figure 1. Segregation model.
Segregation Model As is the case with the previous screening model, the following is an idealized description of the segregation process according to a screening layer model proposed here. When the binary mixture of spherical particles of equal size and different solid densities is poured into the vessel, the solids heap yields the angle of repose to the horizontal. Thus, the feed flows down on the inclined heap surface, eventually reaching the vessel wall to form a stable, stationary layer. During this flow, the local interstices continually expand and contract at random. The denser component migrates under gravity toward the bottom of the flowing layer, since during the expansion phase a space large enough to accommodate a denser particle will be “framed” momentarily by a set of neighboring particles. During the time such an empty frame exists, it is somewhat more likely that a dense particle rather than a light particle will fill it before it collapses, since the dense particle will accelerate faster under gravity. Thus the bottom of the moving layer will become predominantly populated by dense particles and will be called the “separated” sublayer. The middle portion will be called the “segregating” sublayer, whereas the top layer will be called the “depleted” or “remaining” layer. In the proposed model, it is supposed that the heap diameter is so large that the segregation process reaches a steady state over the largest part of the layer’s travel, and that the effects at the ends can be ignored to a first approximation. One can then make a material balance over a short segment of the heap, taking it to be typical of the entire heap surface. The total flowing layer can be divided into segments, or blocks, of differential thickness along the heap line, as illustrated in Figure 1. Along a diameter of the conical heap, one can define stationary differential “zones”, through which the blocks pass as they descend on the heap line. Blocks are numbered from the leading edge of the moving layer inward toward the axis of the heap; zones are numbered outward from the axis of the conical heap. Since each sublayer of the flowing layer is supposed to be many particles deep, one can describe it by epecifying its height, h, at any point on the heap line and the velocity, u, with which the particles (assumed to be same size and shape but of different density) are moving parallel to the heap line. UR denotes the descending velocity of the remaining layer, um that of the segregating layer, and us (< um) that of the separated layer, all of which vary along the heap line. When the mth block from the head of the flowing layer comes to the nth zone from the central feed point, the
Here, the first and second terms on the right-hand side of the above equations indicate the bulk volume flowing in from the ( n - 1)th zone with the mth block and that flowing out from the nth zone with the (m- 1)th block, respectively. The third term represents the bulk volume of the remaining, VR, and/or separated, Vs, particles derived from the segregating layer. Thus, Vs is given by the penetration rate of the denser particles into unit overall area of the segregating layer on the net volume basis, Q, as Vs = QhLwAt/(l- e)
(4)
where E is the void fraction of the particles. VRis obtained from the corresponding Vs of the separated denser particles by the definition of the initial mixing ratio, Mi, as VR =
VS(1-
MJ/Mi
(5)
Provided that the total thickness, hT, and the average velocity, u,, of the overall flowing layer are almost constant during flow along the heap surface hR
+ hm + hs = hT
(6)
F = U,hTW(1 - t)
(7)
where F is a half of the total feed rate, F,,based on the net volume of particles. Hence, summing up each side of eq 1 to 3 for zero increment of the total bulk volume of the flowing layer and rewriting it gives
+ +
hR(m,n - l)UR(m,n - 1) + hRs(m,fl - 1)URs(m,n- 1) hs(m,n - l)Us(m,n- 1) = hR(m - l,n)UR(m - 1,n) hRs(m - l,n)URs(m - 1,n) + hs(m - l,n)Us(m - 1,n) (8) Referring to eq 7, both sides of the above equation are equal to hTua, and thus eq 8 is rewritten with respect to U R as uR(m - 1,n) = hTU,,/[hR(m - 1,n) + hRs(m - 1,n)
R.hs(m (9)
where two more relations are assumed to solve eq 1-3, that is, UR = um and that the velocity ratio, R, is nearly constant during flow, as given below.
Segregation Curves In order to get h and u for each sublayer in each zone computation is carried out one by one for each At. As h(m - 1,n) and h(m,n - 1) are to be given in each step at the preceding time, t - At, h(m,n) in each sublayer is calculated
Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 3. 1984 425
Feeding Bunker
Sampling Tar
i
4
1-500 I
I
Figure 2. Experimental apparatus.
F A * Figure 4. Photograph of density segregation pattern with mixture of glass beads and ball bearings. d
Table I. Particle Propertiesu spheres glass iron lead
Mixture 0 Glass- Lead @Glass-Iron p I2.48 d = 0165cm
d , cm 0.165
0.165 0.165
p,
g/cm3 2.52 7.79
11.34
uav = 15 cm/s; $C = 22". v9. distance from the central feed point along the
-05
0
05
In(D-d)
Figure 3. Calibration curve of feed rate with orifice.
from eq 1-3 by substituting eq 9. The boundary conditions are as follows.
hR(m,O)= 0 hm(m,O) = 0 hs(m,O)= 0
(11)
In addition, there are two restrictions to h and Q: no particles flow down ahead of the first block, that is, hR= hw = hs = 0 for m = 0. In the critical zone, n,, where the mixing ratio reaches the least value consistent with its feed inlet value, Mi,the value of Q becomes equal to hw(m,nJ(l - e)Mi/At. When a new stable layer is just formed, the volumetric mixing ratio, M(n),of the denser particles in the stable layer consisting of three sublayers is evaluated along the heap line by
M(n) =
+ hs(m,n) hdm,n) + hRS(m,n)+ hs(m,n)
(12)
Thus, the segregation curve is drawn as the mixing ratio
heap line.
Experimental Section An apparatus was constructed, as illustrated in Figure 2, to examine the patterns of density segregation exhibited by filling. It consists of a feeding bunker and a two-dimensional hopper. The bunker supplies solids mixture of the known initial mixing ratio a t a constant feed rate through an orifice at the bottom of a lifting tube. Figure 3 indicates a single calibration curve for varying Mi. The cone angle of the bunker, 30°, is small enough to avoid segregation during feeding. The hopper is made of transparent acrylate plates to visualize the segregation pattern, as illustrated in Figure 4. The inside distance between the vertical walls of the hopper, UI, is 3 cm. The hopper has 25 sampling taps on the front side to measure the distribution of the mixing ratio in the horizontal direction which is converted into the distribution along the heap line by making use of the angle of repose, = 22". The collected solids with a sampling tube of 1.5 cm inside diameter are elutriated for separation to calculate the mixing ratio. To minimize spreading of the fed particles, the distance between the orifice and the heap top is steadily controlled throughout feeding to be about 12 cm by rotating the wheel. Particles used for experiments are glass beads, ball bearings, and lead shots of equal size and different solid
426
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984 10
L z270cm F i 6 75ccruc MI i 0400 0. = 0 1OOcmlsec
lo
20 DISTANCE FROM CENTRE ALONG HEAP LINE
,
Figure 5. Effect of velocity ratio, R, on segregation curve.
--
DISTANCE FROM CENTRE ALONG HEAP LINE
cm
,cm
Figure 8. Segregation pattem for different initial mixing ratio with feed mixture of glass beads and ball bearings.
10 L = 270cm
10 20 DISTANCE FROM CENTRE ALONG HEAP LINE
,cm
I
m 10 15 DISTANCE FROM CENTRE ALOW HEAP LINE , cm
0
5
Figure 6. Effect of penetration rate, Q, on segregation curve.
Figure 9. Segregation patterns for different feed rate with mixture of glass beads and lead shot.
i / DISTANCE FROM CENTRE ALONG HEAP LINE
,
,cm
Figure 7. Segregationpatterns for different initial mixing ratio with feed mixture of glass beads and lead shot.
densities, the physical properties of which are listed in Table I. The heavy particles in the mixture behave as the segregatingor penetrating ones in the present model. The segregation patterns were measured by changing the initial mixing ratio and the feed rate of two kinds of mixture, ball bearings and lead shots-glass beads.
Results and Discussion Based on the proposed segregation model, various characteristics of the density segregation in filling vessels were computed and examined by experiments. The denser component behaves just like smaller particles in the size segregation or angular ones in the shape segregation, as mentioned above. It settles near the central feed line and forms a V-shaped narrow zone within the bed of lighter component. The penetration process within the upper flowing layer of the mixture is the same as that in the other kinds of particle segregation. However, since the particles are of equal size in the density or shape segregation, no packing occurs into the interstices in the lower stationary layer, which differs from the size segregation. Density segregation differs from shape segregation also in that it
o.oo
,
,xu
10 20 DISTANCE FROM CENTRE ALONG HEAP LINE
, cm
Figure 10. segregation patterns for different feed rate with mixture of glass beads and ball bearings.
*’ Mi =0.400 0 0.100 cmisec R = 0.05 0.5
4 0
0
6 12 18 24 DISTANCE FROM CENTRE ALONG HEAP L I N E , cm
Figure 11. Segregation curves for different flow length of solids heap.
occurs in a layer of constant total thickness, but with varying velocities in the three sublayers along the inclined heap line. The effecta of flow properties of bulk solids and the operational variables on the segregation patterns are illustrated in Figures 5-11. The fitted curves in the ob-
Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 3, 1984 427
0051
0' 0
02
04
J
06
Mi [ - I
Figure 12. Variation of penetration rate with initial mixing ratio of particle feed.
served patterns are drawn according to the present model by varying Q for each Mi and F. The following trends are revealed in these figures. As the velocity of the lower separated particles approaches that of the upper remaining and segregating layers the velocity ratio, R, increases. The deviation of the mixing ratio from the initial value, or in other words, the degree of segregation, tends to decrease, and the segregation zone, where the separated denser particles are depmited into the heap around the central feed line, expands gradually to indicate the lower peak of the mixing ratio farther from the center, as shown in Figure 5. The degree of segregation clearly increases with the penetration rate of the denser particles, Q; that is, the peak of the segregation curve gets higher and the zone becomes narrow with increasing Q, as shown in Figure 6. An increase of the initial mixing ratio results not only in the expansion of the segregation zone but also in the higher mixing ratio, as shown in Figures 7 and 8. Decreasing the feed rate, the peak of the mixing ratio goes up and the zone shrinks somewhat, and thus the segregation tends to be conspicuous, as shown in Figures 9 and 10. With a longer heap line corresponding to a larger vessel, the segregation zone steadily expands and the peak goes up a little, as computed and shown in Figure 11. The penetration rate of the heavy particles, Q,into the interstices among framing light particles increases with the initial mixing ratio of the denser component and the feed rate of solids mixture, as shown in Figures 12 and 13, respectively. It can be described by eq 13 on the basis of the hypothetical hoppers model (Shinohara, 1979), as follows. The net volume of penetrating denser particles into unit cross-sectional area of flowing mixture layer is written as the product of the number of discharge openings among framing light particles and the volumetric flow rate of the penetrating particles through each opening by (13) where eR is the void fraction of the segregating mixture layer on the basis of the framing component, 6 is the fractional number of the effective openings in proportion to the number of the framing particles, and a and P are constants in the flow rate equation of the penetrating particles. De is the equivalent opening diameter of the
Figure 13. Variation of penetration rate with feed rate of mixture.
effective void spaces among the framing particles. It is, therefore, obtained from
as
where j3 is the fraction of ineffective spaces for separation in the voids, Q. The following relationships exist according to the definitions of the overall void fraction, E, and the mixing ratio, M, in the segregating mixture layer. e
= tR(1 - p ) e S
+ 86R
(15)
where ts is the voidage of only the segregating denser particles filling the effective voids. Hence, from eq 15 and 16 ER = (1 - t)M + t (17) In the present case, d = dR = ds and M = Mi, and putting eq 14 and 17 into eq 13 6dp-2
Q = -A(1 a
- MJ(1 - E)
X
where A = a6
Expressing E in terms of F by using eq 7
where K is a function of F as K = U , W ~ = F/(1 - E)
(22)
428
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
r
I
2oL 0
I
5
10
15
F [cclsecl
Figure 14. Relationship between packing density and feed rate of particles.
The penetration rate, Q, is determined at each feed rate and initial mixing ratio so as to give the least variance between the experimental data of the mixing ratio and the calculated value from eq 12. A and B are considered to be constant for the same mixture and K to vary with F , but independent of Mi. The most appropriate constants are estimated from simultaneous equations generated by putting into eq 21 more than three sets of data of Q vs. Mi at the same F. Next, with A and B thus obtained, K is evaluated for each F using another sets of data of Q vs. F. c is readily calculated from K by eq 22. Assuming the same relationship between (1- c) vs. F for both mixtures, eq 23 is experimentally yielded, as shown in Figure 14. c = 1 - 0.625/exp(O.lOlF) (23)
As a result, the penetration rate, Q, is expressed by eq 21 as functions of the feed rate, F, and the initial mixing ratio, Mi. Because of a few data availableon Q and their spreads, it is not so clear but seems that Q for the steel balls mixture is a little bit greater than Q for the lead spheres one under the same operational conditions of F and Mi. It might stem from the smaller coefficient of friction due to hard, smooth, and spherical surface of ball bearings as compared with soft, frictional, and a bit deformed spheres of lead shots. In other words, the frictional effects on penetration might have exceeded those due to density difference between ball bearings and lead shot.
Of course, the degree of segregation cannot be discussed without taking into account the velocity ratio, R. Particle mixtures with greater Q and smaller R will segregate to a large extent, and on the contrary the mixtures with smaller Q or larger R will result in slight segregation during filling of the vessel. The present case lies between the two extremes. Concluding Remarks The screening layer model developed here was successfully applied to the analysis of the mechanism of density segregation of solids mixture in filling the vessels. It will be possible to extend this model to the size and shape segregation as well and to generalize three kinds of the segregation mechanisms. The patterns of density segregation are similar to those of size segregation, but their segregation mechanisms are different from each other in the interaction of separated heavy particles with the stationary layer of the solids heap. While the patterns of shape segregation look different from those of density segregation, their mechanisms are nearly the same and seem to be little different in the values of the penetration rate and the velocity ratio of their components. In order to compare the effects of different particle properties such as size, density, and shape, the degree of segregation should suitably be defined in relation to the segregation patterns. The correlation between the feed rate and the void fraction of the flowing layer of the mixture is also left to be analyzed in the future. Acknowledgment Mr. K. Muraoka deserves the authors’ thanks for his experimental preparation of the segregation patterns. Literature Cited Fowler, R. T. Awt. the" Eng. Aug 1061, 17. Harris, J. F. G.;Hildon, A. M. Ind. Eng. Chem. Process Des. Dev. 1070, 9 ,
363.
Shinohara, K.; Shojl, K.; Tanaka, T. Ind. Eng. Chem. Process Des.
Dev.
1970, 9 , 174. Shinohara, K.; Shoji, K.; Tanaka. T. Ind. Eng. Chem. Process Des. D e v . 1072, 7 7 , 360. Shlnohara, K. Ind. Eng. Chem. Process Des. D e v . 1070. 78, 223. Tanaka, T. Ind. Eng. Chem. Process Des. D e v . 1971, 70, 332.
Received for review August 6, 1980 Revised manuscript received June 24, 1982 Accepted September 13, 1983
