ro
nuclei radius, L particle radius, L interfacial area of soap present in 1 cc of water, L2/L3 T = induction period, t t = time, t V = particle volume, L3 V o = nuclei volume, L3 V T = cumulative particle volume, L3/L3 V’ = latex volume, L3
r S
=
= =
GREEKLETTERS = dimensionless time q7 = dimensionless population density based on radius qr0 = dimensionless nuclei population density based on radius q o = dimensionless population density based on volume ‘loo= dimensionless nuclei population density based on volume pm = monomer density in a particle p p = polymer density in a particle p , = dimensionless radial growth rate p o = dimensionless volumetric growth rate U, = dimensionless particle radius a,’ = dimensionless nuclei radius u o = dimensionless particle volume :U = dimensionless nuclei volume T = residence time, T T~ = mean latex residence time, t +m = monomer volume fraction in a particle, L3/L3
e
literature Cited
American Society for Testing Materials, “Recommended practice for analyses by microscopical methods for particle size distribution of particulate substances of subsieve sizes,” ASTM Designation E20-68, ASTM Standards, Part 30, 1968.
Boundy, R. H., “Styrene,” Reinhold, New York, NY, 1952. Bovey, F. A., Kolthoff, I. I f . , Rfedalia, A. I., Rlechan, E. J., “Emulsion Polvmerization,” Interscience, Xew York, NY, 1955. DeGraff, A. W.. DeGraff. W., Poehlin. Poehlin, G. W.. W., 65th Yational Meetinn. Meeting, AIC)hE, Cleveland, Cleveiand, OH, >fay 4-7, 1969. Ewart, R. H., Carr, C. I., J . Phys. Chem., 58,640-4 (1954). Funderburk, J. O., PhD thesis, Iowa State University, Ames, 1.4, 1969. Gardon, J. L., J . Polym. Sci., Part A-1, 6,623-41 11968). Gardon, J. L., ibid., pp 643-4. Gardon, J. L., ibz Gardon, J. L., ibz Gardon, J. L., i b i , Gardon, J. L., ibid., 2859-79. Gerrens. H.. Kuchner. K.. Brit.Polurn. J.. 2. 18-24 11970). Gershberg, D. B., Longfield, J. E., &th Bnnbal Meeting, AIChE, New York, NY, December 2-7, 1961. Grancio, M. R., Williams, D. J., J . Polym. Sci., Part A-1, 8, 2617-29 (1970). Harkins, W. D., J . Amer. Chem. Soc 69, 1428-44 (1947). Harkins, W. D., J . Polym. Sci., 5, 2?7-51 (1950). Hartley, G. S., “Aqueous Solution of Paraffin-Chain Salts,” Hermann et Cie, Paris, 1936. McBain, J. W., “Colloid Science,” Heath, Boston, l I A , 1950. Medvedev, S. S., Ricerca Scientijca, Supplemento, Siniposio Internacional Chim. Macromol.. Milan-Turin 26, 897-905 (1954): Chem. Abstr., 54, 12631h (1960) O’Toole, J. T., J . A p p l . Polym. Sci., 9, 1291-7 (1966). AIChE J . , 8 (5), 639-45 (1962). Randolph, A. D., Larson, M.8., Sato, T., Taniyama, I., Kogyo Kagaku Zasshi, 6 8 , 1, 106-9 (1965). Smith, W. V., Ewart, R. H., J . Chem. Phys., 16 (6), 592-9 u,
(1948).
Stockmayer, W. H., J . Polym. Sci., 24, 314-17 (1957). Vanderhoff, J. W., Vitkuske, J. F., Bradford! E. G., illfrey, T., Jr., ibid., 20, 225-34 (1956). RECEIVED for review January 11, 1971 ACCEPTI:DApril 25, 1972 Work supported by National Defense Education Act Fellowship awarded to ,J.O.F., Jr.
Mechanism of Size Segregation of Particles in Filling a Hopper Kunio Shinohara, Kazunori Shoji, and Tatsuo Tanakal Department of Chemical Process Engineering, Hokkaido Cniversity, Sapporo, Japan
Mechanism of size segregation of particles in filling a hopper was theoretically analyzed on the basis of a screening or percolation model. According to the model, when mixed particles of different size are fed in the center of the hopper and flow down on solids heap surface, smaller particles within the mixture pass through interspaces of larger particles during flow and then are packed in interspaces of stationary large particles of the heap surface. By the analysis and experiments, a V-shaped zone, where smaller particles are contained as a result of segregation, wqs found to expand with smaller fraction of large particles in the feed, longer distance of flow along the heap surface, and larger feed rate of particles.
F r o m the operational point of view, there are two types of segregation phenomena in storage and supply of particulate materials in bins and hoppers. One of them occurs in filling hoppers, and the other in emptying hoppers. The mechanism of segregation and blending of particles flowing out of a mass-flow hopper was theoretically analyzed T o whom correspondenre should be addressed.
in our previous paper (Shinohara et al., 1970). There are few papers presenting a theoretical or quant,itative analysis on segregation phenomenon in filling hoppers. Most of the papers dealt with esperiment’al or qualitative esplaiiatioris of the segregat,ion phenomena during filling and emptying. Recently, Matthee (1967) qualitat’ively explained the effect of part,icle size on t,he segregation phenomenon during filling by describing the motion of a single particle rolling or sliding Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
369
. -
Figure glsec
4. Segregation pattern in M , = 0.81 5 and F = 50
Figure 5. Segregation pattern in Mt = dsec
0.71 3 and F = 50
can he written as the product of the number of openings per unit area and mass flow rate through one opening by
where
Subscript rdenotes a relative value, and a' and n are constants defined by the flow rate equation proposed in our previous paper (Shinohara e t al., 1970) as
F
- d)"
= a'@
(6)
Average relative velocity of small particles through the interspaces formed hy large ones, V,,', is, therefore, expressed by Equation 7 from Equation 4
- e l l ) a'D' TdL2 . p ( l - e$')
6(1
.~ + 8'
(7)
where p is the density of particle. Formation of Solids Heap
When particles are poured onto solids pile, they flow down on the pile, and its slope forms an angle of repose, +h. From visual inspection, it could be assumed that the particles stream has constant thickness, h', and its velocity, V , is constant throughout the heaping process. Mass flow rate of large particles flowing down on either side of the pile is written as
30 I
20'
10' 0' ~10 20 ' DISTANCE FROM CENTRE,Fm
static layer of particles. This is one cycle of heaping process, and the successive motion builds up bed of particles. On the other hand, the weight of large particles supplied to form one stable layer for time, tH, is
. F-2 .
tx
=
h
. W . Lx
. p(1 -
ti)
(10)
where Lx is the distance from the heap top of height, H, above the hopper apex to the hopper wall along the heap surface, h i s the thickness of the static layer formed, and e~ is the void fraction of stationary layer of large particles. Since the velocity, Ti, of the head of particles flowing down on the heap surface is assumed to be constant, IX is given by
Combining Equations 10 and 11 gives the thickness of the stationary layer of large particles as
h= wherePisthemassrateofiillingand Wisthewidthof a twodimensional hopper. Hence, from Equation 8
"30
Figure 6. Calculated segregation zones
MI
1
ell
O
FMi 2 WVp(1 - e 3
Ono (1967) stated that the velocity and thickness of the particles stream flowing down an inclined channel becomes constant throughout the channel when the inclination approaches the angle of repose of particles. Segregation Process
When the head of flowing particles layer reaches the hopper wall, the particles stop moving and thus form one stable
Now, let us consider the segregation process in filling the hopper with the binary mixture of particles differingin size. Ind. Eng. Chem. Procerr Der. Develop., Vol. 1 1 , No. 3, 1972
371
DISTANCE FROM CENTRE ALONG HEAP LINELm
Figure 7. Segregation pattern 0.873 arrd F = 50 g/sec 1.0
.r
Im
along heap lines in Mt =
--0-0-
_- _Figure IO. Segregation pattern in 120 g/sec
Mi
=
0.713 and F =
.. DISTANCE FROI CENlRE I\LoIcI HEAP LINE.cm
8. Segregation
Figure
pattern along heap lines in
Mr
=
or 02
Figure
a
5
9. Segregation
10
6
XI
25
M
pattern along heop lines in
Mt
=
0.71 3 and F = 95 g/sec The process can ideally be described as follows: When the particles mixture is poured and flows down on the solids heap, small particles initiate to be separated from the mixture by passing through the interspaces of large particles framing the flowing layer of particles, I, as shown in Figure 1. Then the small particles which have passed through the flowing layer and reached the stationary heap surface flow down on the surface together with and under the separated flowing large particles and drop into the gaps formed by the stationary layer of large particles under the flowing layer. This segregation process continues during one cycle of the heaping process, and the stationary layer of large particles is gradually packed to be full of small particles separated from the flowing mixture. The same heaping with segregation is repeated while filling the hopper. When one particles layer has just formed, small part,icles having passed through the layer, I, are packed into two regions, 111,and IV, small particles passing through the layer, I, are left in a triangular zone due to constant velocities; V and V,,', and a very thin layer of small particles, 11, remains between t h e two stationary layers of large particles. The weight of small particles passing through the particles screen of area, le'' W , for time, ,t is, therefore, equal to the sum of weight of small particles involved in these four regions, I-IV. Then, the following equation holds for small particles in the stationary &ate: 372
Ind. Eng. Chern. Process Der Develop., Val. 11, No. 3, 1972
+ L).ah.W.p(l
(1s
0.767 and F = 50 g/sec
-
€8)
(13)
where 1,' is the distance from the filling point along the heap surface in which small particles supplied a t the top surface of the filling point can reach the heap surface of lower stationary layer; 1. is the length of uniformly packed portion with small particles, 111, of stationary large particles from the filling point; and 1, is the length of stationary layer, IV, in which the quantity of small particles per unit length along the heap line decreases linearly owing to the constant velocity of the particles stream. The second term on the right-hand side i s introduced to deduct the weight of small particles involved once in a triangular region, I, from that in a region, 111, which gives the net weight of small particles newly packed in a region, 111, during one cycle of heaping process. Then, 1,' and 1, are given as functions of relative passing velocity of small particles, Va7'and V,,, by Equations 14 and 15, respectively:
(15) where V,, is obtained by putting c I and e,' instead of e,' into Equations 5 and 7, respectively, as
el'
and
ea" is the void fraction of small particles falling into the interspaces of stationary large particles, and a and 0 are constants for the stationary layer corresponding to a' and p', respectively. Now, focusing attention to the degree of segregation, the dimension of the zone where small particles are contained is to he known. Let be the farthest distance along the heap line from the heap top to the point where small particles can reach. Then L, is given by
L,
=
I,
+ 1,
(18)
10
8 0
0
.
5 10 15 20 OiSTANCE FROM CENTRE ALONO HEAP LINE,cm
Figure 14. Segregation pattern along heap lines with chisel-shaped hopper in Mi = 0.70, F = 63 g/sec and dJd, = 1.53
0
5
K
Initial Mixing Ratio
04
'
I
0.0
0
5 10 15 20 DISTANCE FROM CENTRE ALONG HEAP LINE.cm
Figure 15. Segregation pattern along heap lines with chisel-shaped hopper in M i = 0.23 and F = 71 g/sec
Concerning the distribution of mixing ratio, M , of particles stored within the hopper, in portion 111, 0 < L 5 I,, small particles are uniformly packed in the interspaces of large particles, so by Equation 2
M
=
1-
1-
€2
E l ' E,
I n portion IV, 1, < L IL,, the quantity of small particles per unit length along the heap line decreases linearly, so 1114 =
1 I n portion V, L,
El€,
-
€1
L - I, -* e d l 1,
(30)
dz - (= 14.4)> ds
E*)
= 1.0
M i>
(31)
Experiments and Results
Schematic diagram of experimental apparatus is shown in Figure 2. It consists of a two-dimensional hopper and a feeding bunker. The hopper has two vertical walls which are parallel to and 3.0 cm apart from each other. A transparent plane in the front side of the hopper has 51 sampling taps (8 mm in i d . ) assembled for measuring the mixing ratio of particles. The bunker is divided into two parts in which two particle components of different size were separately stored, and a feeding tube was located centrally in the two-dimensional hopper. When two kinds of particles discharge together from the bunker, all the particles fall through two orifices and three baffle plates, and thus brought about good mixing of both components of different particles. The mixing ratio and the mass flow rate of particles from the bunker were varied by using a slide shutter with screw to adjust the flow rate of small particles and three kinds of orifices different in the opening diameter. Eng. Chem. Process Des. Develop., Vol. 11, No. 3, 1972
(= 8.28)
3(1
< L IL H , there are no small particles, so 1 v
374 Ind.
After the two-dimensional hopper was filled, a photograph was taken to record the visible zone in which small particles were included. The mixing ratio of particles in the hopper was measured by collecting the particle sample with a cylindrical pipe inserted into the sampling hole and by screening each sample. Glass balls were used as particle samples in this experiment. Three of five photographs of the segregation patterns are shown in Figures 3-5 for comparison, which were taken for five kinds of mixing ratio, as listed in Table I, a t a constant feed rate F = 50 g/sec. They are compared with Figure 6 in which the calculated lines, L,, based on Equation 28 are sketched in the same way. Figures 7 and 8 show the variations of the mixing ratio of particles along the heap line within the particles bed, corresponding to the different initial mixing ratio, M i = 0.873 and Mi = 0.767, a t the same feed rate, F = 50g/sec. Figure 9 illustrates the variations of the mixing ratio within the bed for F = 95 g/sec a t IC,= 0.713. A photograph of segregation pattern for F = 120 g/sec is shown in Figure 10 for the comparison with Figure 5 (F = 50 g/sec). The calculated lines, L,, for three kinds of feed rate, F , with X i= 0.713 expressing the V-shaped zone are also indicated in Figure 6. Figures 11 and 12 show photographs of segregation pattern in the hopper with e = 30" and e = 90" a t F = 58 g/sec, and M, = 0.70, which could be compared with Figure 5. Figure 13 is a photograph of segregation pattern in a bin with a parallel-sided section on e = 60" hopper for F = 58 g/sec and Mi = 0.69. Since e l , E,, eg', e,", p, and p' cannot directly be measured, some reasonable values were estimated from the experimental results as listed in Table 11. CY and CY' were evaluated by the flow rate equation proposed in our previous paper (Shinohara e t al., 1968). Series of tests were carried out under the conditions that satisfy Inequalities 1 and 3 by substituting values of d t = 0.171, d, = 0.0119, E,' = 0.56, C = 3, and X i= 0.920, and values of el' = 0.75, E,' = 0.38, and M , = 0.715, corresponding t o the critical conditions, respectively
(*)1
- €,')Mi (= 0.35)
EZ'E,'
erit
Further, i t was checked that critical size ratio and critical mixing ratio of the initial binary mixture exist a t which size segregation in filling a hopper does not take place, as shown in Figures 14 and 15, respectively. Experimental results were satisfactory enough to reproduce and confirm derived equations quantitatively. Discussion
The present consideration of heaping and segregation processes is formulated rather for the static state of particles. Therefore, the complicated dynamical behavior of the particles might not be described in detail, but the segregation pattern in filling a hopper can quantitatively be explained fairly well. According to Equation 28 the following facts can be seen: The shape of the zone where small particles are piled u p is determined by the length of flow path of particles, in other words, the geometrical shape of the hopper. This is indicated by L H of the first term on the right-hand side of Equation 28.
Table
Initial mixingratio Feed rate, g/sec Cone angle, degree Hopper shape Figure
1. Operational Conditions
0.920
0.873
0.815
0.767
0.713
0.713
0.713
50
50
50
50
50
95
120
60 V 3
60 V 7
60 V
60 V 8
60 V
60 V 9
60 V 10
4
5
I n fact, segregation patterns obtained except those in Figures 14 and 15 show that in the V-shaped hopper small particles are packed in the V-shaped zone and in the chisel-shaped hopper the zone is chisel-shaped so long as the filling rate is relatively small. The dimension of the zone increases with the hopper cone angle and the height of particle bed, as shown in Figures 5, 11, and 12. This is also represented by L H as a function of 8 and H in Equations 26 and 28. As the initial mixing ratio of fed materials decreases, the zone spreads as shown in Figures 3-5, This is mainly due to the effect of the first term on the right-hand side of Equation 28. Concerning the feed rate, F , Equation 28 indicates that L,increases with F , that is, the degree of segregation may tend to decrease with increasing feed rate. Within comparatively small range of feed rate in this experiment ( F = 50 120 g/sec), though the effect of the feed rate is not so remarkable, the increase of F certainly causes enlargement of the V-shaped zone, as shown in Figures 5 and 10. This is due t o the direct contribution of the second term in Equation 28, which is proportional to the feed rate. Strictly speaking, the observed zones are a little narrower than the calculated zones, and the curves of the mixing ratio measured are not so clean-cut as the theoretical curves shown in Figures 7-9. These are due to the wall effect that the velocity of particles has small distribution in the direction of the depth of the two-dimensional hopper and due to the slight adhesion of small particles to the vertical wall. Additionally, the latter fact might also be due to neglect of the last term on the right-hand side of Equation 13 for simplicity of derivation. Furthermore, the mixing ratio measured in lower parts of the particles bed deviates from the theoretical curve to small extent, that is, the boundary lines indicating the V-shaped zone are obscure and small particles tend to be piled up near the hopper wall, too, as illustrated in most of the photographs. This phenomenon is due to the effect of impact and bouncing of particles falling freely from a relatively high position, when small particles which collide a ith large particles on the surface of the bed scatter toward outside. If a feeding unit had been assembled to keep the falling height considerably short and constant, small particles vould have ideally segregated from large particles without scattering. -1s shown above, under such conditions that the size ratio .tnd the initial mixing ratio of particles satisfy Inequalities 1 and 3, respectively, the actual dimensions of the V-shaped Tone and the experimental variations of the mixing ratio along the heap line fairly accord with the theoretically calculated lines. However, in case the critical conditions of the binary mixture fed do not hold, i t could be said t h a t the segregation phenomenon ceases to take place in filling a hopper, as shown in Figures 14 and 15. for the size ratio of narticles even for d , l d , = 1.53, slight segregation can be :seen and it is considerably sensitive. This would be due t o
-
0.70
0.70
0.69
0.70
0.23
58
59
58
63
71
30 V 11
90 V 12
60 Chisel 13
60 Chisel 14
60 Chisel 15
Table II. Material Properties of Glass Balls
d 2 = 0.1705 ern d, = 0.0119 ern p = 2.52 g/cc
4
=
€2
= 0.55
E,
0.40 0.56 0.40 = 3
= e,' = eS" =
c
25'
a = 29.4 a' = 32.6
p p' n
= = =
1.2 2.4 2.7
fluctuation in local voidage of large particles even in stationary layer under the influence of shear stress caused by the flowing particles. As the result, small particles could have a chance to pass through the interspaces of large ones. These facts support the present analysis of the mechanism of size segregation based on the similar screening model to that in emptying hoppers. Therefore, if the degree of segregation in filling a hopper could be, for instance, defined as Equation 32, it can directly be estimated as functions of the hopper geometry, material properties, and the operational conditions by using the derived Equations 21 and 22. Degree of segregation
=
LS 1- LH
(32)
Then, Equation 28 suggests some ways to minimize size segregation from the theoretical viewpoint. One of them is to increase the feed rate, F, and, if possible, to decrease the initial mixing ratio, A I t , of fed particles, as described before. Another way is to divide a hopper into several parts to lessen the flow path of particles. However, segregation always occurs to a certain extent even when F is largely increased, because as long as particles flow down on the slope, small particles pass through the interspaces of large particles during flow. To prevent segregation, solid materials should, therefore, be charged into a hopper by such a method as cannot cause the heap to form but to keep the level. For instance, a movable conveyor system Lvhich works in the horizontal direction for feeding materials may be a realistic way in practice. Furthermore, a mass-flow hopper that usually has a smaller cone angle less than 60' should be used to minimize size segregation, though the size segregation occurs even in filling the mass-flow hopper. Because, in the case of a core-flow hopper the central core of small particles formed during filling flows out first, and large particles near the hopper wall flow out last, which causes serious segregation during emptying. Combining equations for both mechanisms of filling and emptying a hopper might produce a better result in the degree of segregation. Conclusions
On the basis of the screening model, the pattern or degree of size segregation in filling a hopper was theoretically analyzed Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
375
and formulated. Experiments with the two-dimensional hopper verified the derived equations. Consequently, the following facts were known: The segregation pattern of binary mixture of different sized particles can generally be expressed by describing the variation of mixing ratio of particles along heap lines in the hopper and the dimension of the zone where small particles are piled up among the interspaces of large particles. The shape of the zone depends on the hopper geometry, and then a longer heap line makes a larger zone of mixed particles. The zone becomes larger with decrease of the initial mixing ratio of large to small particles fed. Smaller particles spread to a larger extent within a hopper by increasing the mass rate of filling, and thus the degree of segregation tends to decrease. Nomenclature
C
n
by Equation 6 tH = time taken for particles to reach hopper wall along heap line from a filling point of height, H , above apex, sec V = velocity of particles flowing down on solids heap, cm/sec V,, = relative passing velocity of small particles through interspaces of large ones, cm/sec W = width of two-dimensional hopper, cm GREEK LETTERS CY
p E
e,’
p
constant relating t o arching diameter of conical hopper outlet, cm Do d = diameter of particle, cm F = filling rate of particles in a hopper, g/sec Fsr’ = relative mass flow rate of small particles through large ones per unit flowing area, g/sec cm2 H = height above hopper apex, cm h = thickness of particles layer, cm Ah = thickness of small particles layer, 11, in Figure 1, cm L = length along heap line from central feed point, cm LH = length of heap line from the feed point of height, H , above apex, cm = length of the zone containing small particles from the feed point along heap line, cm = length of flowing zone, I, shown in Figure 1, cm = length of packed zone, IV, shown in Figure 1, cm = length of uniformly packed zone with small particles, 111, shown in Figure I, cm = mixing ratio of particles defined by Equation 2 = =
= constant relating to mass rate of flow defined
e 6
constant relating to mass rate of flow defined by Equation 6 = ratio of ineffective void which cannot be used for equivalent hopper openings to effective void = void fraction of particles = void fraction of small particles falling into interspaces of stationary large particles = density of particle, g/cc = cone angle of a hopper, degree = angle of repose of mixed particles, degree =
SUPERSCRIPT
’
=
dynamic state
SUBSCRIPTS
i I s
= initial = =
large particle small particle
literature Cited
Matthee, H., Powder Technol., 1, 265 (1967). Ono, E., Appl. Phys. (Japan), 36, 347 (1967). Shinohara, K., Idemitsu, Y., Gotoh, K., Tanaka, T., Ind. Eng. Chem. Process Des. Develop., 7, 378 (1968). Shinohara, K., Shoji, K., Tanaka, T., ibid., 9, 174 (1970).
RECEIVED for review March 4, 1971 ACCEPTED February 10, 1972
Integrated Approach to Design and Control of a Class of Countercurrent Processes Hong Hie Lee,’ Lowell B. Koppel, and Henry C. Lim2 School of Chemical Engineering, Purdue University, Lafayette, I N 47907
An integrated approach to design and control of chemical processes, in which the optimization of process design and process control is made simultaneously, i s proposed. Design charts which allow the process designer to take into account the process controllability during the design phase are given for a class of frequently occurring countercurrent processes, heat exchangers and packed absorbers.
Typical present practice of process design is primarily based on miiiimizing the sum of fixed and steady-state operating costs. Given a process and specification of product quality and production rate, the fixed cost is typically traded off
against the operating costs to give the minimum sum of the Present address, Process Control Research Laboratory, Westvaco, North Charleston, SC 29405. * T o whom correspondence should be addressed.
376 Ind.
Eng. Chern. Process Des. Develop., Vol. 1 1 , No. 3, 1972
two. After a flow sheet specifying equipment size and instrumentation is prepared along with decisions on what input variables are to be manipulated to control what output variables, an attempt is made during start-up to optimize the transient behavior. During this trial-and-error tuning, a decision is made on the type of control mode and corresponding controller settings which appear to give optimal response a t the time.
