MECHANISM OF SEGREGATION AND BLENDING OF PARTICLES FLOWING OUT OF MASS-FLOW HOPPERS KUNIO
SHINOHARA,
KAZUNORI SHOJI,
AND
TATSUO TANAKA
Department of Chemical Process Engineering, Hakkaido University, Sapporo, Japan The mechanism of segregation and blending of a binary mixture of particles flowing out of a mass-flow hopper is theoretically analyzed on the basis of the screen or hypothetical-hoppers model, where small particles pass through the interspaces of large particles during flow. As a result, the patterns of segregation and blending are described by material balance within the hopper, depending on variations of the mass flow rate and the mixing ratio with the elapse of discharge time. Satisfactory agreement between experimental and calculated results verifies this model and the analysis.
FROMthe industrial viewpoint of homogeneity of products or raw materials, segregation and blending phenomena of particulate solids are troublesome in filling and emptying hoppers in practice. The mechanism of segregation and blending has, however, not been investigated theoretically or quantitatively, although Peacock (1938), Brown (1939), Rose (1959), Kawai (1959), Seaton (1959, 1960), Denburg and Bauer (19641, and Williams (1965) have suggested empirical or qualitative explanations and preventive methods. Furthermore, Miwa (1965) pointed out that his equation on sieving was applicable to size segregation in filling, and recently Williams (1967) treated the segregation phenomenon as related to vibration. The degree of segregation or blending is governed mainly by material properties such as particle size, particle density, and particle shape, a hopper geometry which directly relates to the flow pattern, mass flow or plug flow, such as the diameter of a hopper outlet, a hopper cone angle, and the height of a powder bed, and operating conditions such as the filling rate and the method of ing particles. I n this paper, based on the model of a screen or hypothetical hoppers formed by the interspaces of large particles, the mechanism of segregation and blending of binary mixture of particles flowing out of a mass-flow hopper is analyzed, and by material balance variations of mass flow rate and mixing ratio of particles with the elapse of discharge time are theoretically calculated and described. The experimental results in use of a twodimensional hopper confirm this model. Mass Flow Rate and Velocity of Particles
is the void fraction. F can also be given by Equation 2 empirically and theoretically, as discussed and suggested in our previous paper (Shinohara et al., 1968). 6
F
=
a(Do- d,)”
where CY and n are constants, Do is the diameter of a hopper outlet, and d, is the particle size. Equating Equations l and 2 gives the average velocity of particles a t a certain cross section of a hopper in the general form as
The same flow mechanism of particles as of particles having uniform size from a hopper can be applied to the sieving of small particles through the interspaces of large particles, by which segregation and blending phenomena during flow occur. Let De be the effective opening diameter of the screen or the hypothetical hopper, and write the total area of the screen openings as the product of the number of screen openings which could be defined as the number of large particles and the area of one screen opening.
where the number of large particles is obtained by use of the concept of the average projected area-that is, a cube is regarded as a cylinder having an equivalent volume. Hence, from Equation 4
Mass flow rate of granular materials from a hopper,
F , is generally expressed by Equation 1
F = SVp(1 - t)
(1) where S is the cross-sectional area of a hopper, V is the velocity of particles, p is the particle density, and 174
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
where (3 is the ratio of ineffective void which is independent of solids discharge in the interspaces of large particles to an effective void which forms openings in a screen
or hypothetical hopper and subscript 1 denotes the large particle. The relative mass flow rate of small particles through the interspaces of large particles, Fir, is then obtained from Equations 2 and 5 as
where subscripts s and r denote the small particle and the relative value, respectively. Equations 3, 5 , and 6 give the average relative velocity of small particles through the interspaces formed by large ones, V:,
Variations of Mass Flow Rate and Mixing Ratio with Elapse of Discharge Time
Separate or mixed small and large particles in a massflow hopper flow out, when segregation and blending phenomena occur. This mechanism can be analyzed by the screen or hypothetical-hoppers model on the following assumptions: After the initiation of flow, small particles can pass through the screen or hypothetical hopper formed by the interspaces of large particles-that is, Inequality 13 must be satisfied
De > Cd,,
where C is a constant relating to the arching of particles. Replacing Equation 5 in Inequality 13 leads to the critical particle-size ratio of two kinds of particles differing in particle size.
dps
where e l is the void fraction of small particles falling through the effective interspaces of large particles. Then, the mass flow rate of a binary mixture of large and small particles, R,, is obtained as Equation 8 from the sum of the flow rate of large particles, F L , and the absolute flow rate, Fi, of small particles through the interspaces of large particles.
+
h ( 1 8)
where
(9)
Substituting Equations 2, 3, 5 , and 7 into Equation 8 and rearranging gives a t the hopper outlet
where subscript o denotes the opening. On the other hand, the mixing ratio, M , is defined as
M=
> mixture
- ti)
weight of large particles (11) total weight of binary mixture of particles
The mixing ratio is, therefore, rewritten by use of the mass flow rate as
.-
I
-,,I
The level of the particle layer uniformly descends in a mass-flow hopper with a small cone angle-for instance, 0 = 30". The falling velocity of particles in the vertical direction can therefore be regarded as equal over the same horizontal plane. The void fraction of uniformly sized particles is constant within the hopper during flow, except for small particles falling through the void of large particles. Small particles do not interfere with the motion of large particles during flow. Hence, in the case of a complete binary mixture of particles, the fact that the interspaces of large particles are perfectly filled with small ones during flow gives the critical mixing ratio of the mixture.
W,
D+(l
(13)
(1 -
tr)p,
ti(1 -
tJPr
(15)
where W is the weight of particles. The mass flow rate is independent of the height of the particle bed. To describe the segregation and blending phenomena, consider the discharge time, t , with which the mass flow rate, F, and the mixing ratio, M , vary a t the hopper outlet. The following flow-rate equation holds by material balance at the hopper outlet and a certain cross section of the horizontal plane of the hopper.
F
= V,Sop(l-
t)
= VSp(1 -
t)
(16)
From Equation 16, we have
v = s-a S
XV,=- dh dt
Therefore, the time required for the particles to reach the opening is generally represented by
1 hS t=-l dh sa -
v o
where h is the height of the particle bed from the level of the hopper outlet. For a conical hopper, substituting Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
175
Then, the mass flow rate F = Fi, is given by Equation 10, and the mixing ratio is M = Ft/Fls by Equation 12. At last, only small or only large particles left within the hopper corresponding to the cases mentioned above, are discharged to empty the hopper. In the former case, time t3 required for the remaining small particles to leave the hopper is
= So= Do2 4
and 0
,S = ( h tan
+5 )'. 2
into Equation 17 and integrating it gives
h 4 tan20/2 2 tan 012 t=-(h2 + h V, 3 0," DO
+ 1)
(18)
where 6 is the hopper cone angle. Blending. Suppose that layers of small particles are separately placed upon layers of large ones, as shown in Figure 1, and are discharged from a mass-flow hopper. An appreciable amount of blending occurs within the hopper according to the screen or hypothetical-hoppers model. As a result, only large particles are discharged until time t ~ when , small particles reach the opening. tl is therefore obtained by putting h = h~ and V , = Vio + Virointo Equation 18 as
t, =
4 tad012
hi
Vi, +
!
Vk 3
Dt
hi'
+
2 tan 012
DO
hi+ 1)(19)
where Viois the velocity of large particles a t the opening. Then, during this interval the mass flow rate F = Fl is given by Equation 2 and the mixing ratio is M = 1.0 according to Equation 12. Next, the solids mixture begins and continues to flow until all the large or small particles are discharged. In the former case-small particles are so numerous that the upper surface of small particles cannot catch up with the descending surface of large particles within the hopper-time t 2 when the mixture finishes flowing is obtained by putting h = hi and V , = Vl,, into Equation 18 as
hi
t2=-(
4 tan20/2 2 tan 012 h: + hi 0," D"
KO 3
+ 1)
(20)
I n the latter case-when all the small particles have passed through the layer of large particles before the upper level of large particles reaches the opening-time t; is given by
t; = ti
ws +-
Then, the mass flow rate F = F, is given by Equation 2 and the mixing ratio is M = 0 according to Equation 12. I n the latter case, time t,i when all the large particles have left the hopper is obtained by putting h = hi and V , = Vi, into Equation 18 as
t; = te
hi 4 tan28/2 h: Vio ( 3 D2
= -
+
2 tan 012
DO
hi + 1) (23)
Then, the mass flow rate F = Fi is given by Equation 2 and the mixing ratio is M = 1.0 by Equation 12. Hence, the blending phenomenon can be described as the variations of the mass flow rate and the mixing ratio with discharge time by use of Equations 19, 20, or 21, and 22 or 23, as illustrated in the experimental results. Segregation. Suppose that the binary mixture which satisfies Inequalities 14 and 15 is discharged from the hopper. Segregation takes place on the basis of the same model. The solids mixture continues to flow until time t l which is obtained as
where his is the height of the solids mixture bed. Then, the mass flow rate F = Fis is given by Equation 10 and the mixing ratio is M = F,/ Fi, according to Equation 12. Only large particles that remain within the hopper take the additional time ti given by Equation 25 after time ti to empty the hopper.
t; = t r
where
t? =
-
tl
(25)
hl, 4 tan20/2 { - DZ h%+ 2 tanDO 0 1 2 hi, + 1) (26) Vio 3
-
Then, the mass flow rate F = FI is given by Equation 2 and the mixing ratio is M = 1.0, according to Equation 12. The segregation pattern is consistent with that of the latter part of the blending given by Equations 2 1 and 23.
F,'
Experimental and Results Small Partd e s
Large
0
co
70 Figure 1 . Symbols of two-dimensional hopper 176
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
A two-dimensional hopper (Figure 1) was used for the flow test. I t has a transparent plane on the front and an apex angle, fl = 30°, which is small enough for the solids to flow through in mass flow. The slit width, D,, of the hopper is variable and D, = 0.52, 0.66, and 0.75 cm. were used. The average mass flow rate, F , was measured by sampling the particles flowing out of the hopper. Figure 2 plots F us. ( D , - d,) for four kinds of particle size and three kinds of slit width. I t gives the values of n = 1.5 and 01 = 210 in Equation 2. The flow-rate equation for the slit of the two-dimensional hopper is therefore written as
F
= 2lO(D, - o!,)'~
(27)
Table 1. Material Properties and Data Used for Calculation
Blending
U,, em. Wl, g. W,, g.
Wil w, hi, em. Li fr fr
Fi, g./sec. Vi,,, cm./sec. F,, g. i see. pi=
2.52g. cc., d,
p. =
Segregation
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 8
Fig. 9
0.66 2 100 2100 1 39.3 0.390 0.390 0.780 69 22.7 110
0.66 4000 200 20 54.7 0.390
0.52 2100 2100
0.52 4000 200 20 54.5 0.365
0.52 3780 420 9 53.5(=hi,) 0.365
0.52 1400 2800 0.5 53.0(=hi,)
=
1
39.1 0.365 0.380 0.780 44 17.6 82.5
...
0.824 69 22.7
...
0.1705em., d,,
=
0.0119cm., n
=
2.7,
CY,
=
...
0.824 44 17.6
0.582 44 17.6
...
CYI
=
140, B
1.0'
=
1.20, C
... ... ... ... ... ...
...
... =
5, 0
=
30"
G G 0
-
0 0
I
- / / I /
200
v
0,
u) > m
1
100
80 LL
0.2
c
60
1
60
0 1
,o
n
40 (12
a4
06
( D r 4 Figure 2. Plot of F
-
OB
1
crn.
vs. (D, -
d,)
For a conical hopper, n is generally reported to be equal 2.7 (Shinohara et al., 1968). n = 2.7 was adopted to 2.5 in the flow-rate equation of the screen or the hypothetical hopper formed by large particles. The mixing ratio, M , was obtained by screening tests of the particles sampled. The material properties and the data used for calculation are listed in Table I. ti was measured by taking a picture of particles remaining in the hopper during flow. Since t: and p cannot be measured directly, appropriate values were adopted from the experimental data. The flow-test results with regard to blending phenomenon are depicted in Figures 3, 4, 5, and 6, varying the conditions presented in Table I , such as the opening diameter and the initial weight ratio of large particles to small ones placed upon the large particles. Figure 7 illustrates the flow pattern when large particles were put on small ones and discharged. Segregation during flow was investigated by filling the hopper with a small amount of mixed particles, which were spread as uniformly as possible in layers, and discharging them. Care was taken that no segregation occurred in filling the hopper. Figures 8 and 9 show the results under the conditions of Wi/ W , = 9 and %, respectively. Figure 10 indicates the segregation pattern during flow.
Discussion
Good agreement between the straight line and the experimental data in Figure 2 proves the validity of the flow rate equation (Equation 2). I n Figures 3, 4, 5 , and 6, the experimental data of the mass flow rate spread a little around the calculated line, compared with those of the mixing ratio, and the experimental results qualitatively deviate from the theoretical line in the stepwise variation of the mass flow rate and the mixing ratio with discharge time. The former arises mainly from the error in sampling procedures and the flow characteristics of the particles flowing out of the hopper. The latter is due to the discrepancy of the assumptions made in the theoretical consideration and to the adhesion of small particles to the hopper wall. However, the experimental results are satisfactorily consistent with the theoretical as a whole. Figure 7 indicates that no blending occurs in the case of large particles laid on small ones. The reason may be that the ratio of particle size given by Inequality 14 is not satisfied-that is, substituting t s = 0.380, p = 1.20, and C = 5 which is determined by the flow of particles through an orifice
L3(1 + p)
.-
1-
e,]
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
177
1.0
--
O O O
0 00 0
0.9
*I
II ~ - L ~ ~.. .~L~-. . LI 0
20
40 TIME
-~J 80
60 sec.
Figure 6. Blending pattern in WdW, = 20 with D, 0.52
=
r ?
Figure 7. No blendir
I
which does not satisfy Inequality 15. Large particles are therefore involved in small particles and are discharged as one kind of particle. I n the former case (Figure 8), W,/ W, = 9, which satisfies Inequality 15. All these results confirm the analysis of the mechanism of segregation and hlending during mass flow for the hinary mixture of particles which satisfies Inequalities 14 and 15. This indicates
1.0
as 08
t
el
= void fraction of particle bed during flow = void fraction of small particles falling through
effective interspaces of large particles 0 = cone angle of a hopper p
= true density of particle, g./cc.
SUBSCRIPTS
1 = large particle s = small particle Is = mixture of large and small particles o = opening
Kawai, S., Bull. Fac. Eng. Kanazawa Uniu. 2, 187 (1959). Miwa, S.,“Funtai Kogaku Handbook,” p. 448, Asakura Publishing Co., Tokyo, 1965. Peacock, H. M., J . Inst. Fuel 12, 230 (1938). Rose, H. E., Trans. Inst. Chem. Engrs. 37, 47 (1959). Seaton, T. W., Foundry 87, 86 (1959). Seaton, T. W., Metal Ind. 96, 67 (1960). Shinohara, K., Idemistu, Y., Gotoh, K., Tanaka, T., IND. ENG.CHEM.PROCESS DESIGNDEVELOP.7, 378 (1968). Williams, J. C., Chem. Processing April, S6 (1965). Williams, J. C., Powder Technol. 1, 134 (1967).
Literature Cited
Brown, R. L., J . Inst. Fuel 13, 15 (1939). Denburg, J. F., Bauer, W. C., Chem. Eng. 71, 135 (1964).
RECEIVED for review June 3, 1968 ACCEPTED September 22,1969
HEAT TRANSFER IN A CONTINUOUS VI BRATION-PROMOTED TURBULENT FILM E N R I Q U E R O T S T E I N , M A R T ~ NU R B I C A I N , M I G U E L E L U S T O N D O , AND
NUMA
J .
CAPlATl
Planta Piloto de Ingenieria Quimica, Uniuersidad Nacional Del Sur, Bahia Blanca, Argentina
Water flows through a tube subjected to a rotary space vibration, as a thin rotating turbulent film. Its heat exchange characteristics were studied at frequencies between 800 and 1600 r.p.m., amplitudes between 6.5 and 8.9 mm., viscosities between 0.5 and 32.5 cp., thermal conductivities between 0.21 8 and 0.540 cal. hr.-’ min.-’ O C l , surface tensions between 43 and 73 dynes per cm., and residence times between 4.8 and 9.9 seconds. Film thickness is a function of axial and vibrational Reynolds numbers. A design correlation adequate for this type of heat exchanger i s Nu = 1.175 Pros. The smaller error corresponds to a correlation with fluid properties at bulk temperature, compared with “film” temperature or the viscosity correction factor, p u / p b . The axial Reynolds number has no significant influence on the Nusselt number. Heat augmentation is discussed with several criteria. Heat transfer is controlled by the liquid thermal conductivity rather than by viscosity, A theoretical model based on the penetration theory agrees reasonably well with the experimental correlation.
WHEN a tube is subjected to a rotary
vibration, which describes a surface of revolution such as a cylinder, instead of flowing past the tube, eventually filling it, the liquid flows as a thin film adhering to the tube wall. The bulk of the film rotates at practically the same frequency as the vibration. The film is turbulent and by using a stroboscope it can be seen moving as a sea-like wave, rotating with changing instantaneous thickness, showing a high rate of surface renewal. The objectives of this investigation were to evaluate the augmentative effects of this turbulent film on heat 180
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
transfer, t o obtain design data for this type of heat exchanger, and to gain knowledge as to the fundamentals of its performance. Previous Work
Some work has been done on the augmentation of heat transfer coefficients through both heat transfer wall vibration and liquid vibration, which included acoustic vibration and lower frequency pulsations. Bergles and Morton (1965) have reviewed work on these and other augmen-
