Continuous Shape Separation of Binary Mixture of ... - Science Direct

Powder

Technology,

Continuous

54 (1988) 31- 37

Shape Separation

31

of Binary Mixture of Granular Particles

M. FURUUCHI and K. GOTOH Department

of Energy Engineering,

Toyohashi

University

of Technology,

Tempaku-cho,

Toyohashi

440

(Japan) (Received November 26, 1986; in revised form September 15,1987)

SUMMARY

Using a rotating cylinder with blades previously reported for the batch mode operation, binary mixtures of spherical and nonspherical particles are classified continuously according to shape. Newton’s sepamtion efficiency is calculated for various operational conditions and the optimum conditions are found for the continuous shape sepamtion. The performances of the classifier run in batch and continuous modes are compared with each other and the superiority of the continuous operation to the batch mode is shown for the rate of shape separation.

spherical and non-spherical particles flow in opposite directions to each other, so that one can classify the particles according to shape. In the previous paper [4], the shape classification of binary mixtures of spherical and non-spherical particles was discussed for the batch mode operation. In the following, the continuous shape classification is discussed for binary mixtures of spherical and non-spherical particles. The separation performance and the optimum condition’ are investigated theoretically and experimentally. The performances of the classifier run in batch and continuous modes are compared with each other. SHAPE CLASSIFIER AND TEST SAMPLES

INTRODUCTION

Shape classification of particles makes products of granular materials of high quality, since bulk properties are strongly affected by particle shape [ 1,2]. One can obtain, for example, a packed bed of higher density from particles with a higher degree of sphericity. The grinding process may be improved by adjusting the shape of the abrasive grains. The authors previously proposed an inclined rotating cylinder with a row of blades [3, 41 as one of the shape classifiers which make use of the difference in the descent velocity of particles on an inclined wall [ 5 151. For each particle fed into the classifier, there exists a corresponding critical rotational frequency of the cylinder NC, which determines whether the particle stays in or flows out of the classifier. Spherical particles have higher values of NC, whilst nonspherical particles have lower values. Hence, if the rotational frequency of the cylinder is set between these two critical values, the 0032-5910/88/$3.50

The experimental apparatus, shown in Fig. 1, consists of a shape classifier similar to that used in the previous paper [4] and a particle feeder. The mixture of spherical and nonspherical particles is fed into the cylinder by the electric vibratory feeder. The feeding position can be adjusted by sliding the feeder. Both ends of the cylinder are open so that the spherical particles flow out from the lower end and the non-spherical particles from the upper end. Specifications and operational conditions of the classifier are listed in Table 1. The mixture of spherical silica gel and elliptiecylindrical polystyrene pellets is used

particles

roller

particles

Fig. 1. Experimental apparatus. 0 Elsevier Sequoia/Printed in The Netherlands

32 TABLE 1

I

1

I

,

Specifications and operational conditions of classifier Inside diameter of cylinder, D Cylinder length, L Blade length, 1 Blade width, b Number of blades, n Inclination angle of blade, $ Inclination angle of cylinder, 0 Rotational frequency, N

90 mm 840 mm 85 mm 20 mm 14 45” 2” or 4” 0.2 - 0.65 s-’

. spherical y”

I

I

I

I

103

101 MSo.

TABLE 2

.

o non-spherical

10’

MN,,

Cparticlesl

Fig. 2. The rate constant of spherical particles is determined as KS which is independent of Mso.

Properties of particles Particle

Density (kg/m3)

Average diameter (mm)

Aspect ratioa (-)

Silica gel pellets Polystyrene pellets

1720b 1090

3.10 2.55’

0.951 0.561

aAspect ratio is defined as (minimum length)/(maximum length). b Apparent density. CIntermediate diameter.

the test sample. Properties of the particles are listed in Table 2. In order to eliminate the size effect on shape classification, the particles are sieved by screening. The difference in density between silica gel and polystyrene is small and hence the density effect can be ignored. The number ratio of the silica gel pellets (spherical) and the polystyrene pellets (non-spherical) in the mixture is selected from 1:2,1:1 and 2:l. as

RESIDENCE

.

y’

TIME OF PARTICLES

IN CLAS-

SIFIER

Since the residence time of particles in the classifier is an important factor governing the separation performance, one must know it under various operational conditions of the classifier. The residence time of particles can be obtained from the multi-staged cascade model [4], in which the cylinder section containing one blade is regarded as one stage and the classifier is modeled as a series of the stages. Results shown in Fig. 2 are from the batch mode operation and demonstrate the effect of the amount of particles initially fed into

the classifier on the measurement of the rate constants of spherical and non-spherical particles (data for the non-spherical particles are cited from Fig. 2 of the previous paper [4]). The rate constant is equal to the inverse of the residence time. As can be seen in Fig. 2, the residence time of the non-spherical particles can be determined independently of the particle hold-up, but this is not the case for the spherical particles. The main reason is that non-spherical particles flow upward due to the mechanical action of the blades, while spherical particle roll down the cylinder wall due to gravity. In principle, the residence time of the spherical particles can be measured from a single particle. However, it changes from particle to particle because exactly identical particles do not exist. Hence, the rate constant of the spherical particles is determined as Ks in the range Mso < 50, which is free from the effect of particle hold-up. Figure 3 depicts the rate constants Ks for the spherical particles and K, for the non-spherical particles in relation to the rotational frequency N of the cylinder at the angle of inclination 8 = 2”. The sign of the rate constant is taken as negative for the downward flow. Open circles and triangles designate experimental results. The theoretical results are obtained from consideration of the particle trajectory in the classifier [ 41. Each solid curve is the average for 200 K uersus N relations obtained for 200 different values of C; in the range shown in Fig. 3. For detail, see reference [4]. From Fig. 3, the critical rotational frequency Nc, of the spherical particles at which KS becomes zero

33 lower

u”

0.2

0.4

0.3 N

0.5

[S-‘I

is read as about 0.49 s-l. For 0 = 4”, similar results are obtained and Nc? = 0.70 s-’ , whereas the non-spherical particles flow upward only under the conditions of the present study unless they are mixed with the spherical particles.

EFFICIENCY

The separation performance of the classifier run in the continuous mode can be predicted in a similar way to that for the batch mode. After leaving a blade, particles spread over the inner wall of the cylinder to form a mono-particle layer, if Q & 0.9 g/s at 8 = 2” and N = 0.3 s-l, for example. In the section above the feeding stage, some spherical particles are trapped by neighbors and move upward with non-spherical particles, while in the lower section some non-spherical particles are trapped by neighbors and move downward with spherical particles. Following the previous paper [ 41, the probability P(i) for a particle to leave the main stream of the mixture at the ith stage is assumed as follows. Ps(i) = exp]-

SS {MS(i)

+

qs’=KsPs(f)Ms(f)

5:

KsPs(f+lHvfs(f+l)

6:

KNCI-Ps(f)lMs(f)

: KsPs(f+Z)Ms(f+2)

1

: KsCl-PN(f-l)lM,,(f-1)

7

2:

KNPdf--2hf,.,(f-2)

6:

3

: K.s.C~--P~(~)~M~(~)

h(i)

= ew[--h

i = 1, 2, . . . . n

KNC~-Ps(f+l)lMs(f+1)

KNPdf-f)fvfN(f--I)

Fig. 4. Flow chart illustrating population balance of particles at feeding stage.

The subscripts S and N respectively denote spherical and non-spherical particles, ts and & are constants, and M(i) is the number of particles in the ith stage. The stage number is counted from the lower end. The population balance of the binary mixture in the feeding stage at steady state is illustrated in Fig. 4. In the feeding stage, the amount of the spherical particles [l -Ps(f)]Ms(f) is trapped by the mixture flowing upward and moves to the (f + 1)th stage at speed KN. The remainder Ps(flMs(f) flows to the (fl)th stage at speed KS. In the section below the feeding stage, the spherical particles always flow downward so that the flow rate through this section is expressed as qs’ = KsPs(f)Ms(f). Hence, the population balance of the spherical particles for each stage becomes

+ KNU - f’s(R)3

WfJ[KsMf)

= W’df

+ l)&(f

= K,P,(i + K,{l

i=f+l

(1)

C%(i) + %(Oll (2)

(3)

+ 1) + qs

+ KN (1 -

h.(i) [K&(i)

Pdi)ll

+ l)Ms(i + 1)

- Ps(i - l)}M,(i , *-*,

1) (4)

n

MS(i) = MfWdfl

MdOH

i = 1, 2, . . .. n

section

0.6

Fig. 3. KS and KN in relation to N (6 = 2”). Each solid curve depicts the theoretical value obtained from the particle trajectory model [4]. The sign of KS is taken negative for downward flow, but only in this figure.

SEPARATION

upper

%+qN

qN’=KNPdf)M,,(f)

4: 0.1

section

i = 1, 2, . . .. f-1

(5)

qs is the feed rate of the spherical particles at the fib stage and P,(n + 1) = 0. Similarly, the population balance of the nonspherical particles yields

in which

34

MN(f)

IKSI1 -

+ KNPN(f)]

pN(fl)

= KNPN(f -

- 1) + qN

l)MN(f

I

I

(6)

+ KN&(i

-

i = 1, 2, . . .. f MN(i)

+ l)}MN(i l)MN(i -

qN

Cs-‘I

1)

1

(7) i=f+l,...,n

= pN(flMN(fl

KNPN(n)MN(n)

[s-‘1

z e

(8) 0

in which QNis the feed rate of non-spherical particles at the fib stage and PN(0) = 0. In the upper section, the non-spherical particles always flow upward. Newton’s separation efficiency q becomes n=

Cs-‘I

-___-

K,=0.074

+ 1)

(a:

N=0.2

1 Ks=1.72

= KS (1 -PN(i

r-1

e=2

I

I

_

KNil

0.1

--PSl(n)MS(n)

9s

1

10

1

10

1

(9)

The set of eqns. (1) - (8) is solved numerically by the relaxation method so that one can calculate r) from eqn. (9). Figures 5(a) - (c) depict the relation between the total feed rate Q and Newton’s separation efficiency 17for the fixed angle of inclination (0 = 29 and rotational frequency N of the cylinder. The number ratio of qs to qN is varied from 2:1 to 1:2 and the mixture is fed at the 7th stage. Empirical values of KN and KS are used for the calculation of q. Solid curves in these figures are the theoretical results for q, :q N = 1:l. The constants f‘s in eqn. (1) and & in eqn. (2) are adjusted to give the best fit to the experiments. Theoretical results for other mixture ratios are very close to the curve for qs:qN = l:l, so they are not shown in the figures. The sharp decrease in the separation efficiency starts at a Q-value which scarcely depends on the mixture ratio. The theoretical curves show good agreement with the experiments. Similar results are obtained for another angle of inclination of the cylinder (0 = 49. The values of ts and ENare ahOSt constant. However, es and ENmight depend on the particle properties such as shape, size and friction coefficient and the geometry of the classifier. Relationships between Newton’s separation efficiency 17and the rotational frequency N of the cylinder are shown in Fig. 6 for four different feed rates of the mixture at the angle of inclination 8 = 2”. Theoretical curves in the figure are obtained from eqn. (9),

K~=o.l36

‘;

[S-‘I

c

0.1 Q=qS+qN

hisI

-__--

1

z

c

Ks=0.844

[s-‘-J

K~r0.196

Cs-‘I

ts=0.00813

C-l

t,,=0.00020

c-1

0 1 o=qS+4,,

[9kd

Fig. 5. Relation between Q and q at 0 = 2”, f = 7, and N = (a) 0.2, (b) 0.3, (c) 0.4 a-‘. The solid curves are obtained from eqn. (9).

using the theoretical relationships between the rate constant and N shown in Fig. 3 and the average VdUeS Of & = 0.00809 and EN= 0.00019 for the calculation. The separation efficiency shown in Fig. 6 is quite different from that for the batch mode [ 41. For a

35

I ’ ’ ’ ’ ’ ’ ‘I e=2

PI

qS:qN=l:l

0 Cglsl

-

0.385

--a----0.528 . ..O . . . . . . . 0,739 +---

0

0.1

0.2

OPTIMUM CONDITIONS

0.939

SHAPE SEPARATION

I

i

I

0.3

In the example shown in Fig. 7, the number of effective stages is about 4 from each end. The theoretic.al curves explain the decrease in separation efficiency near both ends.

I

I

I

I

0.4

0.5

0.8

0.7

N cs-‘I

Fig. 6. q versus N relations. The solid curves are obtained from eqn. (9) by using the theoretical rate constants shown in Fig. 3. .$ = 0.00809 and EN= 0.00019.

constant feed rate, the hold-up of the nonspherical particles decreases with increasing rotational frequency, but that of the spherical particles increases because its rate constant approaches zero. Therefore, a peak or plateau appears in the separation efficiency as shown in Fig. 6. The calculated q uersus N curves exhibit the trend of the experiments fairly well. The effect of the feeding position on the separation efficiency q is shown in Fig. 7, in which the solid curves are obtained from eqn. (9). For each feed rate, the separation efficiency remains constant until the feeding stage approaches the upper or lower end of the cylinder, where the separation efficiency decreases. No increase in separation efficiency can be expected by increasing the number of stages above an effective value. I 9=2

I

I

[“I

I

I

I

I

I

[.?.-‘I

N=0.4

I

I

I

I

1

I

I

I

1

0.1

e=4

1-

qS:qpl:l

I

lower

4

5

6

7 f

I

N~~‘0.7 [?.-‘I .................................. I tl C%l 0 90-100 A 80-90

C%l 0

70-80

0 50-70

I

I

I

0.5

1

5

a 3

(b)

[“I

z

a 2

I 1

i

1

-50

Q [Q/S]

I

;,;),l,zll]

(a)

.50-70

I!! E

I

.

1

T

CONTINUOUS

Figures 8(a) and 8(b) show the relationships between the feed rate Q of the mixture, the rotational frequency N of the cylinder and Newton’s separation efficiency r) at the angles of inclination 8 = 2” and 4”, respectively. The mixture ratio of spherical to non-spherical particles is varied from 2:l to 1~2. The solid curves in the figures are calculated from eqn. (9), using the theoretical relations between the rotational frequency and the rate constant shown in Fig. 3. The dotted line in the figures indicates the critical rotational frequency NC, of the spherical particles. As

I

1

FOR

OF BINARY MIXTURES

8

9

Cstagel

1011121314

upper

Fig. 7. Effect of feeding position on ~7.The solid curves are obtained from eqn. (9), where .$‘s = 0.00809 and .& = 0.00019.

CSlSl

Fig. 8. r) in relation to Q and N at 8 = (a) (b) 4”. & and & are (a) 0.00809, 0.0019 0.00810, 600025, respectively. The solid are obtained from eqn. (9). ....e.., No2 ; - mum operational condition.

2” and and (b) curves -, opti-

36

the feed rate of the mixture increases, the hold-up of particles in each stage increases, giving rise to a decrease in the separation efficiency. It should be noted that the decrease in separation efficiency is very sharp. The theoretical results are in good agreement with the experiments. In the theoretical counter-line map, there is an obvious valley at the right-hand side of Fig. S(b), below which, i.e., at lower N values, the non-spherical particles can flow out from the lower end of the cylinder. The optimum condition for the shape classification of the binary mixture of particles may be defined in such a way that the product 977 becomes maximum. This condition can be satisfied on the broken line in Figs. S(a) and S(b), so that one can obtain the optimum values of Q and N for a prescribed separation efficiency. However, attention should be focused on the fact that there is a sharp decrease in separation efficiency.

COMPARISON TINUOUS

BETWEEN

BATCH

AND

CON-

MODES

In this section, the continuous shape classification is compared with the batch mode operation reported in the previous paper 141. Figure 9 shows the maximum Newton’s separation efficiency qrnax calculated from eqn. (16) of the previous paper [4] in relation to the total number of stages II in the classifier for the batch mode, in which W is

e=2

I

1

I co1

N=0.4

the total mass of the mixture. Since in the batch mode operation Newton’s separation efficiency increases with time and reaches its maximum after a certain time, vmax is used in place of 7. For a fixed W, qmax approaches unity with increasing II i.e., in the batch mode, an increase in the number of stages always leads to an increase in separation efficiency. But this is not the case in the continuous mode, as shown in Fig. 7. be the measure of the Let ( Wltrnaxhmax separation capacity of the classifier in the batch mode operation. W is the total amount of particles to be classified and t,,, is the time at which the maximum separation efficiency 77max is attained. The theoretical relations between the number of stages n and are shown in Fig. 9. (W/t,,,) ( w/tmaxhnax is the average of the separating rate so that it can be compared with Q in the continuous mode. A theoretical comparison is made in Fig. 10 of the continuous and batch mode operations, in which Newton’s separation efficiency q is depicted as a function of the rate of separation and the rotational frequency N of the cylinder. The solid curve for the batch mode is obtained from eqn. (16) of the previous paper [4] and that for the continuous mode from eqn. (9) along with the theoretical rate constants shown in Fig. 3. The number of stages n in the classifier is 10 for both cases and the mixture is fed at the fifth stage in the continuous mode. Es = 0.00809 and & = 0.00019 are used for the calculation. As can be seen in Fig. 10, the 1

I

I

Cs-'I

I

I e=2

C"1

qs:qN=l:l

I n=10

f=5

I

0.5

1

';: 0.4

2 :

7

0.3 1 :

: :

0.2

=' z

0.1

I 0 10

1 n

0 100

[stages]

Fig. 9. &ax P lotted against n for the batch mode, calculated from eqn. (16) of the previous paper [ 41. The separation capacity ( W/tmax)qma, is also shown in relation to n ES = 0.00809.

I

I

0.1

1 Q,(Wltm..)

Cekl

Fig. 10. Comparison of shape separation performances between continuous and batch mode operations. q vs. Q and vmax vs. W/t,,, relations are calculated from eqn. (9) and eqn. (16) respectively in ref. [4 1, where .$s = 0.00809 and .$‘N= 0.00019.

37

rate of separation in the continuous mode is higher than that in the batch mode. However, one should also take into account the simplicity and convenience of the batch type equipment, especially for small laboratory use without any feeder. CONCLUSION

The separation efficiency and the optimum condition for the continuous shape separation of binary mixtures of spherical and non-spherical particles are investigated theoretically and experimentally using the rotating cylinder with blades. The performances of the batch and continuous modes are compared. As a result, the following conclusions are drawn: (1) As in the case of the batch mode [4], Newton’s separation efficiency in the continuous mode can be obtained from the multi-staged cascade model. (2) There exists a minimum number of effective stages in the continuous mode, beyond which no increase in separation efficiency can be expected. (3) An optimum condition exists under which the product of the separation rate and the separation efficiency becomes maximum. (4) It is shown numerically that, on the basis of the rate of shape separation, continuous operation is superior to the batch mode. ACKNOWLEDGEMENT

P(i) Q 9 t max

W

Greek symbols Newton’s separation efficiency in con77 tinuous mode, %mx maximum Newton’s separation efficiency in batch mode, inclination angle of cylinder, o e %E constant in eqns. (1) and (2), inclination angle of blade, o J/ Subscripts number of feeding stage f stage number L spherical particles N non-spherical particles REFERENCES

4

We thank the referees for their valuable comments. LIST OF SYMBOLS

b c; D K M(i) Mso N k Nc2

blade width, m constant in eqn. (26) of the previous paper 141, m/s internal diameter of cylinder, m rate constant, s-l number of particles in ith stage, number of spherical particles initially fed into classifier in batch mode, rotational frequency of cylinder, s-’ total number of blades or stages, critical value of N, s-’ critical value of N, above which spher-

ical particles flow upward, s-’ probability for a particle to escape from stream of flow at ith stage, total feed rate of mixture, particles/s or g/s feed rate of particles, particles/s or g/s time at which Newton’s separation efficiency becomes maximum in batch mode, s total mass of particles fed into classifier in batch mode, g

10 11 12 13 14 15

K. Ridgway and R. Rupp, J. Pharm. Pharmac., 21 (1969) 30. J. K. Beddow, Particle Characterization in Technology, vol. 2, CRC Press, Florida, 1984. M. Nakagawa, M. Furuuchi, M. Yamahata, K. Gotoh and J. K. Beddow, Powder Technol., 44 (1985) 195. M. Furuuchi, M. Nakagawa, M. Suzuki, H. Tsuyumine and K. Gotoh, Powder Technol., 50 (1987) 137. F. G. Carpenter and V. R. Diets, Nat. Bur. Standards Jour. Res., 47 (1951) 64. G. S. Riley, Powder Technol., 2 (1968/1969) 315. M. Sugimoto, K. Yamamoto and J. C. Williams, J. Chem. Eng. Japan, 10 (1977) 137. K. Viswanathan, S. Aravamudhan and B. P. Mani, Powder Technol., 39 (1984) 83. S. Aravamudhan, N. Premkumar, S. S. Yerrapragada, B. P. Mani and K. Viswanathan, Powder Technol., 39 (1984) 93. E. Abe and H. Hirosue, J. Chem. Eng. Japan, 15 (1982) 323. W. H. Glenzen and J. C. Ludwick, J. Sedimentary Petrology, 33 (1963) 23. B. Waldie, Powder Technol., 7 (1973) 244. Ph. H. Kuenen, Sedimentology, 28 (1964) 207. K. Shinohara, T. Maenami and T. Tanaka, Part. Sci. Technol., 4 (1986) 161. K. Shinohara, Powder TechnoL, 48 (1986) 151.