Separation Based on Shape Part I: Recovery Efficiency of Spherical Particles

Powder Technology. 39 (1984) 83 -

83

91

Separation Based on Shape Part I: Recovery Efficiency K.

VISWANATHAN.

Department

(Received

21.1983;

Engineering. in revised

Indian form

Institute

November

of Technology,

Hauz

h-has. Xezz

Delhi-I

10016

(India)

3.1983)

SUMMARY

The use of an inclined rotating disc apparatus, empzoyed to separate spherical from nonspherical particles, is studied theoretically and experimentally_ The effects of feed rate -US, speed of rotation of the disc n, angle of inclination of the disc 0, angular co-ordinate of the feed point (Y, and the feed point position on the efficiency of recovery of spherical particles rl+ are investigated. A ratetype equation is proposed to describe the variation of qS with F, and n_ The ‘rate constant-K is theoretically related to 8 by a linear equation_ The experimental results generally confirm the features of the model. The results indicate F,, n and 8 to be critical variables. Further, it is found that qS is critically affected by the cngular co-ordinate of the feed point CY, whereas the radial coordinate of the feed point is a non-critical variable_

INTRODUCTION

Particles are characterized by their mechanical and morphological properties_ Density, hardness, elasticity, etc., fall under the mechanical properties category, and these have been extensively studied_ Such properties do not vary greatly from particle to particle, and once determined can be used with sufficient confidence in further studiesOn the contrary, morphological characteristics, namely, particle size and shape, vary from particle to particle_ There exist many methods for measuring particle size distributions [l, 23_ However, the other morphological property - particle shape -has not received much attention; only recently, a method of analysis 0932~5910/84/33_OQ

Particles

and B. P. MAN1

S. ARAVAMUDHAN

of Chemical July

of Spherical

of particle shape by Fourier transform has been developed [3 - 51. Characterization of particle shape is important as in some industries it is required to separate differently shaped particles. For example, it is important to separate chaff and adulterants from whole grains. Here, the wanted material (whole grain) is smooth and more rounded than the unwanted material (chaff and aduiterants)On the other hand, in the abrasive industry, the wanted material is in the shape of triangles (used in making grinding wheels) and blocks (good for grits and abrasive sheets), which has to be separated from the unwanted smooth and rounded particles. A novel technique of separating differently shaped particles from a mixture was recently suggested by Meloy IS]_ He pointed out that the time of arrival of particles at the bottom of a stack of sieves all of same aperture size would be different for differently shaped particles, with spherical particles arriving the quickest_ This may be a potentially effective method for characterizing t-he shape of the particles based on the arrival times at the sieve stack bottom (from a feed of same size particles). However, this may not be so useful as a separation device due to the following two reasons: (1) The mixtr;re will contain different size particles too, and the arrival time of a smaller non-spherical particle will be equal to that of a larger spherical particle. (2) Power consumpt.ion is expected to be uneconomically high_ A simpler apparatus - an inclined rotating disc - suggested by various investigators [7 - S] may be potentially useful as a separation device which may not have the abovementioned drawbacks of Meloy’s ‘sieve cascadograph’_ It uses the principle that on an inclined rotating disc, spheres will roll in a @

Elsevier

Sequoia/Printed in The Netherlands

84

straight line and can be collected at the ‘near end’, whereas non-spheres move in a circular path along with the disc and have to be removed using a scraper at the ‘far end’. In the initial studies [7,8], no attempt was made to study the effect of operating conditions on the efficiency of separation nor to explain the mechanism of the separation process In a more recent work [ 91, the efficiency of separation was correlated to a limited number of variables, namely, the feed flow rate and the speed of revolution of the disc. Considering the potential usefulness of the apparatus as an efficient separation device, it was decided to study the apparatus comprehensively, both theoretically as well as experimentally_ An attempt is made in the present paper (1) to develop a theoretical model for hold-up which leads to the identification of the proper ‘independent variable’, (2) to present experimental results on the ‘efficiency of recovery’ of spherical particles, (3) to develop a mathematical model for the efficiency of recovery of spherical particles, and (4) to compare the predictions of the model with the experimental data and to identify the critical variables.

DEFINITIONS

AND

GENERAL

NOMENCLATURE

The terms ‘spherical’ and ‘non-spherical’ are relative only and have no absolute significance_ The feed rates of spherical and nonspherical particles are represented as F, and F,_ The hold-up of spherical and nonspherical particles on the disc are represented as W, and W,. The position at which the spherical particles are collected is called the ‘near end’ and the position at which the non-spherical particles are collected is called the ‘far end’. Subscripts 1 and 2 to variables refer to near end and far end respectively_ The feed rates F, and F, divide themseives into F,, and Fnl at the near end, and Fs2 and F n2 at the far end. The efficiency of recovery of spherical particles is defined as

(1)

It must be noted that q= is only the recovery efficiency of spherical particles and not the separation efficiency. The overall separation efficiency would also involve the recovery efficiency of non-spherical particles_ Though T], may be very near to unity, the separation efficiency of the overall process may be very small if non-spheres also get collected at. the near end. Nevertheless, in applications where the desired material is the valuable spherical particles, one may not mind some contamination of unwanted non-spheres at the near end with spherical particles, but any loss of valuable spherical particles may be economically unacceptable_ Similarly, in applications where the desired material is the non-spherical particles (as in the abrasive industry), one may not mind losing some nonspheres at the near end, but contamination of spherical particles at the far end with the wanted material may be technically unacceptable- In such cases, the objective function would be to maximize the recovery efficiency of spherical particles vs, and not the overall efficiency_ In the present paper, therefore, rls is studied experimentally and theoretically_ The angle of inclination of the disc with the horizontal is denoted by 19.The ‘scraper angle’ is defined as the angle covered by the non-spherical particles in travelling from the feed point to the scraper where they are removed, and is denoted by 0. An angle Q(is defined such that cY=2p?r---jJ

(2)

wherep equals the number of times the nonspheres pass the same angular co-ordinate as the feed point_ For a straight scraper, p equals 1, whereas for a spiral scraper, p may be greater than 1. For the scraper used in the present study, p equals 1, whereas for the scraper used by Sugimoto et nl. [ 91, p equals 2.

HOLD-UP

OF NON-SPHERES

ON THE

DISC

Ideally, the spheres are supposed to roll and be collected at the near end, whereas the non-spheres are supposed to move with the disc in a circular path and be collected by scraping at the far end. The hold-up of spheres and non-spheres at steady state on the

disc would depend upon their feed rates as well as their speeds of movement on the disc. For a weight feed rate of F,, of nonspheres, the number feed rate is Number

feed rate =

F,

(3)

fv,dn3&m

where f, is the volume shape factor of nonspheres corresponding to a sieve aperture diameter d_ Non-spheres at a radius rf corresponding to the feed point have a tangential velocity wrf_ These particles, by assumption, do not roll or slide and hence remain at rf till they reach the scraper. The arc length travelled is Arc length travelled

by non-spheres

= flrf

(4)

The duration for which a non-spherical particle stays on the disc is given by Arc length travelled

Time of stay =

Speed of travel

Pf

WTf

P = -2nii

hold-up

The mass hold-up IV,., = (Number From

=

FIX

(6)

P

1+;71dn3ppn 2nir

(7)

is therefore

hold-up)(f,d,3p,,)

(8)

eqns. <7) and (S), one obtains (9)

It must be noted that eqn. (9) gives hold-up on the disc for an ideal scraper_ A non-ideal scraper will be unable to scrape all the particles coming to it and therefore would lead to accumulation of particles near it. A correction factor for non-ideality can be incorporated in eqn. (9) as w,=CPF, 2~

n

4

i

io-

f r” 5:

B

i; z

4-

Non

The particles fed during this time period will stay on the disc. The number of particles staying on the disc at any instant from eqns. (3) and (6) equals Number

20-

(5)

Since the speed of travel is wrf, eqns. (4) and (5) lead to Time of stay = -

z

(10)

where C would be greater than unity_ The experimental data of Sugimoto et al. 191 are shown in Fig. 1. For these data, 0

-

s&eres

Feed

eafi: P& 7iu3Lution, i$&

[sj

Fig. 1_ Comparison of experimental and predicted variation of hold-up with feed rate per revolution of non-spheres_

equals 3~. The predictions based on an ideal scraper can be seen from Fig_ 1 to be lower t.han t.he data. The best-fit line to the data indicates a non-ideality correction factor C equal to 1_2_ It must be not.ed that rhe dependence of W, on F, and n is correctly predicted by t-he model For the experimental results to be presented in this paper, the correct.ion factor C was found to be an order of magnitude higher. This implies high non-ideal@ as far as hold-up is concerned. But, as discussed below, this does not. mean ineffective separation of spheres from non-spheres-

THE PROPER

IXDEPENDENT

V_4RIABLE

It has been suggested by Sugimoto ef aZ_ 191 that the separation efficiency depends on the hold-up of nonspherical particles on the disc. But this may not be entirely correct_ Though the overall hold-up may be high, the hold-up in the separation zone would be given by eqn_ (9) only_ Because the accumulation of particles near a non-ideal scraper will not cause any hindrance to the movement of spherical particles, the efficiency depends

only on hold-up in the separation zone. Therefore, the proper independent variable is F/n; the term F&z would beneficially affect the efficiency whereas the effect of F&z would be the opposite.

MODEL

FOR

RECOVERY

dx

-

EFFICIENCY

n&nxm--1

(11)

where x = F&z

K = K,(sin 8 -p

cos 19)

where p is the coefficient given as

The approach used is very similar to the grinding model [ 10 - 121 developed recently_ It has been shown that the proper independent variable that beneficially affects the recovery efficiency of spherical particles is F&z_ Since the exact dependence is not known, it is postulated that a differential increase in 71~is proportional to a differential If m equals 1, this is increase in (FJn)“. equivalent to a first-order chemical reaction (or first-order grinding kinetics) where time is the independent variable- Here m is kept as a parameter to be estimated. Furthermore, the ‘driving force’ for the increase in qs is (1 -q,), Le. the greater the ‘scope’ for increase in n)?,the greater will be the increase m q+, which means that dq, is proportional to (1 - rls). Therefore, the basic equation is

drls = K(l

greater temperatures). If it is postulated that the recovery rate constant is proportional to the acceleration of particles down the disc, one obtains

(12)

constant K in eqn. (11) is called the ‘recovery rate constant’ for spherical particles. This is again analogous to the rate constant of chemical reactions and the breakage rate function (selection function) of comminution kinetics [lo - 123. Integrating eqn. (11) leads to

(14) of rolling friction

Jl=tane8,

(15)

where 8, is the minimum angle of inclination of the disc below which spheres will not roll. From eqns. (14) and (15), one obtains K = K,(sin 8 -sin

0,

cos

e/cos13,)

(16)

Since the angles of inclination involved, 8 and 8,, are very small, cos 8 and cos 9, tend to unity, and sin 8 and sin d, tend towards 8 and 8, respectively_ Therefore, eqn. (16) becomes

~=~,(e-e8,)

(17)

It may be pointed out that in eqn. (17) 8 and 8, can be taken to be in degrees as the constant K, will absorb the proportionality ratio (n/180) between radians and degrees. One can also express eqn_ (17) as K=K,B-K,,

(18)

which indicates that a plot of K versus 8 will yield the values of K, and K, (and thus also 0,).

The proportionality

qs = 1 -

exp[-K(FS/n)m]

(13)

which indicates that a plot of ln[-ln(1 - TQ] versus ln(FJn) would yield the values of m and K. It must be noted that in arriving at eqn. (13), the boundary condition assumed is 77, = 0 at F&z = 0 Dependence of K on 8 The recovery rate constant

Wa)

would be greater for greater angles of inclination (similar to greater reaction rate constant at

EXPERIMENTAL

Apparatus The apparatus is based on that used by

previous investigators [ 7 - 91, but with considerable modification in the details and with provision for changing the inclination and the speed of rotation of the disc_ The schematic diagram of the experimental set-up is shown in Fig_ 2_ The rotating disc was made from 6 mm thick perspex sheet and was 50 cm in diameter_ The disc was mounted on a pivot. The inclination of the disc could thus be changedA protractor was attached to the supporting rod (and was stationary). A needle was welded to the sleeve which rotated along with the pivot- Thus, the inclination of the disc 0 could be measured quite easily.

67

v I-1

II

2

1

I

\( 3

‘*

Fig. 2. Schematic diagram of experimental set-up1, hopper for material; 2, electric vibratory feeder; 3. feed hopper funnel; 4. inclined rotating disc; 5, stationary straight scraper; A, B, feed points.

The disc was coupled to a motor with a variac in series. The speed of the disc could be varied from 6 to 150 r-pm. A straight scraper was used which was kept just touching the disc. The vibratory feeder used was capable of feeding up to about 150 g/min_ It was coupled with a feed hopper funnel so that the materials could be made to fall on the required spot on the discMuteriais used The binary mixture used in the present experiments was mustard-silica gel, with mustard being the more spherical of the two_ The mean particle size of both materials was roughly l-4 mm_ The specific gravities of mustard and silica gel were 1.1 and 1.07 respectively. Since the particle sizes of the two components were the same, the separation could not have been effected by sieving. Further, since their densities were also nearly equal, methods based on settling velocities also would have been ineffective in separating the components. Opemting conditions The fractions by weight of spherical and non-spherical particles in the feed were kept

equal for all the experimental runs, so that

R = FJF,

= l/2

The total feed rate Ft was varied from 50 to 100 g/min_ The speed of rotation was varied in the range 8 to 20 rpm. The angle of inclination was varied in the range 5 to 9 degrees The values of angle cx in in Fig. 2 studied were 32.5 and 63-S degrees. That is, the scraper angle p assumed values of 327-5 and 2965 degrees. The effect of feed point position was also studied for these scraper angles For a given scraper angle, two positions, A and B in Fig_ 2, l-5 cm above and below the centre of t.he disc respectively, were used as the feed points. Procedure The angles 0 and a were set as needed. T’ne mixture was taken in the feed hopper and fed onto the disc using the vibratory feeder and the feed hopper funnel. The total feed rate and the speed of rotation of t.he disc were adjusted to required values. After steady state was attained, the product streams at the near end and far end were collected_ The spherical and non-spherical particles in the two streams were manually separated and weighed. The recovery efficiency of spherical particles was t’nen calculated for each run using eqn. (1) A total of 72 runs was performed to study the effect of the variables on the recovery efficiency of spherical particles_

RESULTS

AND

DISCUSSION

The esperimental results of the recovery efficiency of spherical particles are shoxvn in Figs_ 3 and 4_ The estimated m-values for the various sets of runs are given in Table 1 where the correlation coefficients p between In[ -ln(l - ns)] and In x are also given (refer eqn (13)). It can be concluded from the p-values given in Table 1 t.hat the model representation of the separation process is adequate_ Furthermore, the mean m-value is seen to be nearly independent of feed point position_ For each feed point position and each value of (L’,K-values were calculated using the mean m-values in Table 1 and these are summarized in Table 2. The K-values are plotted against 0

0

2

L ‘=k‘”

6

193

Fig_ 3_ Comparison of experimental and predicted variation of efficiency feed rate per revolution of spherical particles, feed at A_

%/”

Cgl

Fig. 4. Comparison of experimental and predicted variation of efficiency feed rate per revolution of spherical particles, feed at B. TABLE

of recovery of spherical particles with

of recovery of spherical particles with

1

Summary

of estimated

6

5

P

m-values for the various sets of runs

In

P

7

‘i-2

8

9

m

P

m

P

m

P

m

P

m

63-5 32.5

Case 1: Feed at A 0.97 0.39 0.89 1 0.48 1

0.23 0.45

1 0.99

0.44 0.40

0.93 0.99

0.531: 0.44

0.96 1

0.40 0.33

0.97 0.98

0.36 0.27*

63.5 32.5

Case 2: Feed at B 0.91 0.35 O-92 1 0.46 1

O-30 O-47

O-96 0.99

O-45 0.47

0.95 0.99

0.37 0.51

O-22 0.93

o-02* 0.53

0.91 0.97

o-13* O-27

*.Not considered

in obtaining the mean m-value.

89

TABLE

2

Summary of the estimated the v-lrious sets of runi

0.86 0.94 1.06 1.07 l-31 1.45

1.23 1.35 1.60 1.85 2.48 2.25

5 6 7 7-2 8 9

recovery rate constants

s” 7 T-2 8 9

l-19 1.47 1.83 2.04 2-78 2.47

for

0.94 1.00 l-14 1.24 l-68 1.68

Final uaZues of the parameters Final values of the parameters m, K,, K, , and 0, are given in Table 3_ It, can be seen from Table 3 that the parameters are significantly different for the two different ~-values. Thus, it can be concluded that the angular co-ordinate of the feed point critically affects the recovery efficiency, whereas the radial co-ordinate of the feed point does not_ Predictions of recovery efficiency from eqns- (13) and (18) using the parameter values given in Table 3, are shown for cx equal to 63.5 degrees in Fig. 6 and for (Y equal to 32.5 degrees in Fig_ 7_ It can be seen in Figs_ 6 and 7 that the comparison between t.he esperimental and predicted Jalues of pS is excellent.

2d r

=

63.5

0

Feed

at

A

.

Feed

at

B

lines in Fig_ 5_ It can be seen that the model representation of the recovery rate constant as given by eqn. (17) is valid to an excellent degree. The minimum angle of inclination 19, is simply given by the intercept on the O-axis. K and hence T], is preFor 6 less than 8,, dicted by the model to be zero. The value of 0, can be seen from Fig. 5 to be higher for an higher cx-value_

dqrees

l-

0

TABLE

3

Summary

0

1

2

3 6

Fig_ 5 disc.

Variation

L [degrees]

I 5

1 6

7

of K with angle of inclination

0

of

in Fig_ 5 to test the relationship given by eqn. (18). Best straight lines are fitted using the least-squares technique, and the resultant equations are given as dotted lines in Fig_ 5. The predicted recovery efficiency values using these dotted lines are included in Figs_ 3 and 4 for comparison. It can be seen that the comparison is excellent.

9

of final values of the parameters

a

m

h-o

h-1

em

63.5 325

0.37 0.45

0.63 0.11

0.35 0.15

1-S 0.6

1 0

o-9--

O-E -

K=Y 0

-6 6

.

7-2

p .

a

D

o.z-

63-5 dpgrpes

of feed position and 9 on K It can be seen from the K-values shown

Effect

in Fig. 5 that the effect of feed position, A or B, is insignificant. For a given c+ therefore, a mean straight line can be drawn between K and 8_ Such mean lines are shown as solid

O-6

o-7

G-6 ‘I%

o-9

=P

Fig. 6_ Comparison of experimental with mean Ko and Kl values.

and predicted

1

I?+

90 1

e

O-9 -

-9

Kay

6

l

7 7-2

A A

a

0 1

06

Fig. 7. Comparison of experimental xvith mean K, and K1 values.

and predicted .v*

CONCLUSIONS

The possibility of using an inclined rotating disc for recovering spherical particles from a mixture of spherical and nonspherical particles was explored_ A mathematical model for estimating the hold-up of non-spherical particles on the disc at steady state was derived. The predictions were shown to compare well with the published data [9] _ The correction factor accounting for non-ideality was found to be 1.2 for the spiral scraper [9] whereas it was much higher for the present straight scraper. The above analysis indicated that the proper independent variable that affects the recovery efficiency of spherical particles beneficially is F&z_ A rate-type equation was proposed for relating the recovery efficiency with the flow rate per revolution of spherical particles, F&z_ A theoretical expression was derived for the ‘rate constant’ K in the rate equation as a function of the angle of inclination of the disc- It was shown that there exists a minimum angle of inclination below which the recovery rate constant and the recovery efficiency would be zero. Experimental results were presented on the variation of recovery efficiency with feed flow rate, speed of rotation of disc, angle of inclination of disc, scraper angle, and feed

point position. The experimental data confirmed the various features of the model. From the results obtained, the effect of the variables on the recovery efficiency can be summarised thus: (1) The effects of F, and rz on q, are significant with rlSincreasing with increasing F,fn. (2) A greater 0 yields a greater vS. The rate constant K is proportional to (6 - 0,) (3) The effect of the radial co-ordinate of the feed point position on nS is insignificant, whereas t), depends critically on the angular co-ordinate of the feed point (a-value). A greater (Yyields a greater g. There exists great scope for further work on this apparatus. For example, the effect of R on q, needs to be studied. Also, the effect of scraper type, straight uersus spiral, needs to be examined. The power consumption of the entire experimental set-up was around 100 WV_ Considering the fact that with more efficient design the power consumption can be brought down, this device may be potentially useful for cleaning agricultural materials. Work is in progress to ‘quantify’ the ‘quality’ of rice and also to effect removal of unwanted materials such as broken rice grains, husk, etc., from unbroken rice grains.

ACKNOWLEDGEMENT The authors are grateful to N. Premkumar for having helped in the construction of the experimental set-up.

LIST OF SYMBOLS

non-ideality correction factor for scraper, particle diameter_ mm volume shape factor of particles, feed flow rate, g/min recovery rate constant, g-” parameter in eqn (18), g-” parameter in eqn. (18), g-” exponent in the rate equation, speed of rotation of the disc, revimin number of times the non-spherical particles pass the same angular coordinate as the feed point, radial co-ordinate, cm

91

W

radial co-ordinate of the feed point, cm mass ratio of spherical to total particles in feed, hold-up of particles on the disc, g

x

equals F,ln,

2

g

Greek symbols equals 2ps - p, degrees scraper angle, degrees ; recovery efficiency of spherical parti77, cles, 0 angle of inclination of the disc, degrees minimum 0 below which K and qS equal 0,

zero, -

P

correlation

coefficient between - qS)] and In x, dimensionless particle density, g/cm3 angular velocity of the disc, rad/min

ln[-ln(l PP

w

Subscripts n non-spherical particle S spherical particle 1 near end 2 far end

REFERENCES 1 K_ Viswanathan, Ind.

Eng_ Chem.

P. V_ R. Iyer and B. P_ Mani, Des_ and Dee_. 21 (1982)

Process

345. 2 K. Visxvanathan and B. P_ Mani. ind. Eng_ Chem. Process Des. and Dev.. 21 (1982) 7i6. 3 K. Beddow and G. Phillip. Planseer. 23 (1975) 1_ 4 R. Ehrlich and B. Weinberg, J. Sed. Petrol.. 10 (19iO) 205. 5 T_ P_ Meloy, Powder TechnoL. 16 (1947) 233. 6 T. P_ Meloy, Paper presented at the International Symposium on Recent Advances in Particulate Science and Technology. Madras. India. Dec. S - 10, Paper AlO, 1982. 7 F. G. Carpenter and V_ R_ Deitz. J_ Res. _VQ~_ Bur. Sfd. 4i (1951) 139_ S G. S_ Riley, Powder TechnoL. 2 (1968) 31% 9 M_ Sugimoto. K_ Yamamoto and J_ C_ Williams. J. Chem_ Eng_ Japan. 10 (19X) 13’7. 10 K. Viswanathan, Boject Report to Dept_ of ChemicaZ Engg.. Indian Institute Xew Delhi. 19S0, pp_ i3 - 111.

11

K. Viswanathan Proc..

12

9 (1982)

and B. P_ Mani.

of Technology.

Id_ J_ _11iner_

385.

K_ Visxvanathan and B_ P_ Mani. Paper presented at International Symposium on Recent Advances in Particulate Science and Technology, aladras, India. Dec. S - 10, Paper Bl. 1982.