Mechanism of Sieving under Tapping Based on Solids Flow

Literature Cited B-1,1.

Shinohara, K., Tanaka, T., Kagaku Kogaku, 32, 88 (1968). Shinohara. K.. Tanaka, T., Chem. €ng. Sci., 29, 1977 (1974). Shinohara, K., Tanaka, T., Ind. Eng. Chem., Process Des. Develop., 14, 8 (1975). Taneya, s., sone, 7.. Oyobutsuri, 31, 483 ( 1 9 6 2 ) .

Kawakita, K.. J. SOC.Mater. Sci. Jap., 13, 427 (1964). Kuno, H.. Proc. Fac. Eng. Keio Univ., 11, 1 (1958). Shinohara, K.. Idemitsu, Y . , Gotoh, K . , Tanaka, T.. Ind. Eng. Chem., Process Des. Develop., 7, 378 (1968).

Receiuedfor reuieul September 21,1973 Accepted July 1,1974

Janssen, H. A . , V.D.1.Z..39, 1045 (1895). Jenike, A . W., Johanson, J. R . , J. Eng. lnd., 91, 8-2, 339 (1969). Jenike. A. W., Johanson. J. R., Carson, J. W.. J. Eng. lnd.. 95, 6- . 13 119731 -.-, ~

Mechanism of Sieving under Tapping Based on Solids Flow Kunio Shinohara* and Tatsuo Tanaka

Downloaded by UNIV OF NEW SOUTH WALES on August 26, 2015 | http://pubs.acs.org Publication Date: January 1, 1975 | doi: 10.1021/i260053a002

Department of Chemical Process Engineering, Hokkaido University, Sapporo, Japan

The mechanism of sieving of particles under tapping is theoretically investigated from the viewpoint of solids flow through screen apertures. Stratification of larger particles gradually shields the apertures and thus the flow rate of smaller particles decreases. Various sieving characteristics could be expressed by use of the flow-rate equation.

There have been many works on sieving (Fowler and Lim, 1959; Jansen and Glastonbury, 1967/1968), but only a small number of papers are reported on the nature of sieving process from theoretical consideration (Miwa, 1960; Hudson et al., 1968/1969). There does not yet seem to be a definite law of sieving established. The sieving process could roughly be separated into the following, judging from the predominant stages: segregation and mixing of small particles within the particles layer, stratification of oversize particles on the screen, and blinding (Roberts and Beddow, 1968/1969) of the screen opening with particles of similar size. From the industrial viewpoint, the cut-off size of particles is definitely determined by the screen opening, and capacity and efficiency of sieving are of main concern which are governed by the dimension of the screen and the operational conditions or the type of sieving machine. Estimation of the end point of sieving time by extrapolation is also necessary in the large scale of classification. In this paper, as one of the applications of flow characteristics of particles through apertures, the mechanism of sieving is investigated by setting up a simple model of stratification on the screen and applying a flow-rate equation. Here only such solids systems of mixture are ideally dealt with as are composed of considerably larger and smaller particles than the screen aperture and the effective opening of the particles layer. Thus, no blinding of the screen and no blockage within the layer take place. Moreover, particles are subjected to tapping in a batch operation for simplicity of analysis. As a result, the sieving process is formulated to express various kinds of sieving characteristics. Sieving Process The sieving process of particles under tapping can ideally be considered as follows. The bulk space formed by small particles having passed through the screen due to one tapping is filled with a raw mixture of particles supplied from above the sieving layer on the screen. The ma8

Ind. Eng. Chem., Process Des. Develop., Vol. 14,No. 1 , 1975

terial balance between the (n - 1)th and the nth tapping is, therefore, written for the first bottom layer of particles concerning sieving, as is generally illustrated in Figure 1, by AsZ$p(l - c l , l , n - i ) + A s ~ l , i , ~ - i l b-~ ~< , ,(I ) l A s ~ 1I,, n . l E s v O p ( l - ~ s i), + A S C I , l,n-1csvlt~p(1

- ~

f

)

= As1bpD(1 - E ~ , I , ~+) AsEl,i,nTbPp(l - € , , I ) (1) where A , is the cross-sectional area of the screen plate, lb is the effective bed thickness on the screen subjected to sieving under tapping, and subscripts, 1, 1, and n - 1 denote larger particles than the screen aperture, the first layer, and the ( n - 1)th tapping, respectively. Thus, the first term on the left-hand side of eq 1 represents the weight of larger particles contained in the first layer after the ( n - 1)th tapping, the second term that of the smaller particles, the third term that of smaller particles discharged from the screen in the nth tapping, the fourth term that of raw mixture moved from above the layer to fill the bulk space of the smaller particles, and the first and the second terms on the right-hand side indicate the weight of larger and smaller particles remaining on the screen after the nth tapping, respectively. The void fraction, e l , is defined as the volumetric fraction of bulk space excluding the net volume of larger particles, cs the void fraction of smaller particles passing through the screen, e f the void fraction of raw mixture, and csv is the fraction of openings of the screen plate. Rewriting eq 1 with respect to c i , i , n

Hence, the void fraction of larger particles in the mth layer on the screen after tapping n times, c l , m , n is written in general as

05 raw mixture

m-th layer - (rn-1Xh !dyer e

w=: 0 1 -,screen

plate

005

I tapping

Figure 1 . Model of stratification of larger particles on a screen.


where the subscript av denotes the average value, and k , is a proportional constant. Rewriting eq 4 with respect to Zm -1

Since the screen openings for particles mixture in the mth layer are the interspaces of the (m - 1)th layer of larger particles, the weights of smaller and larger particles contained in the mixture are written, respectively, by

J

0 2

I

where t I , r n - l , n - 1 is the minimum voidage in the (rn 1)th layer, and it gives the effective screen opening for smaller particles in the mth layer to pass through, as discussed later. tl,,,0 is the voidage of larger particles involved in the raw mixture, and t g , m is the voidage of smaller particles in the mth layer. Let Z , _ 1 be the size of the effective screen opening of larger particles in the (m - 1)th layer, whose cut-off size is Z m - 2 . Provided that the number of openings of the (rn - 1)th layer of larger particles is proportional to the number of larger particles

001

0

10

5

n Figure 2. Variation of void fraction of larger particles with the number of tapping.

where u s p , , . n is the discharge velocity of smaller particles in the mth layer during the nth tapping; tl and t z are discharge times during one cycle of tapping which are determined by the operational condition of tapping. Thus, the term of integration indicates the flow length or the bed thickness relating sieving, l b . Putting eq 3, 8, and 9 into eq 10 gives

i

Jz2]:f(ndZ

1

-

(1 - cf)el,m-*,nm"l Jz:

f(2)dZ)"'l

(11) Here, the maximum number of tapping, nm,is required to complete the mth layer after the (rn - 1)th layer has just formed. n, can be related to the minimum voidage of larger particles in the mth layer and be determined by the operational condition of tapping. Thus, from eq 3 5

El,m,nm

{l

-

(1 - c f ) e ~ , m ~ , , , m ~ l ~ z m a x f ( Zx) d 2 ) n m m.1

(Ef

+

(1

-

cflJZm-'f(%)dZ] Zmin

or 111

11,

Hence, from eq 6 and 7

cs,m

=

(9)

c f + (1 - cf)Jzm-'f(Z)dZ 'min

cl.m.n,

cy

+

(1 - c I ) ~ z m -f(Z)dZ i Zmin

5 111

f

'

i

1

-

(1 -

cf)cl,m-1,nm-,

I (12)

lzmax f(Z)dZ]

Zp-1

There are various ways to express sieving characteristics. Among them, the following, derived on the basis of eq 11, are representative and useful in practice. The rate of sieving or the passing rate of smaller particles is obtained from differentiation of eq 11 with respect to the number of tapping, n.

where f(Z) is the size-distribution function of the raw material by weight. Figure 2 shows straight lines of t l . 1 . n on a semilog paper by using eq 3. Sieving Characteristics The weight of smaller particles discharged from the rnth layer just after tapping n times, F,,,,n is expressed by fi.s,m,n

-

t, AsEl,m,n-Icl,m.l,n,-~~p(l

- c5,rn)J

-lsp,m,n

df

tA

(10) Ind. Eng. Chem., Process Des. Develop., Vol. 14, No. 1 , 1975

9

n

10 5

h

0

v

Ld

1

0.5


0-1

0

10

5

/

n

Figure 3. Variation of discharge rate of smaller particles with the number of tapping.

The cumulative undersize by fractional weight, Du, is written by

I 1

5

O B !

2

6

0.6

1

13

7

0 4

1

14

8

02

’

,

~

1

5

10

n

Figure 4. Variation of cumulative oversize and undersize with the number of tapping.

n=n +n +*“+nm-l+t

I ?

where Wsv is the total weight of raw mixture subjected to sieving on the screen. If u ~ ~is nearly , ~ the , ~same in the mth layer for varying number of tapping, as discussed later, from eq 11and 14 C t 7

Then the cumulative oversize, Ro, is given as 1 - D,.

Discussion Because of discrepancies between the actual experimental condition of flow test and the ideal analytical one, as mentioned in our previous paper (Shinohara and Tanaka, 1975), it is not attempted here to correlate our experimental data with derived equations directly. However, most of their trends are well known empirically and easily found in the literature cited below. Figures 3 and 4 depict the above-mentioned sieving characteristics in the binary system of particles during stratification of the first layer. They are plotted against the number of tapping, n, corresponding to the sieving time. Table I presents the numerical values used for calculation. It is found in these figures that the weight of undersize particles passing through the screen gradually decreases with an increasing number of tapping. Thus, the rate of 10

Ind. Eng. Chem., Process Des. Develop., Vol. 14, No. 1, 1975

Table I. Material Properties of Binary Mixture Used for Calculation = 350 g p , = 2.52 g/cc E~ = 0.35 Zb = 0 . 3 c m As = 176.6 c m ? E,, = 0.45 2,= 0 . 1 c m Zav,t= 0 . 1 5 cm Z s , , = 0 . 0 3 c m sieving and the cumulative oversize decrease with increasing sieving time. In addition, it could be explained by eq 15 that the so-called exponential law of sieving (Miwa and Ito, 1957; Hudson, et al., 1968/1969) would hold within the range of a relatively short period. In order to verify the two-stage theory (Heywood, 1938; Whitby, 1958) in the long run of sieving, further experimental investigation of the maximum number of tapping, n,, in which the closest packing of larger particles in each layer on the screen is attained, is required. However, it could easily be inferred that both Fs,,,, and n , get smaller with increasing m and thus characteristic curves of sieving such as D, given by eq 14 might be approximated by more than two straight lines on semilog or log-log paper, as suggested in Figure 4. According to the proposed model of sieving, the stratification process of larger particles on the screen controls the flow rate of particles. Then it is only necessary to know which layer has the minimum opening size. In the present analysis the effective opening size is generally considered to become small with an increasing number of complete layers piled up on the screen. The newest layer always predominates the flow of smaller particles. However, when the screen opening is smaller than the effective openings formed in the layers, for instance, due to violent vibration, the screen plate regulates the solids flow, as is the case with the early stage in the formation of each layer. The operational condition of tapping determines the value


of ~ , , , - l , ~ , - Within ~. a short period of sieving time this latter case would be common in the industrial operation. ~ not independent of the The discharge velocity, u , ~ . , , is diameters of discharge opening and the interspace among large particles which actually form the wall of the container (Shinohara and Tanaka, submitted for publication). However, as the effective opening of a completed layer is usually much smaller than the bulk space which permits the smaller particles to flow out, u ~ ~could , ~ roughly be regarded as constant during tapping n times in the rnth layer. In such a solids system involving a relatively small amount of larger particles, the above-mentioned discussions would lead to more simple treatment of the derived equations such as eq 15. No. 1 in Figure 4 could better be approximated by a straight line than no. 2, 3, and 4. Needless to say, Ib or u S p are given as functions of sizes of particle and screen aperture relating to an effective opening area, as in solids flow from a vessel (Shinohara, et al., 1968). It is also quite clear that larger W,, and smaller F, give smaller D , in eq 14 or 15, which suggests appropriate operational conditions. The main purpose of this paper is to analyze the sieving process bv considerations similar to the analysis of solids flow through the aperture of the vessel. In the sieving process, the number of openings, and possibly the opening size, change with the cumulative number of tapping, the particle-size distribution, the aperture size of the screen, and other operational conditions and material properties. The stratification of particles is the only point differing from the homogeneous solids flow from a storage vessel. In this sense, the sieving process could be regarded as one sort of segregation process (Shinohara, et a1 , 1970, 1972). Nomenclature

.4, = area of a screen plate, cm U , = cumulative undersize by fractional weight, dirnensionless F , = mass rate of flow of smaller particles, g/sec f(Z) = size-distribution function of raw mixture of particles on weight basis, cm-1

k , = proportional constant, dimensionless lb = effective thickness of particles layer which substantially relates to sieving under tapping, cm m = the number of layers of larger particles on the screen plate, dimensionless n = the number of tapping, dimensionless n, = maximum number of tapping to complete the rnth layer of larger particles, dimensionless Ro = cumulative oversize by fractional weight, dimen~ sionless . ~ t = discharge time during one cycle of tapping. sec usp = discharge velocity of smaller particles, cm/sec Wsv = total weight of raw mixture subjected to sieving on the screen, g Zav,, = average size of particle in the mth layer larger thanZ,-1, cm 2, = size of effective screen opening formed by larger particles in the rnth layer, cm Greek Letters t f = void fraction of raw mixture of particles, dimensionless C I = fraction of void space excluding the volume of larger particles, dimensionless t S = void fraction of smaller particles, dimensionless tSv = void fraction of a screen plate, dimensionless pp = particle density, g/cm3 Literature Cited Fowler, R.T., Lim, S.C.. Chem. Eng. Sci.. 10, 163 (1959). Heywood, H., lnst. Mech. Eng. (London). 257 (1938). Hudson, R.B., Jansen, M.L., Linkson, P.B., Powder Technol.. 2, 229 (1968/1969). Jansen, M.L., Glastonbury, J.R., Powder Technol., 1, 334 (1967/1968). Miwa, S., Kagaku Kogaku, 24, 150 (1960). Miwa, S.,Itoh, H., Kagaku Kogaku. 21. 374 (1957). Roberts, T.A., Beddow, J.K., Powder Techno/., 2, 121 (1968/1969) Shinohara, K., Idemitsu, Y . , Gotoh. K., Tanaka. T.. lnd. Eng. Chem., Process Des. Develop., 7, 378 (1968). Shinohara, K., Shoji, K., Tanaka. T., lnd. Eng. Chem.. Process Des. Develop., 9, 174 (1970). Shinohara, K.. Shoji. K., Tanaka, T., Ind. Eng. Chem.. Process Des. Develop., l l , 369 (1972). Shinohara. K., Tanaka, T.. Ind. Eng. Chem. Process Des. Develop.. 14, 1 (1975). Whitby, K.T.,A.S.T.M., STP, 234 (1958)

Received for reuieu, October 5, 1973 Accepted July 1,1974

Ind. Eng. Chem., Process Des. Develop., Vol. 14, No. 1, 1975

11

Mechanism of Sieving under Tapping Based on Solids Flow

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