A model for continuous grinding in a laboratory hammer mill

Powder Technology. @ Elsevier Sequoia

22 (1979)

S.A.,

199 - 204

Lausanne -Printed

199 in the Netherlands

A Model for Continuous Grinding in a Laboratory Hammer Mill L. G. AUSTIN Department

of Materials

Science,

The Pennsylvania

State

University,

University

Park.

Pa. 16802

(U.S.A.)

V. K. JINDAL Asib

Institute

of Technology.

Bangkok

I202

(Thailand)

C. GOTSIS Mineral

Processing

(Received

Section,

May 4,197s;

The Pennsylvania

State

in revised form September

University,

University

Park,

Pa. I6802

(U.S.A.)

29,197s)

SUMMARY

The equations of fully mixed, steady continuous grinding with rapid ideal classification through a discharge screen have been modified to allow for non-rapid removal of material less than the screen size. Values of specific rates of breakage, daughter fragment distributions and the variation of specific rates with mill hold-up determined in prior batch tests on a hammer mill were used in the equations, to predict hold-up uers~(sfeed rate and product size distribution uer.ws feed rate. The predictions were in reasonable agreement with the experimental values from continuous grinding. There is a maximum production rate which can be cbtained, and energy utilization is inefficient at low production rates.

INTRODUCI’ION In a previous paper Cl] we described the results of kinetic studies of the batch grinding of maize in a small laboratory hammer mill. In this paper we will examine the steady-state continuous grinding of this material in the same hammer mill with the 74n.-diam. cylindrical steel case replaced with a similar case of screen containing l&-in. perforations.

EXPERIMENTAL

by a screen having perforations $-in_ diam. T’ne ground product, discharged by gravity and induced air flow, was collected in a cloth bag fastened directly under the mill which reduced the loss of fines to minimum and let the air filter out. After steady-state operation of the mill was obtained, the product collected over a period of time gave the feed rate into the mill. To check the size distribution of the material retained in the grinding chamber, the vertical slide gate of the feed hopper was pushed down rapidly, which prevented the entry of feed into the grinding chamber, and the motor power was turned off. The power required in grinding was indicated by a recording wattmeter. The material retained in the mill was analyzed by sieve size analysis using d/2 U. S. standard sieves. Sieve analysis was also performed on a representative 100-g sample of mill product. The mill was run with a single sharp-edged hammer at 3000 r.p.m. to correspond to the conditions at which most kinetic test work was performed [l] _

THEORY

The equation of fully mixed, steady-state continuous grinding has been derived by Austin and Gardner 121, and extended to rapid ideal size classification through a screen exit by Klimpel and Austin [3] _ The equation is i-l

A steady feed rate of whole maize (oven dried to 5.6% moisture) into the grinding chamber was maintained by a vibratory feeder through the feed hopper while the mill operated continuously. The whole area around the periphery of the grinding chamber was covered

pi=

k--SirrLir

+ T C bisiSimi

i = 1, 2, _ . . , n

i=1 i>l

(1)

where pi is the weight fraction of materiai of size interval i in the product, fi that in the feed; Si is the specific rate of breakage, time-‘; mi

200

is the weight fraction of size interval i in the material retained in the mill; bi.i is the primary breakage parameter, i-e_ the fraction of material of size j which appears in size i after breakage; T is the mean residence time defined by mass in the mill divided by feed rate, W/F. Although the mill used in the tests reported here was certainly fully mixed, it was found that the assumption [ 31 that all fine material rapidly left the mill was not capable of explaining the results. Alternatively, then, it was assumed that the rate of material being thrown against the screen was a function of the mass loading W and that the probability of particles of a size i passing through the screen was proportional to the relative amount of size i, that is, m,. Then piF = r(l

-si)miW

-Si)rmj

(2)

where 1 - si is the fraction of size i which passes through the screen per presentation so that si is the fraction returned to the mill, and r is the fraction of mass W presented to the screen per unit time. It is expected that Si will be zero for sizes much smaller than the screen opening, and one for sizes larger than the screen opening. However, if the effective size of the screen opening to a near-size particle striking the screen is lower than its true size due to angle of contact, then r (1 - si) may be of comparable magnitude to Si and thus be important in the rate balance for these particles_ Combining eqns. (1) and (2) to eliminate pi gives i-7 i=l

yi =

(Simii)

=

;rni

= 1.0,

or

This is the more general form of eqn. (4) of ref. [3]. The values of mi are then determined from

or pi =r(l

classification, r(1 - Si) equals zero for i < i, and equals infinity for i > ic, where i, is the size interval whose upper size corresponds to the screen opening; the equations then reduce to the equations given before [3]. Obviously, pi = 0 for all sizes greater than the screen size. Further, from eqn. (3), since

i>l

1 + r(l

mi = yi jSir

(6)

It should be noted that both r(1 - Si) and Si are functions of the steady mill loading W achieved at a given feed rate, which is not known a priori_ When a given steady feed rate F is established into the mill, the value of W will vary to bring the process into balance, if a stable set of conditions exists for the F considered. To summarize, if for a known W the values of Si and r(1 - Si) are known, then y values follow from eqn. (3), F follows from eqn. (5) and pi follows from eqn. (4). Unfortunately, if F is given it is not usually possible to calculate W because of the complex nature of the relation between S and W, so it is necessary to calculate F values from W values and interpolate to a desired F.

(3)

-Si)/Si

RESULTS

where y values are determined sequentially starting at i = 1, using ~YIOWJI values of Si, b,.i and r(1 - si). Note that the 7 values do not depend on mi. Substituting into eqn. (1) to eliminate mi gives the form pi=fi--_i+

i-1 x i=1 i>l

bi_jrj

i=1,2

,_._,

n

(4

Equations (3) and (4) are more general forms of those given previously [3]. For rapid ideal

Table 1 gives data obtained from three continuous grinding tests. The data were used with eqn. (2) to calculate r(1 - Si) values, which are shown in Fig. 1 and Table 2. As expected, the finer sizes pass through the screen most rapidly, but the fraction passing per minute decreases as the feed rate increases. It can be assumed that si is_virtually zero for sizes smaller in size than the upper size of the sixth interval, so that r(1 - ss) is r and, hence,

201 TABLE

1

Experimental size distributions (5.93% d-b.) at 3000 r.p.m. Feed rate, F Weight in the mill, Residence time . IV

U.S. mesh number

Size

&in.

X 3

8000

6 8 12 16 20 30 40 50 70 100

6730 4760 3360 2380 1680 1190 850 600 425 300 212 150

3X4 4x 6x 8x 12X 16 x 20 x 30 x 40 x 50 x 70 x -Cl00

TABLE

(wn)

(cumulative

percent fess than si?e) from continuous

= = =

510 65 0.127

glmin min -1 g

763 102 0.139

glmin min -1 g

1076 0.267 290

gfmin g min _-l

\/2 Size interval i

Product % < size

Mill charge % < size

Produc.% < siz

Mill charge 5%< size

Product % < size

Mill charge % < size

1 2 3 4 5 6 7 8 9 10 11 12 13 (sink)

_

100.00

lOa_ 00

87.4 63.0 19.7 4.3 2.5 1.9

89.9 66.1 20.6 3.7 1.5 0.5

100.00 89.0 48.8 30.9 21.4 15.0 10.9 7.4 5.4 4.0 2.2

100.00 89.0 47.5 29.7 20.2 14.0 10.1 7.2 5.4 2.8 2.2

100.00 88.3 63.2 16.5 4.5 3.2 2.6

100.00 86.5 49.4 34.6 25.7 18.7 13.9 9.9 6.9 4.9 3.6

2

Experimental

size distributions

in the mill and i-(1 -

J2 size Size (pm)

1 2 3 4 5 6

mi

r(l

0.126 0.244 0.430 0.154 0.018 0.006 O-02

0 0 2 20 79 125 (125)

-Si)

r(1 - s,) is also r, etc. Cross-plotting Wr (1 - Si) versL(sW for a given i indicates that the data point at 65 g, i = 6, seems to he high; it is difficult to get accurate values at this size because the value of ma is small_ Equation (3) was arranged to i-l fi +

IZ bi,iYi

L

- r(l

1076

763

510

i

6730 4760 3360 2380 1680 1190

si) values

Feed rate, glmin

interval

8000 x 6730 x 4760 x 3360 x 2380 x 1680 x ==1190

si=

grinding tests with maize

-Si)

?lliT

and values of Si calculated, using the bi,i values determined previously Cl], see Table 3. Note that S, = fi/m, r, yl = Slml r; S2 =

mi

O-102 0.238 0.455 OS69 0.022 0.01 0.05

r(1 -s& 0 0 1.7 17 57

(fi + b2,1yl)/m2~,

a

mi

0.117 0.251 0.467 0.120 0.013 0.006 0.026

~2 = S2m,

7, and

-

si)

0 0 1.2 12 43

so on,

so

that 7 values are calculated sequentially. The values for S are shown plotted in Fig. 2: in eqn_ (‘7), Si is a small difference between large quantities as i increases and, hence, the cakulation is subject to wide errors beyond the first three or four intervals. However, it appears that the slopes of the log S versus log x plots in Fig. 2 are the same as those obtained previously [l] from direct batch tests, SO that Si can be estimated from extrapolation of the lines orS- =axa3. &ano&er re’port [4] we have calculated the mean values of S for whole maize based on

202 TABLE

3

Values of primary daughter Breakage Size intervali 1

2 3 4 5 6 7 8 9 10 11 12

Bi.1

fragment

distribution bi.1

Bi.2

0.53 0.21 0.10 0.05 0.034 0_022 0.016 0.012 0.009 0.006 0.021

1 1 0.48

distributions

determined

from batch grinding tests [l]

fractions bi.2

Bi.3

bi.3

Bi.4

bi.4

Bi.5

bi.5

Bi.6

bi.6

Bi.7

bi.7

0.52 0.21 0.09 0.06 0.033 0.023 0.017 0.012 0.01 9.028

1 1 0.49 0.28 0.18 OS.3 0.093 0.069 0.059 0.039

0.51 0.21 0.10 0.05 0.034 0.024 0.017 0.013 0.039

1 1 0.49 0.29 OS9 0.13 0.096 0.072 0.054

0.51 0.20 0.10 O-06 0.036 0.024 0.018 0.054

1 I 0.50 O-29 0.19 0.14 0.10 0.075

0.50 0.21 0.10 0.06 0.04 0.025 0.075

1 1 O-51 0.30 0.20 0.14 0.10

0.49 0.21 0.10 0.06 0.04 0.10

1 1 0.51 0.31 0.21 0.15

0.49 0.20 o-10 0.06 0.15

1 -I

G-47 0.27 0.17 0.12 O-086 0.064 0.048 0.036 0.027 0.021

0.2’7

0.18 OS2 0.90 0.067 0.050 0.038 0.028

---ASSUMED FOR INTERPOLATION

SIZE

INTERVAL

UPPER

INTERVAL

SIZE.pm

Fig_ 1. Rate of mass transfer of material of size interval i passed through the %h_ screen casing; fraction 16 of size i mass passed per minute_

Fig. 2. Values of S determined from steady-state grinding tests. plotted against upper size of interval (Jz intervals). A W = 290 g, n W = 100 g, c W = 65 g.

batch experimental data using whole maize, as a function of W, see Fig. 3. The values are not directly comparable to the S values in Fig. 2 because whole maize includes two size intervals, but we can assume that values of S for a &en size interval vary with W in the same ratio- Table 4 gives the results as ratios based on W=3OOgasunity. Table 5 gives the results of measurements of hold-up as a function of feed rate, at steady state, and the corresponding wattage of the mill motor.

Fig_ 3. Variation of breakage rate of the sum of the top two sizes as a function of hold-up in the mill, estimated from batch tests on whole maize.

203 TABLE

4

TABLE

Factor for variation of Si aa a function W: S,(W) = k(W)Si(300)

of mili load

6

Values of (I-

ai) estimated

by interpolation

r(1 -.Q)

W(g)

k(W)

50 100 150 200 250 300 400

1.58 1.52 1.35 1.24 1.11 1.0 0.71

TABLE

Wiz) =

i

50

3 4 5 6 7 etc.

5

Experimental

2.5 25 89 109 (109)

100 1.8 18 63

150 1.5 15 54

3oDco

results of steadystate

440 510 680 740 910 1080

48 65 90 102 190 290

7 (mm)

Power (kw)

Kilowatt min per gram x 10s

0.109 0.127 0.131 0.139 0.209 0.267

0.615 0.64 0.65 0.66 0.75 0.87

1.4 1.25 0.96 0.90 0.83 0.81

20,ooo

1.4 14 49

1.3 13 46

300 1.2 12 43

1

-

i i

z I J

I

11I 1 w.g 4clo

2

COMPUTATIONS

It is now possible to compute the expected feed rates for a range of W values, using eqn. (3) to determine Ti values and, hence, eqn. (5) to determine F. In order to do this it is necessary to A~OW for the variation of Si and r (1 - si) with W. The val-ue of Si was taken as Si(W) = k( W)Si (300)

250

St2e lntervol

grinding L

F (glmin)

200

Fig. 4. Plots of experimental vahes of Wr( 1 - Si) us. Wr the solid lines were used for interpolation_

RESULTS

(3)

DISCUSSION

Figure 5 shows fairly close agreement bethe experimental and predicted size distributions- The computations show that higher feed rate and mass loading give slightly coarser product size distribution and a finer mill charge than for lower feed rates. However, the differences are less than the scatter in the experimental data for the three flow rates shown. The rates of breakage *k the mill must exactly match the feed rate into the mill for all sizes greater than the exit opening. This means that the hold-up in the mill increases as feed rate is increased, to give higher rates of breakage. However, beyond a certain hold-up, the rates of breakage start to decrease (see Fig. 3) due to overloading of the mill. Consequently there is a m aximum stable production rate

tween

where the k(W) values are given in Table 4. The set of values Si(300 j was calculated from (~,/8000)~.~ Z&(300), where Xi is the upper size of interval i; Si for W = 300 g is about 18 _ -‘. Table 6 gives the values for r(1 - s-) %kinecl by interpolation from the Wr(1 -ii) versus W plots (shown in Fig. 4) for a given i; the origin can be used as a point in this plot. It was found that the sixth and higher terms in eqn. (5) made a negligible difference to the residence time. Equation (6) was used to calculate mi and eqn. (4) gives the product size distribution. The results are shown in Figs. 5 and 6.

AND

204

Fig_ 7. Variation of specific mm/g, with mill load, g.

750 g/min would be inefficient utilization_

L-

-_ li--

Fig. 5. Experimental and computed size distributions for three feed rates. c 510 glmin, . 763 glmin, A 1076 glmin, -computed.

‘500Y _C E 2”

1000

I; if ,”

0

I

!

0

2

oo

z 5000

grinding energy,

0

100

200

HOLD

300

UP

400

w.g

Fig. 6. Comparison of computed and experimental steady feed (= product) rates US. mill hoId-up IV. computed, o experiment.

which can be obtained_ Figure 6 predicts that the maximum flow through the mill is about 1250 g/min at a Ioading of 370 g. Loadings to the right of this maximum represent an unstable condition, because any small increase in feed rate would lead to a higher W value, but the production rate decreases so that W would increase, and so on. The maximum feed rate tested experimentally was 1080 g/min, giving a W of 290 g_ Higher feed rates brought the mill too near to the maximum and to unstable operation. The agreement between the experimental and predicted F uersus W results is reasonable, bearing in mind the experimental difficulty of getting an accurate value of W. Figure 7 shows t.hat the specific grinding energy does not change much from a hold-up of 100 to 300 g. Production rates below about

kW

in energy

CONCLUSION

The equations of fully mixed, steady continuous grinding with rapid ideal classification through a discharge screen have been modified to allow for non-rapid removal of material less than the screen size. Values of specific rates of breakage Si, daughter fragment distributions 6ii, plus the variation of Si with mill hold-up W had been determined in prior batch tests. These values were used in the equations to predict hold-up uersus feed rate and product size distribution uersus feed rate. The predictions were in reasonable agreement with the experimental values from continuous grinding. ACKNOWLEDGEMENTS We thank Professor N. N. Mohsenin, Department of AgriculturaI Engineering, for permission to publish these data REFERENCES V. K. Jindal and L. G. Austin, Kinetics of hammer milling of maize, Powder Technol.. 14 (1976) 35. L. G. Austin and R. P. Gardner, Prediction of sizeweight distribution from seIection and breakage data, Proc. 1st Eur. Sy.rnp. on Size Reduction, Verlag Chemie, DusseIdorf, 1962, pp_ 232 - 248. R. R. Klimpel and L. G. Austin, Mathematical modelling and optimization of an indus-rial rotarycutter milling facility, Proc. 3rd Symp. on Size Reduction, DECHEMA Monogr., 69 (252 - 1326) (1971) 449 - 473_ L_ G. Austin, T. J_ Trimarchi and N. P. Weymont, An analysis of some cases of non-fist-order breakage rates, Powder Technol., 17 (1977) 109.