Powder Technology. 38 (1984) 255 - 266 255

A Simulation Model for An Air-Swept Ball Mill Grinding Coal

L. G. AUSTIN, P_ T_ LUCKIE

Deportment of Mineral Engineering. The Pennsylvania State University. University Parh. PA (U_S._.l.)

K. SHOJI

h-ure Research Laboratory (formeriy h’ennedy Van Saun Corporation), Hiroshima (Japan)

R. S. C. ROGERS and K_ BRAILLE

Kennedy Van Saun Corporation. Danuille. PA (USA.)

(Received February 3.1983;in revised form May 6.1983)

SUMMARY

The conventional model for grinding in tumbling ball milk was modified to allow for air-sweeping, for the case where all the material is carried out of the mill in the air stream. It was shown that this type of mill can be treated as a single fully mixed reactor. The values of the internal classification numbers given by the air-sweeping were determined for a 1 m by l-5 m pilot-scale mill grinding coal. The resuits showed that only 1 to 270 of the mih charge are exposed to the sweeping action per mill revolution_ S and B values determined in a laboratory mill were

scaled-up for use in the continuous miiZ modei and the simulations gave product size distri- butions and mill capacities which agreed with the pilot-scale mill data within the experi- mental accuracy of the pilot-scale data.

INTRODUCTION

The most widely used equipment for grinding coal to the sizes prescribed for pulverized coal firing of utihty boilers are the ball-race and roll-bowl types of mill. However, there are some advantages to the use of tumbling ball mills for coal grinding and several manufacturers supply such systems. It has been known for many years that coals do not grind well if fines are allowed to buiid up in the grinding zones. This has been recently confirmed [l, 21 by studies of the specific rates of breakage of coals in ball-race and tumbling ball mills. For this reason,

industrial dry grinding of coal is always carried out with a high rate of air-sweeping to remove fines from the mill_ The inlet air is heated to give drying in the mill so that the product is fine dry coal- Surface moisture prevents the fines from being swept out and greatly reduces grinding efficiency-

The simulation of tumbling ball mills wit.h mathematical models has reached a fairly advanced stage 13, 4, 51 but the concepts have not been applied to coal grinding be- cause the models have not included the effect of air-sweeping- In this paper, we will modify the models to allow for air-sweeping, assum- ing that the removal of fines allows the frac- ture to proceed in a first-order manner [2]. The characteristic of an air-swept ball mill is that the entire flow of powdered coal from the mill is in the air stream- The model incorporates closed circuit via an estemal classifier and t.he simulation is validat.ed by tests on a pilot-scale mill_

THEORETICAL TREATMENT

Consider the mill receiving a feed of known size distribution: as customary [3], the size intervals of the size distribution are chosen in \/2 geometric sequence corresponding to the standard sieve sequence, with the masimum size interval denoted by 1, the second size interval by 2, and so on_ The nth interval is the ‘sink’ interval containing all material less than the smallest size measurement, 2’70 mesh (53 pm) for the tests described here_ The feed is denoted by the set of numbers fi, where f, is the weight fraction in size interval 1, etc_ It

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256

will be assumed that breakage obeys the firs& order hypothesis

rate of breakage of size i = SiWiR’ (1)

where W is the mass of coal acted on by the mill, Wi is the fraction of the mass of size i and Si is the specific rate of breakage of size i, time-‘_ It will also be assumed that primary breakage of size j produces a suite of daughter fragments described by the set, of numbers b,, where b, is the fraction of material which is of smaller size i after primary breakage of Iarger size i (i > i); note that

5 b, = 1 i=n

’ The size distributions produced from a tumbling ball mill also depend on the resi- dence time distribution (RTD) of material in the mill. A particularly convenient method of allowing for the RTD is to consider the mill as a series of fully mixed stages. For the purpose of this paper, equal sizes of the stages are assumed, but the equations developed can be readily extended to stages of unequal size_ Figure 1 illustrates the system and nomen-

clature.

Fig_ 1. Illustration of air-swept mill as a series of stages_

Austin and Gardner [6] have given the mass-rate balance of a single fully mixed mill operating at steady state as

rate of removal of size i material in product = rate of addition of size i material in feed

+ rate of production of size i material by breakage of all larger sizes - rate of breakage of size i material

or, using eqn. (l),

i-l

PiF = fiF + W C b,SjWi - SiWi W (2) i=1 i>l

where F is the feed rate and pi the product size distribution. If there is no classification

action at the mill exit, pi = Wi and eqn. (2) becomes

i-l

pi = + T C bi$ijp~ i=1 i>l

l=Gi<n (W

where -r is the mean residence time defined by 7 = W/F. To modify this to allow for air- sweeping, the left-hand side of the mass-rate balance becomes

rate of removal of size i in the air stream + rate of removal of size i in the powder

flow

Let the rate of removal of size i by air- sweeping be ri, then the total rate of removal of powder by air-sweeping is R = Zyri and eqn. (2) becomes

i-l

ri +Pi(F- R) = fiF + W C bi$jWi -SiwiW i=l i>l

(3)

It is assumed that she powder acted on by the grinding action is the flowing powder and that the suspended powder is not subject to breakage action; then, Wi =pi_ It is also assumed that a fraction 77 of the powder maSS W is exposed to the air stream by tumbling, per mill revolution. For a mill rotation speed o, the amount of size i material in W exposed per unit time is pivOW_ It will be assumed that the exposed material of size i has a probability Ci of being retained in the hold-up and hence the fraction swept up will be 1 -Ci or

ri = pi??0 W(1 - Ci) (4)

Inserting into eqn. (3) and rearranging using 7 = W/F,

i-l

fi + TC b&jpi j=1

pi = (1 - E) + Si7 + qO(l - Ci)7 l<i<n

(5)

where

E = R/F = n)w-&l - cj)pj (54 1

E is thus the fraction of the feed rate removed in the air stream; if the RTD of the powder flow in the mill corresponded to one fully- mixed reactor, E = 1_

The treatment is extended to m fully mixed stages in series as follows (see Fig_ l)_ A mass-rate balance on the second stage is identical except that the product piF(1 - E) is the feed into the second stage. Since it is known that the hold-up level along this type of mill does not change very much, it can be assumed that the hold-up is equal in each stage, at W/m_ For the relatively short mills used in coal grinding, the balls are fully mixed throughout, SO the same values of Si and b, can be used in each section_ It will also be assumed that particles in the air stream are rapidly swept out and do not fall back into the bed of powder. Then, for the first stage

i-l

fi * 71 C bi+jPi. 1 i=1

Pi-l= t1 -El) + SiTI + qO(l -Ci)Tl

T/m - - Is2 - (1 _Ei)(l --El) --- (1 -Eg_,)

= 7/m

l<k<m @a)

il = 7/m where -r is the overall mean residence time defined by IV/F and

Ek = Tk,&l - ‘+)&.k lGk<m (6b) 1

There is no powder flow out of the mth reactor, so E, = l_ For a given set of 1 - ci vdue~, and given 77, W, w and fi, the value of F must be such as to lead to E, = l_ The size distribution of the air-swept product is

or

pi’ = wyCi) 2 pi_k l=si<n (7)

k=1

l<i=Gn If the values of n and Sj are assumed constant along the mill, a mass balance on size i over the total mill gives

E 1= r1T-J i: (1 --jlPi 1 1

i=1

gri_, = fiF + ‘i ‘g’ bijSjgpi_& W m

- si - x:Pi_ L_

1 ?7lj=, 1 m 1

i>l

For the second stage or

i-l

Pi.1 +~2 C bijSjPj.2

i=l

piy2 = (1 - fZ*) + Sira + 7jW(l -Ci)T*

ldidn

i-l

pi = TJwFV(I - ci jwi = fiF + tV 3 bijSjU?j - j=1

i>l

- Si WWi (8)

where

r2 = (W/m)/(F -R,) = 711(1 - EI)

E2 = T27& (1 - CilPi.2 1

In general [ 73, then,

m wi=xPi.k m

1 I

is the mean size distribution in the mill, ri is the total flow of size i in the air stream and IV is the total mass of powder in the mill_ Thus, since r = TV/F,

i-l

Pi. k - 1 -t Tk c b$$P~.k i=1

i-l

fi -i- i C bi$TiWj

Pivk = (1 - ek) + Sirk -I- n~(l -Ci)Ti;

l<ibn

i=l Wi =

SiT ~ ‘I;lO(l -Cj)T l=Gi<n (9)

lCk=Gm (6) and

258

Pi ’ = mW(l - Ci)Wi ldi<n (IO)

The conclusion is thus reached that for these assumptions the mill behaves as if it were a single fully mixed mill containing a mean overall size distribution. Since the flow rate of material presented to the internal air classification is VOW, and the rate of fine material leaving is QWWCi(l -Ci)UJi.

the equivalent internal circulating load is l/Xi(l -Ci)Wi-

DISCUSSION OF MODEL Laboratory ball mill test

In this type of system, a large particle in the feed entering the mill must stay in the mill as long as necessary to break it down in a

z series of steps to sizes small enough to be swept out. A large particle which moves to the discharge end of the mill will strike the end wall and back-mis into the mill contents. On the other hand, a small feed particle can soon be swept out. The concept of residence time distribution starts to lose meaning under these circumstances_ The assumption that particles will not fall back into the bed once they are entiained is clearly an over-simpli- fication which cannot apply to a long mill [ 73 _ However, the experimental measurement of size distributions along the axis of a pilot- scale mill (see later) showed that there was only a minor variation along the mill. In addition, the treatment of the closed cir- cuit system (Appendix 1) shows that the effective overall classifier action is given by (1 - Ci)(l -.Q) where 1 -si is the fraction returned by the external classifier. Examina- tion of the classifier data show that terms in Si dominate over those in Ciy SO that the final result of closed-circuit operation is not highly sensitive to the values of ci_

For these reasons, the simple model of a fully mixed mill was used, with the values of Ci being mean effective values over the mill as a whole_ This gives a particularly simple system for experimental analysis. The com-

putational procedure is given in Appendix 1. For ~IIOWII VdUeS of Si, bif, Ci and T)O, there is a unique value of 7. In practice, the air flow rate is usually adjusted to match the coal feed rate so that a correct filling level of powder is obtained in the mill, using the sound of the mill as a guide. Since IV is thus

fixed, the coal feed rate follows from F = llr,- Because the values of Ci vary with air flow

rates, it is necessary in the simulation to: (a) specify the air-flow rate; (b) calculate Xi, 7, pi’; (c) calculate F for a correct value of W; (d) repeat for different values of air flow rate until the desired value of F is obtained by trial-anderror search or interpolation.

DETERMINATION OF BREAKAGE PARAMETERS

4ND SCALE-UP RELATIONS

In addition to the air-sweeping action, ball mills grinding coal have another characteristic which is somewhat different from the usual ball milling of mineral rocks. The coal feed to the air-swept ball is typically a crushed feed with a top size of about 1 in, as shown in Fig. 2. The presence of appreciable quantities of such large sizes means that care must be taken to describe the breakage of these large sizes with reasonable accuracy_ The descrip- tion of the breakage of small sizes which are nipped and crushed in a normal breakage by the balls has been well studied 15, S], but the kinetics of breakage of larger sizes have not been so fully investigated [ 9] _

The normal values of S and B were mea- sured on a coal g-round dry in a cylindrical steel laboratory ball mill of 195 mm diameter and 5.22 litre volume- The ball charge used was 4810 g of 1 in dia. steel balls, correspond-

w N

P/CIRCUIT PRODUCT

r.0 L

1290 ACFH 970 IbS~h.

c =,‘I8

Fig. 2. Experimental size distributions for coal ground in 0.98 rn I.D. pilot-scale air-swept ball mill fitted with a twincone classifier_

ing to a fractional filling of the mill volume by the bed of balls of J = 0.2, using a formal porosity of 0.4 for the ball bed (290 lb/ft3). The dry mass of coal used was 122 g; assum- ing an apparent specific gravity of coal of 1.3 and a formal porosity of O-4, the formal

bulk density of coal is 770 kg/m3, giving a fractional filling of the mill volume of f, = 0.03. The experimental methods have been described in detail elsewhere 110 J_

For particles which are small with respect to the ball size, it has been fGund that the values of Si vary with particle size -Xi, ball diameter d and mill diameter D according to simple power functions

Si(d) a XiQDNIjdNo .W)

where Ni is close to O-5, and (Y is characteristic of the material, 0.83 for this coal. The value of iv0 is still in question due to conflicting results of various tests but was taken [ll] as l-O_

In the usual way [ 123, the primary daugh- ter fragment distribution calculated using the BII method was expressed in cumulative form

Bii = 2 b,i i>j k=n

and the results fitted to

259

P

(12)

It was found that the cumulative daughter fragment distribution Bii was normalized [12] for the coal (that is, 4:_ = a constant for different breaking size j), and ‘1~~ = O-36(5), y = O-90, p = 2.8. The values of S and B are shown in Figs. 3 and 4_

The experimental values of the normal values of Si can be converted to other powder

s oo~L_-__ ___._-_L;1__ _. _ .~_ ~. _.

002 or :j

RELATIVE SlZE xi /xi

Fig. 4. Cumulative daughter fragment distribution of Belle Ayre coal (d2 inierval).

/O 0=195mm 4

.I =0.20 P I

f,= 0.03 d =25mm i

3

2

A i I I I1ll‘l 1 rrr,.,l I t I.,.

0.1 I IO loo

PARTICLE SIEVE SIZE, mm

Fig. 3. S values used in the simulations: Belle Ayre coal (d2 size interval, values plotted at upper size of intervai).

260

and ball filling conditions using the empirical equation developed by Shoji et OZ. [S]:

1 si =

1 -i 6.652.3 exp(-1.2U) (13)

where U is interstitial filling of balls defined by tl = f,/O.4J.

For larger sizes, the power function of eqn. 11 is modified [5] by an empirical function Q(w) which makes the values of Si pass through a maximum and decrease at large sizes :

D NIXiQ S,(d) a N, Q(x)

d (14)

where

1 Q(x)=

1 + (xi/P)* A> 0 (15)

The value of ~1 is the size at which Q(x) = 0.5 and A is an index of how steeply the values fall with increasing particle size. The experi- mentally determined values of M and A for the test coal were ~1 = 3.7 mm, A = 2.75 for 1 in balls in the laboratory mill. It was assumed that u varies with ball diameter and mill diameter according to

p a d*Dh’z (16)

where Nz is about 0.2. This espression allows for the effect of ball diameter and mili diameter on the position of the maximum in S values, via P, assuming A to be constant_

The overall values of S for the mixture of ball sizes in the mill were assumed to be the weighted sums of the S values for each bail size,

Z?i= 5 mhSi(d,) (17) k=1

where ml is the weight fraction of balls of d, = 1 in diameter, m2 that of d2 = 1 l/4 in, m3 that of d, = 1 l/2 in, etc., (m, = 0.25, m2 = O-375, m mill

, see Fig 3; = 0.375 for the pilot-scale

- .

Pilot mill tests: Batch grinding Iu order to test the scale-up relations of

eqn. (ll), a pilot-scale mill of 0.98 I.D. by 1.53 m long was run as a batch mill with the same filling conditions as for continuous operation (see next section). A feed of coal predominantly in the size intervals 4 X 6 and

6 X 8 was prepared and 75 kg (dry basis, moisture content 15%) was loaded into the mill. The mill was run for 112 min, stopped. a sample of about 500 g taken by random scoop sampling, and the mill run for addi- tional grinding times, with sampling for each time, to give grinding periods of l/Z, 1 l/2, 3,4, 5 and 16 min. The size distributions at each time were determined by screening of a measured amount of dried coal, followed by redrying and weighing: weight loss was assumed to be material of minus 270 mesh. The results are shown in Fig. 5.

The first-order plot for the top size interval is shown in Fig. 6. The mean value of gI calculated from the slope of the curve was 4.0 min- ‘_ Figure 5 shows the results of simulation of this batch grind using the laboratory S parameters scaled to this mill diameter and ball mixture and assuming the same normalized B values, using exponents N, = 0.5, N2 = 0.2. It was clear that the break- age in the pilot mill gave the same basic shape of the size distributions as those in the 195 mm dia. laboratory mill and the simula- tion was satisfactory.

For grind times beyond 5 min, the slowing down of breakage rate observed with small mills was also seen in the pilot mill results. In addition, a similar batch test in the pilot mill on the same coal without prior air-drying to 15% moisture gave a significantly different result, especially for breakage of the larger sizes. The coal used in these tests was a sub- bituminous coal (Belle _4yre South) with an as-received moisture content of 30%. It is clear that for this type of coal it is necessary that the laboratory batch test be performed at moisture conditions corresponding to those in the continuous test. The mechar.isms by which moisture content affects the breakage of larger sizes remains to be investigated-

Pilot mill tests: Continuous grinding The test mill was a cylindrical air-swept

mill of 0.98 m mean ID_ by 1.53 m long, with inlet and discharge trunnions of approxi- mately 0.3 m dia. The ball load corresponded to 24% mill filling (J = O-24), with a ball mix of 25% 1 in diameter, 37.5% of 1 l/4 in, and 37-5% of 1 l/2 in. The mill was run in closed circuit at steady state, at 80% of critical speed, with the feed rate adjusted by an experienced operator to give the correct

MINUTE

3

4

3

-I’! 1-L_I-I_L_I-lI- -I

-_LLl-d 100 1000 loooo

SIEVE SIZE, pm

Fig. 5. Rcsulls of batch grinding 75 kg Belle Ayrc coal (J = 0.25, fc = 0.083, moisture contrn~ of cod = I 5%) in pilot-scale mill.

Fig. 6. First-order plot for the 4 x 6 mesh top size in the batch pilot mill.

sound level. The mass flow rate of recycle from the classifier was measured by diverting the flow to a collecting bin for a short time- interval, and the make-up feed rate was measured by a calibrated, variable-speed belt feeder_ The exit air temperature was close to 150 OF, ensuring coal of low free-moisture content. The test system is described in detail in another place [13 J_ Samples were taken for feed, recycle and product size analysis The mill power, feed and air flow rate were then

suddenly and simultaneously stopped, and the mill came to rest in less than one-third of a revolution_ The contents were tipped out. weighed and mixed, and sampies taken for size analysis. Four tests at various coal flow rates gave mill hold-ups of 157, 160, 157 and 175 lb of coal (dry basis), that is, a value of about 160 lb. This is a fractional formal mill filling of f, = 0.081, which is a fractional filling of the ball interstices of U = 0.83.

Knowing the steady mass flow rate to the classifier, the mass flow rate of recycle, and the recycle and product size distributions, the size distribution of the mill discharge, pi’, is readiIy calculated. r is calculated from the mill feed rate and hold-up (both dry basis)_ Thus the values of qw(l -cci) in eqn. (10) (pi’/TUri = 7)0(1 - Ci)) GUI be calculated_ It was found that these values became constant for sizes smaller than about 60 pm. Since the

air velocity in the open space in the mill is

about 1.2 m/s, it can be assumed that ci is zero for these small sizes. The constant value is thus VW and since w is known (34.6 rpm), the value of r/ for the test conditions follows_

The results are shown in Fig_ 7_ The effect of air flow rate on 71 is given by

7) = 6.2 X 104Vo-s

that is

(18)

q = 1.2 X lo-‘(u, m/s)“-s (I8a)

262

gi [

1

gr : 0

2% &

i

-! ILlI.

0001 i 0 1 ,,:.,i 1 :

IO 50 loo 400

APPARENT SPACE VELOCITY. V miri’

Fig. 7_ Value of 7 for different fiow rates of air through a 0.95 m 1-D. by l-5 m long airswept ball mill.

where V is the apparent space velocity (vol- ume changes per minute, based on an empty mill fraction of 1 -J) and u is velocity through the mill based on a cross-section of 1 -J_ It is concluded that the fraction of the mill powder which could be swept up in the air stream per mill revolution is only 1 to 2% of the hold-up and that it increases with air velocity_ The actual amount swept out of the mill depends, of course, on the values of ci and Wi- Since ~wW is the mass iate of material exposed to the internal air-sweeping classification action, and F is the mill mass flow rate, the air-sweeping action gives an apparent internal circulating load of

1 -L C’ = QoWJF = ~WT (19)

where C’ is the apparent internal circulation ratio- For a value of i- of 5 min and q = 1% per revolution, this gives C” = 0.75.

Since q is now known, the value of ci can be calculated from the values of qo(l - ci)- The smoothed set of cc values for an air flow

rate of 1290 actual cubic feet per minute (ACFM) at the exit temperature of 150 “F are shown in Fig. 8. Particles larger than about 2000 pm fall back into the bed and cannot leave the mill- The individual values of Ci for different air flow rates were too scattered to give a definite relation to air flow rate- However, it. appeared that the curves were of the same shape but shifted along the size scale to larger sizes as flow rate increased. The change of ci with air velocity was esti- mated by a simple settling velocity treatment

UPPER SIEVE SIZE. ym

22 20 IS 16 14 12 10 8 6 4

SIZE INTERVAL NUMBER

Fig_ 8. Smoothed values of internal c?assification numbers cc (for d2 intervals. plotted at upper size)_

(see Appendix 2), giving the results shown as broken lines in Fig. 8. _4lthough this treat- ment is not very sophisticated, it gives results which are consistent with the esperimental data and which aBow interpolation and extrapolation in a logical manner. In addition, the external classification values dominate over those in ci, so that the final result of closed circuit operation is not highly sensitive to the values of ci_

COMPARISON OF SIMULATIONS AND PILOT

MILL DATA

The model discussed above was pro- grammed and simulations performed for coal and air flow rates corresponding to contin- uous tests run in the pilot mill_ The samples of make-up feed and recycle in the contin- uous tests enabled the size distribution of the mill feed to be calculated_ These were used as feed to the mill simulation (see Table l), giving the typical result shown in Fig. 9. The simulation correctly predicts the long plateau in the size distribution of the powder in the mill, which is due to the shape of the feed size distribution and due to the larger feed sizes being to the right of the maximum of S values, as shown in Fig_ 3. The mill products shown are passed through a classifier to prepare the standard pulverized coal grind, see Fig. 2, and the mill feed shown is, of course, made up of fresh feed and recycle. Table 2 shows the comparison between simulated 0erssu.s experimental results for two air-coal

TABLE 1

Data used in simulation: air flow rate = 1290 ACFM (0.61 ma/s. u = 1.04 m/s).

Interval Upper Feed size cf Si number i size of distribution (mm-‘)

interval (% < size)

Wm)

1 38100 100 1-o 0.77 2 26900 99.15 1.0 1.71 3 19000 9519 1-o 2.98 4 12500 92-56 1-c; 4.18 5 9510 85.77 1-o 4.86 6 6730 83.86 1.0 4.93 7 4760 SO-31 1.0 4.26 8 3360 75.37 1.0 3.36 9 2350 72.54 1-o 2.68

10 1680 69.72 0.99 2.03 11 1190 67-00 0.96 l-52 12 841 64.86 0.92 1.14 13 595 63-19 0.85 0.86 14 420 61.04 0.76 0.64 15 297 5756 O-66 O-48 16 210 52.32 O-54 0.36 17 149 44.22 0.42 0.27 18 105 32.75 O-30 O-20 19 74 22.32 0.17 0.15 20 53 16.01 0.0 0.11 21 38 11.00 0.0 O-09 22 26 7.80 o-0 0

rates. The simulations are in reasonable agreement with the experimental results.

To obtain an indication of the level of experimental variability of the pilot-scaIe data, a series of closedcircuit tests on the same coal was performed at varying flow rates_ The size distributions around the twin- cone classifier were used to ca.Iculate si values fitted by the form [14]

1 &=a+(l--a)

1 + (X</dsc)-” I (191

where the classification action is defined by the three parameters of a, dsO and A: a is the

Fig. 9. Comparison of experimental results and computed results; mill hold-up 160 ib (73 kg), air flow rate 1290 ACFM (6.61 m3/s) at 150 ‘F (66 ‘C).

by-pass fraction {O =G a G LO), dgn is for the classifier curve, and h is related to Sharpness Index, SI, by In SI = -2_1972/h. The result- ing values of si were used in t-he closed-circuit simulation model to give product size distri- butions and capacities for comparison with the experimental results, as shown in Figs. 10 and 11. It was concluded that the simula- tion model predicted the correct mill capacity and fineness of product wit.hin the scatter of the experimental results.

In t.his particular pilot mill, the drive and bearing friction losses were 5.9 kW and the net mill power was 7.5 kK;, giving a total motor power of 16.2 kW, as the motor efficiency was 82% The fan power at 1500 ACF’M was 2-6 kW and at 2000 ACF41 was 6 kW_ It is clear that fan power is significant compared with net mill power. The vaiue of J = O-24 is lower than in conventional non- swept ball milling because the trunnions must be large enough to pass air without excessive pressure drop and fan power.

TABLE 2

Comparison of simulated uerszzs experiment for continuous pilotscale mill tests

Air flow (ACFhI)

Mill feed rate (d-b.) Ublh )

Exp. Sim.

AIill product

(% < 200 mesh)

Exp_ Sim.

(‘% c 250 mesh)

Esp. Sim_

1 890 1680 1685 64-S 63.1 50.7 48.7 2 1290 2380 2385 55-4 52.9 41.6 39.6

264

i

i

- SIMULATION

E

f5 I

CIRCUIT PRODUCT RATE, Ibs/hr Fig. 10. Comparison of experimental and predicted fineness of grinding for Belie Ayre cod ground closedcircuit in the pilots&e mill.

1400 Iscm laoo 2000

AIR FLOW RATE. ACFM

Fig. 11. Circuit capacity versus air flow rate for Belle Ayre coal ground in the pilot-scale mill.

Appendix -42 gives a method of scaling the internal classification to larger mills_ At the moment, however, it is not possible to scale- up the total model to larger mills because the variation of 71 with mill diameter is not kIlOWIl.

CONCLUSIONS

If it is assumed that the grinding conditions are constant along the mill, an air-swept mill in which all of the mill product leaves in the air stream can be treated as a single fully mixed mill. The results from a 0.98 m diam- eter by 1.53 m long pilot-scale mill showed that the percentage of mill hold-up exposed to the air-sweeping action per revolution of the mill was about 1 to 2%, at a ball fiiing of 24%, an optimum powder filling of 8%, and 80% of critical speed. The optimum powder filling represents a value of U = 0.83. This can be compared with the optimum value of 0.84

given by eqn. (13), based on results from a laboratory mill of 195 mm I.D. Scale-up of experimental breakage rates from a 195 mm dia. laboratory mill gave values which, when used in the air-swept simulation model, correctly predicted the size distributions and mill capacity for the pilot-scale mill, within the experimental scatter of the pilot-scale mill results

ACKNOWLEDGEMENTS

This work was supported by research contract No_ EX-76-C-01-2475 from the U-S. Department of Energy, contract monitor Mr. T. K. Lau_ A number of people have been involved in the test work on which the results are based, but we wish to thank especially Mr. Ken Gardner, Chief Engineer, and Mr- James Wilver, Manager of Testing Services, the Kennedy Van Saun Corporation, Danville, Pa_, U.S.A.

REFERENCES

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L. G. Austin and P_ Bagga, Powder TechnoL. 28 (1981) 83. L. G. Austin, J_ Shah. J. Wang, E. Gallagher and P. T_ Luskie, Powder TechnoL. 29 (1981) 263. L. G. Austin, Powder TechnoL. 5 (1971/72) 1. J. A_ Herbst and D_ W_ Fuerstenau, Inf. J_ of Min. Proc. 7 (1980) 1. L. G. Austin, R R Khmpel, P. T_ Luckie and R. S. C. Rogers, Symp. on MiZZing, AIME, Hawaii, Sept_ 1982. pp_ 301 - 324_ L. G. Austin and R. P. Gardner, in H. Rumpf and D. Behrens (Eds.), Proc 1st European Symposium in Size Reduction, Verlag Chemie, Weinheim. 1962, pp_ 232 - 248_ L. G. Austin, N. P_ Weymont and 0. Knobloch, in K_ Schonert, W. Gregor and F. Hofmann (Eds_), Preprints. Particle Technology 1980, Dechema, Vol_ B (1980) 640 - 655_ K_ Shoji, L. G_ Austin, F_ Smaila, K_ Brame and P_ T. Luckie, Powder TechnoL. 31 (1982) 121. T. Miles, I. Shah and L. G. Austin, Breakage Rates and Breakage Dish-butions of Large Particles in Ball Mills. in preparation. L. G. Austin and V_ K. Bhatia, Powder TechnoL. 5 (1972) 261. F. Smaila, K_ Brame, L. G. Austin and P_ T. Luckie, The Effect of Ball Diameter on Breakage Rates and Breakage Distributions in a Laboratory Mill, in preparation_ L. G. Austin and P_ T. Luckie, Powder TechnoL. 5 (1972) 215

. 265

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P. T. Luckie and L. G. Austin, Coal Grinding Technology: A Monuol for Process Engineers. U.S. Department of Energy, FE-2475-25, avail- able National Technical Information Service, Springfield, Va_. (1979) p_ 92. L. G. Austin and R. R. Klimpel, Powder TechnoL. 29 (1981) 277.

APPENDIX 1 I COMPUTATIONAL PROCEDURE

As discussed previously, the internal sweeping action can be treated as an apparent classification action- However, the programs we have used previously for mill circuits [Al] choose a value of mean residence time r = F’/W and calculate C’ and, hence, F/W- For the air-swept system, the values of ci and q depend on the air flow, so it is more con- venient to set the air flow, which then defines r_ The computation is performed as follows. Equation (9) can be put as

i-1 fi + z biirj

i= 1 i>l

Yi =

1+ Bw(l --Ci)

n> iz= 1 (Al-l)

Si

where Yi = WiSir_ The Y ~23~2s are calculated sequentially starting at i = I, for known values of b,, Si, Ci and VW_ (For example, the value of Ci will probably be 1 for size 1, SO y1 = fI_) Then, since CiWi = 1,

n-1

Note that Y,, = 0 because S, = 0, but w,r is not zero_ It is given by

where C, is the mean value of c for the sink interval (fortunately, z = 0 in most cases)_ Thus, there is a unique value of T_ The values of wi follow from Wi = Yi/SiT and eqn. (10) gives the product size distribution.

When the mill circuit is closed via an external classifier, it can be assumed that the external classification action can be described by a set of classifier numbers Si- The make-up feed size distribution is gi and the actual feed

size distribution to the mill is related to the make-up and recycle by fiF = giQ + tiT where F = Q + T, Q being the make-up feed rate and T the recycle rate_ Because the circulation ratio is defined as C = T/Q, then 1 + C = F/Q_ Since si is defined as the fraction of Fpi’ which is recycled, that is, tiT = FSipi’, then fi(l + C) =gi + (1 + C)sipi’- Substituting into eqn. (Al-l) to eliminate fi gives

i--l

yi* =

gi + jTl bijYi*

lGi<n 1 + l?O(l -si)(l - cil

si (X1-3)

where yi* = (1 + C)WiSir = WiSi7*- AS before, the y* values are computed sequentially and

n-l

n--l G -r- C %Y,*

i- 1 7 * = c (Yj*/S,) -I-

i=l r?w(l -s,)(l -Z) (Xl_‘Z)

Then wi = yi*/Sir*, from eqn. (10) pi* =

T*vc;(~ - Ci)Wi, (where pi* = p;‘(l + C)) and

l-?-C= gpi*= 7*wJ 2 Wj(1 - Cj) (X1-5) i=1 i=1

Knowing C, the values of pi’ follow. The size distribution of the circuit product is given by

4i = (1 + C)pi’(l -St)

The circuit capacity Q is W/r*-

(_41_6)

It is useful to note that the closed circuit behaves like the open-circuit model but with the value of 1 - ci replaced with an effective value of (1 - ci)(l -si). The computation program for closed circuit can thus be used for open circuit by setting Si = 0 and gi = fi-

APPENDIX 2: VARIATION OF INTERNAL CLAS-

SIFIC-TION WITH AIR R_TE

The concept used is that a particle of a given size (and density) ent.rained in the air stream will follow a falling path determined by its starting position, settling velocity and the time for the air to reach the end of the mill. If the path does not reach the bed before the particle is swept to the end of the mill, it will leave in the air stream- When the velocity of air flow through the mill is increased, an

266 .

entrained particle wilI follow the same path and have the same probability of leaving if its settling velocity is higher than the original particle, so that its time for settling is de- creased in the exact ratio that the time for the air to reach the end of the mill is de- creased. Thus, as air velocity is increased the size of particle which leaves the mill by a particular path also increases.

In quantitative form, the treatment is as follows. For spheres of 40 to 6000 pm, the variation of settling velocity us in air with particle radius r given by Fuchs [A23 was converted to the air viscosity and density at 150 ?F, corresponding to typical air tem- peratures in the mill_ The result was fitted by the empirical expression

log u, = 4-352 - 0.796f”-*” (A2_1)

where r is in cm, ZJ, in cm/s. At a given air- sweeping veiocity u in the mill, the experi- mental value of ci represents the mean effect of all possible paths for powder of radius ri with a flow time for air in the mill of t. At anjr other flow rate L(‘, t.he time is L-’ = h/u’. In order for particles of radius r’ to go through the same set of paths to give the same ci, the settling velocity must increase proportionally to l/C. Thus, u,‘t’ = u,t and v,‘Iv, = u’ju. Using eqn. (A2.1),

J- = l/[ ($)-O-24 + 1.256 log(u’/u)]‘-” (A2.2)

The equation is arranged in this form so that setting a value for r’ enables the calculation of r for a given u’/u. The value of c for this value 0% r can be interpolated from an esperi- mental Ci versus ri CUNe.

In the case given here, u a 0.61 m3/s for the ci curve of Fig. 8 (at 1290 ACFM). This was fitted by the empirical curve

(A2.3)

where x was in ym, that is, _r = 2r X 104. Thus, for LX’ in the desired t/2 sequence, eqn. (A2.2) allows the calculation of x, and eqn. (A2.3) then gives ci-

With respect to mill diameter D, the same set of geometrically similar settling paths would be espected for the same L/D at the same gas flow velocity. The ratio of LL’ to u is calculated from volume flow rates V by

V-‘/(D’)7

l’r’i’ = VW* (A2_4)

If the c relation of eqn. (A2.3) is to be used, V = 0.61 m3/s and D = 0.98 m, for L/D = 1.5, that is, u = 1.04 m/s.

REFERENCES

Al

A2

P. T. Luckie and L. G. Austin, Min. Sci. and Engng.. 4 (1972) 24. N. A. Fuchs, The Mechanics of Aerosols, Mac- millan, New York, 1961.