Effects of mill feed size on product fineness and energy consumption in coarse grinding

Minerals Engineering, Vol. 4, No. 5/6, pp. 599-609, 1991

0892-6875/91 $3.00 + 0.00 © 1991 Pergamon Press pie

Printed in Great Britain

EFFECTS OF MILL FEED SIZE ON PRODUCT FINENESS AND ENERGY CONSUMPTION IN COARSE GRINDING YIGEN ZENG and E. FORSSBERG Division of Mineral Processing, Lule~t University of Technology, S-951 87 Lule~t, Sweden (Received 30 August 1990; revision accepted 14 November 1990)

ABSTRACT The effects of some grinding parameters on the product fineness and the energy consumption under dry batchwise coarse grinding conditions are studied on rod and ball mills. A parabolic-sine equation is suitable to describe the particle size distribution. The constants in the equation are related to the grinding parameters, which permits a mathematical analysis. A relation between energy consumption and grinding parameters is also established, and the grinding effects in the rod and ball mills are compared. Keywords Rod grinding;, ball grinding; energy consumption; parabolic-sine equation INTRODUCTION For a conventional comminution flowsheet, consisting of crushing and grinding stages, coarse grinding generally means that the top feed size to the mill is about 5-20 ram. This size range is usually obtained by fine crushing, since crushing usually consumes less energy than grinding to produce these size ranges. Coarse grinding plays an important role in comminution as an intermediate stage between fine crushing and fine grinding, because it is a key step both for liberation of minerals and energy consumption. If satisfactory liberation is achieved in this process and the liberated particles are immediately separated from the coarse grinding product, the risk of overgrinding valuable minerals in a subsequent fine grinding stage would then be minimized. The top size of raw material and its effect on energy consumption in rod milling has been investigated on a commercial scale mill [ 1]. Considerable energy saving in the comminution process was achieved by simply reducing the feed size to the mill. In an ordinary comminution plant, coarse grinding is usually performed by either a ball or a rod mill. In this paper, the effects of mill feed size and batch grinding time on product fineness, as well as the energy consumption, are studied in both a rod and a ball mill under dry coarse grinding conditions. The grinding results are then compared. MATERIAL, EQUIPMENT AND TEST METHOD This study was performed on a fine crushed quartzite, which consisted of about 90% quartz, 5.4% galena and 1.6% sphalerite, with barite, fluorite etc. making up the remainder. After crushing to -40 mm and screening of excessive fines (-2 mm), the ore was crushed to four different top sizes from 16 to 3.35 mm. The density of the ore was 2.80 t/m 3. Its bulk density was about 1.7-1.8 t/m 3 in the given size ranges. The moisture content was < 1%. u.E. 4-s/6-E

599

600

YIGEN ZENG and E. FORSSBERG

A 600x900 mm rotary mill supported by rubber rollers was used to perform the grinding tests. It was driven, via a gearbox and belt, by an electrical motor with 4 kW rated power. Twenty "wave-shape" lifters of about 8 mm in height were evenly spaced around the inside of the mill shell. To determine the net energy consumption, the torque input to the mill was measured at the outgoing axle of the gearbox and transmitted to a processor. The torque was continuously recorded by a plotter during each test. As the speed (rpm) was measured by a tachometer, the energy consumption could be evaluated for a set of period of time. The mill was operated at 68-73% of critical speed. The largest rod diameter was calculated by the equation given by Rowland [2] and then adjusted to the rod sizes available, i.e., diameters of 75, 65, 55 and 55 mm for top mill feed size 16, 10, 7 and 3.35 mm respectively. At 25% of the charge volume, three gradations of rod charges were composed with a formula derived by Bond [3]. To completely fill the void space between rods with quartzite, about 26-28 kg of the mass was used for one batch rod grinding test. The solid density of the balls was 7.75 t/m 3 and the bulk density was about 4.646 t/m 3 [2]. At 25% of the charge volume, the ball charge was 278 kg and consisted of 26 (103 mm diameter) and 73 (82 mm diameter) balls, corresponding to 40 and 60% in weight respectively. To fill the void space between the balls, the mass needed for one batch ball grinding test was about 41-44 kg. The mill was just charged with grinding media, then the ore sample was evenly spread over the surface of grinding media. The mill was then run for a set period of time. When the test was completed, the mill was opened and the grinding media were gently removed and the grinding product was carefully collected and sampled by means of rotary and riffle dividers, taking 0.5-2.0 kg of the mass (max. 3% of total mass) for particle size analysis. If necessary, the remainder was then weighed and returned to the mill for further grinding, the missing weight of the material being adjusted for evaluation of the energy consumption. Particle size analysis was performed by wet-and-dry sieving, removing -0.075 mm fine particles by wet sieving on a Retsch vibrator, then drying the oversize material and sieving for 20 minutes on a laboratory "Ro-tap" screen shaker. RESULTS AND DISCUSSIONS Particle Size Distribution

The idea of finding an equation which is generally valid for the description of particle size distribution of comminution products has attracted the interest of many researchers. The well-known 2-parameter equations proposed by Gates-Gaudin-Schuhmann and RosinRammler are widely used. Other methods such as those recommended by Svensson [4] and Peleg [5], which are based on mathematical derivation, have been used in describing particle size distribution. These methods, however, are not satisfactory in the case of crushing and coarse grinding. By choosing certain graphic scales, many workers tried to obtain straight lines by plotting cumulative percent against particle size. Straight line plots present simple pictures of general relationships, while choosing a suitable formula to present size data gives the advantages of being highly accurate, reliable and easy to use, especially in process control. A parabolicsine equation (the P-S equation), easily displayed on log-log graph paper, is given by log Y = a + b*log X + c*(log X) 2 + d'sin(log X)

(1)

where Y is the cumulative percent undersize (%) at an arbitrary particle size X (mm), and a, b, c and d are corresponding constants. Using multiple regression to fit the sizing data on the P-S equation for the rod and the ball mill tests, the results are shown in Tables 1 and 2. The squared correlation coefficients (R 2 ) between the P-S equation fitting and the experimental particle size data are >99.78%. All the terms in the P-S equation are significant

Effectsof mill feed size in coarse grinding

6Ol

at 95% confidence level. Figures 1 and 2 compare the results from the particle size distributions predicted by the P-S equation (curves) and the data from screen analysis (points) for the rod and the ball mills respectively. It can be seen that the equation describes the particle size distributions very well, since the differences between predictions and observations are very small in the given size ranges. TABLE 1 Fit results by the P-S equation for the rod mill tlme (mln)

1 2 4 6 1 2 4 6 1 2 4 6 1 2 4 * log(Y)

Constants in the P-S equation * a b c d mill feed size 1.7089 0.3287 1.9287 - 0 . 9 4 5 0 1.9898 - 3 . 1 6 5 8 1.9410 - 4 . 8 4 6 4 m111 feed s i z e 1.7839 0.1261 1.9432 -1.1127 1.9897 -3.4511 1.9297 - 5 . 0 8 6 2 m i l l feed s i z e 1.8499 - 0 . 3 1 5 0 1.9700 - 1 . 6 2 8 2 1.9846 -4.0081 1.9037 -5.7855 mill feed size 1.9591 -1.2134 1.9962 - 2 . 4 5 8 6 1.9923 - 2 . 9 8 0 9 = a + b*log(X)

= 11.37 mm -0.1126 0.1471 -0.4599 1.2787 -1.0639 3.0618 -1.5484 4.2994 = 7.70 mm -0.1763 0.3287 -0.4983 1.4296 -1.1373 3.3084 -1.6341 4.4586 = 5.11 mm -0.3069 0.7764 -0.6479 1.8916 -1.3058 3.7542 -1.8501 4.9949 = 1.89 mm -0.5447 1.4778 -0.8808 2.5106 -1.0834 2.7600

+ c*(log

X) 2 + d ' s i n ( l o g

Rz %

99.93 99.94 99.89 99.81 99.93 99.93 99.88 99.78 99.94 99.93 99.87 99.78 99.94 99.93 99.82 X)

Relations between Grinding Parameters and Product Fineness It is known that grinding time plays a major role in particle size distribution of ground product, both in rod and ball mill grinding. The effect of grinding time dominates strongly during the first minute, but decreases rapidly over longer grinding periods [6]. The coarser the feed size to the mill, the larger the reduction ratio in the first minute, especially in the rod mill. Thus, after 8 min ball grinding or 2-4 minutes rod grinding, the effect of prolonging the batch grinding period on the product fineness is rather small. Stepwise regression of the constants of the P-S equation in Table 1 for rod grinding (indicated by subscript r) upon the grinding time (t-min) and 80% passing size of the mill feed (l=g0ram), at 95% confidence level yields eqs (2-5): a r = 1.9895 - 0.0467"Fso + 6.296x10-3 *t*Fso

(2)

b r -- 1.1433 - 0.2348"t - 2.3948/F80

(3)

c r = 0.0618 - 0.0633"t - 0.6105/Fa0

(4)

d r = -0.5990 + 0.1896"t + 2.0977/F80

(5)

602

YIGEN ZENG and E. FORSSBERG

TABLE 2 Fit results by the P - S equation for the ball mill time (mln)

Constants In the P-S equation * a b c d ml]] 1.4986 1.6249 1.8408 1.9798 ml]] 1.6092 1.7485 1.8930 1.9891 ml]] 1.7586 1.8401 1.9392 1.9981 ml]] 1.8868 1.9266 1.9762 2.0000

I 2 4 8 1 2 4 8

I 2 4 8 1 2 4 8 * log(Y)

100 N L

10

feed s i z e = 11.1 mm 0.7351 -0.0375 -0.3157 0.4720 -0.1126 -0.0353 0.2173 -0.2033 0.1407 -1.0597 -0.5223 1.2250 feed s i z e = 7.7 mm 0.6067 -0.0586 -0.1763 0.2969 -0.1559 0.1297 -0.1038 -0.2742 0.4580 -1.1069 -0.5372 1.2515 feed s l z e = 4.3 mm 0.3624 -0.1528 0.0616 0.1259 -0.2182 0.2697 -0.5682 -0.3936 0.8636 -1.2138 -0.5666 1.3263 feed s i z e = 2.06 mm -0.3549 -0.3146 0.7211 -0.5108 -0.3744 0.8368 -0.8122 -0.4479 1.0438 -1.8680 -0.7272 1.8630

= a + b'log(X)

~

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99.91 99.93 99.81 99.92 99.93 99.95 99.90 99.91 99.95 99.93 99.92 99.88 99.92 99.93 99.88 99.91

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Fig.l Particle size distributions of rod mill products o m feed; • o + v m 1,2,4 and 6 minutes in observation; corresponding equation fitting Equations (2-5) account for 90.5, 97.9, 98.1 and 96.7% of the variances for constants a, b, c and d in Table 1 respectively. Similarly, stepwise regression of the constants in Table 2 for ball grinding (indicated by subscript b) upon the relevant grinding parameters under the same confidence level yields:

603

Effects o f mill feed size in coarse g r i n d i n g

a b = 1.8717 + 0.1211"t - 0.02153"t 2 - 0.02695"F80 + 0.006475*t'F80

(6)

b b = -0.09988 - 1.1023"t + 0.1125"F80

(7)

c b = -0.2432 - 0.2847"t + 0.0341'F80

(8)

d b = 0.6181 + 0.7925"t - 0.09035"F80

(9)

Equations (6-9) account for 87.0, 95.7, 96.7 and 94.1% of the variance of the constants a, b, c and d in Table 2 respectively. Closer inspection of the relationships in eqs (2-9) reveals the fact that constants b, c and d with both mills are linearly related to the grinding time, while the e f f e c t of mill feed size varies. In rod grinding, the constants are inversely proportional to the mill feed size. In ball grinding however, they are directly proportional to this parameter. The effect on the constant a is complicated since the grinding products will all be <1 mm after a certain grinding time. By combination of eq.(l) with eqs (2-5) or eqs (6-9), the sizing data could be estimated in batchwise rod or ball grinding under given conditions.

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Fig.2 Particle size distributions of ball mill products o m feed; • o + v m 1,2,4 and 8 minutes in observation; - - corresponding equation fitting Comparison of Product Fineness between Rod and Ball Mill To make a comparison between the rod and the ball mills, a ratio of cumulative percent below a certain size in the ground products from rod and ball grinding (using the ratio for short) is defined as R = Yr/Yb, that is, log R = log(Yr) - log(Yb) , where Yr and Yb are the cumulative percent of products at a given size in the rod and the ball mills respectively. Firstly, the ratio of the observation data is analyzed. Figures 3 and 4 demonstrate the effects of the feed size and the grinding times on the ratio. Figure 3(A) shows that after two minutes of grinding, the local maximum points which correspond to a certain product size appear on the curves. These points are independent of the mill feed size. As the grinding time reaches four minutes, (Figure 3(B)), no local maximum points appear on the curves. The ratio decreases with increase in particle size of the mill product. If the grinding time is maintained constant, the coarser the feed size, the higher the ratio. Figures 4(A) and 4(B) illustrate the effects of the batch grinding time on the ratio, at mill feed size -16 and -7 mm respectively. The local maximum points decrease when grinding times are prolonged. The effect of grinding time on the ratio varies with the feed size; the coarser the mill feed size, the larger the influence of grinding time on the ratio. Hence, in coarse grinding, the rod mill could produce a finer size product than the ball mill.

604

YIOEN ZENO a n d E. FORSSaEgO

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Fig.3 Comparison of the sizing data between the rod and the ball mill products under 2 and 4 minutes grinding time o • zx - - 7-0, 10-0 and 16-0 mm mill feed size

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Effects of mill feed size in coarse grinding

605

To compare the product fineness between the rod and the ball mills quantitatively, the mill type is described by an index, i.e., M i ffi 0 and 1 for the rod and ball mills respectively. Stepwise regression of the constants in Tables 1 and 2, together with the grinding time, the mill feed size and the mill type index, yields eqs (10-13): a

- 1.9908 - 0.02624"Fao + 0.005164*t*Fao - 0.01636"Fao *M i

(lO)

b

= -0.07557 - 1.0072"t + 0.1061*Fao + 0.7706*t*M i

(ll)

c

= -0.2463 - 0.2764"t ÷ 0.02970"Fa0 + 0.2093*t*M i

(12)

d

= 0.5243 + 0.8141"t - 0.0898'F80 - 0.6362*t*M i

(13)

Equations (10-13) account for 77.0, 97.3, 97.6 and 96.4% of the variance for a, b, c and d respectively. All the terms in these equations are full of significance at 95% confidence level. The cumulative percent will be 100% passing 1 mm after 2 and 8 minutes in rod and ball grinding respectively, (Figures I and 2), so that the regression of constant a on the grinding parameters did not therefore have such good correlation as was obtained for other constants. Equation (2) and eq.(6) do not share a similar equation form. The constants b, c and d, which are the shape factors for the sizing curves, are all closely related to the grinding parameters. Moreover, it is found that the variation for all constants with these parameters is similar. According to the P-S equation, the higher the constants, the finer the product within the prediction range (0.053-20 mm). Substituting eqs (10-13) into eq.(1), the cumulative percentage undersize in rod (log Yr) and ball grinding (log Yb) could be derived as functions of the grinding parameters. Subtracting log(Yr) from log(Y 0 and rearranging yields log R -- l o g ( Y r / Y b ) = log,(Yr) - log!Yb) 0.01636"F80 + t [-0.7706 log X - 0.2093"(1og X) 2 + 0.6362*sin(log X)]

(14)

Therefore, the coarser the feed size, the larger the ratio under a certain grinding time. The e f f e c t of grinding time on the ratio is dependent on the product size, (Figure 4).

Energy Consumption Estimation Figures 5 and 6 illustrate the observed energy consumption on total mass as a function of the mill feed size for the rod and ball mills respectively. Comparison of these two figures indicates that the effect of mill feed size on the energy consumption is larger in the rod mill than in the ball mill. With a stepwise variable selection method at 95% confidence level, the energy consumption may be related to the grinding time (t-min) and the mill feed size (Fs0 -mm): E r = 1.0761"t - F80 *(0.02684 - 0.04468"t)

(15)

E b = t*(0.7150 + 0.006301"t + 0.002321"F80 )

(16)

where E r and E b are the energy consumption on total mass in the rod and ball mills respectively. Equations (15,16) account for 99% of the variances observed in the energy consumption. Equation (15) also shows that the energy consumption is mainly dependent on time. The mill feed size has a small effect and affects the energy consumption only together with the grinding time. The longer the grinding time, the larger the effect of mill feed size on the energy consumption. Equation (16) shows the relation between energy consumption and the relevant parameters. However, the effect of grinding time is greater than in the rod mill, and the effect of mill feed size is small. Although the rod mill has a higher energy consumption on the total mass product than the ball mill, it produces a finer product than a ball mill with the same mill feed size and the same batch grinding period of time. With a stepwise variable selection method at 95%

YIGEN ZENO and E. FORSSBERG

606

confidence level, the relationship between energy consumption, 80% passing size of the ground product (P8o -mm) and mill feed size could be established: E r = -2.8240 + 2.3815/P8o + 0.3094"F8o

(17)

E b = 2.8038 + 3.0445/P8o - 6.9875/x/F8o

(18)

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Fig.5 Energy consumption on total mass product in rod grinding o • • ¢ - - for 1, 2, 4 and 6 min. grinding time

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Fig.6 Energy consumption on total mass product in ball grinding o . • ¢ - - for 1, 2, 4 and 8 min. grinding time

Effects o f mill feed size in coarse grinding

607

Equations (17,18) account for 98 and 99% of the variances observed in the energy consumption. Figures 7 and 8 illustrate the residuals between the energy consumption predicted by eqs (17,18) and the observed energy consumption (circles). The residuals are within 0.7 kWh/t for rod grinding and 0.4 kWh/t for ball grinding. Therefore, these equations may describe the energy consumption satisfactorily. The energy consumption is inversely proportional to the product fineness, the effect of the mill feed size varying with the mill type. Hence, the effects of feed size on energy consumption can be estimated, based on eqs (17,18) when the product fineness is determined in the rod and the ball mills. If the mill feed size is reduced from 13 mm to 3 mm by fine crushing, the net energy consumption will be about 1.38 kWh/t. To achieve the same ground product fineness with those feed sizes, that is, substituting Fs0 = 13 and 3 mm in turn into eq.(17) and subtracting, the rod mill may save 3.1 kWh/t. Similarly, using eq.(18) the ball mill will save 2.1 kWh/t.

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Fig.7 Residuals of energy consumption by two ways in rod grinding o - empirical equation; • - - the Bond formula

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608

YIGENZENG and E. FORSSBERG

The Bond formula for estimating energy consumption is used to estimate the total inefficiency factor of the mills. The energy consumption under laboratory grinding conditions is shown by the following equation: E = k*W = k*[Wi*(IOA/P - IOA/F)]

(19)

where k is total inefficiency factor, W is the power needed in a Bond mill, Wi is work index (here W i l l 2 ) , P (/~m) and F (/~m) are 80% passing size in product and feed respectively. With the aid of a non-linear regression method, eq.(19) is fitted with data from the rod and ball mills in turn, and the results are k r =1.25 and k b =1.51 for the rod and ball mills respectively. Hence, in batchwise dry grinding conditions, the rod mill is more efficient than the ball mill. Considering the inefficiency factor, by reduction of the mill feed size from 13 to 3 mm, the energy savings are 2.5 and 1.4 for the rod and ball mills respectively. Figures 7 and 8 indicate the fitting results by the Bond formula (black dots) against the predicted energy consumption. Since there is only one variable in eq.(19), the residuals between prediction and observation by the Bond formula are larger than that of the empirical formula, (eqs (17,18)). Hence, to make further comparison between the rod and ball mills, the empirical equation fitting method is employed. By stepwise variable selection at 95% confidence level, the energy consumption is related to the mill feed size, product fineness and mill type index: E = 0.5303 + 2.4802/P80 + 0.1661"F80 - 3.4524/v/Fs0 + Mi*(1.6311 - 0.1649"Pa0 )

(20)

Equation (20) accounts for 98% of the variance in the observed energy consumption. Figure 9 shows the energy consumption in the rod and ball mills to achieve the same product fineness. It can be sent that the rod mill consumes less energy than the ball mill. On the other hand, substituting M i = 0 and 1 into eq.(20) for the rod and the ball mills respectively and subtracting yields: zxE -- E r - E b = - 1.6311 + 0.1649"P8o

(21)

Statistically, the rod mill consumes less energy than the ball mill to obtain the same ground product fineness. It is in coarse grinding that the rod mill is more efficient than the ball mill.

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Effects of mill feed size in coarse grinding

609

CONCLUSIONS Through large laboratory scale grinding investigations, the following conclusions have been derived: a.

A parabolic-sine equation can satisfactorily describe the particle size distribution of products from rod and ball mill grinding. The equation fitted by multiple regression method accounts for >99.7% of the variance observed in particle size analysis.

b.

In both rod and ball mill grinding, the effect of the mill feed size on the ground product size varies with the mill type, i.e., the product fineness is more affected by the mill feed size in rod grinding than in ball grinding.

C.

It is in coarse grinding that the rod mill produces finer particles than the ball mill with equivalent energy consumption. In other words, to achieve the same ground product fineness, rod grinding consumes less energy than ball grinding.

d.

If the mill feed size is reduced from 13 mm to 3 mm, it would be expected that 2.5 kWh/t would be saved by rod grinding and 1.4 kWh/t by ball grinding under similar operating conditions. ACKNOWLEDGEMENT

This research project is financed by the Swedish Mineral Processing Research Foundation (MinFo Research Project 20). The authors are gratefully to Dr B. Sk61d for his encouragement and advice. REFERENCES

.

Wunderlin M., Energy saving with reduced mill feed size, Proceedings of 53rd annual Minnesota section AIME meeting, 12:1 (Jan. 1980).

.

Rowland C.A. & Kjos D.M., Rod and ball mills; Mineral Processing Plant Design, ed. by Mular & Bhappu, 2nd Edn. 239, SME/AIMMPE, New York, (1980).

3.

Bond F.C., Crushing and Grinding calculations. The Canadian Mining and Metallurgical Bulletin, 466 (July 1954).

.

Svensson J., A new formula for particle size distribution of products produced by comminution. Transaction of the Royal Institute of Technology, Stockholm, 88 (1955).

.

Peleg M., Noemand M.D. & Rosenau J.R., A distribution function for particle populations having a finite size range and a equation independent of the spread. Powder Technology, 46, 209-214 (1986).

6.

Hukki, R.T., Trans. SME/AIME, 220, 403-408 (1961).