## grinding mill power

The powerrequired to drive a tumbling mill is of interest both to the designer and to the mill operator: to the former as a basis of design for the determination of the necessary size of the elements of the machine; and to the latter because all other factors being equal, the most economical machine is that which demands a minimum power for driving.

The power required to drive a mill depends, to some extent on every one of the physical dimensions defining the mill shell and ball charge and on many of those defining the properties of the powder charge. Thus the number of variablesinvolved is very large. Since even a moderately completetheory for the internal dynamics of the ball mill, in which all these variables are given due importance, has not been propounded, the calculation of the power requirements of a mill, from theoretical considerations, cannot be made. Similarly, owing to the great number of variables, no complete experimental investigation of the power demands of ball, tube and rod mills has been made, the amount of work required for an analysis of all of the variables being prohibitive.

Probably one of the most comprehensive experimental investigations of the power demands of the ball or tube mill is that carried out by one of the present authors, with a worker, Rose and Evans by use of small-scale models; the test apparatus being that shown in Fig. 3.1. By use of small-scale models, a very large number of tests may be carried out in a reasonable time and, furthermore, variables which would be difficult to change on a large-scale test may easily be studied. No valid major objection to the use of small-scale models exists since by use of the method of dimensional analysis the results may be generalised to be applicable to mills of any size. The valid generalisation of the results of the model test demands that there should be a complete dynamical similarity between the systems, and, in the case of the ball mill, the pressures in the bed of material in the mill cannot be made strictly similar in a large and a small mill. Even so the satisfactory results obtained by use of model tests suggest that errors arising from this source are small. In the present chapter the question of the power to drive a mill will be approached from the point of view of these model tests and the results of previous workers studied by comparison with these data.

The power, P, to drive the mill would be expected to depend upon the length of the mill, L, the diameter, D, the diameterof ball, d, the density, e, of the ball, the volume occupied by the charge (including voids), expressed as a fraction of the total mill volume, J, the speed of rotation, N, the acceleration due to gravity, g, the coefficient of restitution of the material of the balls and mill, e. The power, P, would also be expected to depend upon the following characteristics of the powder; the representative diameter of the particles b, the energy necessary to bring about unit increase in the specific surface of the powder,E and the volume, V, occupied by the powder charge including voids), expressed as a fraction of the volume between the balls in the mill. Furthermore, the power, P, would be expected to depend upon the effective kinematic viscosity of the mixture of powder and fluid, v, the effective if density of the mixture, , and, in the case of wet milling, by the ratio of the volume of the solid material to the volume of the liquid in the pulp, U. Finally, when the interior of the mill is fitted with lifters it would be expected that the power, P, would depend on the number of lifters, n, and upon the height of the lifters, h.

Theoretically, the dimensionless groups can be combined in any way and it is the function of experiment to determine these relationships. In the present work, however, it has been found that, in most cases, the functions of the dimensionless groups may be multiplied together and the results of the experimental analysis have been arranged in this way.

The first variable to be studied was the effect of the length of the mill on the power requirements: that is, the determination of 2(L/D). The experimental results of Rose and Evans fall on a series of straight lines passing through the origin and so it follows that 2(L/D) is a linear function. Thus,

It alsofollows that in order that dynamical similarity, as regards this group, shall hold between a number of mills of different diameters, it is necessary that they shall all run at the same fraction of their respective critical speeds. This isin accordance with the findings of Gow, Campbell and Coghill and other workers.

The relationship between the power group (P/D6Ne) and the group (Nc/N), as determined by experiment, is shown in Fig. 3.2, and it is seen that for speeds less than 80% of the critical, the relationship 1(Nc/N) can be replaced by

where K2 = 3.13for lifter mills and 3.66 for mills without lifters. Within this range these relationships can be used directly in the equations for the determination of power required to drive a mill. The deviation from linearity shown by these curves is exactly what would be predicted from theoretical consideration since, ideally, when the whole ball charge is centrifuging, the centre of gravity of the charge would coincide with the axis of rotation and the power required to drive the mill would be zero.

In theory, the centrifuging of the charge takes place in layers and each layer forms at a definite speed. In practice however, the process is complicated by interaction between the balls, lifters, etc., and centrifuging occurs more orless gradually over a wide range of speeds, this leading to the deviation from linearity, shown in the figure. Since these curves are plotted on logarithmic scales, the curves should be asymptotic to the negative branch of the y axis, mid in fact the lines show this trend.

Since as has already been mentioned, the critical speed is a criterion for the occurrence of many phenomena in a mill, it is perhaps of interest to mention that during the course of tests at various speeds, the configuration of the balls within the shell was observed through a perspex end cap, and, in the case of the smooth mill, centrifuging of the outer layer of the balls was not observed at speeds up to 1.2Nc, the highest speed reached during the tests, but when effective lifters were fitted to the shell, the outer layers of balls commenced to centrifuge when the speed ofrotation exceeded the critical value Nc, by a very small amount. For a smooth mill, cataracting of the charge generally commenced at speeds of about 0.8Nc, while, when lifters were fitted to the shell, cataracting commenced at about 0.6Nc. This is roughly in accord with the results of Chapter 2, illustrated in Fig. 2.19, that if a mill with a 30 % to 40 % filling is taken as representative, the boundarybetween cataracting and cascading is when the speedof rotation is about 0-7Nc.

Coming now to a study of the effect of the density of the grinding media upon the power required to drive a mill it follows that since the mass of grinding bodies has a definite configuration, as suggested by Fig. 3.3, then the torque to drive the mill will be proportional to the weight of the charge; that is, without powder, proportional to the density of the material of the balls or rods. The correctness of this view is established by Fig. 3.4a, based on the work of Rose and Evans, in which the measured power is plotted against power calculated on the basis of this assumption for charges of glass, steel and lead balls; the relative densities of these materials being given in Table 3.1. It is at once seen that an extremely good linear relationship holds between these variables and so it must be concluded that the suggested relationship is valid.

If now the material undergoing grinding occupies the spaces between the balls of the ball charge, it follows that the centre of gravity of the powder mass will coincide with that of the ball mass. Thus, if the mill charge retains the same configuration as it has when balls only are present, that is, the configuration is unaltered by the inclusion of the powdered material, then it follows that the torque to drive the mill will be proportional to the total weight of the charge in the mill. Thus it would be expected that, at least as a first approximation, the power to drive a mill containing a powder can be calculated from the equation applicable to a mill containing balls only, provided that the density of the balls, in the latter case, is increased by an amount sufficient to include the effects of the additional mass due to the powder charge. If this point of view is adopted, it follows that the mill charged with powder become a special case of the mill containing only a ball charge, and effects arising from the particle size and other characteristics of the powder charge may be expressed by corrections to the equations applicable to the mill without powder. On this basis, for a mill containing a powdercharge.

That these terms are simply additive is supported by the curves of Fig. 3.4b in which are plotted data given by Coghill and Devaney, for the power required to drive a rod mill, in which the rods are tubular and the effective density of the rods is varied by filling the tubes with various materials. For these tests the mill was grinding chert, for which 3.0, and, since the void space between the rods is about 15% of the mill filling, it follows that, if the simple addition is valid, the power is given by the expression of the form

The validity of this relationship has been further examined by tests carried out using powders consisting of emery, iron powder and silica sand; the density of these materials being given in Table 3.2.

Examination of these data shows a scatter over a range of about 5% with no consistencies in the deviations. Thus, it appears that these differences arise from random errors mid so the functional relationship 10(/q), given by equation (3.10), again appears to be satisfactory. It is also possible that some of the scatter arises from the actual porosity of the powder and ball charge differing from the assumed value of 40%, but such deviations are scarcely calculable. Thus it is concluded that the function 10(/q) may be replaced by (1 + 0.4/q), with an accuracy sufficient for practical purposes. The remaining functional relationships cannot be represented by simple algebraic expressions and so are best presented by graphs.

In the first place it follows from Fig. 3.4a that, since materials having a wide range of values of coefficient of restitution are used and yet the points suffer no significant scatter, the effect of the coefficient of restitution on the power requirements of the mill is very small. Thus, at least for practical purposes, the function 8(e) may be neglected.

This result would be expected since, although the coefficient of restitution of the metals would affect the conditions of rebound when clean solid particles make contact, the effects of changes in the coefficient of restitution of the metals are, in the case of a ball in a mill, masked by the energy dissipating capacities of the loose bed of balls and powder upon which the ball falls and by powder between the contact surfaces.

If, on referring to Fig. 3.3, it is assumed that the curve representing the free surface of the charge is a straight line and also that the angle is unaltered by variation of the quantity of the charge, then it is clear that the power to drive the mill is proportional to the product of the cross-sectionalarea of the charge and the radius of the couple arm; both of which depend on the mill filling J. On the basis of these assumptions it is a simple matter to calculate the torque in terms of the mill filling and, when this is done the theoretical line of Fig. 3.5 is obtained. This method of treatment is a gross over-simplification, however, since, as explained in Chapter 2, the curve representing the surface of the charge depends on the filling and frictional characteristics of the material. Furthermore, the balls are not mathematical points and so there is mutual

interferencebetween the balls. There is also interference between the balls and lifters, etc., and, when the relationship is determined by experiment, a vastly different curve, as is also shown in Fig. 3.5, is obtained. The importance of this difference will become apparent later, when various simplified formulae, which have been proposed for the power requirements of a mill, are discussed.

It has already been explained in Chapter 2, in which the motion of a mill charge is discussed, that the coefficient of friction between the ball surfaces and the number of balls in the charge are, from the point of view of motion of charge, not independent variables. For this reason the functions 4(d/D) and 7(f) must be considered together In Fig. 3.6a are shown the results of tests on a smooth mill containing clean steel balls and in Fig. 3.6b of tests

on a mill with lifters. Examination of these curves shows that, for the practical case of (D/d) greater than 20, the value of the function does not differ significantly from unity. For values of (D/d) appreciably less than 20, however, the value of the function is not independent of the filling of the mill.

The coefficient of friction between surfaces treated with molybdenum disulphide is about 0.05 as against 0.1 to 0.2 for oiled surfaces and 0.3 for dry steel on steel. The coefficient of friction between the surface contaminated with powder is unknown but is probably very high, say 0.5 to 1.0.

In order to investigate the effect of small quantities of powder, quartz powder from the sample passing through a 200-mesh sieve was added to the ball charge under investigation and normal tests carried out. The quantity of powder used for this purpose was 5.0 % of that required to fill the void spaces between the balls. This quantity of charge was sufficiently small to ensure that, for all practical purposes, the change in torque, from the corresponding case of the mill charged with clean dry balls only, was due to the change in the coefficient of friction and not to the increased density of the charge. As a result of these tests, no significant difference was found between the results for the cases (1), (2) and (3) above, and the combined results for these tests can be presented by the full line of Fig. 3.7, this curve, of course, corresponding to that for (J) = 0.5 on Fig. 3.6. It is probable that this drooping characteristic arises from the comparative freedom of rotation of the balls when the coefficient of friction is low and when the number of contact points between the surfaces is small; as is bound to be the case when (D/d) is small.

The effect of adding a small quantity of powder is shown by the broken line on Fig. 3.7, this curve being adequate for all values of (J) and, furthermore, a study of Fig. 3.6 suggests that this curve is adequate for lifter mills. Thus, it may be concluded that for smooth mills grinding materials having a low coefficient of friction 4(D/d) should be read from Fig. 3.6a. For lifter mills, or mills grinding materials giving a high coefficient of friction, the functional relationships should be read from the dotted line of Fig. 3.7.

It has been suggested by Bond, that for values of the ratio (D/d) greater than about 80 the ball charge slumps and, thus, the curve of Fig. 3.7 falls for values of the ratio greater than this figure. It is also stated that this effect is particularly marked in wet milling, but, since it is rather unusual to use balls so small that the ratio of (D/d) exceeds 80, it is probable that this effect is not of great practical importance.

In view of this comment the present authors have, by use of the small-scale apparatus, subsequently extended the range of Fig. 3.7 to (D/d) ~200 and, in no case, with mills chargedwith iron shot or glass balls, has slumping of the charge been observed during dry milling.

The results of tests carried out to determine the effect of the number of lifters, n, on the power requirements to drive amill, that is to establish 5(n), are plotted in Fig. 3.8 The function 5(n) is influenced by a large number of variables to a small extent, and the curves presented in this figure are the lines best representing the whole of these data. This method of presentation is adopted since it is felt at the errors introduced by the use of a single line, to replace many, are not significant but that the simplification obtained is worthwhile.

From the figure it can be seen that, provided the number of lifters is greater than about 6 and the ratio of (D/d) exceeds about 20, the function has a constant value of unity. Thus it follows that, provided there are sufficient lifters to effectively key the charge to the mill shell, the number of lifters is unimportant. For a number of lifters less than 6, but with (D/d) less than 20, there is some dependence upon the value of mill filling as shown by the branching of the curve to the left of the figure. For practical purposes the value of the function, for values of (D/d) less than 20, can be calculated by multiplying the relevant value read from the main curve of Fig. 3.8 by the appropriate correction factor read off from the graph inset in that figure.

The effect of the height of the lifters is given by the function 6(h/D) and the results of tests to determine this function are given in Fig. 3.9, from which it is seen that for mills in which (D/d) is greater than about 20, the height of the lifter has little effect on power requirements. For the less usual case, in which (D/d) is less than 20, there is a considerable deviation for these limited cases, but a straight-line interpolation between the two curves of Fig. 3.9 is probably adequate. These conclusions are valid only for lifters of reasonable height, that is, for lifters in which h is not greater than, say, 0.2D and not less than about d/2. If h exceeds 0.2D the lifters unduly interfere with themotion of the ball charge, while if h is much less than d/2 the balls ride over the lifters and so locking to the shell does not occur. This at once suggests that wave, ship-lap, and other such liners cannot act as effective lifters.

The effect of the particle size on the power requirements of a mill would be expected to be small, provided that the particles were not so small that powerful aggregation takes place. If this were to occur, the charge could become jelly-like and the motion of the ball charge would be seriously impeded; with a consequent effect on the power demands of the mill.

The results of tests, carried out on smooth and lifter mills, to determine the form of the function 9(D/b) are given in Fig. 3.10; these tests being such that the value of the group (D/d) exceeded 19, (J) varied from 0.1 to 0.75 mill in the case of the lifter mill, (h/D) = 0.1 and (n) = 6, whilst the group (D/b) was varied from 77 to 740; this range of values of the group (D/d) largely representing the practical range covered by the operating conditions of a large number of mills. The points relating to different test conditions are seen to suffer a considerable amount of scatter. This scatter probably arises from the interdependence of the variables which has been discussed earlier, and itis felt that, unless a very great complication in the form of additional correction factors is introduced, this must be accepted. For practical purposes such complication is not justified, and, in the present work, mean curves have been drawn through the points of Fig. 3.10. When this is done, the curves for the two cases are seen to be identical and give the functional relationship 9(D/b) shown in Fig. 3.11.

Decreasing the value of the (D/b) ratio in a given mill is seen to decrease the power necessary to drive the mill. This may be explained on the basis that the larger particles separate adjacent balls to a greater extent than do the smaller particles, and, therefore, increase the dilatation of the charge. The centre of gravity of the combined charge is thus displaced towards the vertical through the axis of

rotationand so the torque is decreased. In the small mills usedfor these tests, the presence of very fine powder, such thatthe ratio (D/b) greatly exceeded 800, resulted in separation of the powder from the balls, but this only occurred at speeds in excess of 0.8Nc. The powder formed a continuous layer around the cylindrical wall of the shell, and thepower required to drive the mill decreased by a largeamount when this occurred. This phenomena would probably not occur with large industrial mills, but data is notavailable on this point. The question is bound up with the, absolute size of the particle and a ratio of (D/b) of 800in mills used in the tests corresponds to a charge consisting of particles of 0.003-in. average diameter. Generally, however, it would be expected that, except in the case of very fine grinding, this effect is unimportant.

Throughout the series of tests, the value of the powder filling group (V) has mainly been held constant at unity; this being the condition that the powder charge is theoretically just sufficient to fill the void space between the balls. In fact, the powder increases the total volume of the charge and a layer of balls may occur at the surface of the charge, the void space between which is unfilled with powder. This restriction on the value of the parameter (V) has been made because, in practice, less powder than is necessary to fill the void space is rarely used since a part of the ball charge is then inoperative in the grinding process; even though the power necessary to maintain the centre of gravity of the inoperative portion of the ball charge in the displaced position must still be supplied. Conversely, if the powder is excessive, choking of the mill occurs, and a portion of the powder charge is remote from the ball charge at any instant and so cannot be effectively ground. Thus the decision to restrict the value of the parameter (V) to unity is based upon the practical importance of this case and the necessity, in view of the large number of variables involved in the problem, to limit the amount of experimental work.

Although this restriction is useful from the point of view of limiting the number of variables to be investigated, it imposes the restriction that the results are strictly applicable only to batch mills and not to the industrially important grate mills and overflow mills. Thus the results of the present investigation, which has been carried out for the case of (V) = 1.0, are strictly applicable only to a batch mill. The results have, however, been extended directly to the cases of the trunnion overflow mill and by inference to the grate discharge mill. Consider now the extension of the work to these cases; the mills together with a batch mill have already been illustrated in Fig. 1.4.

In the grate mill the powder surface probably slopes from the inlet end of the mill to the discharge end in the manner indicated in Figs. 1.4b and 1.4d. In this case the moment of the powder charge about the axis of rotation of the mill is probably much the same as in the ball mill; the greater moment at one end of the mill being roughly balanced by a smaller moment at the other end. Thus, the method of the present work would be expected to be applicable to a grate mill with an accuracy sufficient for most purposes.

In the case of the overflow mill, in order that the product shall overflow through the hollow trunnion, it is necessary that the general level of the powder surface must correspond to about a 45-50% filling of the mill. Then when the ball filling is about 45-50% the overflow mill corresponds to a batch mill, and so the power input should be directly calculable from the present work. For small

valuesof ball charge, however, a correction factor is necessary and this has been determined by experiment and is given in Fig. 3.12. In the use of this factor the power is calculated on the basis that the mill has a ball charge equal to that in the actual overflow mill and the normal correction for powder charge is introduced. This calculated power is then multiplied by a factor, corresponding to the value of the actual ball charge; the factor being read from Fig. 3.12. The shape of the curve of Fig. 3.12 would be expected, since for very small ball charges the powder standing above the ball charge level contributes a relatively large moment, and so increases the power required over that calculated on the basis of a batch mill. For intermediate fillings, however, the contribution to the moment by this additional powder is small, but there is, in addition, a loss of moment due to this additional powder displacing the centre of gravity of the relatively heavy ball charge towards the centre of rotation.

Strictly, there should be a correction curve for each value of the density of the charge material, but, in fact, the correction is small and when regard is paid to the overall accuracy which can be expected in calculations of this type, interpolation between the curves of Fig. 3.12 is all that is justified.

The parameter E is the characteristic of the powder which is a measure of its resistance to grinding. Thus the quantity (E/DNQ) would not be expected to affect the instantaneous power input to the mill, although it would be expected that the rate of grinding is dependent upon it. The question of rate of grinding will, however, be treated in a later chapter.

The tests carried out with different materials in the mill, in order to determine the effect of the density of the powder, show no significant variations which can be attributed to the difference in the value of the parameter E between the materials. Thus it appears that in connection with the power input to the mill, the dimensionless group (E/DNq) is not a significant variable, and so 12(E/DNq) is eliminated from the equation.

In order to compute the power required to drive a mill it is necessary to include the values of all the various relevant factors previously discussed and on the basis of this foregoing work, the equation for the power becomes

3(J) is read from Fig. 3.5,
4 ( d/D) is read from Fig. 3.6 or Fig. 3.7 as required by mill parameters,
5 ( n) is read from Fig. 3.8,
6 (h/D) is read from Fig. 3.9,
9 (b/D ) is read from Fig. 3.11,
and 11(V) is read from Fig. 3.12 for a trunnion overflow mill or is taken as unity for a batch mill or
grate discharge mill.

Theequation (3.11) has been applied to the computation of the power requirements of large industrial mills, based on the published data given by Taggart, and the resultsof these calculations are shown in Fig. 3.13. From thiscurve it is seen that the points are scattered along good straightlines, even though the ratio of the horse-power to drive the largest mill to that required to drive the model is of the order of 30,000 to 1.0. Thus it appears that there in no systematic errors in the results of the model tests.

In connection with the scatter of the points, it is perhaps ofinterest to mention that the data quoted by Taggart, whichhave been apparently obtained by circulating a questionnaire to a large number of mill owners, are, for the following reasons, subject to some uncertainties.

In view of these uncertainties, it is believed that the agreement between the calculated and measured powers is as close as can reasonably be expected, and it may be concluded that the results of this work upon small-scale models may be applied with confidence to the calculation of thepower input to industrial mills of even the largest sizethat is to machines absorbing even several thousand horse-power. The method of calculation developed in this work will be demonstrated by application to the data, relating to a large mill, published by Carey and Stairmand.

A lifter mill of 6 ft diameter, revolving at 18.7 r.p.m., is grinding coal. The body of the mill is divided into two sections: one section, 7.5 ft long, being charged with 5 tons of steel balls, of 2 in. average diameter, and the other section, 15 ft. long, being charged with 6 tons of Cylpebs of 5/8 in. average diameter.

Volume of balls = 38 cu. ftandVolume of chamber = /4 x 6 x 75 = 212 cu. ft
Thus (J) = 018
From equation (3 .5), the critical speed for both sections in 31 r.p.m.
ow from the appropriate graphs,
1(c/) = 92, 3(J) = 065, 4 (D/d) = 10,
5 (n) = 10, and 6 (h/D) = 10
andsince the particle diameter, b, is probably less than about 1/8 in., 9 (D/b) = 10.
Thenfrom the equation

From the published data, the measured power to the motor terminals is 103 kW, and so the power demand of 86 kW by the mill leads to a combined efficiency of motor and transmission of 83%, which is reasonable. Thus the agreement is as close as can be expected.
Consideration of equation (3.11) will show that for speeds less than 80 % of the critical,

From Fig. 3.7, however, it would be expected that 4,(D/d) varies approximately as (D/d)0.1 and in Fig. 3.14 is shown the curve (D/d)0.075 plotted on the same axes as the dotted curve of Fig. 3.7. It is seen that the fit is reasonable.

Thus it follows that in tests in which the ball diameter is constant the ratio (D/d) varies directly as D, and so the power to the mill, which varies with 4(D/d), will vary as D0.1approximately. Thus, it follows that, if the ball diameter is maintained constant, the equation (3.14) must be multiplied by D0.1, which brings about agreement with equation (3.15).

One of the earliest attempts to derive, from theoretical considerations, a formula for the power to drive a mill is that of Davis, who starting from the basis of the ball trajectories discussed in the previous chapter, calculated the striking velocity and so the kinetic energy of the elements of the charge in a given trajectory and, by integration, the kinetic energy to the whole charge; this kinetic energy clearly being supplied to the mill from the external source.

Theequation (3.16) is not directly comparable with equation (3.11), since for a given composition of charge, the weight of the charge, W, cannot be varied independently of the diameter of the mill and similarity maintained at the same time. Thus, the functional relationship3(J) of the previous work is absorbed in the form of equation (3.17). Furthermore, the equation of Davis is derived for the conditions of maximum efficiency and so is not general.

where P is the power draft in kilowatts per ton of grinding balls. The equations of both Davis and Bond are based only on the mill diameter, the mill filling and the speed of rotation expressed in terms of the critical speed; the forms of the equations arising from the empirical expressions defining the relationships between several variables. Since the weight of the ball charge enters both these equations directly, it follows that the power is proportional to thelength of the mill; a conclusion which is in accord with the equation of Rose and Evans.

where L is the length of the mill and K is a constant having the value of 0.9 for mills of length less than 5 ft and 0.85 for mills of length greater than 5 ft. This function is plotted in Fig. 3.15, and it is seen that for long mills the power is considerably smaller than is demanded by a linear function.

The higher power requirements of a short mill would, perhaps, be expected, since the end plates of the mill might cause a piling-up of the charge at the ends of the mill. Such an effect would, however, be expected to be negligible in a long mill. Thus, this deviation for a long mill is surprising and it is believed that this formula should be accepted with caution.

where C is the length in feet of the chord defined by the surface of the mill charge, L is the length of the mill in feet, is the mean specific gravity of the charge and n is the speed of rotation in r.p.m. This formula approximates to the formula of Rose and Evans insofar that the term q(1 +0.4 /q) is proportional to the mean density of the pulp, and so to of equation (3.20), and in that the power is proportional to the length of the mill.

For values of mill filling greater than 50%, however, the agreement is not good but, in practice, high mill fillings are perhaps not so frequently used as the lower lines so it may be concluded that, provided that the correct value is assigned to the coefficient, the equation of Hancock, in spite of its simpler form, is, over a practical range of variables, in good agreement with those of other workers. Thus this formula includes, approximately, the dimensionless group 3(J) of the previous work.

The power required to drive a rod mill does not appear in have been extensively studied. As a first approximation, it would be expected that the value of the function 10(/q) would be different to that for a ball mill but that the values of the remaining functions would not be seriously altered.

It is probable, however, that the action of the powder to dilate the charge, and so to reduce the horizontal displacement of the centre of gravity of the charge, would be more marked with rods than with balls. This would lead to a reduction in the actual power to below that which would be calculated by the use of the formulae of the ball mill, as modified by the use of equations (3.23) or (3.24). This effect would, however, probably be small and so could be neglected.

A further case which requires consideration is that in which the grinding media is of some form such as Cylpebs; off-cuts from scrap drill rods sometimes being used in the mining industry. Also in some cases, though rather rarely, large pieces of ore are used as a grinding medium. It appears, however, that only in the case of extremely high costs for the carriage of replacement balls can the use of such lumps of ore be justified on economic grounds; since the grinding efficiency is adversely affected thereby and the milling cost correspondingly increased.

The use of such non-circular or non-spherical materials wouldlead to a locking of the elements of the charge, and so to an increased displacement of the centre of gravity of the charge, with a relatively increased power demand. The power demand would, furthermore be modified by the tightness of packing of the grinding bodies, which is different to that for balls, and by the density of the material. The power to drive mills charged with such materialsdoes not appear to have been studied, however, and with the present state of knowledge, little more than roughestimates of the power calculated on the basis of theprevious equations modified on the lines suggested, can bemade.

The foregoing work deals entirely with the power necessaryto keep the charge within the mill shell inmotion. Clearly, in order to determine the power to be supplied by the driving motor, it is necessary to add to this figurethe power lost in the bearings supporting the mill shell in the bearings supporting the intermediate speed shaftsand in the reduction gearing. The estimation of these losses, is, however, a matter of general engineering knowledge, and is not specific to the tumbling mill, and suchmatters are specifically excluded from the present work.

## effect of grinding media charge on mill power draw kw

Maximum power is drawn by a mill when the charge occupies approximately 50% by volume. The power curve becomes very flat in the range above 45%. As a result, mills are seldom run with charge levels greater than 45%.In rod mills, the charge is swollen by particles of feed which separate the rods. If the mill is shut down immediately after the feed is shut off, the charge level will be greater than if the mill had been ground out prior to shutdown. Rod mills are normally operated with a 32 to 40 percent charge by volume which during operation, becomes a 40 to 50% charge, with a bulk density considerably lower than that of stacked rods. This chart shows the impact a mills grinding ball load has on the mills motor power draw.

## ball mill design/power calculation

The basic parameters used in ball mill design (power calculations), rod mill or anytumbling millsizing are; material to be ground, characteristics, Bond Work Index, bulk density, specific density, desired mill tonnage capacity DTPH, operating % solids or pulp density, feed size as F80 and maximum chunk size, productsize as P80 and maximum and finally the type of circuit open/closed you are designing for.

In extracting fromNordberg Process Machinery Reference ManualI will also provide 2 Ball Mill Sizing (Design) example done by-hand from tables and charts. Today, much of this mill designing is done by computers, power models and others. These are a good back-to-basics exercises for those wanting to understand what is behind or inside the machines.

W = power consumption expressed in kWh/short to (HPhr/short ton = 1.34 kWh/short ton)
Wi = work index, which is a factor relative to the kwh/short ton required to reduce a given material from theoretically infinite size to 80% passing 100 microns
P = size in microns of the screen opening which 80% of the product will pass
F = size in microns of the screen opening which 80% of the feed will pass

Open circuit grinding to a given surface area requires no more power than closed circuit grinding to the same surface area provided there is no objection to the natural top-size. If top-size must be limited in open circuit, power requirements rise drastically as allowable top-size is reduced and particle size distribution tends toward the finer sizes.

A wet grinding ball mill in closed circuit is to be fed 100 TPH of a material with a work index of 15 and a size distribution of 80% passing inch (6350 microns). The required product size distribution is to be 80% passing 100 mesh (149 microns). In order to determine the power requirement, the steps are as follows:

The ball mill motorpower requirement calculated above as 1400 HP is the power that must be applied at the mill drive in order to grind the tonnage of feed from one size distribution. The following shows how the size or select thematching mill required to draw this power is calculated from known tables the old fashion way.

The value of the angle a varies with the type of discharge, percent of critical speed, and grinding condition. In order to use the preceding equation, it is necessary to have considerable data on existing installations. Therefore, this approach has been simplified as follows:

A = factor for diameter inside shell lining
B = factor which includes effect of % loading and mill type
C = factor for speed of mill
L = length in feet of grinding chamber measured between head liners at shell- to-head junction

Many grinding mill manufacturers specify diameter inside the liners whereas othersare specified per inside shell diameter. (Subtract 6 to obtain diameter inside liners.) Likewise, a similar confusion surrounds the length of a mill. Therefore, when comparing the size of a mill between competitive manufacturers, one should be aware that mill manufacturers do not observe a size convention.

In Example No.1 it was determined that a 1400 HP wet grinding ball mill was required to grind 100 TPH of material with a Bond Work Index of 15 (guess what mineral type it is) from 80% passing inch to 80% passing 100 mesh in closed circuit. What is the size of an overflow discharge ball mill for this application?

## analysis of ball mill grinding operation using mill power specific kinetic parameters - sciencedirect

Effect of operating variables on the energy efficiency of ball mill analyzed.Rates of particle breakage and production of fines per unit power input considered.Both the parameters exhibit significant variation with operating conditions.Effect of variables found to be different under different operating conditions.Rate of production of fines parameter better suited for mill design and scale-up.

With a view to developing a sound basis for the design and scale-up of ball mills, a large amount of data available in the literature were analyzed for variation of the two key mill performance parameters: power specific values of the absolute breakage rate of the coarsest size fraction, S*, and absolute rate of production of fines, F*, with some of the important operating and design variables such as the mill speed, ball load, particle load, ball diameter and mill diameter. In general, values of both the mill performance parameters were found to vary significantly with the mill operating conditions. The nature and relative magnitude of variation for the two parameters also differed significantly. Moreover, the effect of any particular variable on the S* and F* values was found to be significantly different for different sets of operating conditions. It has been emphasized that, as the purpose of grinding is to produce fine particles, the mill design and scale-up work should be based mainly on the F* parameters. Moreover, it is not correct to regard the S* values to be independent of the mill design and operating variables as a general rule, especially for a fine analysis of the performance of the grinding systems.