## to verify the laws of size reduction using ball mill and determining kicks rittinger's bond's coefficients power requirement and ## rittinger - an overview | sciencedirect topics

As a rule, size reduction operations are heavy in energy consumption (Loncin and Merson, 1979; Hassanpour et al., 2004). As an example, the cost of energy is the single largest item in the total cost of wheat milling. Milling of one bushel (approx. 27kg) of wheat requires 1.74kwh of electric energy (Ryan and Tiffany, 1998).

The total energy consumption of a mill consists of two parts: the energy imparted to the milled material and that needed to overcome friction in bearings and other moving parts of the mill. The energy transferred to the material corresponds to the work of deformation and is stored in the particle as internal stress. When the particle fractures, the stored energy is released. Part of it provides the increment in surface energy resulting from increased surface area but most of it is released as heat. Eventually, friction losses also generate heat. Consequently, size reduction may result in considerable increase in the temperature of the treated material. Temperature rise as a result of size reduction may be an important technological issue, particularly with heat-sensitive products, thermoplastic substances and materials with high fat content. When necessary, this problem is addressed by air- or water-cooling of the machine or using cryogenics such as liquid nitrogen (cryo-milling).

Mechanical efficiency m of a size reduction device is defined as the ratio of the energy transferred to the material to the total energy consumption W of the device, per unit mass of material treated.

A different expression for energy requirement of size reduction has been proposed by Kick. Kick assumes that the energy needed to reduce the size of the material by a certain proportion (say by half or by one order of magnitude) is constant (first order relationship). Kick's law is written as follows:

Sugar crystals were ground from an average Sauter diameter of 500m to powder with an average Sauter diameter of 100m. The net energy consumption was 0.5kWh per ton. What would be the net energy consumption for grinding the crystals to 50m powder:a.according to Rittinger's lawb.according to Kick's law.

a.Rittinger's law:E=K(1x2-1x1)K is calculated from the first milling data and applied to the second milling:K = 0.5/(1/1001/500) = 62.5kwh.m/tonE = 62.5 (1/501/500) = 1.125kwh/tonb.Kick's law:E=Klog(x1/x2)K = 0.5/log (100/500) = 0.715E = 0.715* log (50/500) = 0.715kwh/ton

Bond [BON 52] made the hypothesis that the exponent of dp is the arithmetic mean between Rittinger and Kicks values, being 0.5. In addition, it returns energy not to the volume but to the mass being processed and, for the specific energy, he writes:

Bond calls Wi the work index, literally meaning energy index. Naturally, Wi depends only, in principle, on the nature of the milled body. It is the reason why he provides Wi values (as well as for the true density) for most commonly used ores in Table IIIA on page 548 of [BON 61b].

In Table IIIA, Bond [BON 61b, p. 548] provides the standard specific energies expressed in kilowatt hour which he calls a short ton (whose value is 2,000 pounds).kWhshortton=3600kilojoule0.90718metircton=3.968.103kilojoule.ton11kWhshortton1=3.968kJ.kg1

The concept of measuring Wi is the following. We put a total mass MT of ore to be milled in a small laboratory grinder. After a determined number of rotations NT, we empty the grinder and sort the mass MT on a sieve with an opening P1. The mass of the underflow is MP=MTMR, where MR is the mass of overflow. We complete the mass MR with fresh ore to obtain MT that we reload into the grinder.

Assuming that the mass MT placed in the grinder has, at the start of each of its last cycles, 80% underflow with a size of df microns and that this same mass has, at the end of each of these cycles, 80% underflow with a size of dp microns, Bonds standard specific energy Wi will be:

Then, 700g of this output, compacted following a standard procedure, are dry milled in the vessel (=305mm and L=305mm). The rotation speed of the vessel is 70 rev.mn1, indicating 85% of the critical speed. The balls load weighs 20.125kg and is made up of a specific number of balls with sizes ranging from 12.7 to 38mm.

The method consists of grinding the load for a short period of time (100300 revolutions). We then sieve the load with a screen size xT set in advance (that is, 300m for example) and replace the downflow with an equal mass of fresh feed. We repeat the operation until there is a constant ratio equal to 2.5 between the overflow and the underflow and the mass MF of underflow obtained for the content of the mill is equal to a constant.

The length of time required for each operation must be determined by trial and error and the number of operations may, depending on the situation, vary from 7 to 15. The method is therefore not simple.

For a ball mill, if the reduction ratio becomes less than 3 (target grinding of concentrates), the energy index Wi must be multiplied by a given coefficient given by the authors equation 27 [BON 61b, p. 545].

If we want to reduce a very large compact solid (df=) into particles with diameter dp=100m, the required specific energy becomes equal to Bonds energy index Wi that is measured in kilowatt hour per short ton (907.18kg) of solid. Values for this energy index will be provided for many ores [BON 60].

Note that carbon is missing from Bonds list. This issue was dealt with by Chandler [CHA 65]. The applicable standard method for carbon is Hardgroves [HAD 32] test that Chandler [CHA 65] describes.Note IIn practice, it is very difficult to make use of a bulk solid when the size of all its particles is equal to dp. This is why Bond defines the sizes df and dp as screen openings that let through 80% of the feeds solid mass and grinder output, respectively.Note IIBond [BON 54] proposed correspondent relationships between his energy index and magnitudes characterizing the capacity during grinding given in other texts written by Bond himself [BON 49].

In practice, it is very difficult to make use of a bulk solid when the size of all its particles is equal to dp. This is why Bond defines the sizes df and dp as screen openings that let through 80% of the feeds solid mass and grinder output, respectively.

This coefficient is only applicable when R<6.ExampleLet us consider by dry grinding 40 ton.h1 of ore in a ball mill that enters at 5mm and exits at 200m. The apparent density of the ore is 1.5 and its energy index Ei is equal to 15. We estimate that the diameter of the mill is less than 3.8m.C1=1.3C2=1C3=2.4430.2=0.96xo=40001315=3724mC4=1+15750003724372450.2=1.11C5=1C7=1E=151.110020010050001.30.961.11E=22.860.710.14=13kWh/tonPa=4013=520kW

Let us consider by dry grinding 40 ton.h1 of ore in a ball mill that enters at 5mm and exits at 200m. The apparent density of the ore is 1.5 and its energy index Ei is equal to 15. We estimate that the diameter of the mill is less than 3.8m.C1=1.3C2=1C3=2.4430.2=0.96xo=40001315=3724mC4=1+15750003724372450.2=1.11C5=1C7=1E=151.110020010050001.30.961.11E=22.860.710.14=13kWh/tonPa=4013=520kW

We could use the same reasoning for a rod mill but the energy needed to raise a rod will be proportional to DCB2LD and the number of rods will be DCB2LD and the number of rods would be D2L/DCB2L . We obtain:

There is a need to describe the relationship between the capacity of the mill and the properties of the milled material. Appropriate methods are based on various comminution theories, the most common of which are Rittingers , Kicks  and Bonds .

The commonly used method to evaluate the grindability of coal in medium speed pulverizers is the Hardgrove Grindability Index (HGI) . The HGI test is based on Rittingers theory. It allows to predict the mill output, performance and energy requirements, and (qualitatively) also the particle size distribution after milling . As the value of HGI increases, the capacity of the mill increases as well. Numerous experiences show that if the HGI test is a good indicator of milling performance for medium speed mills when grinding coal, it is poor for other materials such as biomass. Another disadvantage of HGI is that the tester is a batch device and does not reflect the continuous grinding process.

Broad dissemination of biomass burning in PF boilers caused the search for other indicators better reflecting the comminution of such materials . The studies show that in this case better results give the methods based on Bonds theory.

Although it is impossible to estimate accurately the amount of energy required in order to effect a size reduction of a given material, a number of empirical laws have been proposed. The two earliest laws are due to Kick(7) and von Rittinger(8), and a third law due to Bond(9,10) has also been proposed. These three laws may all be derived from the basic differential equation:

which is known as Kick's law. This supposes that the energy required is directly related to the reduction ratio L1/L2 which means that the energy required to crush a given amount of material from a 50 mm to a 25 mm size is the same as that required to reduce the size from 12 mm to 6 mm. In equations 2.3 and 2.4, KR and KK are known respectively as Rittinger's constant and Kick's constant. It may be noted that neither of these constants is dimensionless.

Neither of these two laws permits an accurate calculation of the energy requirements. Rittinger's law is applicable mainly to that part of the process where new surface is being created and holds most accurately for fine grinding where the increase in surface per unit mass of material is large. Kick's law, more closely relates to the energy required to effect elastic deformation before fracture occurs, and is more accurate than Rittinger's law for coarse crushing where the amount of surface produced is considerably less.

Bond terms Ei the work index, and expresses it as the amount of energy required to reduce unit mass of material from an infinite particle size to a size L2 of 100 m, that is q = . The size of material is taken as the size of the square hole through which 80 per cent of the material will pass. Expressions for the work index are given in the original papers(8,9) for various types of materials and various forms of size reduction equipment.

Hardgrove Indexbased on Rittinger's Law, which states that the power consumption is proportional to the new surface created. A prepared sample receives a definite amount of grinding energy in laboratory ring-roll pulverizer. The sample is compared with a coal chosen as having 100 grindability (Pittsburgh Seam coal).Index=136.93w where w = wt of material passing 200 B.S. sieve (obtained from orig. wt of 50 g-wt retained on sieve). Usual range of indices 25 to 75. For details see A.S.T.M. D409.

Finally, a look should be taken at coal elasticity, hardness and strength. However, a particular matter of importance which arises from those consideration is the ease of coal grinding, an important step in whatever coal preparation efforts for further processing. The more fundamental material properties are covered reasonably by Berkowitz (1994), so the discussion here will be limited to coal grindability. For that purpose, use is made of two different indices, both determined experimentally with the material to be ground. One is the Hardgrove grindability index and the other the Bond work index.

The Hardgrove index is determined using the ASTM method D 40971. It involves grinding 50g of the material, e.g. coal, of specified size (1630 mesh cut) in a specified ball-and-race mill for 60 revolutions. The amount of 200 mesh material is measured (w grams) and the index is defined as I = 13+ 6.93w. Thus, the higher the index, the easier is the grinding task. This method loosely assumes that the specific energy consumed is proportional to the new surface generated, following the concept of Rittingers law of comminution.

Berkowitz (1994 p.96) gives a generalized variation of the Hardgrove index with coal rank. According to the variation, anthracites are hard to grind, bituminous coals the easiest, and the subbituminous more difficult, with lignites down to the same low index level as anthracites. It is suggested that the decrease in the index below daf coal of 85% is caused by plastic deformation and aggregation of the softer coal particles, hence reducing the 200 mesh fraction generated by the grinding test.

The Bond work index (Bond, 1960) is based on Bonds law, which states that the energy consumed is proportional to the 1.5 power of particle size rather than the square of Rittingers law. Accordingly, the energy consumed in reducing the particle size from xF to xp (both measured as 80% undersize) is given by

We should note that the higher the value of the work index, the more difficult it is to grind the material. A compilation of data is available, for example, in Perrys Chemical Engineers Handbook (Perry et al., 1984). For coal, one average value is given, with Ei = 11.37 for = 1.63. Bonds law is useful because of the extensive comparative database.

Interestingly, Hukki (1961) offers a Solomonic settlement between the different grinding theories (rather than laws). A great deal of additional material related to grinding, or size reduction, comminution, is available in handbooks, e.g. by Prasher (1987) and research publications in journals such as Powder Technology. A very brief overview of grinding equipment is given in Section 1.5.3.

This family of models is the oldest of the comminution models and they continue to find widespread use (Morrell, 2014a). Energy-based models assume a relationship between energy input of the comminution device and the resultant effective particle size of the product. Many rely on the feed and product size distributions being self-similar; that is, parallel when cumulative finer is plotted in log-log space (Chapter 4). The energy input is for net power, that is, after correcting for motor efficiency and drive train mechanical losses. Typically, energy is measured as kWh t1 or Joules, depending on the model.

The oldest theory, Von Rittinger (1867), stated that the energy consumed in size reduction is proportional to the area of new surface produced. The surface area of a known weight of particles of uniform diameter is inversely proportional to the diameter, hence Von Rittingers law equates to:

As Lynch and Rowland (2005) note, the means to make measurements of energy and size necessary to validate the Von Rittinger and Kick models did not exist until the middle of the twentieth century when electrical motors and precision laboratory instruments became available. The literature from this period includes work by a group at the Allis Chalmers Company who were trying to calibrate Von Rittingers equation to industrial rod mills (Bond and Maxson, 1938; Myers et al., 1947).

Often referred to as the third theory, Bond (1952) stated that the energy input is proportional to the new crack tip length produced in particle breakage. Bond redefined his theory to rather be an empirical relationship in a near-final treatise (Bond, 1985). The equation is commonly written as:

where W is the energy input (work) in kilowatt hours per metric ton (or per short ton in Bonds original publications), Wi is the work index (or Bond work index) in kilowatt hours per metric ton, and P80 and F80 are the 80% product and feed passing sizes, in micrometers.

Solving Eq. (5.1b) for n=3/2 gives the same form as Eq. (5.4) with the constant 2 K ahead of the bracket. In effect the 2 K is replaced by (10Wi), which is convenient because Wi becomes equal to W in the case of grinding from a theoretical infinite feed size to 80% passing 100m. The Bond model remains the most widely used, at least for the conventional comminution equipment in use at the time Bond developed the model and calibrated it against industrial data. It is one reason that the 80% passing size became the common single point metric (mean) of a particle size distribution.

A modification of Eq. (5.1a,b) was proposed by Hukki (1962), namely substituting n by a function of particle size, f(x). This provoked debate over the size range that the three established models applied to. What can be agreed is that all the models predict that energy consumption will increase as product particle size (i.e., P) decreases. Typical specific energy values (in kWh t1) are (Morrell, 2014b): primary crushing (i.e., 1000-100mm), 0.1-0.15; secondary crushing (100-10mm), 1-1.2; coarse grinding (10-1mm), 3-3.5; and fine grinding (1-0.1mm), 10.

Fine grinding tests are sometimes expressed as a signature plot (He et al., 2010), which is an experimentally fitted version of Eq. (5.1a,b) with n=f(x). A laboratory test using a fine grinding mill is conducted where the energy consumption is carefully measured and a slurry sample is extracted periodically to determine the 80% passing size. The energy-time relationship versus size is then plotted and fitted to give (in terms of Eq. (5.1a,b)) a coefficient K and a value for the exponent f (x).

The problem that occurs when trying to solve Eq. (5.5) is the variable nature of the function g(x). A pragmatic approach was to assume M is a constant over the normal range of particle sizes treated in the comminution device and leave the variation in size-by-size hardness to be taken up by f(x). Morrell (2009) gives the following:

where Mi is the work index parameter related to the breakage property of an ore and the type of comminution machine, W is the specific comminution energy (kWh t1), P and F are the product and feed 80% passing size (m), and f(x) is given by (Morrell, 2006):

The parameter Mi takes on different values depending on the comminution machine: Mia for primary tumbling mills (AG/SAG mills) that applies above 750m; Mib for secondary tumbling mills (e.g., ball mills) that applies below 750m; Mic for conventional crushers; and Mih for HPGRs. The values for Mia, Mic, and Mih were developed using the SMC Test combined with a database of operating comminution circuits. A variation of the Bond laboratory ball work index test was used to determine values of Mib. This is similar to the approach Bond used in relating laboratory results to full scale machines. The methodology continues to be refined as the database expands (Morrell, 2010).

Morrell (2009) gave a worked example comparing the energy requirements for three candidate circuits to illustrate the calculations. Taking just the example for the fine particle tumbling mill serves that purpose here (Example 5.1).

From the Mi data the relevant value is Mib=18.8kWh t1. Noting it is fine grinding then the feed F80 is taken as 750m. Combining Eqs. (5.6) and (5.7) and substituting the values:W=18.84(106(0.295+750/1,000,000)750(0.295+750/1,000,000))=8.4(kWht1)

Milling or grinding can reduce coarse fly ash particles to fine particle size in a similar range as particles separated out from air classification. But milling does not directly compete with air classification, as milling is not a filtration technique. Milling reduces the fly ash particle size by breaking up large spherical particles into smaller irregularly shaped particles that can have a negative impact on rheology. Milling also consumes more energy than air classification to obtain the fine particle size distribution and has a size limitation, as it is difficult to reduce the particle sizes down to less than 10m. The advantage of milling is that the entire milled quantity will consist of only one defined particle size distribution instead of fine and coarse particle size distributions.

Milling breaks solid materials into smaller pieces by grinding, crushing, or cutting by attrition, collision, or compressive forces. For particles less than 50m, the energy needed to grind the material down to the desired size follows the Von Rittinger grinding law, while the particle sizes less than 50mm but greater than 50m follows the Bond grinding law, as listed here:

Three types of millings have been used to grind fly ash to smaller particle size: ball milling, vibration milling, and plate (pan or plane) milling. Ball milling is typically loaded with particulate materials at its 30%40% capacity. A higher rotation speed, longer processing time, greater ball density, or greater impact force produces finer particle size distribution. The accumulated volumes of Class II fly ash, which has no more than 25% by weight higher than 45m, and its ball-milled samples at different times of 15, 30, 45, 60, 90, and 120min are shown in Fig. 10.12. A longer grinding time produces finer particle size. After 15min of grinding, all milled fly ashes become Class I fly ash, which is no more than 12% by weight greater than 45m, according to fly ash classification under the Chinese standard, GB/T 15962005.

Vibration milling is the core technology for energetically modified cement (EMC) technology, which was patented in 1993 by Dr. Vladimir Ronin. Plate milling was introduced in 2015 at the World of Coal Ash conference by Professor Li Hui from Xi`an University of Architecture and Technology. Her paper showed the energy consumption to grind down fly ash with D50 of 21.57m down to 3m is 1019, 1323, and 120kWh/ton for ball, vibration, and plate milling, respectively.

Air-dried limestone was crushed continuously in a laboratory single-toggle Blake jaw crusher designed to provide a throw of 228mm. The lower opening close set was 102mm and the maximum bottom opening was 330mm. The gape was 813mm and the width of hopper 1067mm. 90% of the ore commenced crushing 200mm from the bottom of the crusher. The Bond index was estimated as 15 kWh/t. Assuming that the density of the limestone was 2.6t/m3, determine1.the optimum RPM of the toggle,2.the maximum annual capacity of crusher with 99% availability,3.power consumption at the optimum speed.

A single toggle Blake jaw crusher with 22.8cm 47.7cm receiving hopper crushed gold ore at the rate of 85t/h with closed setting at 2.54cm and maximum opening of 3.8cm. The work index of the ore was 13.5kWh/t.

The feed to a jaw crusher was 60mm+40mm and the product analysed:Screen Size (mm)Product (% Retained)Screen Size (mm)Product (% Retained)810.0+0.3510.1+421.8+0.255.5+216.3+0.1256.2+0.7520.10.12510.0

The compressive strength of the mineral was 20MN/m2. The crusher was next used to crush a second mineral of compressive strength 55MN/m2 at 5kg/s. The feed size of the second mineral was 55+40mm and yielded a product whose average size was 0.4mm. Estimate the change in power required during the second operation.

A Blake jaw crusher had the following dimensions: Gape=160cm, open set=24.4cm, close set=5.0cm. The width of the hopper was 1.5 times the gape. The ore contained 20% material minus 4.0cm. The bulk density of the rock was 1.75t/m3 and the nip angle 22.8.

A cement manufacturer needed to produce lime at the rate of 140,000t/year in a rotary kiln operating 360days in the year. Limestone for the purpose contained 30% CaO. The S.G. of limestone was 2.7. The mined material had a top size of 40 100cm after screening through a grizzly. The kiln accepted top size of 10cm. A single toggle Blake jaw crusher was available for crushing. Assume the shape factor of the feed and the product were the same.

A jaw crusher was used to crush a chert ore. The top size of the ore was 25cm and the moisture content was less than 3%. It was required to produce a product 100% of which would be less than 4cm. The shape factor of feed and product was 1.7. Assume that the cumulative weight-size curve was a straight line, determine:1.crusher size,2.rate of crushing (QT).

A jaw crusher had a gape of 685mm. It was charged continuously by a conveyor belt to keep a charge level constant at 46cm from the bottom of the jaws. A reduction ratio of 7.5 was desired. If the maximum opening between the jaws at the discharge end was fixed at 20cm for a material of density 2.8, compute1.the angle between the crusher faces (assume flat),2.operating speed and critical speed of operation,3.the rate of crushing when the angle between plates is increased by 2.

The angle between the straight faces of a Blake jaw crusher was progressively altered from 22 to 28 in steps of 2. 1200kg of a material of bulk density 1460kg/m3 was crushed each time. Indicate1.the adjustments to the set that would be necessary each time to maintain same production rate,2.the mathematical relation between the angle of nip and the set.

Iron ore was crushed in a jaw crusher. The average sizes of the feed (F80) and product (P80) were 50 and 10mm, respectively. The energy consumed during crushing was found to be 5kWh/t. The top size of the material was then altered to an average size (F80) of 75mm when the product size (P80) of 5mm was required. Estimate the energy to crush in the altered condition.

A single toggle jaw crusher crushed limestone having an average size of 75mm. The size analysis of the product wasSize (mm)Mass % RetainedSize (mm)Mass % Retained12.50.201.515.07.58.00.755.25.051.00.402.12.513.00.205.5

The closed set of an operating jaw crusher was 125mm. A continuous stream of ore was fed at the rate of 30t/h. On an average, 10% of the ore was less than the set. The F80 was 410mm in size. The crusher was initially operated at 200rpm at a reduction ratio of 1:4, but the toggle speed was to be increased. Calculate1.the maximum speed, C, at which it can be operated,2.the maximum capacity at the maximum operating speed of the toggle.

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