kicks law with the help of jaw crusher

jaw crusher - sciencedirect

jaw crusher - sciencedirect

Designs of different types of jaw crushers such as Blake, Dodge with single and double toggles used for initial comminution of minerals, as received from mines, are described in detail. The method of calculating operating variables such as the critical speed, toggle frequency and throw and power consumptions are explained and illustrated with practical examples with solutions. Further, inter-related mathematical relations between variables such as critical speed, toggle throw, frequencies of operation and crusher power and throughput, as derived by different workers, are indicated.

rittinger - an overview | sciencedirect topics

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 [2], Kicks [3] and Bonds [4].

The commonly used method to evaluate the grindability of coal in medium speed pulverizers is the Hardgrove Grindability Index (HGI) [5]. 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 [6]. 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 [79]. 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.

how to kick faster? improve kicking speed law of the fist

how to kick faster? improve kicking speed law of the fist

Kicks are a thing of beauty except theyre devastating. The sound of shattered ribs and a broken jaw as your opponent tries to hold it on together after the crunching kick that you just landed is simply a work of art.

For a kick to be devastating, it needs to be lightning quick along with being powerful and measured. If it lacks speed, your opponent will easily predict the movements and will manage to outmanoeuvre you every single time.

Thus, this blog is about helping you to learn how to kick faster with the help of several drills and techniques which will make your kick venomous. By the end of this read, you will have the basic knowledge essential to knock someone out before they even get close to you.

So having learnt about the kick types, we can understand that hips, waist and core are crucial in delivering a venomous kick. The question is how do you make it quick? How do you improve your kicking speed?

There are numerous drills which can help improve your kicking speed; the primary one is stretching and flexibility. There are three muscle groups that we need to work to get faster kicks. The first one is the hamstring group followed by adductor (inside portion of your leg) and your upper legs.

By slow, I meant slow motion. Shadow your kicks, while performing your kicks, try slowing them down to check the flow of it along with your body form. This will help you understand kicking abilities better including your kicking speed. Muscle fibers increase endurance and help develop rhythmic pace essential for speed kicks.

This drill is great to build your leg muscle. To perform lateral box jumps, you need to stand on a sturdy platform or a box (not the cardboard one).Now you need to stand on the left side of the box and jump towards the right and vice-versa.

Keep building up the speed as you keep jumping. (You can take this one step ahead by jumping forward and converting this stance into a squat, this explosive drill will help build your legs and improve the overall speed).

Easy visually, difficult to emulate. Long jumps are quite effective and help you generate speed quickly due to your thrusting hips. To do this, swing your arms and jump as forward as you can. Do this at least 10 times daily to witness effective results over some time.

Stand up on stairs, preferably a couple or triple. Step off the stairs and upon touching the ground, jump up with both feet. In this drill, the emphasis should be on how quickly you can jump the moment your foot touches the ground. Once, you master this exercise; your front kick will improve drastically.

For a speed kick exercise, you need to have a partner. Herein, you hit the punching bag held by your partner and ask him/her to give a call randomly. This will sharpen your alertness and will increase your instincts to kick at the right time. Not only will it improve the speed but also the precision of your kick.

Lateral leg raise is a great exercise to strengthen and improve the overall speed of your kicks. It requires you to lie down flat on the ground and raise both your legs together towards the ceiling. Take it as far as you can. Not only will it build the core but also enhance the overall kicking speed.

There are many exercises which you can do with the help of a wall. These include kick holds, figure 8, and kick stretch. It requires you to touch a wall with your left hand or vice-versa and practice the hold.

All these drills require you to hold your kicks mid-air as it improves the overall speed and precision of your kicks. Figure 8 involves drawing an 8 in the air after holding your kick mid-air and kick stretch as the name suggests refers to stretching as far as possible.

You might practice all that you want but the true abilities can only be tested during a sparring session. Youll improve your overall movement, reflexes and kicking speed when you have a sparring partner. If youre not much into sparring then you can try kicking a bag and keep a track of the progress.

Practice disguised kicks. Often utilized by professional martial artists, disguised kicks are quite exciting. Herein, you try to distract the opponent with your punches while gearing up for a knock-down kick, out of nowhere. The key point in such kick is to not let your body posture give away your next move. Disguised kicks are rapid and would help you take down your opponent in a flash if delivered correctly.

Stretching helps you train the muscles to relax in a certain position. As long as you hold the stretch, the muscles learn to loosen when you go back to the given position. You should put a tremendous amount of energy and efforts in stretching.

Its all personal preference. Its quite hard to state which ones the fastest as there is many and each one equally good. It all depends upon the individual. But if I were to pick one, I would say Taekwondo.

Ive seen 6-7 kicks in less than 1 second from Taekwondo practitioners, but that doesnt state that Muay Thai or other martial art forms cannot achieve this. As is said, its all personal preference, and it comes down to an individuals skill.

However, I can surely point out one thing, though Karate is great at offence, its not as fast as Taekwondo when it comes to kicks. Having learnt about the importance of stretching, let us check out drills which can improve your kicking speed. Note that, kicking speed has got a lot to do with your core strength. Thus youll find exercises which emphasize on your core and control along with the striking exercises.

Unless youre bulking up and gearing to become huge, its not going to affect the speed of your kicks. Many coaches around the world recommend adding weight training to the drill to strengthen the core and your legs which can result in faster kicks.

However, theres a proper proportion which needs to be maintained while weightlifting. You must ensure not to go overboard and exert unnecessary pressure on your legs. Moreover, you must play safe with explosive drills because they can get you injured sooner than you expect and might also result in a threatening blow.

Remember, to kick fast; you need to start slow. Keep practicing with patience and do not get overwhelmed with results. You have a better kicker standing right next to you. Also, do not forget to warm up before you approach the explosive drills. Negligence will only lead to harm.

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rock crushing theory and formula using kick & rittinger's law

rock crushing theory and formula using kick & rittinger's law

The work done in crushing is proportional to the surface exposed by the operation; or, better expressed for this purpose, the work done on a given mass of rock is proportional to the reciprocal of the diameter of the final product, assuming that all the mass has been reduced to one exact size, which is only theoretically possible.

Kicks law is:The energy required for producing analogous changes of configuration of geometrically similar bodies of equal technological state varies as the volumes or weights of these bodies.In other words, the energy expended is porportional to the volume reduction, instead of the diameter reduction.

That these two laws would give widely different results may be shown by a simple imaginary case. A ton of 16-in. cubes is broken to 1-in. cubes at the first operation, and these are broken to 1/16 -in. cubes at a second operation. Since the first operation produces only one-sixteenth as much new surface as the second, the ratio of energy expended in the two operations would be by Rittingers law as 1 to 16. By Kicks law, since the volume-reduction ratio is the same in both cases, the ratio of energy would be 1 to 1. This discrepancy is great enough to challenge a test by actual experiment, of which there has been, hitherto, but little compared with the amount of argument. The present paper is the record of such a test, and is offered as showing that Rittingers theory more nearly represents the actual facts that any other proposed hitherto.

Mr. Stadler, in the paper already cited, applying to rock fracture Kicks law (which I accept as applicable to the deformation of elastic bodies), proves that the stamp mill, working on relatively coarse feed, is several times as efficient as the tube mill on fine feed, this relation being expressed in what he calls relative mechanical energy per horsepower. He distributes his energy units among the various sizes produced, according to Kicks law. But my results show, among other things, that many more of Stadlers energy units are obtained per foot-pound-applied, in coarse than in fine crushing. This suggests a doubt as to the correctness of his unit, since one would expect a foot-pound-applied to produce the same number of energy units, whether the feed and resulting product were coarse or fine. When Rittingers theory is applied, the number of mesh-tons (my unit of surface produced) is nearly proportional to the foot-pounds-applied, whether the product be coarse or fine; hence my conclusions favor, the simple Rittinger theory that surface produced is proportional to energy applied.

It is the purpose of this paper to record and interpret a series of experiments made for the purpose of studying the consumption of energy in the crushing or fracturing of rock particles both coarse and fine. At the start, the expectation was to determine constants according to the Rittinger theory for several rocks, whereby it would be possible to calculate absolute crushingefficiencies of machines crushing such rocks, or to predetermine what a machine would accomplish in the way of tonnage and screen analysis upon a given rock, knowing what the machine accomplished upon another rock whose constants were likewise known. While approximate constants for a few rocks were determined, the results indicated that an amount of work (perhaps with the microscope upon sizes too small to be screened) greater than could be accomplished by a single investigator would be necessary to determine these constants with reasonable accuracy. For this reason no constants are submitted as such in this paper.

It is possible, however, to compare by means of these experiments! the two antagonistic laws of crushing already mentioned. The fracture of cubes by compression in a testing machine produces, in nearly all cases, pyramids of the same general shape (Fig. 1), indicating that the stresses are greatest along certain planes of fracture; and, since the relative movement must have been greatest along these fracture or shear planes, the greatest amount of energy must have been expended there. While I know of no tests that show that the several pieces return to thedimensions existing before the break, it seems reasonable that they do, and that they do not retain the deformation. All that has resulted from the pressure and consequent movement is cracks. They are new, while the particles resulting are of the same form and specific gravity as they were before separation; and if put together accurately the dimensions of the mass would be the same.

Rittingers theory is the law of the cracks. It is well explained in an article by E. A. Hersam in the Mining and Scientific Press, an abstract of which is given in the third volume of Richards Ore Dressing, and to which I acknowledge my indebtedness for some of my ideas and,methods.

Assume a cube of rock (Fig. 2) of side D, divided into smaller cubes by planes passing through it in three directions, as shown. The surface of the original cube being 6D2, the aggregate surface of the smaller cubes will be 6ND2, and the new surface exposed will be 6(N 1)D2. The theoretical mesh or reciprocal of diameter of new particles, M, is N/D hence N = MD; and the aggregate final surface, S = 6ND2 = 6MD3.

With D constant, that is, starting from the same size of cubes in both cases, S : S: : M : M, or the total surfaces of equal volumes of rock, each composed of particles of uniform size and shape, are to each other as the theoretical mesh or the reciprocal of the diameter of the particles composing the respective masses.

that is, the summation of the weights by the theoretical meshes, where k is a constant, six times the volume of unit weight of the rock. If different rocks were considered, specific gravity would enter into the value of k. Likewise,

For a given screen analysis, weights of the different theoretical meshes, we may plot on co-ordinate paper x against M and x against M, and obtain a crushing-surface diagram (Fig. 3) representing the equation

above given for new surface, the area between the steps being proportional to the difference in surface area: If we take closer divisions of M, smaller differences in x will result, and the limits of such divisions will result in the smooth curves, the area between which is a true measure of the surface produced.

Thus on the Rittinger theory that surface produced is proportional to energy appliedthe area of the diagram is a measure of the energy going into the crushing. (It should be distinctly understood that bearing- and gearing-friction losses should not be included in the energy-applied measured by the crushing-surface diagram, and that it is therefore impossible to compare the two laws by. tests upon commercial machines, unless the friction losses are known.)

The crushing-surface diagram has the advantage over any other graphic method of comparing screen analyses, that it shows at a glancethe classification of the pulp, the distribution of the surface, and (according to the Rittinger theory) the energy spent in its production. As the cards of a steam engine enable us to determine the power developed, and to locate faulty operation or design, so the crushing-surface diagram enables us to determine the power consumed, and to locate points at which the crushing practice may be improved and faults remedied in design.

100 Mesh Cube 200 Mesh Cube Area one section = A Area one section = A/4 Average resistance to shear per sq. in. = F Energy = FAD Energy = F A/4 D Surface Produced = 2 A Surface Produced = 2 A/4

Suppose it were possible to hold two similar cubes between the two offset faces as shown, and that forces were applied until the deformation shown by the dotted lines was obtained, it will be seen that only the molecules along the vertical center line

[it should read plane] are stressed and deformed, the mass of the cube away from this surface receiving practically no pressure or deformation. The energy in this case required to produce rupture will be the product of the average resistance to shearing per square inch by the area along which rupture takes place, and by the distance the two offset faces move together. The average resistance to shearing is a variable quantity, as the deformation increases up to rupture. To reduce to cubes, this amount of energy must be multiplied by three, as three similar fracture planes must be made to produce cubes. And it will not be hard to see that the distance through which the offset faces must move in either case must be the same and not proportional to the thickness of the piece. To break the molecular bond between adjacent particles would require the same movement, regardless of the thickness of the piece.

It will be noted that if eight of the half-diameter cubes be sheared to produce cubes of half their size, the new surface presented will be double that formed when the single large cube is sheared into half-size cubes, and also that the energy required in the case of the smaller cubes is double that required in the case of the larger cubes. This should demonstrate that energy applied, to crushing is proportional to the surface produced.

New units are apparently necessary when using the crushing- surface diagram. The term mesh-grams as a proposed unit for crushing constants and measurement of surface I have defined.as the surface produced by a theoretical crushing operation in which one gram of particles of the same diameter are all reduced to a diameter whose reciprocal is one greater than before reduction. The unit mesh-ton i have similarly defined as the increased surface produced by crushing

As the crushing contemplated in the definitions is only theoretically possible, a couple of examples may explain them better. Shapes remaining the same, preferably cubes for arguments sake, during all stages of the crushing, if 1 g. of 0.1-in. rock (reciprocal, 10) be crushed until it is all 0.05-in. (reciprocal, 20), the new surface produced is 10 mesh-grams. Or if 10 g. be reduced to exactly 0.01-in. from exactly 0.1-in. particles, there will be an increase of 10 times 100 minus 10 times 10,. or 900 mesh- grams. Mesh-pounds, mesh-ounces, mesh-percentages, etc., are similarly justified and defined.

The term mesh for a measurement has been objected to, as ambiguous in meaning. Some other short term may be desirable to take its place in forming these compound names for surface produced. But nothing better has been suggested. Reciprocal (of diameter) is too general and too long; theoretical mesh-tons is likewise too long and is also liable to misinterpretation; dia-ton has been suggested; but it would hardly do to speak of dia-grams in connection with a diagram. Perhaps pitch-grams, pitch-tons, etc., would be satisfactory, pitch being defined as the theoretical mesh or reciprocal of diameter(probably in inches). I would here again point out that these quantities are measures of surface, and that the weight element is only permitted to enter on the basis of equal specific gravity; where the specific gravity varies, a correction must be made for it.

100 Mesh Cube 200 Mesh Cube Area one section = A Area one section = A/4 Average resistance to shear per sq. in. = F Energy = FAD Energy = F A/4 D/2 = FAD/8

This law does not apply so much to crushing as to deformation of bodies before rupture takes place. In Fig. 6 are represented two particles of ore of equal technological state, shown as cubes between the faces of a crushing or testing machine. Assuming the theoretical mesh, equivalent to the reciprocal of diameter, and using concrete values, we have a 100-mesh particle with eight times the volume of the 200-mesh particle, and with an area per face four times that of the 200-mesh particle. The dimensions are as two to one, and the bodies being similarly deformed within the elastic limit without fracture, the energy that must be applied in each case to produce this deformation is the product of the average resisting force per square inch, the same in both cases, by the area worked against and by the distance through which this average force works. As shown in the figure, in this particular instance the energy absorbed is proportional to volume, and it can be similarly shown for the general case. On the gradual release of the external pressure the energy absorbed is given back to the machine producing the deformation [better, which produced the deformation] and the body returns to its original shape. It should be noted that the body has been deformed only by a gradually increasing pressure, the first increment of deformation not requiring so much pressure as the last. In case the body has been deformed beyond its elastic limiteither the whole mass of particles have been reduced to the molecular state by the freeing of their bonds with adjacent particles, which never happens, or fracture takes place along a few surfaces by the breaking down of some of the weaker bonds, and the particles thus formed are free to resume their original shape in so far as they are not held between the machine surfaces. The energy given up by them is probably used in some sort of lever action in making fracture planes. So the energy absorbed according to Kicks law does not stay in the particles after pressure is released and therefore this does not govern to any great extent the amount of energy absorbed in crushing.

There is no incompatibility between the Rittinger theory and the laws of elastic bodies such as glass, steel, rock, etc. Up to the elastic limit, deformation is proportional to stress, and absorbed energy is

proportional to the volume of the particle stressed, according to Kicks law. In Fig. 7 the area Oyd is a measure of this energy. When pressure within the elastic limit is released the body returns to its former shape, there having been no interior breaks so far. When, however, the elastic limit has been exceeded, certain local ruptures have taken place within the body at points weaker than others; these breaks occur consecutively or simultaneously at various places within the body, and upon release of pressure the body is found upon careful measurement to have taken on a permanent set. The first fracture noticeable occurs after several ruptures have weakened the body at one point, which, may be along a plane of greatest resultant stress, or of structural weakness. Referring back to Fig. 7, the elastic limit having been exceeded, as pressure is gradually, released, its relation to the deformation is shown by the line af, the area afn being a measure of the energy given back to the machine(by Kicks law), while the area Ofa is a measure of the energy absorbed by the local breaks, which may or may not have become visible. In the case of a wrought-iron or mild-steel test bar undergoing tension between the jaws of a testing machine, local ruptures are probably occurring throughout the bar, reducing the section at those points of weakness, and putting greater load upon adjacent particles which may be then able to carry the load better.

The final rupture of the steel bar occurs when a number of local breaks , at one particular section have so weakened the bar that all the work the machine is doing is concentrated at that section, as shown by the necking in just before failure takes place. Unquestionably in spite of the permanent set in the test specimen away from the point of failure, the majority of the crystals composing the specimen are in the

same form as before the test, but some have slipped away from others and there is a new adjustment due to the failure of the few which could not hold on, the others at that section being in position to support the load for a further period. If it were wrought iron, we could say the fiber slipped along the slag or impurities between fibers until the slipping fibers were crowded so that they obtained a sort of strangle hold.

In the case of the shearing of iron and steel bars, it has been shown that the depth of penetration (the distance traversed by the cutter before separation takes place) is not proportional to the thickness of the metal, as it would be in case Kicks law applied, but is proportionally greater in thin sheets than thickabout 75 per cent, of the thickness in the case of sheets 1/8 in. thick, and about 25 per cent, in the case of sheets 1 in. thick.On the face of it, this fact does not uphold the theoretical argument for Rittinger any more than for Kick, but let us look farther.

In a commercial shearing machine we do not get true shear, but a condition shown in Fig. 8. The material is compressed under the blades of the shear, which are themselves somewhat distorted during the operation, and failure is due to tension in a line inclined to the plane of the cutting edges. In the case of a thick plate there would be more deformation of the cutting edges due to the higher resistance of the material, and more of a width concerned than in the case of thin sheets. When a hole is punched in steel, there is a volume of material around it that must be removed for first-class work; this volume is probably full of minute fractures the area of which

cannot be measured. I am told that rock crushed for concrete, is seen upon microscopic examination to be full of minute fractures, due to the crushing, which cause each piece to be weaker than a natural pebble of the same mineral and size. If the rock were uniformly full of these invisible fractures; Kicks law would apply in spite of all our theoretical reasoning. Just as in the case of the hole punched in steel, I believe that these minute fractures in the rock are close to the fracture planes.

A. F. Taggart, has discussed Kicks law thoroughly from the standpoint of its advocates and there is no reason for a repetition of his work. However, taking it at their valuation, and using data obtained from Stadlers papers already referred to, we can by reasoning similar tothat, already employed develop a diagram for energy absorbed similar to the crushing-surface diagram. (Fig. 9.)

D and d being initial and final dimensions, V and v being initial and final volumes. For example: Reducing 1-in. cubes to one-half volume, 1E. U. has been expended; reducing to one-fourth volume, 2 E. U.; to one- eighth volume, 3 E. U.; to one-fourth dimensions, 6 E. U.; to eighth dimensions, 9 E. U.; and so on. Reducing from 100 in. diameter to 1 in. the same number of E. U. are expended as from 1 in. to 0.01 in., or as from 0.01 in. to 0.0001 in.(all on quantities of the same volume, both here and in the previous sentence). Of course various sizes are produced in crushing, and each grade or size has a different value, the same reasoning applying as in section 4, above.

The new mechanical value, W W =c 1n(x E. U. x E. U.).Plotting x against E. U., we have, when the limiting curves are drawn, a diagram (see Fig. 10) similar to the crushing-surface diagram, the area of which is a measure of the mechanical value of the crushing operation or the energy units (E. U.) absorbed. (Percentages should be expressed decimally in the above calculations.) The mechanical value of sands given in E. U. from the diagram is multiplied by the tonnage per day (or hour) and divided by the horsepower, giving the relative mechanical efficiency.

The E. U. per foot-pound applied upon 100 g. as found from these mechanical-value diagrams compared with mesh-grams per foot-pound as found from crushing-surface diagrams for the same crushing operation will, if covering a range of operations show which of the two laws under consideration is consistent with the facts. This is the method used incomparing my results. (In Fig. 9, it will be noted that the theoretical mesh or reciprocals of diameter are designated at the corresponding Ordinal numbers of energy units, which is self-explanatory, and perhaps permits easier comparison with the crushing-surface diagram for the same crushing operation.) The following comparisons based upon the experimental work performed show that the Rittinger theory applies to crushing operations and that Kicks does not. There are discrepancies when we apply the Rittinger theory; butthese can be explained reasonably by considering the expenditure of energy upon the sizes finer than our screen permits our measuring, which must present considerable

In this work one or many pieces of rock were compressed betweenthe faces of the Amsler-Laffon hydraulic testing machine in the Laboratory for Testing Materials of Purdue University. (This machine is described in the appendix.) The movement between the faces (similar to those of a crusher) was measured by a deflectometer having a vernier on which movements of 0.001 in. could be read, while the pressures between faces was read upon a dial to a maximum of 150,000 lb:. The product of the average pressure (or rock resistance) by the distance through which that pressure was exerted is work or energy, and easily reduced to foot-pounds.I found it easier to plot my results as obtained, pressure against distance, as. shown in Fig 10, the area of the closed curve, completed by the record of pressure and distance when pressure was gradually released, being a measure of the energy in inch-pounds (divided by 12 equals foot-pounds). This energy as foot-pounds, or as horsepower-hours, can be compared with the surface produced as shown by the crushing-surface diagramof the screen analyses of initial and final products of the operation; and it can likewise be compared with the energy units (E. U.) of Stadler, as shown by mechanical-value diagrams similar to Fig. 9. By crushing the same amount of rock from coarse to medium, from medium to fine, from fine to very fine, etc. (coarse and fine being merely relative terms), the two laws may be compared through a sufficient range to detect any abnormal results or inconsistencies.

To each test, whether it involved one or several consecutive crushing operations, was given a letter by which it is designated throughout this paper. My records are in the form of crushing-surface diagrams with the weights marked to place as plotted. From these the mechanical- value diagrams must necessarily be derived, as this work was not undertaken primarily to settle the case of Kick vs. Rittinger. The pressure- deformation curves were plotted directly in a similar manner without recording numerical readings. The advantage of recording by plotting the observations directly upon co-ordinate paper is the speed at which it is done, and the check afforded by the shape of the curve. It is hoped that the work will be acceptable in its present form, as numerical tabulations would add nothing of value to this paper. So far as possible the results of calculations are given in the diagrams, and hence it will be unnecessary for the reader to refer back and forth from text to diagram to any great extent.

As the crushing-surface diagram does not depend upon any definite system of screens so long as the aperture be known, all the screens used were measured with a Bausch & Lomb microscope having a scale reading to 1/1500 in., and the reciprocals of diameter of these apertures were used in plotting.

Certain of the finer sizes were screened through bolting cloth held in embroidery frames. I have suggested specially prepared plotting paper,i. e., with vertical lines drawn in at the reciprocals of diameter corresponding to the apertures of the screens used, for such cases as the Tyler standard screen-scale sieves, or the I. M. M. standard. Since this paper was first submitted to the Institute and of course since the work was done, I have discovered that cumulative direct diagram sheets with Tyler S. S. S. sieves are desirable for this work and that there is no need of any other sheets for crushing-surface diagrams. Fig. 11 shows a crushing-surface diagram on this paper, the plotting running in the opposite direction from that for which it was designed; 1 sq. in. on the regular sheet, with, screen sizes located as indicated, representing 8.85 theoretical mesh-tons of surface per ton of ore crushed.

All results have been modified to a basis of 100 g., so that percentages required for use with E.U. are readily obtained, about one-third of the tests having been made upon a definite 100 g. In none of the

tests was any attempt made to account for the losses occasioned by fine dust floating away, or by coarser pieces shooting off as they were fractured. In no case could these be more than 2 per cent., which would not materially affect the results. Extreme accuracy was not particularly necessary in this work. It was rather a reconnaissance; and if more than three, significant figures are given at any point, too much importance should not be attached to the last figure. At the same time the work was carefully done and the record carefully made, but there is so much unknown that I would not care to defend the fine details of my results.

In Fig. 12, A and B are results obtained upon a tube-mill feed, a fairly siliceous gold and silver ore from Mexico, of which 100g. of 5/16-in. + 4-mesh pieces were crushed by two movements of the machine, without

intermediate screening, the purpose being to crack every piece. The energy is represented by the areas A-1 and A-2. It is of interest to note how the release of maximum pressure is usually accompanied by some movement of the jaw, as in this case, indicating a return, of energy to the machine, shown in the diagram by the slight curve back to the zero-pressure line. After the particles adjusted themselves upon release of the pressure it took more movement between the crushing faces to get up to the previous pressure, as indicated by the gap between A-1 and A-2. After sizing and plotting the screen analysis in the form of a crushing- surface diagram A, the particles were re-crushed eight times, each time being screened through a screen of about 30 mesh to remove the fines and eliminate the condition of choke crushing as far as possible, the oversize only going back to the machine. The energy areas for these eight crushings are B-3, B-4, B-5, B-6, B-7, B-8, B-9, and B-10;the screen analysis is plotted as a new side of the crushing-surface diagram, B. The areas of the energy diagrams A-1 and A-2 were found by planimeter to be 1.40-and 2.38 sq. in. respectively, a total of 3.78 sq. in. One square inch, for the scales.used, being equal to 250 in.-lb. or 20.83 ft.-lb., 3.78 times 20.83 or 80.5 ft.-lb. of energy was expended in making the crushing A. By similar reasoning, the area of the crushing-surface diagram A, as found by planimeter (as far as screening was carried but, to the 200-mesh sieve), multiplied by the value of its unit area in mesh- grams, shows that 1,400 mesh-grams of surface was produced. Fromthese two quantities, a value of 1400/80.5 = 17.3 mesh-grams per foot-pound(+200-mesh sieve) was obtained, as the constant to be used in calculating efficiencies, etc., for this rock. Note, however, that the mesh-grams entering into this constant are only those plus 200-mesh sieve, and that there are perhaps more mesh-grams in the minus 200-mesh than in the plus, and they must at some time be taken care of in calculations. It is these minus mesh-grams which will account for the discrepancies in results by Rittingers theory, and which will not account for discrepancies in results by Kicks law.

For B, the total energy expended was 401 ft.-lb., producing 6,900 mesh-grams ( + 200-mesh sieve) giving a unit value of 17.21 mesh- grams per foot-pound plus. Fines were screened out after each jaw movement, to reduce choke crushing.

Tests E, F, G, H, J, K, L, M, and N were made upon Bedford limestone, the first in July, 1912, and the rest in July and August, 1913. This is one of the best known building stones in the United States. It occurs in great beds in southern Indiana, is oolitic in texture (weathering brings out myriads of fossils) and, because of its apparent uniformity, is probably as good a rock as could be selected for a standard. (I reserve the right to change this opinion later.) The particular specimens came from stone furnished for the Purdue Library building.

The crushing surface diagrams of E and F (Figs. 5 and 4 in my original article in the Engineering and Mining Journal of May 24,1913) arenot shown here. In E, practically all the material was broken in the machine, whereas in F, a great many pieces between 30 and 80 mesh were not broken, but simply had their edges rubbed off. My records as to F

Test G was on a block of the dimensions shown in the smallsketch in Fig. 13, having a superficial area of 63 sq. in. and a weight of approximately 1,031 g. The energy curve is rather long drawn out after the first break, and the irregularities in this curve may be explained

as due to frequent ruptures of fairly large pieces where the load drops back almost to zero. A total movement of nearly 2 in. is recorded: here only one or two pieces perhaps were resisting, while in some of the other cases a great many pieces were resisting at one time. At the end of the first in. of movement the finer pieces were screened out, and the larger returned to the crushing zone; these pieces being rather flat; about 2 per cent, of 200 mesh being produced.

H, shown in Fig. 14, covers the results of crushing four pieces, having a total superficial area of 30.25 sq. in. and a total weight of 175.5 g. After crushing, eight pieces weighing 163 g. had an area of 30.25 sq. in., just, offsetting the original area, while the remaining 12.5 g.were screened, and the results were plotted as a crushing-surface diagram. 12.66 ft-lb. were expended in crushing the 175.5 g., equivalent to 7.2

ft-lb. per 100 g.; the eight pieces noted above were estimated to be the equivalent of 60 per cent, of 1 in., 32.5 of -in., and 7.5 of fines, as recorded in the crushing-surface diagram. The energy curve is irregular asin the case of G and for the same reason, movement was 0.4 in.; there was no intermediate screening.

In J (Fig. 15) a single rectangular piece about 1 by 1.25 by 1.25 in., weighing33 g., was crushed to pass a 10-mesh sieve, with one intermediate screening. After a deformation of about 0.6 in. was recorded

without great-pressure, the pressure goes up very rapidly, probably by reason of the packed condition of the particles, a great number resisting crushing. After screening out the 10 mesh, a similar energy curve is obtained on the + 10-mesh pieces. The crushing-surface diagram represents the combined product from both crushings. In both cases, the

Fourteen very flat pieces weighing 55.6 g., with about 20 sq. in. of surface, averaging in size about 0.75 in. sq. by 1/8 in. thick, were subjected to four separate crushing operations, K, L, M, and N, as shown in Fig. 16. The first crushing K was continued until every piece was found to have been broken at least once; screen analysis was then made and recorded in the crushing-surface diagram, and the material +100 mesh was recrushed as L. M is the result of crushing in a similar manner the +80 mesh from L; and N is similarly the result from the +80 mesh of M. Each crushing was carried as far as the record shows without attempting to remove any fines, although in the case of N pressure was released once and then reapplied, as recorded, in an endeavor to apply more energy. The base lines of L, M, and N separate from the discharge or product lines of the preceding operations, showing the smaller quantities of material crushed in each case.

In Fig. 17 the results obtained in A and B on the siliceous Mexican ore have been plotted in a mechanical-value diagram. For the coarser crushing A, 0.041 E. U. per foot-pound-applied on 100 g. was the result,

while in the case of B only 0.018 E. U. (etc) resulted, 125 per cent, more being produced when the coarser crushing was done. If graduated crushing had not been done, more energy would have been expended upon the 200 mesh; and I believe this discrepancy would then be greater.

In Fig. 18, E, F, H, and J (Bedford limestone) have been plotted as mechanical-value diagrams. Note how the E. U. per foot-pound applied on 100 g.,decrease consistently from coarse to fine: H, 0.375; J, 0.238; E, 0.051; and F, 0.011. Similarly in Fig. 19, K, L, M, and N are plotted to E. U. diagrams with results decreasing consistently from coarse to fine: K, 0.135; L, 0.071; M, 0.0544; and N, 0.0372 E. U. per foot-pound-applied on 100 g.

On another ore, an Arizona porphyry, a series of crushings similar to K, L, M, and N, gave results from coarse to fine as follows: O, 0.055; P, 0.049; Q, 0.0296; R, 0.0149; S, 0.0127; all E. U. per foot-pound- applied on 100 g. The range of sizes was small; the initial feed averaged

Fig. 20 shows the results of plotting an approximate average of feed and discharge for all tests against both the E. U. per foot-pound- applied on 100 g. and the mesh-grams per foot-pound (+350 reciprocal).

The estimated average, taken as near to the center of gravity of the mechanical-value diagrams as possible by ocular observation, gave the horizontal position of each point, while the calculated E. U. and mesh- grams gave the two vertical positions.

In the upper diagram the inclined line (Bedford limestone) connecting the various points shows that for different average sizes of material undergoing crushing, the production of E. U. per foot-pound-applied on 100 g.decreases from coarse to fine; according to Stadlers interpretation of Kicks law there should be no decrease, they should be the same at all sizes. From the values tabulated in Table I, which is a summary of all these tests and calculated results, it will be found that for the Bedford limestone, the coarsest crushing gave 34 times the number of E. U. per foot-pound-applied on 100 g. that the finest crushing gave. For the other materials, the graphs incline similarly and in sufficient degree to show the fallacy of Stadlers application of Kicks law.

In the lower diagramproduce the same number of mesh-grams at all sizes, then the Rittinger theory is correct, and a horizontal line represents it in this diagram. If some of the discrepancies be explained as due to choke crushing, producing an excess of unmeasurable 200 mesh ( 350 reciprocal), the majority of points will come nearer to a horizontal line, and better confirm our belief in the Rittinger theory. The results clearly show that the Rittinger theory is so much more nearly correct for crushing operations, that I believe we can accept it and use it for the determination of crushing constants, for machine comparisons, for efficiency tests, etc.

Since it is obvious that the sizes finer than our ordinary screens will separate are important if the Rittinger theory be correct, finer screens or classification followed by microscopic measurements will probably be necessary for accurate determinations. To some extent at least, we can approximate the quantity of the finer sizes; there seems to be a law connecting the quantities of different sizes, that the discharge line of a crushing-surface diagram from a uniform crushing operation will be in the form of a hyperbola, and, when plotted on logarithmic paper, as in Fig. 21, will appear as a straight line. In Fig. 21 the product line of N indicates that it would have some 4 per cent, of 1,000-reciprocal size and some 0.6 per cent, of 10,000-reciprocal size; they have surfaces respectively equivalent to 40 per cent, of 100 reciprocal, and 60 per cent, of 100 reciprocal, taken separately; and if these values are correct, these sizes are fully as important in the consideration of energy disposal as all of the + 100-reciprocal material. The law of the finer sizes might be something of this form:

R being the reciprocal of the diameter (theoretical mesh) under consideration; W, the weight of the material passing through a screen of this diameter opening; and x, a constant depending upon the character of the

ore, the type of machine in which crushing is done, and possibly upon the screen analysis of the feed; and C, the number of crushing units.(mesh- grams, mesh-tons, etc.). in the undersize, considered as all of that one size, and proportional to the area between the co-ordinates of the point on the curve (intersection of C and W) and the zero lines.

Table I gives a summary of results. It will be noted that in most cases the high values for mesh-grams per foot-pound were obtained with low values in foot-pounds, and vice versa. This seems to indicate that the highest values and therefore the most efficient crushing would be obtained when each particle was just cracked; a thing that is generally understood to be true in connection with rolls, for example, free crushing being, within limitations, more efficient than choke crushing. Choke crushing here gave uniformly, low results, as might be expected. In the case of B, the several crushing operations involved small amounts of power, but they were lumped together for final screen analysis. If all the power had been applied at one time, without intermediate screening, there would not have been such a nice agreement in the mesh-grams per. foot-pound, between A and B.

As the writer sees the situation, after the acceptance of the substantially correct theory of crushing, there must be recognition of the fact that rocks of varying characteristics are met in crushing operations, that these rocks offer different resistances to crushing, and that it is unfair to compare the operations of two crushing machines upon dissimilar ores (even if they look alike) until some values are given to the resistance or energy absorption of these rocks under crushing conditions. Probably these units will be determined practically by applying a definite amount of energy to a definite weight of the rock, all of a predetermined standard size. The work of determining these constants should be done thoroughly by competent hands. The matter of checking my conclusions can be accomplished perhaps in several of the school testing laboratories of the country. No elaborate apparatus, and apparently no very high degree of accuracy, will be required to prove that Kicks law is of no use to mill operators and engineers.

This paper has been prepared under some disadvantages which may account for the possible lack of logical arrangement. The matter was originally submitted to the committee of the Institute in May, 1914, and upon their suggestion the paper was enlarged, corrected, and rearranged. The writer acknowledges his indebtedness to their suggestions. Acknowledgement should also be made of the courtesies extended byDr. W. K. Hatt, head of the department of Civil Engineering, and Professor Schofield, at Purdue University, who allowed the writer free run of the laboratories for Testing Materials.

The Amsler-Laffon hydraulic testing machine, with which these experiments were made, is shown in Fig. 22. The material is crushed between the faces of the jaws J by pressure applied against the ram R, contained in the cylinder C. The head H carrying the upper jaw can be moved up and down the fixed screws S by nuts having worm-wheel threads cut on their outer surface into which mesh two worms driven by the hand crank shown to the left.

The pump P for supplying the pressure to the ram cylinder C is operated by a hand screw on the side of the frame carrying the registering and auxiliary mechanism. Suitable piping connects the pump with the cylinder, the supply tank O and the registering cylinder X. From X suitable links and levers connect to the pendulum and adjustable weight W, the swing of the pendulum, which is proportional to pressure exerted, being recorded on the drum or indicated upon a dial D. In all of this work the readings of the dial were used. With different settings of the weight W, four different scales of pressures having maximum values of. 15,000, 50,000, 100,000 and 150,000 lb. may be used, suitable dials being provided.

The deflectometer, which is a part of the regular laboratory equipment, consists of a lever carried at its fulcrum in a suitable base. At the short end of the lever a fine-thread vertical adjusting screw can be brought into contact with the upper jaw J of the machine, while the long end of the lever fits against a vertical scale, andmoves a vernier with it as it moves up and down, thus permitting the measurement to 0.001 in. of the movement between jaws.

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