ABSTRACT

The energy size relationship is examined, either as the specific energy required breaking a particulate material from an initial size d₁ to a final size d₂, where d is usually the d₈₀ size, or as the specific energy required to break a single particle. The present work uses the results obtained using a controlled frequency centrifugal crusher to crush particles of a predetermined size class under different rotation frequencies related to the kinetic energy of the particles at the moment of crushing. The paper calculates the relationship between the rotation frequency and the kinetic energy of the particles before crushing and examines the size distribution of the products. The study results allow presenting the relationship between the kinetic energy of the particles and the mass of particles produced below the initial size class. The work also produces the optimum mathematical model that describes this relationship among three proposed ones. According to this model one can calculate the energy required breaking half of the initial mass below the initial size class and the corresponding specific energy is appointed to the average size of the class. The parameters of the mathematical model can be used to compare the grindability of the different materials. The process can be used as an alternative to the drop weight technique used so far for the study of the breakage energy of minerals and rocks.

1. Introduction

Comminution of minerals is an energy consuming operation and is responsible for the main energy cost in mineral processing plants. Modern mineral uses demand finely ground materials for building and chemical Industries and environmental applications. The most modern application is the use of magnesium bearing minerals for the capture of carbon dioxide, P. Renforth et al. [1].

It is well known that the specific energy, energy per unit mass, required to break a mineral particle increases rapidly as the particle size is reduced. Based on grinding data several theories of the energy-size relationship have been proposed and the most common ones are those of Rittinger [2], Bond [3], Kick [4], Charles [5], Stamboliadis [6] and Stamboliadis et al. [7], to name some of them. According to these theories the energy size relationship refers to the specific energy e_1,2 required to grind a material from an initial size x₁ to a final size x₂ and is given by Equation (1), where x₁ and x₂ are not the sizes of a specific particles but the screen sizes at which a predetermined fraction of the material passes. Usually, x is the screen size at which 80% of the material will pass.

(1)

In Equation (1), C is a constant and the difference between these theories is the value of the exponent n and the way it can be measured.

This work refers mainly in the study of the breakage energy required to break single particles and this consists a different approach than the one mentioned above. The first experiments to determine the energy required to break a single particle were made using the drop weight technique. According to it a weight of mass M is allowed to fall on a mineral particle of mass m, from a height h. The initial potential energy E_p of the falling weight is E_p = M·g·h, where g = 9.81 m/s² is the acceleration of gravity. At the moment the weight strikes on the particle its potential energy has been transformed in to kinetic E_k = MV²/2, where V is the velocity obtained by the falling weight, obviously, E_p = E_k. Initially the energy was determined by measuring the height but in our days one can also measure the velocity at the moment of impact, thanks to technological developments in high speed cameras [8].

The present work uses a new method to provide the energy required to break a particle. This is done using a centrifugal crusher that accelerates particles on a rotating disc that escape from the disc with a kinetic energy that depends on the rotation frequency [9]. The particles strike vertically on a specially designed wall and break.

The size distribution of the daughter particles depends on the strength of the material and the kinetic energy of the initial particle at the moment of impact. Such data have been presented recently by E. Stamboliadis et al. [10], but here the analysis is going a step further to provide a mathematical model to determine the energy-size relationship for breakage. The model one should look for is one that gives the mass fraction of the feed material that breaks below the size class of the feed as a function of the kinetic energy that the feed particles have obtained. The maximum fraction that can break is unit (100%) and the model one should look for is unit model that varies from zero to one. The proposed model can be used to compare the grindability of different types of rocks and minerals using the parameters of the model.

This work also provides the relationship between the rotation frequency of the disc and the kinetic energy that the particles acquire when leaving the disc. As it will be shown below the specific energy, energy per unit mass, of any particle leaving the disc is independent of its size and depends only on the rotation frequency and the disc diameter that is standard for the particular equipment used. The rotation frequency is used as the parameter that influences the kinetic energy.

Three different models have been tested to find the one that fits the results obtained. They all have a parameter ΔH_x = kJ/kg that indicates the specific energy required to break a particle of size x. The inverse of this parameter k_x = 1/ΔΗ_x gives the breakage rate k_x = kg/kJ that shows the mass of the particles of size x that break per unit of specific energy provided.

Each model gives a curve that differs from the data obtained. The sum of the squares of the differences between the measured and the calculated value gives the accuracy of the model. For each case one can vary the value of the parameter ΔΗ_x, and choose the one that gives the least value to the sum of squares. This sum is the best accuracy that a certain model can provide. Comparing the sums of least squares for the three models tested one can select the best one that gives the minimum sum of least squares.

As will be explained later, the model chosen can help to answer the question, when does a particle break, and obviously the answer is a statistic one because for practical reasons the particles tested are not all of equal mass or size but they have been chosen to belong to the same size class that is as narrow as possible. Consequently when one speaks for the size of a particle he actually denotes the average size of the size class and only when one refers to the size of a screen below which certain particles will pass the size is absolute.

2. The Crusher Used

2.1. Description of the Crusher

The equipment used is a locally made centrifugal crusher described in detail by D. Stamboliadis [9]. It consists of a horizontal rotating disc, 500 mm in diameter, surrounded by a homocentric, cylindrical cell 900 mm in diameter. The disc rotation axis is vertical and is linearly and directly connected to the axis of an electric motor through a cobbler. The rotation frequency of the motor and consequently of the disc is controlled by an inverter in the range of 700 to 2500 rpm. Radialy on the disc there are two symmetric, vertical blades that oblige any particle on the disc to rotate. The particles are introduced at the center of the disc, through a vertical shaft, and are obliged to rotation by the radial blades. As a result of the rotation, a centrifugal force acts on the particles and drives them to the periphery of the disc along the blades. As the particles move from the center to the periphery of the disc their rotation velocity, which is vertical to the radius, increases continuously and so does the centrifugal force that gives them a velocity on the direction of the radius. At the moment the particles reach the periphery of the disc they escape with the two velocity components that are vertical to each other and equal in magnitude, as calculated below. Their resultant is the vector sum of the two velocities and its direction is at 45 degrees to the radius of the disc at the moment of escape. This means that the resultant velocity vector is not vertical to the homocentric cell surrounding the rotating disc and the particles will not crush on it at an angle of 90 degrees. In order to ensure that the particles leaving the disc will crush on a surface vertical to the direction of their velocity the inner side of the surrounding cell is lined by blades of hard steel at an angle of 45 degrees to the radius. Figures 1 and 2 give an outside and an inside view of the crusher.

2.2. Calculation of the Kinetic Energy

The calculation of the kinetic energy of the particles at the moment of impact has been described by D. Stamboliadis [9] and is as follows. Let R be the radius of the disk and N the rotation frequency. Assume a particle of mass m been at a distance r from the center of rotation. The peripheral velocity V_p at this point it given by Equation (2):

(2)

A centrifugal force acts on the particle that is related to its peripheral velocity according to Equation (3):

(3)

The centrifugal force moves the particle to the perimeter with an acceleration calculated by Newton’s law given by Equation (4)

(4)

Substituting (2) and (3) into (4) one obtains Equation (5)

(5)

From the laws of motion one has the relationship between the centrifugal velocity V_c, the time t and the centrifugal acceleration given by Equation (6), as well as, the relationship between the centrifugal velocity, the time and the radius given by Equation (7).

(6)

(7)

Equating and deleting dt from (6) and (7) one has Equation (8)

or       (8)

Substituting (5) into (8) one has the differential Equation (9) that relates the centrifugal velocity to the distance of the particle from the center of the rotation.

(9)

The integration of (9) gives Equation (10)

(10)

For r = 0 then V_c = 0 and consequently C = 0.

At the moment when the particle escapes from the disc r = R and the centrifugal radial velocity is given by (11).

(11)

At the same moment the peripheral velocity is given by Equation (12) as is equal but vertical to the centrifugal velocity.

(12)

The vector sum of these two velocities is the actual escaping velocity V that is calculated from Equation (13)

(13)

Taking into consideration (11) and (12) the final velocity is given by Equation (14) and has a direction of 45˚ relative to the radius of the disc at the moment of escape.

or    (14)

where D is the disc diameter D = 2R.

The kinetic energy E of a particle with velocity V is given by Equation (15)

(15)

Substituting (14) into (15) the kinetic energy of the particle at the escape point from the disc is given by Equation (6)

or    (16)

The specific energy e = E/m is then given by Equation (16) and it is independent of the particle mass.

or        (17)

Applying the above formulas to the present case one calculates that for the particular crusher, with a disc 500 mm in diameter, the frequency required to achieve a specific energy e = 3600 (J/kg) or the same 1 (kWh/ton), which a usual specific energy required , is 2293 rpm and is independent of the size of the particle. This frequency is within the capacity of the machine manufactured.

3. Experimental

3.1. Materials Used

Two different rocks are studied, namely 1) microcrystalline limestone from the operating quarry of Hordaki near Chania, in the island of Crete, Greece and 2) serpentine from the area of Mantoudi in the Island of Euboea, Greece.

There are two reasons for the selection of these materials. The first is that limestone is more or less homogenous compared to serpentine that is weathered and these rocks are expected to have different behavior regarding the energy size relationship. The second reason is that limestone is used widely as a building material, while serpentine is a source of MgO that is studied for the capture of carbon dioxide in environmental applications [1]. In both cases the energy cost for size reduction is very important.

The microscopic structure of the samples appears in polished sections presented in Figures 3 and 4 respectively. One can see that limestone is microcrystalline and the crystals are not distinguished giving a homogeneous appearance at the scale of the particles tested. On the other hand serpentine crystals can be distinguished but they appear to be weathered and the space between them consists of the weathering product that is expected to be weaker than the healthy crystals.

3.2. Experimental Procedure

The experimental procedure followed during the test work

is described by E. Stamboliadis et al. [4]. A quantity about 30 kg of each material tested was crushed to minus 30 mm using a laboratory jaw crusher. The material is then classified into the following size fractions of very narrow size range (16 - 22.4 mm), (8 - 11.2 mm), (4 - 5.6 mm), (2 - 2.8 mm) and (1 - 1.4 mm). The geometric average size of each size fraction is calculated to be (18.93 mm), (9.47 mm), (4.73 mm), (2.37 mm) and (1.18 mm) respectively. Each size fraction is crushed in the centrifugal crusher at different rotation frequencies using 1 kg of the particular feed size fraction at a time. The frequencies used and the corresponding specific energies are presented in Table 1.

The crushed product of each test is collected and classified in size fractions using the screens 16, 8, 4, 2, 1, 0.5, 0.25, 0.125 and 0.063 mm. The mass distribution of each product is the cumulative mass % finer than the corresponding screen and is plotted versus the screen size in the same figure for all the specific energies used for the same feed fraction. From such a figure one can see the effect of the specific energy to the size analysis of the products.

The results are also presented in a way that for the same feed size, it gives the mass fraction of the material broken below the feed size as a function of the specific energy. The results obtained from this kind of presentation are used to derive the mathematical model that fits them. This model gives the energy required to break each feed fraction size and contains parameters that allow comparing the different materials tested.

3.3. The Results Obtained for Limestone

The results obtained for each feed fraction of limestone tested at different rotation frequencies are tabulated in tables. Table 2 presents the data obtained for the fraction 16 - 22.4 mm of limestone. It is reminded that each rotation frequency corresponds to a certain specific energy as shown in Table 1. The top row of Table 2 shows the screen size (mm), while the left column gives the specific energy of the feed particles (J/kg). The values presented in the table give the measured mass fraction of the product that passes through the corresponding screen for the indicated specific energy.

The horizontal plot of the values of Table 2 gives Figure 5 that shows the mass fraction of particles finer than the screen size for the different specific energies. The

higher the specific energy the curves move to finer sizes.

The vertical plot of Table 2 gives Figure 6 that presents the mass fraction of the product that passes the indicated screen size, for all specific energies applied. All curves tend to 1, which is the maximum mass fraction that can be produced below any size. Obviously the coarse particles are produced at a higher rate than the finer ones and as shown in Figure 6 only a small fraction of the fines is produced at the maximum specific energy tested (4500 J/kg),which actually is the limit of the machine used. The fact that the curves of Figure 6 have a maximum indicates the type of mathematical equations that can be applied to describe the phenomenon. The same type of curves, are obtained for all the feed fractions tested for each type of rock tested. Table 3 presents the results obtained for limestone from all size fractions tested at different specific energy inputs. The data give the mass fraction of the product that passes below the initial feed class. As an example, for the feed fraction (1 - 1.4) mm, the table shows the mass of the feed that passes through the (1) mm screen for all the specific energies applied.

Figure 7 presents the mass fraction of limestone broken below the feed class tested, denoted by its average size, as a function of the specific energy. Figure 8 presents the same results for all energy levels applied as a function of the average size of the feed class. Both figures show, each one in a different way, that for any feed