4. RESULTS AND DISCUSION
4.3 Model for predicting steady state pebble size distribution
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rock had much more sharper edges than the local ore. This may be an influence of the different closed side settings of the crusher from which the respective ore came from. It also may be attributed to the characteristics of the different type of ore.
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Figure 4-14: Size distribution of rocks between 65/35 mm.
Table 4-2: Natural size distribution of rocks between 65/35 mm
Size (mm) Mass (kg) Mass %(mf) % Passing
65 53 7.75 16.94 100
53 44 17 37.16 83.06
44 35 21 45.90 45.90
35 28 0 0 0
28 16 0 0 0
16 0 0 0
45.75
The third parameter used in the model was the shape factor (SF). As discussed earlier the shape factor for a group of rocks in a given size fraction is defined as the ratio of the rocks average mass to that of a sphere that has a radius equivalent to the average of the top and bottom size fractions. This required the average mass of the rocks in a size fraction be determined by counting as many rocks as possible and weighing the total mass. This then allowed a shape factor of the rocks in every feed size fraction to be determined as follows:
The initial average mass (i.e. t=0) of a rock originally from size interval i is given by:
( )
Where
0 10 20 30 40 50 60 70 80 90 100
0 10 20 30 40 50 60 70
Cumulative % Passing
Size (mm) Natural distribution
(Gold rock) 65/35 distribution (Gold rock)
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= Initial average mass of a rock originally from size interval i. (kg)
= Total mass of sampled rocks originally from size interval i. (kg) = Number of rocks in sample
The relationship between the mass of a sphere at time t originally from size interval i and diameter of that same sphere is given by:
( )
( )
Where
Mass of a sphere from size interval i. (kg)
Diameter of a sphere at time t originally from size interval i. (m)
The mass of a reference sphere that has a radius equivalent to the average of the top and bottom size of interval i was determined as follows:
( )
( )
Where
Mass of a reference sphere from size interval i. (kg)
= Density of rock (kg/m3)
= Arithmetic average of the limits of size interval i (m) given by:
( )
Where
= Upper limit of the size interval i (m)
= Lower limit of the size interval i (m)
Assuming that the rock size initially starts at the average size of interval i then:
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( ) Where
Initial average diameter of a rock originally from size interval i. (m) This assumption implies that:
( )
Initial average mass of a rock originally from size interval i. (kg)
The shape factor is now the ratio of the average mass to the mass of a sphere and was determined as follows:
( )
Where
= Shape factor of rocks at time t originally from size interval i.
So the initial SF is given by:
( )
Where
= Initial shape factor of rocks at time t originally from size interval i.
Now that specific wear rate, the feed size distribution, and the SF of the rocks are known it is first assumed that 100 kg of rocks initially enter the mill. This mass enters in different fractions ( ) as determined from the size distribution tabulated in Table 4-2. This allows the total initial mass (t=0) in a given size fraction to be determined as follow:
( ) Where
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= initial total mass of rocks in model originally from size interval i. (kg)
= mass fraction of rocks from size interval i in feed
The total number of rocks in the model at any given time t originally from size interval i is given by:
( ) Where
= number of rocks in model at time t originally from size interval i.
= Total mass of rocks at time t originally from size interval i. (kg)
So the initial average mass (t=0) originally from size interval i as determined from equation 4.1 allows the initial number of pebbles in model originally from size interval i.
( ) Where
= Initial number of rocks in model originally from size interval i.
Using the total mass of rocks at time t originally from size interval i, the specific wear rate (Rs) was used to determine the total mass at next time interval. The specific wear rate (Rs) is a function of time, rock diameter and charge size distribution. The introduction of a specific wear rate correction factor (ks) is used to account for the variation of specific wear with charge size distribution. A detailed explanation of the specific wear rate correction factor is given in Chapter 4.4.
( ) ( ) Where
= Total mass of rocks at time t+1 originally from size interval i. (kg)
= Specific wear rate as determined from rock tagging procedure (1/min)
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= Specific wear rate correction factor.
= Time increment (min)
Assuming the total number of rocks in model remains constant then:
( ) Where
= number of rocks in model at time t+1 originally from size interval i
This then allowed the total mass of rocks at time t+1 originally from size interval i to be converted to an average mass at time t+1 originally from size interval i.
( )
Substituting equation 4.13 into equation 4.14 gives
( )
Where
= Average mass of rocks at time t+1 originally from size interval i. (kg)
A relationship between mass and diameter is quite simple for steel balls which are essentially spheres. When it comes to pebbles the shape of the rock makes this relation much more complex.
In order to convert the average mass of a rock into an average diameter the SF was introduced.
Preliminary shape factor testing revealed that the SF is a function of time and particle size. Using this data a shape factor model was generated which then allowed the average mass to be converted to an average diameter as shown below:
( ) ( ) Where
( ) = Shape factor function for rocks originally from size interval i evaluated at time t ( ) ( )
74 Where
= Shape factor of rocks at time t+1 originally from size interval i.
( ) = Shape factor function for rocks originally from size interval i evaluated at time t+1
Using the definition of the SF (equation 4.7) at the next time increment t+1 gives:
( )
Where
Mass of a sphere (kg) from size interval i given by:
( )
( )
Where
Diameter of a sphere at time t+1 originally from size interval i. (m) Substituting equation 4.19 into equation 4.18 gives:
( )
( )
Rearranging and solving for the average diameter at next time interval gives:
√
( )
This procedure was repeated for i size fractions present in the feed size distribution for a given period of time until average diameter became less than 3.3 mm. This is the size which separates the rocks from the fines. After all size fractions wear modelled the total mass of rocks in each size interval i was determined by summing the total mass of rock in every size interval which has an average diameter between the upper and lower limits of size interval i
∑ ∑[ ] ( )
75 Where
= Predicted mass of rocks in size interval i. (kg)
∑ ( )
Where
= Total predicted mass of rocks. (kg)
( ) Where
= Predicted mass fraction of rocks in size interval i.
This data is put into cumulative for form to give the predicted steady state size distribution of the charge as shown in Figure 4-15. This procedure makes it possible to predict the steady state size distribution of pebbles, given any size distribution of feed pebbles. It can be seen that the prediction is relatively accurate. For the purpose of this work the model was sufficiently accurate however improvements to the model are possible. Future work could investigate the probability of rock breakage and rock travel into smaller fractions.
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Figure 4-15: Predicted size distributions (Gold ore, 65/35mm feed size) 0
10 20 30 40 50 60 70 80 90 100
0 10 20 30 40 50 60 70
Cummalative % Passing
Size (mm) Predicted Steady State
Actual size distribution
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