Summary of calculation:

Part 1: Size reduction result

Product size distribution:

Average size (µm) 6 5 4 3 2 1

Distribution 0.054 0.2083 0.2873 0.1820 0.1449 0.1234

Average particle size:

Feed (µm) 4.64

Product (µm) 3.474

Reduction energy
prediction:
  Rittinger Bond Kick
13743.72 0.00405
Reduction energy (kJ/kg) 1 2 8.673
Appendix
Part A
B)
Table 1: Feed size distribution
Average size (µm) 6 5 4 3 2 1
Fraction 0.18 0.42 0.3 0.06 0.04 0

Table 2: Specific rate of breakage and breakage distribution for ball mill (5 mins grinding
time)
Average size (µm) 6 5 4 3 2 1
Interval no (j) 1 2 3 4 5 6
Sj 0.7 0.6 0.5 0.35 0.3 0
b(1,j) 0 0 0 0 0 0
b(2,j) 0.32 0 0 0 0 0
b(3,j) 0.29 0.4 0 0 0 0
b(4,j) 0.14 0.2 0.5 0 0 0
b(5,j) 0.13 0.2 0.25 0.6 0 0
b(6,j) 0.12 0.2 0.25 0.4 1 0
Assume that S is a rate of breakage. The following equation expresses the rate of change of
the mass of particles in size interval i with time:
d mi j=i −2
= ∑ [ b(i , j) S j m j ] −S i mi
dt j=1

j =i−2
Where ∑ [ b (i , j ) S j m j ]=mass brokeninto interval i¿ all intervals of j>i ¿
j=1

Si mi=mass broken out of interval i

Since mi= y i M ∧m j= y j M , we can write a similar expression for the rate of change of mass
fraction of material in size interval i with time:
d y i j=i−2
= ∑ [ b(i , j) S j y j ] −S i y i
dt j=1

Where M is the total mass of feed material and y iis the mass fraction in size interval i

Change of fraction in interval 1:

dy 1
=−S i y i
dt
dy 1
=−( 0.7 ) ( 0.18 )
dt
dy 1
=−0.126
dt
New y 1=0.18−0.126

New y 1=0.054

Change of fraction in interval 2:

d y2
=b ( 2,1 ) s 1 y 1 −S 2 y 2
dt
d y2
=0.32 ( 0.7 ) ( 0.18 ) −0.6 ( 0.42 )
dt
d y2
=−0.2117
dt
New y 2=0.42−0.2117

New y 2=0.2083

Change of fraction in interval 3:

d y3
=b ( 3,1 ) s 1 y1 +b ( 3,2 ) s 2 y2 −S 3 y 3
dt
d y3
=0.29 ( 0.7 )( 0.18 )+ 0.4 ( 0.6 ) ( 0.42 )−0.5 ( 0.3 )
dt
d y3
=−0.01266
dt
New y 3=0.3−0.01266

New y 3=0.2873

Change of fraction in interval 4:

d y4
=b ( 4,1 ) s 1 y 1 +b ( 4,2 ) s2 y 2+ b( 4,3) s3 y 3−S4 y 4
dt
d y4
=0.14 ( 0.7 )( 0.18 ) + 0.2 ( 0.6 )( 0.42 ) +0.5 ( 0.3 ) ( 0.5 )−0.06(0.35)
dt
d y4
=0.1220
dt
New y 4=0.06+0.1220

New y 4=0.1820
Change of fraction in interval 5:
d y5
=b ( 5,1 ) s 1 y 1 +b ( 5,2 ) s 2 y 2 +b ( 5,3 ) s3 y 3+ b(5,4)s 4 y 4 −S 5 y5
dt
d y5
=0.13 ( 0.7 )( 0.18 )+ 0.2 ( 0.6 )( 0.42 ) +0.25 ( 0.3 ) ( 0.5 ) +0.6 ( 0.35 )( 0.06 )−0.3(0.04 )
dt
d y5
=0.1049
dt
New y 5=0.04+ 0.1049

New y 5=0.1449

Change of fraction in interval 6:

d y6
=b ( 6,1 ) s1 y 1+ b ( 6,2 ) s2 y 2+ b ( 6,3 ) s 3 y 3 +b ( 6,4 ) s 4 y 4 +b(6,5) s 5 y5 −S 6 y 6
dt
d y6
=0.12 ( 0.7 )( 0.18 )+ 0.2 ( 0.6 )( 0.42 ) +0.25 ( 0.3 ) ( 0.5 ) +0.4 ( 0.35 )( 0.06 )+ 1 ( 0.3 )( 0.04 )−0
dt
d y6
=0.1234
dt
New y 6=0+ 0.1234

New y 6=0.1234

To check the answer, the sum of predicted interval mass fractions

¿ 0.0540+0.2083+ 0.2873+ 0.1820+0.1049+0.1449+0.1234
¿ 0.9999 ≈ 1.0000

Assumption made:
1. This calculation is assumed to be able to use the values of S and b determined from
small-scale tests for the prediction of product size distributions on a large scale for
industry purpose.
2. This method is assumed to give fairly reliable predictions.
C)
Find the average size for feed, x 1 :
x 1=0.18 ( 6 ) +0.42 ( 5 ) +0.3 ( 4 ) +0.06 ( 3 ) +0.04 ( 2 ) +0

x 1=4.64 µm

x 1=4.64 ×10−6 m

Find the average size for product, x 2 based on the new fraction in each interval calculated:
x 2=0.054 ( 6 )+ 0.2083 (5 )+ 0.2873 ( 4 ) + 0.182 ( 3 ) +0.1449 ( 2 ) +0.1234 (1)

x 2=3.474 µm

x 2=3.474 ×10−6 m

Prediction of the energy required for the ball mill:

a) Rittinger’s Law
The energy required for particle size reduction is directly proportional to the area of
new surface created.

Energy required α area of new surface created

Assuming shape factor and density are constant,

E=C R ( x1 − x1 )
2 1

Given C R =190,
1 1
E=190
(
3.474 ×10−6
−
4.64 ×10−6 )
E=13743721 J /kg

b) Bond’s Law
The total work input represented by a given weight of crushed product is inversely
proportional to the square root of the diameter of the product particles. 

1 1
E=C B
(√−
x2 √ x1 )
Given C B=120 ,
 1 1
E=120
(√ 3.474 × 10−6
−
√ 4.64 × 10−6 )
E=8673.68 J /kg
c) Kick’s Law
The energy required in comminution process is directly proportional to the ratio of the
volume of the feed particle to the product particle.

volume of feed particle

Energy required α
volume of product particle

Assuming constant shape factor,

x1
E=C K ∈ ( )
x2
Given C K =14,
4.64 × 10−6
E=14∈ (
3.474 ×10−6 )
E=4.052 J /kg

Assumptions made for size reduction law:

1. Rittinger’s Law is for very small particle size (ultra-fine grinding)
2. Bond’s Law is for intermediate particle size, the most common range for many
industrial grinding processes.
3. Kick’s Law is for large particle size (coarse crushing and crushing).
Discuss

Based on stress analysis theory for plastic deformation, Kick’s Law states that
“energy required for comminution process is direction proportional to the reduction ratio”. It
means that the comminution energy required depends only on the reduction ratio and is
independent of the original size of the feed. For an example, this proposal predicts that same
energy is required to reduce 10mm particle to 1mm as is required to reduce 1m boulders to
10cm block since the reduction ratio is same. However, it is very obvious that more energy is
required for the reduction of 10mm to 1mm particle since higher reduction energy is required
for reduction of small particle size to further smaller size. Thus, Kick’s Law is applicable for
large particle size.

Rittinger’s Law states that “reduction energy required is directly proportional to the
area of new surface created”. Rittinger’s Law is for the fine grinding where the increase in
surface area of particles is large. Thus, Rittinger’s Law is applicable when we deal with fine
material since there is more chances to create new surfaces.

Bond’s Law states that “The total work input represented by a given weight of
crushed product is inversely proportional to the square root of the diameter of the product
particles”. Bond’s formula is more suitable for intermediate particle size, the most common
range for many industrial grinding processes.

Kick’s Law is not suitable for fine grinding compared with Rittinger’s Law. Bond’s
Law is an intermediate and more realistic method for prediction of reduction energy.
However, this law is more suitable for intermediate particle size and not applicable for flour.
Thus, Rittinger’s Law is the best prediction method since it is suitable to reduce a small
particle size to a further smaller size.