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American Journal of  
Geospatial Technology (AJGT)

Comparative Assessment of  Comminution Energy Equations (Models) Using Some
Selected Legumes, Tubers, Cereals and Sea Food

Akpan Abasiama J.1, Olosunde William A.1*, Antia Orua O.1

Volume 2 Issue 1, Year 2022
ISSN: 2833-8006 (Online)

DOI: https://doi.org/10.54536/ajgt.v2i1.1969
https://journals.e-palli.com/home/index.php/ajgt

Article Information ABSTRACT

Received: August 23, 2023

Accepted: September 27, 2023

Published: September 30, 2023

The quest to predict more precisely energy that is just enough to achieve closely any desired 
final particle size of  food material subjected to comminution is necessary. This is to avoid 
energy wastage, obtain finer product size(s) that are uniform and can encourage easy mixing, 
dehydration/drying, etc. In this regard, some selected legumes (soybeans and beans), cereals 
(sorghum, millet, corn and wheat), sea food (crayfish) and tubers (cassava and yam) were 
subjected to comminution through the application of  some major energy equations (mod-
els); as they find various applications in food industries. Four energy equations (models) 
namely the Kick’s, Rittinger’s and Bond’s for size reduction and the OruaAntia’s minimum 
energy equation for mass-size reduction were employed. The constant in each equation 
(model) to be applied in grinding of  selected food materials was determined and used in 
obtaining the required specific energy of  comminution needed to accomplish desired final 
product size(s). The corresponding grinding time expected to achieve the desired final prod-
uct were computed and used in operating the grinding machine. Results revealed that the 
OruaAntia’s minimum energy equation for mass-size reduction operation may be applied on 
the selected food materials to achieve very closely any desired final average particle diameter 
with a percentage deviation of  1.66% followed with Rittinger’s equation having average 
percentage deviation of  6.88%. Technical analysis re-affirm that Kick’s equation could only 
achieve coarse particles as the final particle size showed average percentage deviation of  
38.40%, while Bond’s equation may be limited to prediction of  coarse and intermediate 
particles; since the final particle size average percentage deviation was 16.19%. Besides using 
Rittinger’s energy equation to obtain fine particles, the use of  Orua Antia minimum energy 
equation may possibly further achieve desired finer particle size(s).

Keywords
Comminution, Energy Equations, 
Food Materials, Grinding Machine 
Particle Size

1 Department of  Agricultural and Food Engineering University of  Uyo, Uyo, Nigeria
* Corresponding author’s e-mail: williamolosunde@uniuyo.edu.ng

INTRODUCTION
The amount of  energy used for grinding materials 
may depend on the materials physical characteristics, 
machine geometrical and kinematic parameters used for 
the grinding. Size reduction operation is very important 
in food sector as it enhances powder mixing, reduction 
in sedimentation rate among several things (Sushaut & 
Archana, 2013; Song., Mycong., Hyong., & Won., 2014; 
Dabbour., Bahnasawy., Ali., & El-Haddad., 2015; Mulla., 
Hajare., & Doijad., 2016). During size reduction, particles 
may experience elastic and inelastic deformation. Hence, 
requires appropriate energy to produce product that may 
be averagely uniform in size or categorized into distinct size 
ranges. Thus, accurate estimation of  energy requirements 
necessary for size reduction equipment is economically 
essential in many aspect (Antia., Obahiagbon., Aluyor., & 
Ebunilo., 2013; Antia & Aluyor, 2018).
In an effort to minimize cost coupled with achieving 
the desire size range of  material, various equations 
(models) have been proposed and developed based on 
the differential energy (dE) required to produce a small 
change (dX) in size of  a unit of  a material x (Akinoso., 
Lawal., & Aremu., 2013; Ndukwu., Nwakuba., & Henry., 
2016; Antia, 2021, pp. 672-673) as:
dE/dX ∝ x(-n)                (1)
dE= -kx(-n) dX                (2)

Where, k and n are constant. n=1,2,3⁄2for kicks, Rittinger’s 
and Bond’s concept respectively.
These expressions are as follows:
Ek=Kk [ln x1/x2 ]                (3)
ER=KR [1/x2 -1/x1 ]             (4) 
EB=KB [1/√(x2 )-1/√(x1 )]               (5)
Where,Ek,ER  and EB are energy by Kick’s,Rittinger’s and 
Bond’s respectively and may be expressed in J
or   J⁄kg or  kWh⁄kg  or  Ws⁄kg.
Kk, KR  and KB may be expressed in J or J⁄kg
or  kWh⁄kg,Jm or  Jm⁄kg or (kWh.m)⁄kg,Jm(1⁄2)  or
Jm(1⁄2)⁄kg or kWh.m(1⁄2)⁄(kg) respectively as Kick’s, 
Rittinger’sand Bond’s constants while x1 and x2 are 
respectively initial and final particle sizes(diameter(s)) in m.
A new dimension considered by Antia., Obahiagbon., 
Aluyor., & Ebunilo., 2014 in this regard was that energy 
(dE) is required to produce a small change (dA) in area 
of  a unit material. This approach is because it is easier to 
measure effectively any material by mass (m) irrespective 
of  it size and shape. Moreso, mass could be related to 
area (A) as:
m=ρV=ρAT                (6)
Where, V= volume of  the material
ρ= density of  the material 
T= thickness of  the material  
Where, A= m/ρT                (7)



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Hence, the concept was expressed as:
dE/dA ∝ A(-n)                (8)
dE= -KA(-n) dA                 (9)
Emin= Bm(1-n)/(1-n)             (10)
Where, n= power index
E_min=Minimum energy required for mass reduction 
of  the materialThe power index n was experimentally 
determined as n =1⁄2. The minimum energy for the mass 
reduction was obtained as:
Emin=2Bm(1⁄2)              (11)
Where,B = mass index, m= mass of  the material
A combination of  analytical and empirical approaches to 
relate size(diameter) of  the material resulted in one of  the 
expressions (models) for mass-size reduction operations 
(Antia, 2020) as:
EA=K(A3 ) [1/(D2

(3⁄2) )-1/(D1
(3⁄2) )]             (12)

Where, K(A3 ) = Orua Antia Energy constant expressed in 
Jm(3/2) or Jm(3/2)/kg or KWh. m(3/2)/kg 
EA = Orua Antia minimum energy for mass-size reduction 
of  material and may be expressed in J/kg.kWh/kg or 
Ws/kg
D1= Initial diameter of  the material in m.
D2 = Final diameter of  the material in m.
K(A3)= (2Bρm

(-½))/(Cf Mf) (0.2304)= (0.4608 Bρ(-½))/(Cf Mf) (13)
Where, Cf=  crushing efficiency
Mf= Mechanical efficiency
Besides the mentioned energy equations (models), 
various researchers have proposed other models during 
studies on energy consumption for grinding materials. 
Mani., Tabil., & Sokhansanj (2004) worked on energy 
consumption of  switch grass, corn stover and wheat 
straw using hammer mill with screen size of  0.8, 1.6 
and 3.2mm. The data obtained fitted a straight line and 
a second order polynomial respectively at 8 and 12% 
moisture content of  the material as:
E = k1+ k2 d               (14)
E = k1+ k2 d+k2 d

2              (15)
Where,  E = specific energy,(kWh⁄t)
d is the hammer mill screen size (mm) 
k1, k2= constants.
Bitras., Womac., Chevanan., Miu., Igathinathane., 
Sokhansanj et al. (2009) also worked on switch grass, 
corn stover and wheat straw having respectively the initial 
mean sizes of  8.3, 7.1, and 8.3 mm. The feeding rate of  
41.7 g/s and hammer mill with 3.2 mm screen were used. 
The data obtained fitted equation given as:
Ee= KΔXg              (16)
Where,  K is a function of  the rotational speed, N [i.e. 
K=f(N)]
Ee= specific energy (MJ/Mg)  
ΔXg= unit size reduction (mm)
Adapa., Tabilo., & Schoenau (2011) obtained barley straw, 
canola straw, oat straw, and wheat straw for grinding using 
two hammer mills. The first had 30 mm screen while 
the second had 6.4, 3.2, and 1.6 mm screen. The energy 
equation proposed was given as:
E=k1 S

(-k2 )              (17)
Where, E = specific energy, (kWh⁄t)

s= hammer mill screen size (mm) 
k1, k2= constants.
Miao., Grift., Hansen., & Ting (2011) used a knife mill 
with screens that ranges from 10mm to 1mm to grind 
some biomass at two moisture content levels of  15 and 
7%. The energy consumption and particle size were 
related as: 
E=axb               (18)
Where, E= specific comminution energy, J/gx= 
comminution ratio(ratio of  average size of  initial biomass
to average size of  comminuted particles)a and b = 
regression constants.
In another development, the Kick’s, Rittinger’s and 
Bond’s equations were used to grind hard wood and 
soft wood (Naimi., Sokhansanj., Bi., Lim., Womac., Lau 
et al., 2013). It was observed that the predicted specific 
comminution energy fitted best using Rittinger’s equation 
followed by Bond’s and Kick’s equations. Ladan., Flavien., 
Xiaotao., Jim., & Shahab (2016) used rotary knife mill and 
Lingocellulose biomass conditioned at 11.5% moisture 
content coupled with Kick’s, Rittinger’s and Bond’s 
equations. The experimental data generated fitted best 
the Rittinger’s followed by Bond’s and Kick’s energy 
equations for size reduction operations.
In this study, the energy Equations (models) 3,4,5 and 
12 were used to assess the most applicable energy of  
comminution for some selected cereals (corn, millet, 
wheat and sorghum), legumes (soya beans and beans), 
tubers (yam and cassava) and sea food (cray fish).

MATERIALS AND METHODS
Equipment, Sourcing and Preparation of  Materials 
The equipment used includes: vernier calliper, sieves, 
digital weighing balance, polytene bags, sieve shaker, 
stop watch and grinder (Hammer-Attrition mill). The 
selected materials used were cereals (corn, millet, wheat 
and sorghum), legumes (soybeans and beans), tuber 
(cassava and yam) and sea food (cray fish,) obtained 
from Uyo local market, Akwa Ibom State, Nigeria. Each 
material was cleaned manually to remove any unwanted 
material on it; and then air dried at 105℃ till constant 
weight (bone dried mass) was achieved (Antia., Oboh., 
& Olosunde., 2019). Thereafter, each material was stored 
for use in this study.

Evaluation of  Energy Equations (models) Constants  
The grinding machine energy was first determined 
by reading the power of  electric motor used from the 
machine description panel. A constant time for grinding 
was chosen to evaluate all the constants in the energy 
equations (Kick’s,Rittinger’s, Bond’s and OruaAntia’s). 
The energy of  the grinding machine was given 
(MohdRozalli., Chin., & Yusof., 2015; Antia, 2021, pp. 
681-692) as:
Energy (J)= power(J⁄s)x time(s)             (19)
Therefore, the specific energy E1

* of  the grinding machine 
powered by the electric motor may be given as:
E1

*=([Energy (J)])/(mass of  material @subjected to 



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grinding, kg)=(Electric motor power (J⁄s)×@grinding Time 
(s))/(mass of material @subjected to grinding, kg)                         (20)
The specific energyE1

* may be expressed in  J⁄kg or  
kWh⁄kg  or  Ws⁄kg.
The grinding time chosen and the electric motor 
power used were 120sec and 0.746kW respectively. The 
mass (500g) of  each selected material to be used per 
experimental run was chosen and its average initial (feed) 
diameter was determined. The average final (product) 
diameter of  each selected material after being subjected 
to grinding was determined using sieve analysis (Antia, 
2021, pp. 700-710; Okoro, 2001, pp. 31-53). The energy 
constants of  Equations 3,4,5 and 12 were determined using 
specific Energy E1

*, the sample average initial feed diameter 
of  the feed and the average final product diameter.

Estimation of  Specific Energy of  Comminution and 
Grinding Time Applicable to Each Energy Equation
In assessment of  the energy Equations 3,4,5 and 12 
for size reduction operations;three desired final particle 
sizes were chosen as 300μm, 400μm, and 500μm. Sieve 
aperture size that correspond to the chosen (desired) 
final product was used to sieve out each of  the selected 
materials after grinding.The specific comminution 
energyE2

*based on the four energy Equations 3,4,5 and 
12 were computed for each of  the selected samples by 
using each corresponding energy equation constant 
obtain in section 2(ii)coupled with the initial average 
diameter of  the feed and each desired (chosen) average 
diameter of  the final product.Each time expected to grind 
the particles to achieve each of  the three (3) chosen sizes 
(diameters) 300μm, 400μm, and 500μm were calculated 
based on the Equation 21(MohdRozalli., Chin., & Yusof., 
2015, Antia, 2021, pp. 681-692) as:
tg=(E2

*×ms)/Pm               (21) 
Where, E2

*= Expected specific comminution energy 
required in (kWh.Kg(-1)) 
ms=Chosen quantity of  sample to be ground (Kg)
Pm=power of  the grinding machine (kW)
tg=Expected time required for grinding (hr)
The estimated time required to achieve each final 
product size chosen per selected material was used 
to run and grind the sample(s) accordingly. The final 
product of  eachselected sample per grinding time was 
subjected to sieve analysis; in order to determine the 
most appropriate energy equation for application, to 
achieve the expected desired size (diameter) (d2

*) of  the 
final product.

Determination of  Particle Diameter for Comparative 
Assessment of  the Energy Equations
Sieving tests were conducted to determine the initial and 
final average/mean particle size (dp), of  each selected 
material subjected to comminution. Each test was 
conducted by placing a series of  sieves with progressively 
smaller mesh sizes on top of  each other and passing the 
sample through the stacked sieve “tower” using a sieve 
shaker for a given time.The following expressions were 

used for the analysis (Antia, 2021, pp. 700-710; Okoro, 
2001, pp. 54-69):
Rf   =  ᵶ⁄m              (22)
Where, ᵶ=weight of  material on a given sieve
m= mass of  sample
dpi= Si Rf               (23)
dp = ∑dpi               (24)
Where, i=1,2,.....  n is the serial number of  sieve 
aperturesincluding the final bottom material n sieved out
Si=aperture size of  the  sieve used
dpi=average particle size (diameter)  on i sieve
The values of  the average/mean diameters obtained 
using each energy equation on each selected material 
were compared; to determine the equation that could 
be used to predict closely the comminution of  each 
selected material. This may be achieved by computing 
the percentage deviation of  actual final average diameter 
(d2

*) with predicted (chosen) final average diameter (d2) 
of  the selected materials subjected to comminution.The 
equations may be given as:
% deviation (Ep )=[(∑d2

*- d2 )/NT ]×100            (25)
Where, NT=d2 n
d2= desired (chosen)  final average particle size (diameter), m
n = number(s) of  selected material(s) used in testing the 
energy equations
d2

* = actual final average particle size (diameter)  following 
comminution,m 
average % deviation (Epa)=( ∑Ep )/NTS                    (26)
Where,NTS=the total number of  size(s)(diameter(s))
chosen and used in testing the  energy equations(s).
 
RESULTS AND DISCUSSION
Each constant required as applicable to Kick’s, Rittinger’s, 
Bond’s and OruaAntia’s energy Equations 3, 4, 5 and 12 
respectively for each of  the selected sea food (crayfish), 
cereals (sorghum. millet, corn and wheat), legumes 
(soybeans and beans) and tubers (cassava and yam) 
wasobtained and are presented in Table 1.
From Table 1, it is observed that at the chosen grinding 
time 120sec (2min), the grinding machine with power 
of  0.746kWcoincidently produce final average product 
size of  600μm for each 500g of  the selected materials. 
This may imply that at commencement of  grinding, each 
equation accommodate this size within two (2) minutes 
range of  grinding time; and beyond which each energy 
equation may likely start to exhibit its peculiar capability.
In assessing each of  the energy equations, the specific 
energy of  comminution E2

* applicable to each of  the 
energy equation considered was calculated based on the 
initial average feed diameter(s) and chosen final average 
product size(s) (300μm, 400μm, and 500μm) coupled 
with their respective determined constant in Table 1. The 
obtained values of  E2

* using Equations 3, 4, 5 and 12 are 
presented in Tables 2 to 4.
From Tables 2 to 4, each grinding time per selected 
material was obtained from Equation 21. Moreso, the 
desired (chosen) average particle size(s) d2 and actual 
average particle size(s) d2

*obtained after subjecting the 



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Table 2: Specific comminution energy E2
*,predicted grinding timetg, desired (300μm) and actual final product(s) size(s)

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* Specific Energy 
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Time (min:sec)

Actual Final product 
Diameter (d2

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Table 1: Energy equations constants obtained for selected materials with respect to grinding machine power 
(0.746kW) and time (120sec) used.

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(J
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or

 k
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) Energy equation constants

K
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k’
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 K
k 

(J
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 (1

0(-
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, 
K

k (
Jm

(3
⁄2

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g)

(1
0(-

7)
)

Sea food Crayfish 0.02967 0.00060 0.049733 1.274932 3.045596 1.418761 7.330361
Cereals Sorghum 0.00317 0.00060 0.049733 2.987782 3.680654 2.156346 7.965170

Millet 0.00195 0.00060 0.049733 4.223176 4.312190 2.737464 8.816342
Corn 0.02017 0.00060 0.049733 1.414946 3.075503 1.472139 7.346981
Wheat 0.00186 0.00060 0.049733 1.769614 3.175078 1.614205 7.418804

Legumes Soybean 0.00516 0.00060 0.049733 2.311909 3.376890 1.848848 7.611337
Beans 0.00630 0.00060 0.049733 1.311845 3.052907 1.433589 7.334147

Tubers Cassava 0.00140 0.00060 0.049733 6.332049 5.484559 3.750929 10.560293
Yam 0.00196 0.00060 0.049733 5.283426 4.892742 3.245050 9.664068



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Am. J. Geo Spat. Technol. 2(1) 7-14, 2023

Ya
m

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19
6

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0

0.
08

63
55

0.
13

12
79

0.
10

46
08

0.
16

99
63

03
:2

8

05
:1

6

04
:1

3

06
:5

0

4.
76

3.
23

3.
44

3.
10

Table 3: Specific comminution energy E2
*,predicted grinding timetg, desired (400μm) and actual final product(s) size(s)

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 (d
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(m
) E2

* Specific Energy 
Requirement (kWh⁄kg)

tg Predicted Crushing 
Time (min:sec)

Actual Final product 
Diameter (d2

*) (× 10(-4) m)

K
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k'
s 

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B
on

d'
s

O
ru

aA
nt

ia
's

K
ic

k'
s 

R
itt

in
ge

r’s

B
on

d'
s

O
ru

aA
nt

ia
's

K
ic

k'
s 

R
itt

in
ge

r’s

B
on

d'
s

O
ru

aA
nt

ia
's

Se
a 

fo
od

C
ra

yfi
sh

0.
02

96
7

0.
00

04
0

0.
05

49
03

0.
07

51
13

0.
06

27
01

0.
09

14
86

02
:1

3

03
:0

1

02
:3

1

03
:4

1

4.
61

4.
18

4.
41

4.
14

C
er

ea
l

So
rg

hu
m

0.
00

31
7

0.
00

04
0

0.
06

18
48

0.
08

04
05

0.
06

95
18

0.
09

51
02

02
:2

9

03
:1

4

02
:4

8

03
:1

3

5.
14

4.
27

4.
91

4.
03

M
ill

et

0.
00

19
5

0.
00

04
0

0.
06

68
57

0.
08

56
68

0.
07

48
50

0.
09

99
50

02
:4

1

03
:2

7

03
:0

0

04
:0

1

4.
82

4.
25

4.
59

4.
07

C
or

n

0.
02

01
7

0.
00

04
0

0.
05

54
70

0.
07

53
63

0.
06

32
40

0.
09

15
81

02
:1

4

03
:0

2

02
:3

2

03
:4

0

5.
53

4.
18

4.
53

4.
06

W
he

at

0.
00

18
6

0.
00

04
0

0.
05

69
08

0.
07

61
92

0.
06

45
44

0.
09

19
90

02
:1

7

03
:0

4

02
:3

6

03
:4

2

5.
81

4.
46

4.
70

4.
15

Le
gu

m
es

So
yb

ea
n

0.
00

51
6

0.
00

04
0

0.
05

91
07

0.
07

78
74

0.
06

66
97

0.
09

30
86

02
:2

2

03
:0

8

02
:4

1

03
:4

4

5.
43

4.
25

5.
04

4.
06

Be
an

s

0.
00

63
0

0.
00

04
0

0.
05

50
52

0.
07

51
74

0.
06

28
87

0.
09

15
08

02
:1

3

03
:0

1

02
:3

2

03
:4

1

5.
53

4.
26

4.
44

4.
16

Tu
be

rs

C
as

sa
va

0.
00

14
0

0.
00

04
0

0.
07

54
08

0.
09

54
38

0.
08

41
49

0.
10

98
83

03
:0

2

03
:5

0

03
:2

3

04
:2

5

4.
92

4.
39

4.
41

4.
02

Ya
m

0.
00

19
6

0.
00

04
0

0.
07

11
56

0.
09

05
06

0.
07

95
07

0.
10

47
79

02
:5

2

03
:3

8

03
:1

2

04
:1

3

5.
05

4.
46

4.
49

4.
09

selected material to specific energy (E2
*) obtain from 

each energy equation were compared. The comparative 
assessment revealed that the new concept energy equation 
termed as OruaAntia energy equation had the actual and 
predicted (chosen) product size(s) to be reasonably close 
for all the selected samples. However, this new concept 
energy equation capability was followed by Rittingers, 
bond’s and then Kick’s energy equations. This is evident 
on the average percentage deviation of  1.66, 6.88, 
16.19and 38.40% respectively computed in this study and 

presented in Table 5.
Table 5 suggest that the new concept/approach of  
accomplishing any desired final product up to finer 
particle sizes may likely to be more reliable using 
OruaAntia’s energy equation followed by Rittinger’s 
equation. The percentage deviations obtain also confirm 
that using Kick’s equation may only achieve coarse particle 
while coarse and intermediate particle sizes are limited to 
Bond’s equation.Therefore, obtaining the desired coarse, 
intermediate and fine particles may likely be achieved 



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Am. J. Geo Spat. Technol. 2(1) 7-14, 2023

Table 4: Specific comminution energy E2
*,predicted grinding timetg, desired (500μm) and actual final product(s) size(s)

Se
le

ct
ed

 m
at

er
ia

ls
 

(s
am

pl
es

)

In
iti

al
 fe

ed
 

di
am

et
er

 (d
1) 

(m
)

D
es

ire
d 

(c
ho

se
n)

 
Fi

na
l p

ro
du

ct
 

di
am

et
er

 (d
2) 

(m
) E2

* Specific Energy 
Requirement (kWh⁄kg)

tg Predicted Crushing 
Time (min:sec)

Actual Final product 
Diameter (d2

*) (× 10(-4) m)

K
ic

k'
s 

R
itt

in
ge

r’s

B
on

d'
s

O
ru

aA
nt

ia
's

K
ic

k'
s 

R
itt

in
ge

r’s

B
on

d'
s

O
ru

aA
nt

ia
's

K
ic

k'
s 

R
itt

in
ge

r’s

B
on

d'
s

O
ru

aA
nt

ia
's

Se
a 

fo
od

C
ra

yfi
sh

0.
02

96
7

0.
00

05
0

0.
05

20
58

0.
05

98
85

0.
05

52
12

0.
06

54
21

02
:0

5

02
:2

5

02
:1

3

02
:3

8

5.
88

5.
11

5.
48

5.
08

C
er

ea
ls

So
rg

hu
m

0.
00

31
7

0.
00

05
0

0.
05

51
81

0.
06

20
02

0.
05

81
36

0.
06

67
80

02
:1

3

02
:2

9

02
:2

0

02
:4

1

5.
70

5.
13

5.
32

5.
01

M
ill

et

0.
00

19
5

0.
00

05
0

0.
05

74
33

0.
06

41
07

0.
06

04
00

0.
06

86
01

02
:1

9

02
:3

5

02
:2

6

02
:4

6

5.
59

5.
25

5.
37

5.
06

C
or

n

0.
02

01
7

0.
00

05
0

0.
05

23
13

0.
05

99
85

0.
05

54
70

0.
06

54
57

02
:0

6

02
:2

5

02
:1

4

02
:3

8

5.
89

5.
11

5.
60

5.
08

W
he

at

0.
00

18
6

0.
00

05
0

0.
05

29
60

0.
06

03
17

0.
05

60
23

0.
06

56
11

02
:0

8

02
:2

6

02
:1

5

02
:3

8

5.
90

5.
30

5.
64

5.
01

Le
gu

m
es

So
yb

ea
n

0.
00

51
6

0.
00

05
0

0.
05

39
48

0.
06

09
90

0.
05

69
37

0.
06

60
23

02
:1

0

02
:2

7

02
:1

7

02
:4

0

5.
58

5.
21

5.
32

5.
10

Be
an

s

0.
00

63
0

0.
00

05
0

0.
05

21
25

0.
05

99
10

0.
05

53
19

0.
06

54
29

02
:0

6

02
:2

5

02
:1

3

02
:3

8

5.
89

5.
23

5.
56

5.
03

Tu
be

rs

C
as

sa
va

0.
00

14
0

0.
00

05
0

0.
06

12
78

0.
06

80
15

0.
06

43
49

0.
07

23
34

02
:2

8

02
:4

4

02
:3

5

02
:5

5

5.
62

5.
18

5.
51

5.
04

Ya
m

0.
00

19
6

0.
00

05
0

0.
05

93
66

0.
06

60
42

0.
06

23
78

0.
07

04
16

02
:2

3

02
:4

0

02
:3

1

02
:5

0

5.
69

5.
12

5.
51

5.
06

Table 5: % Deviation of  Actual Final average diameter d2
* with predicted (chosen) final average diameter d2 of  

selected materials subjected to comminution
Diameter of  selected materials (m)

C
ra

yfi
sh

 (1
0(-

4)
)

So
rg

hu
m

 (1
0(-

4)
)

M
ill

et
 (1

0(-
4)
)

C
or

n 
(1

0(-
4)
)

W
he

at
 (1

0(-
4)
)

So
yb

ea
n 

(1
0(-

4)
)

B
ea

ns
 (1

0(-
4)
)

C
as

sa
va

 (1
0(-

4)
)

Ya
m

 (1
0(-

4)
)

∑
d 2* -d

2 
(1

0(-
4)
)

E
p

K
ic

k’s
 e

qu
at

io
n d2

* 5.46 4.96 4.24 5.27 5.19 5.10 5.27 5.68 4.76
d2 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
d2

*-d2 2.46 1.96 1.24 2.27 2.19 2.10 2.27 2.68 1.76 18.93 70.11
d2

* 4.61 5.14 4.82 5.53 5.81 5.43 5.53 4.92 5.05
d2 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00



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Am. J. Geo Spat. Technol. 2(1) 7-14, 2023

d2
*-d2 0.61 1.14 0.82 1.53 1.81 1.43 1.53 0.92 1.05 10.84 30.11

d2
* 5.88 5.70 5.59 5.89 5.90 5.58 5.89 5.62 5.69

d2 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
d2

*-d2 0.88 0.70 0.59 0.89 0.90 0.58 0.89 0.62 0.69 6.74 14.99
Epa 38.40

Ri
tti

ng
er

’s 
eq

ua
tio

n

d2
* 3.13 3.54 3.08 3.10 3.27 3.64 3.34 3.23 3.23

d2 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
d2

*-d2 0.13 0.54 0.08 0.10 0.27 0.64 0.34 0.23 0.23 2.56 9.49
d2

* 4.18 4.27 4.25 4.18 4.46 4.25 4.26 4.39 4.46
d2 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
d2

*-d2 0.18 0.27 0.25 0.18 0.46 0.25 0.26 0.39 0.46 2.70 7.50
d2

* 5.11 5.13 5.25 5.11 5.30 5.21 5.23 5.18 5.12
d2 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
d2

*-d2 0.11 0.13 0.25 0.11 0.30 0.21 0.23 0.18 0.12 1.64 3.64
Epa 6.88

Bo
nd

’s 
eq

ua
tio

n

d2
* 3.45 4.17 3.64 3.53 3.75 4.17 3.66 3.58 3.44

d2 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
d2

*-d2 0.45 1.17 0.64 0.53 0.75 1.17 0.66 0.58 0.44 6.39 23.67
d2

* 4.41 4.91 4.59 4.53 4.70 5.04 4.44 4.41 4.49
d2 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
d2

*-d2 0.41 0.91 0.59 0.53 0.70 1.04 0.44 0.41 0.49 5.52 15.33
d2

* 5.48 5.32 5.37 5.60 5.64 5.32 5.56 5.51 5.51
d2 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
d2

*-d2 0.48 0.32 0.37 0.60 0.64 0.32 0.56 0.51 0.51 4.31 9.58
Epa 16.19

O
ru

aA
nt

ia
’s 

eq
ua

tio
n

d2
* 3.03 3.01 3.01 3.06 3.09 3.07 2.97 3.14 3.10

d2 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
d2

*-d2 0.03 0.01 0.01 0.06 0.09 0.07 -0.03 0.14 0.10 0.48 1.78
d2

* 4.14 4.03 4.07 4.06 4.15 4.06 4.16 4.02 4.09
d2 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
d2

*-d2 0.14 0.03 0.07 0.06 0.15 0.06 0.16 0.02 0.09 0.78 2.17
d2

* 5.08 5.01 5.06 5.08 5.01 5.10 5.03 5.04 5.06
d2 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
d2

*-d2 0.08 0.01 0.06 0.08 0.01 0.10 0.03 0.04 0.06 0.47 1.04
Epa 1.66

using Orua Antia’s and Rittinger’s equations but much 
finer particles may possibly be further achieved using the 
Orua Antia’s energy equation.

CONCLUSION
Predicting and achieving closely the desired final particle 
size following comminution of  an initial particle size 
could be achieved more closely using the OruaAntia’s 
energy equation for mass-size reduction operation 
followed by Rittinger’s energy equation. The computed 
average percentage deviation 38.40%, 16.19%, 6.88% 
and 1.66% of  the desired particle size(s) with the actual 
particle size(s) of  material subjected to comminution re-
affirm the fact that kick’s equation may be limited only 
to coarse particle size reduction; while Bond’s energy 

equation may be for both coarse and intermediate particle 
size prediction. Moreso, the likely prediction of  coarse, 
intermediate and fine particles could be achieved using 
Rittinger’s and OruaAntia’s energy equations. However, 
further size reduction to finer particle sizes may be 
better achieved using the OruaAntia’s minimum energy 
equation which is a new concept/approach for mass-size 
reduction operations of  a given material.

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