International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 04, April 2019, pp. 285-299. Article ID: IJMET_10_04_028
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=4
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
Scopus Indexed
COMPARISON OF THE DIFFERENT WAYS OF
THE BALL BOND WORK INDEX
DETERMINING
Vladislav Valerevich Lvov and Leonid Sergeevich Chitalov
Department of Mineral Processing, Faculty of Mineral Processing
Saint-Petersburg Mining University, Saint-Petersburg, Russian Federation
ABSTRACT
When designing concentrating plants, the selection of the technological
comminution scheme for beneficiation, the type, quantity and size of the equipment, the
determination of optimal grinding regimes and the calculation of equipment loads are
carried out by preliminary studies of grinding in semi-industrial or laboratory
conditions. Semi-industrial tests ensure the most reliable information for calculating
loads for equipment. However, this requires a significant amount of ore samples, a lot
of labor and the pilot plants. Fred Bond published an article in 1961, which, described
the procedure for testing ores for a Bond Ball Index. This parameter is still one of the
most demanded tools in the design, evaluation and optimization of ball grinding plants
around the world.
However, the testing methodology for the Bond Dall Index requires about 10 kg of
sample, the standardized equipment and takes an average of 6 to 12 hours. Many
researchers have tried to find alternative methods for determining this Index - to reduce
labor costs, sample weight, or to get one without standard equipment. This paper was
carried out with the purpose of reviewing, classifying and testing the existing methods
for determination the Bond Ball Mill Index. The authors of the considered methods were
Aksani B. and Somnez B., Todorovic D., Berry T. F. and Bruce R. W., Horst W. E. and
Bassarear J. H., Ahmadi R. and Shahsavari Sh., Kapur P. C., Gharehgheshlagh Hojjat
H., JKTech, Lewis K. A., etc. The paper presents the relative errors of the obtained
value from the actual Bond Ball Mill Index, the average working hours for the testing
procedure and necessary equipment.
Keywords: Ball Mill Work Index, Bond Index, Grindability, Ball grinding, Ore
testing, Grinding energy consumption, Physical and mechanical ore properties.
Cite this Article Vladislav Valerevich Lvov and Leonid Sergeevich Chitalov,
Comparison of the Different Ways of the Ball Bond Work Index Determining,
International Journal of Mechanical Engineering and Technology, 10(4), 2019, pp. 285299.
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1. INTRODUCTION
During the development of dressing plants the choice of a technological scheme for preparing
the ore for dressing, the type, quantity and size of the main processing equipment, the
determination of the optimal regimes of grinding and the calculation of loads for the equipment
are carried out by preliminary studies of the grindability in the semi-industrial or laboratory
conditions. Most of the iron ore deposits in Russian Federation are known for showing
comparatively low grade ores and are consequently subjects for a necessity of ore dressing that
requires obligatory intense grinding. The result is that such concentrates are not adopted for
direct utilization in metallurgic separation and require clotting [1]. Semi-industrial tests on
continuous installations provide the most reliable information for calculating unit loads of the
equipment. However, they are associated with significant volumes of ore samples, high labor
costs and the presence of pilot plants.
In 1961 Fred Bond published a methodology for determining the working index of spherical
grinding. This indicator still remains one of the most popular tools for the development,
evaluation and optimization of spherical grinding plants around the world. The methodology
for determining the working index of Bond spherical grinding requires about 10 kg of material
and the availability of a standardized ball mill. The testing process, depending on a number of
conditions (size of the test sieve, homogeneity of material, grain-size composition, etc.), takes
up to 12 hours.
To cut labor costs for determining the Bond spherical grinding working index, to reduce the
required sample mass, to be able to determine the index in the absence of standardized
equipment, many researchers tried to find alternative methods for determining this indicator,
including Aksani B. and Somnez B., Todorovic D., Berry T.F. and Bruce R.W., Horst W.E. and
Bassarear J.H., Ahmadi R. and Shahsavari Sh., Kapur P.C., Gharehgheshlagh Hojjat H.,
JKTech, Lewis K.A. and others.
This work was performed to review, approbate and classify the existing methods for
obtaining the working index of Bond spherical grinding. The paper presents the deviations of
the results of the authors' methods from the actual working index of Bond spherical grinding,
as well as the estimated labor costs for the testing procedure and the necessary equipment.
Changes were proposed for the methods of Kapur and Ahmadi to improve the accuracy of their
results.
The work is carried out under financial support of the Ministry of Education and Science of
the Russian Federation, the project RFMEFI57417X0168
2. METHODS
The key stage in improving the technology of beneficiation of most ores in recent years has
been an increase in the efficiency of grinding operations [2]. During the development of
beneficiation plants the choice of the technological scheme for preparing the ore for
beneficiation, the type, quantity and size of the main process equipment, the determination of
the optimal grinding regimes and the calculation of loads for equipment are carried out by
preliminary studies of the grindability in the semi-industrial or laboratory conditions. Semiindustrial tests on continuous installations provide the most reliable information for calculating
unit loads of the equipment. However, they are associated with significant volumes of ore
samples, high labor costs and the presence of pilot plants.
In 1961 Fred Bond published a methodology for determining the working index of spherical
grinding. This indicator still remains one of the most popular tools for the development,
evaluation and optimization of spherical grinding plants around the world. The methodology
for determining the working index of Bond spherical grinding requires about 10 kg of material
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Comparison of the Different Ways of the Ball Bond Work Index Determining
and the availability of a standardized ball mill [3]. The classification of the ore grinded in the
ball mill in the hydrocyclone, mechanical and hydraulic classifier has shown a possibility of the
material fineness based clean cut separation. Since the underflow consists primarily of darkcoloured (deleterious) minerals, the classification also serves as a dressing process [4]. The
testing process, depending on a number of conditions (size of the test sieve, homogeneity of
material, grain-size composition, etc.), takes up to 12 hours.
To cut labor costs for determining the Bond spherical grinding working index, to reduce the
required sample mass, to be able to determine the index in the absence of standardized
equipment, many researchers tried to find alternative methods for determining this indicator,
which include a lot of methods for assessing the grindability of ore [5–11]. The choice of
methodology depends on the method and approach to the type of the developed scheme. The
most widely used methods are the methods of F. Bond.
Despite widespread use of the Bond spherical grinding working index [12–15], its major
drawback is that it takes up to 12 hours and up to 10 kg of the original sample to determine it.
In this regard, the test procedures to determine the working index of Bond spherical grinding
are difficult to use in projects where one needs hundreds or thousands of tests, for example, the
use of the methods of spatial modeling of the deposits by strength properties or the current
analysis of the ore coming to the processing plant [33]. In this regard, many researchers have
tried to intensify the procedure for determining the BWi. We will consider the most effective of
the existing methods.
2.1. The standard methodology of F. Bond [9]
The procedure for determining the working index of Bond spherical grinding [16, 3]
(hereinafter - BWi) is a periodic dry grinding in a standardized grinding mill at a fixed drum
rotation speed (70 rpm), standardized ball loading (20,125 kg) and feed size (-3.35 + 0 mm).
The BWi index is calculated from the following expression:
𝐵𝑊𝑖 =
1,1023⋅44,5
𝐴0,23 𝐺
𝑏𝑝
0,82
10(
1
−
1
√𝑃80 √𝐹80
)
, kWh / t
(1)
where A is the size of mesh of the test sieve, mkm (usually 106 mkm); Gbp – grindability
parameter in the last three test cycles, g/rev; F80 – theoretical size of sieve mesh, through which
80% of the mass of the initial sample passes, mkm; Р80 – theoretical size of sieve mesh, through
which 80% of the mass of the final product passes, mkm. All of the above-mentioned indicators
are determined by using the following test procedure.
In the first cycle the initial sample is ground at 100 revolutions of the mill. The supply of
the second and subsequent cycles consists of the oversize fraction of the product of the previous
cycle, supplemented with fresh supply to the initial sample mass.
The number of revolutions of the mill in each subsequent cycle is calculated from the
grindability parameter in the previous cycle so that the circulating load in the mill-classifier
node reaches 250%. For this goal the mass of the undersize product after grinding should be
1/3.5 of the mass of the original supply. Grinding cycles can be completed when the grindability
parameter (Gbp, g/rev) and the weight of the finished class will be unchanged for three cycles
(± 3%). The products of these three cycles are combined to find the parameter Р80.
The reproducibility of the results of the standard methodology of F. Bond is in the range of
3-5%.
2.2. Todorovic and others.
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The essence of this work [17] is to use the detected patterns between the indicators of various
cycles in the framework of the Bond test. Todorovik processed the Bond test database and
derived the following average ratios for the grindability parameters in the Bond test:
𝐺𝑒
𝐺2
𝐺
𝐺
≈ 𝑐𝑜𝑛𝑠𝑡 ≈ 1.158; 𝐺𝑒 ≈ 𝑐𝑜𝑛𝑠𝑡 ≈ 1.096; 𝐺𝑒 ≈ 𝑐𝑜𝑛𝑠𝑡 ≈ 1.037,
3
4
(2)
where Ge - – stabilized grindability for the last three cycles of the Bond test, g/rev; G2, G3,
G4 – grindability for the 2, 3 and 4 cycles of the Bond test, respectively.
The first cycle was not used, because the mass of the undersize product depends on the
granular composition of the supply and other characteristics of the material.
The same relationship was found for the product size parameters:
𝑒
𝑃80
2
𝑃80
𝑒
𝑃80
3
𝑃80
≈ 𝑐𝑜𝑛𝑠𝑡 ≈ 1.035;
≈ 𝑐𝑜𝑛𝑠𝑡 ≈ 1.030;
𝑒
𝑃80
4
𝑃80
≈ 𝑐𝑜𝑛𝑠𝑡 ≈ 1.017,
(3)
𝑒
3
2
4
where 𝑃80
is the product size for the last three Bond test cycles; 𝑃80
, 𝑃80
, 𝑃80
– the size for
products in the 2, 3 and 4 cycles of the Bond test, respectively.
Therefore, by using these ratios one can perform two or more grinding cycles in accordance
with the standard Bond procedure, after which it is possible to calculate grindability, to conduct
a size analysis of the product of the second cycle and calculate a theoretically final grindability
and the size of the product. The obtained indicators, as well as the parameter of the size of
supply and the mesh of the test sieve, are further used to calculate the BWi index according to
the standard formula (1).
The stated relative error for this method is within the range of 4%. This method was tested
on a sample of oxidized ferruginous quartzites and showed a relative error of 4.4% and 0.3%
when using the indicators of the second and third grinding cycle, respectively.
2.3. Berry and Bruce
The method of Berry and Bruce [18] makes it possible to use any grinding mill to determine
the BWi index.
The essence of the method consists in obtaining the working index of Bond spherical
grinding from the results of grinding of the investigated and the reference ore (for which the
working index of the Bond spherical grinding is known). The algorithm of action is as follows:
1. A sample of the known mass of the studied ore is ground to achieve the required
product size. Berry and Bruce used 2 kg of ore with a size of -1.7 mm and a wet
grinding mode.
2. A reference ore (with a known working index) of the same mass as in paragraph 1
is ground in the same mill under the same conditions.
3. A sieve analysis of the supply and the products of grinding is carried out for each
sample and the parameters of the size F80 и P80 are calculated. Since the operating
conditions of the mills in both experiments are the same, the energy applied to the
samples during grinding should be equal. Based on this, the Bond spherical grinding
index of the studied ore can be calculated from the equation:
𝐵𝑊𝑖 = (
1
√𝑃𝑢
−
1
√𝐹𝑢
1
Р
√ 𝑟𝑒𝑓
) ≅ 𝑊𝑖𝑟𝑒𝑓 (
−
1
),
𝐹
√ 𝑟𝑒𝑓
(4)
where Wiref is the working Bond index of the reference ore, Fu and Pu – F80 and P80 are the
studied ores, Fref and Pref – F80 and P80 are the reference ores.
The stated relative error lies within the range of 8%. This method was tested on a sample of
apatite-nepheline ore and showed a relative error of 6.3%.
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2.4. Horst and Bassarear
This method [19] is similar to the method of Berry-Bruce in terms of obtaining the Bond index
at any existing grinding mills.
Horst and Bassarear used the original supply with a size of 1,7 mm, similar to the method
of Berry-Bruce for non-standard laboratory mills. The essence of the method is as follows:
1. The granulometric composition of the supply for the reference ore is measured.
2. A sample of the reference ore (with a known Bond index) is ground for a certain
period of time until the desired size is obtained.
3. Three samples of the studied ore are ground under the same conditions with different
time periods including the periods shorter and longer than in step 2.
4. The results of the sieve analyses of products from three experiments are reduced to
the following kinetic equation:
𝑙𝑛( 𝑚𝑖 ) = 𝑙𝑛( 𝑚0𝑖 ) − 𝑘𝑡,
(5)
where mi is the total mass of the fraction remaining in the i-th sieve; moi – the total mass of
the fraction remaining in the i-th sieve at time zero; ki – the ratio of grinding of fraction in the
i-th sieve; t – time.
1. The granulometric composition of the product of grinding of the tested sample is
calculated by using the coefficients of equation from paragraph 4, but with the initial
granulometric composition as with the reference sample.
2. From step 5 80% of the ground product is evaluated.
3. The working index of the unknown ore is determined by the Berry-Bruce equation.
In fact, one determines the granulometric composition of the tested ore if its initial
granulometric composition fully corresponded to the initial granulometric composition of the
reference ore.
This method is considered more accurate than the method of Berry and Bruce, but it should
be borne in mind that not all ores follow a simple first order equation. This method was tested
on a sample of apatite-nepheline ore and showed a relative error of 8.44%.
2.5. The method of the Anaconda Company
Yap et al. [20] proposed their own method for determining BWi, which makes it possible to use
any grinding mill. It is similar to the methods of Berry-Bruce and Horst- Bassarear, but does
not require the reference ore in a single test. Instead, it requires the calibration of a grinding
mill different from the Bond mill, which will be used for later testing.
Its essence is the following: one determines the conditions, which are as close as possible
to the corresponding conditions in the Bond test methodology:
 Dimensions of the mill’s drum;
 Ball loading;
 The size of supply (the finished class is removed from the supply);
 The speed of rotation of the mill
The index is calculated from the following equation:
𝐵𝑊𝑖 =
𝛼𝐸̄
10
⋅(
10
√𝑃80
−
10
√𝐹80
−1
) ,
(6)
where 𝛼 is the coefficient of proportionality, 𝐸̄ — the specific energy of destruction in the
open cycle.
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𝛼𝐸̄
The calibration of the mill consists in the selection of expression 𝐴 =
using the least
10
squares method in such a way as to minimize the average relative error between the actual BWi
index and the index obtained at the calibrated mill according to the following expression:
−1
𝐴=
∑𝑛
𝑗=1 𝐵𝑊𝑖𝑗 ⋅(1/√𝑃80 −1/√𝐹80 )
𝑗
∑𝑛
𝑗=1(1/√𝑃80 −1/√𝐹80 )
(7)
−2
𝑗
Yap and Sepulveda calibrated their mill with 19 samples of various types of ores. For each
of them a standard Bond test was performed, which was followed by grinding in a calibrated
mill for 10 minutes. The diameter of the drum was 25,1 cm with a length of 21 cm. The grinding
media consisted of steel balls (see Table 1), the rotation speed was 92 min-1 or 96% of the
critical, the mass of the loaded sample – 1 kg, particle size – 1,700 + 147 mkm. A wet grinding
was used with a solid content of 50%. The screening of supply and ground product was
conducted. For the chosen mill Yap and Sepulveda received A=0.5031.
Table 1. Ball loading of the mill used in the method of Yap and Sepulveda
Diameter of balls, mm
35.6 – 38.1
31.8 – 33.0
29.2 – 31.0
25.4 – 27.9
24.1 – 25.4
22.9 – 24.1
Total
Number of balls, pieces
11
17
13
10
7
30
88
Mass, g
2316.5
2325.4
1534.8
822.5
449.7
1634.0
9082.9
The stated average relative error for more than twenty samples was 4.1%.
2.6. Ahmadi R. и Shahsavari Sh.
Ahmadi and Shahsavari [21] presented the method of Magdalinovich [22] in their new edition.
Their method uses a standard mill and Bond M ball loading.
The essence of the method is as follows:
1) Two identical subsamples are prepared. They represent 2.5M /3.5= MC of the mass of the
oversize product and 1/3.5 of the mass of the representative original sample;
2) The first sample is placed in the mill and ground at 100 revolutions;
3) After the grinding the entire sample is classified according to the check class. The mass
of the oversize product MOS is registered. This mass should be equal to mass MC during the
circulation of 250%.
4) The constant of grinding of a large class K is calculated from the following expression:
𝐾=
𝑛(𝑙𝑛 𝑀С −𝑙𝑛 М𝑂𝑆 )
,
𝑁
(8)
where n is the number of revolutions of the mill per minute, N is the number of revolutions
of the mill in the first grinding cycle.
5) The total number of revolutions of the mill for the second grinding cycle NZ is calculated
from the expression:
𝑁𝑍 = 𝑛𝑙п(1 + 0,4т0 )/𝑘,
(9)
where m0 = the proportion of the upper class in the original sample;
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6) After the second grinding period the mill product is classified according to the check
class with Р80 determined for the undersize. The oversize product should be approximately
equal to 2.5М/3.5М.
7) The BWi index is determined by the standard Bond equation using the grindability
obtained in the second grinding period.
This method was tested on a sample of apatite-nepheline ore and showed a relative error of
7.0%.
2.7. Kapur
This method [23] uses the same mill and grinding conditions as in the standard Bond test. The
essence of this method is in the use of the following empirical equation:
BWi=K[Pi]a [G2]b [RoM1]c [1- Ro]d,
(10)
where Pi is the mesh size of the control sieve, mkm; G2 – the parameter of grindability in
the second grinding cycle, g/revs; R0 – the mass of oversize product in the original material,
unit fraction, M1 - the mass of the mill’s load, g; K, a, b, c, d – dimensionless empirical
coefficients depending on physical and mechanical characteristics of the ore.
The results of Kapur's equation, when used with general coefficients in practice, have an
average relative error of 9.4%. The Kapur’s method was tested on 37 samples of various types
of ores and showed an average relative error of 8.6% [32].
2.8. Carr
Carr [24] modified Kapur's algorithm and proposed another empirical equation for calculating
the Bond index, which also uses the Bond grindability and only two grinding cycles:
−0,125
𝑊𝑖 = 9,934 ⋅ 𝑃0,308 𝐺2−0,696 𝐹80
(11)
The average relative error declared by the author was 5.0%, which is a better result than in
the Kapur’s method. The Carr’s method was tested on 22 samples of various types of ores and
showed an average relative error of 5.5%.
2.9. Smith and Lee
Smith and Lee [25] conducted a standard Bond test for 8 types of mineral raw materials for five
classes of product sizes and, based on this study, proposed an alternative method of obtaining
the Bond working index by using the following formula:
𝐵𝑊𝑖 =
16
𝐺 0,82
√
𝑃
,
100
(12)
where G is the grindability in the first Bond cycle, g/revs; Р – the upper limit in the size of
the finished class, mkm.
According to the data obtained by the author, the stated average relative error for this
method is 8.2%. This method was tested on 44 samples of various types of ores and showed an
average relative error of 16.6%.
2.10. Gharehgheshlagh Hojjat H.
The essence of this work [26] is to assess the Bond grinding work index by means of classical
tests for determining the parameters of grinding kinetics according to the grinding duration of
0.33, 1, 2, 4 and 8 minutes for all size classes in the same mill and under the same conditions
of grinding as in the standard Bond test. The calculation of the working index of Bond spherical
grinding after the test can be carried out with the same success on any control sieve.
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The testing is carried out according to the following algorithm:
1. A sample with a particle size of -3.35 mm and a mass of about 10 kg is prepared;
2. One determines the mass M0 of the representative sample from the initial sample
with a particle size of -3.35 mm tightly tamped down on a vibrating plate into a
vessel with a volume of 700 ml.;
3. The granulometric analysis of the initial sample is carried out;
4. 5 representative samples with the mass M0 are prepared;
5. Samples are ground for 0.33, 1, 2, 4 and 8 minutes;
6. The granulometric analysis of 5 products of grinding is performed;
7. The time of grinding is translated into the revolutions of the NR mill;
8. The coefficients a and b of the level of dependence between the number of
revolutions of the mill NR and the newly formed class GNe are found:
𝑁𝑅 = 𝑎(𝐺𝑁𝑒𝑡 )𝑏 ;
(13)
1) The coefficients a1, a2, a3, a4 of the polynomial dependence between the indicator of the size
of the product P80 and the number of revolutions of the mill NR are found:
P80= a1NR3 +a2 NR2+ a3 NR+a4;
(14)
2) The required newly-formed class at 250% of circulation GNet(250%) (250%) is calculated by
the expression:
(15)
where Gfeed is the mass of the finished class in the initial sample before grinding;
3) One calculates the number of revolutions in order to reach 250% of circulation NR(250%) by
using the necessary mass of the newly formed class from step 10 in the equation by using
the coefficients of step 8;
4) One calculates the parameter P80 by using the equation and coefficients found in step 9 with
the use of the number of revolutions found in step 11;
5) One calculates the grindability according to the Bond Gbp expression:
GNet(250%)=(M0/3.5)-Gfeed;
Gbp =GNet(250%)/NR(250%);
(16)
6) The BWi index is calculated according to the standard Bond formula.
7) The stated relative error lies within the range of 4.5%. This method was tested on a sample
of apatite-nepheline ore and showed a relative error of 3.5%.
2.11. JK Bond Ball Mill Test
The methodology of testing [27] was obtained by JKTech company after the statistical
processing of data of about 1400 standard tests of the Bond ball grinding. JKTech showed that
in order to obtain the reproducibility of the test it is sufficient to conduct only the first three
Bond test cycles followed by mathematical processing of the obtained results. This testing
method uses a standard Bond mill.
After this the undersize product of the 3rd cycle is sieved. The stated average relative error
is less than 4.1%.
This method was tested on four samples of different types of ores and showed an average
relative error of 3.5%.
3. THE REVIEWED METHODS
3.1. Lewis K. A. et al.
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The Lewis method [28] was formed after obtaining the appearance functions for n size classes
for subsequent use in the cumulative grinding model:
𝑅𝑖 (𝑡) = ∑𝑖𝑛=1 𝑎𝑛𝑖 ⋅ 𝑒𝑥𝑝(−𝑘𝑛 ⋅ 𝑡)
(17)
where with n≠i the following equation is observed:
𝑎𝑛𝑖 =
∑𝑖−1
𝑗=𝑛(𝑘𝑗 ⋅𝑏𝑖𝑗 ⋅𝑎𝑛𝑗 )
(18)
𝑘𝑖 −𝑘𝑛
and with n=i the following equation is observed:
𝑎𝑖𝑖 = 𝑊𝑖 (0) − ∑𝑖−1
(19)
𝑛=1 𝑎𝑛𝑖
where 𝑅𝑖 (𝑡) is the mass of class i after the time of grinding t; ki – a selective function for
size i; bij –the appearance function. These functions are calculated in the following way:
𝑋
𝛼
𝑖
𝑘𝑖 = 𝑆1000 ⋅ (1000
) ;
(20)
𝑏𝑖𝑗 = 𝐵𝑖𝑗 − 𝐵𝑖+1,𝑗 ;
𝑋
𝛾
(21)
𝑋
𝛽
𝐵𝑖𝑗 = 𝛩 ⋅ (𝑋 𝑖 ) + (1 − 𝛩) ⋅ (𝑋 𝑖 ) ,
𝑗
𝑗
(22)
where Xi is the geometric average size for class i, mkm; S1000, α, β, γ – the parameters of
the model, which are necessary for finding each sample. The time of grinding t is calculated
from the number of revolutions.
By using this model after the first grinding cycle it is possible to obtain the appearance
function for a certain set of size classes, then calculate all subsequent periods until the process
stabilizes at the required (250%) circulating loading making it possible to calculate the
grindability in the last cycles and the size of the ground products. At the last stage one can use
the standard Bond formula. The stated average relative error is less than 3,5%.
3.2. MiniBond
This technique [29] is part of the Metsuite test series of the Aminpro company and is aimed at
mass testing to perform a field mapping by using the Bond ball grinding index. The essence of
this technique consists in carrying out one experiment of grinding a sample weighing 600 g in
a mill, which is half a drum of the standard Bond ball mill with the corresponding ball load. To
carry out a series of tests it is necessary to calibrate the mathematical apparatus of the method
by using a standard Bond ball mill with two grinding cycles. After calibration this method
makes it possible to determine the working index of Bond ball grinding of a series of samples
with an average relative error in the range of 3.0%. This technique has not been tested due to
the need to use specialized equipment.
3.3. Aksani B., Somnez B.
The authors of this paper have simplified the calculation and experimental procedure for
determining the working index of Bond ball grinding [30].
The essence of the work is to use a mathematical model based on the following kinetic
equation:
𝑅(𝑥,𝑡) = 𝑅(𝑥,0) ⋅ 𝑒𝑥𝑝(−С ⋅ 𝑥 𝑛 ⋅ 𝑡),
(23)
where R(x,t) is a total yield on the sieve with mesh size x in time t; C and n are constants
depending on the used mill and characteristics of the studied material.
To determine the parameters of the model during testing it is necessary to carry out the
grinding, the conditions of which should correspond to the standard Bond test. The
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Vladislav Valerevich Lvov and Leonid Sergeevich Chitalov
granulometric characteristic of the supply in the first grinding period, the mass of the supply
and the number of revolutions are the initial data for calculating the next grinding cycle with
the help of a specially created mathematical apparatus. The product of unloading of the mill is
used to calculate P80(general), after which P80 of the ready class is also determined. The results of
the next cycle, which will be carried out within the framework of the mathematical model, are
calculated by using the above-described kinetic equation. Later a cyclic calculation is carried
out until the grindability is stabilized in the last three cycles, after which the standard Bond
equation is used to calculate the index. The stated relative error for this method is shown within
4%. This technique was not tested due to the need to use the mathematical apparatus, which is
absent in open sources.
3.4. Armstrong
In his work Armstrong [31] showed that the conversion factor from dry to wet grinding, when
converted from the laboratory index of Bond grinding into the production one, is too rough a
way of calculation and contains an error. The method uses a rod mill with a diameter of 20.3
cm and a length of 25.4 cm for wet grinding (60 rpm, 67%тв) in an open cycle (the load consists
of 25 rods with a diameter of 2.54 cm), which should give a product approximately the same
size as the standard Bond ball mill in the open cycle. The power consumption of the mill was
recorded to exactly determine the power consumed for grinding. One performs two independent
grinding cycles at different times. After that the working index of the Bond ball grinding was
calculated by using the following formula:
𝐵𝑊𝑖 =
0,064
1,1023(
10
−
10
√𝑃80 √𝐹80
,
)
(24)
where 0,064 is the power of the engine used at the mill, kW; 1.1023 – the coefficient of
conversion of short tons into metric ones; F80 – theoretical size of the sieve’s mesh, through
which 80% of the mass of the initial sample will pass, mkm; Р80 – theoretical size of the sieve’s
mesh, through which 80% of the mass of the final product will pass, mkm.
The obtained average relative error from the Bond ball grinding working index was about
6%, however, in his work Armstrong makes it clear that this index does not require a conversion
factor from dry to wet grinding, therefore, it is more suitable for calculations at industrial mills.
This technique has not been tested due to the lack of specialized equipment.
4. RESULTS AND DISCUSSIONS
The studied methods can be divided into four categories [24]:
1. Comparative methods, in which reference ore is used to obtain the working index of
ball grinding (Berry-Bruce, Horst- Bassarear, Anaconda).
2. Empirical methods, in which a periodic test and an empirical equation are applied
by using regression methods and a database of ores of various types (Kapur
equation, Carr, Smith, and Lee).
3. Modeling methods, in which the functions of destruction and selection of ore are
measured (usually with a standard Bond mill) with an introduction of the classifier
model followed by the simulation of a closed cycle as in the standard test
(Gharehgheshlagh, Lewis, Ahmadi).
4. Short methods, which are based, as a rule, on the empirical basis of tests data
obtained earlier (JKTech, MiniBond, Armstrong).
Table 2 and Figure 1 show average relative errors of the results and the labor costs of the
considered techniques. Figure 2 shows a flowchart for selecting the appropriate methodology
for different research conditions.
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Comparison of the Different Ways of the Ball Bond Work Index Determining
Table 2. The methods for determining the working index of Bond ball grinding
Method
*М, kg **∆1, %
***∆2,
%
Bond (standard)
10
Todorovic
2
4.0
0.3
Gharehgheshlagh
Horst
Ahmadi
Ahmadi (mod.)
Berry
JKBBM
10
5
5
5
1
4
5.0
7.0
7.0
3.5
8.0
4.1
3.5
8.4
6.8
3.5
6.3
3.5
Kapur
3
8.0
9.5
Kapur (mod.)
Carr
Carr ( mod.)
Smith and Lee
Lewis
3
3
5
2
2
4.4
8.2
3.5
3.5
5.1
1.7
16.6
-
MiniBond
4
3
-
Armstrong
Aksani
Anaconda
3
2
2
6.0
4.0
4.1
-
Note
Standard methodology, unchanged since 1961
∆ср the smaller, the greater the number of
grinding cycles
Calculation for any size of the finished class
Applicable to all mills, reference ore is required.
The analog of the Magdalinovich method
Database required
Applicable to all mills, reference ore is required.
Patented test
After calibrating the method for a specific type
of ore ∆ср is significantly reduced (up to 3.5%)
Database required
The analog of the Kapur’s method
When using three cycles
Very outdated method
Low ∆ср for homogeneous ores
Patented test; only for mass testing, calculation
for products of any size
The test for wet grinding in a rod mill
Low ∆ср for homogeneous ores
Applicable to all mills, reference ores required.
* M – the approximate required sample mass;
∆1 – the average relative error of the method stated by the author; ***∆1 – the received
relative error;
Figure 1. Alternative methods for determining the working index of Bond ball grinding
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Vladislav Valerevich Lvov and Leonid Sergeevich Chitalov
Figure 2. Block diagram of the choice of alternative methods for determining the working index of
ball grinding of Bond
When analyzing the Ahmadi method, it was found that the desired circulation of 250% is
not always possible to achieve in two grinding cycles, which leads to an increase in the relative
error of the entire test. However, the result was significantly improved [19, 20] when in the first
cycle one used the number of revolutions, which contributed to the approximation to a given
circulation in the second cycle. For this it is necessary to have the results of the standard Bond
tests for ores, which are close to the studied ones according to the genesis [34]. From these tests
it is necessary to use the arithmetic mean of the number of revolutions of the last three grinding
cycles. On the same sample of apatite-nepheline ore the Ahmadi method showed an error of
3.5% versus 7% in the standard edition of the method.
When analyzing the Kapur's method a correlation of error values for the ores of similar
genesis was found. In this regard it was proposed to use the Kapur equation with empirical
coefficients found for individual types of ores that are similar in their genesis. The coefficients
for apatite-nepheline, copper-nickel, gold-bearing ores and oxidized ferruginous quartzites
were found [19, 20]. The average relative error of this method with such approach will be within
3-4%.
When analyzing the Carr method the average relative errors were derived by using three
and four grinding cycles, for which the average relative errors of determining the BWi index
were 1.7 and 1.3%, respectively. At the same time one should use the empirical coefficients of
the equation determined on the basis of standard Bond tests for 22 tests of ores with different
check meshes of the sieve:
0,5362 −0,8821 −0,2711
For the third cycle: 𝑊𝑖 = 9,9317 ⋅ 𝑃100
𝐺2
𝐹80
;
0,5037 −0,8540 −0,2370
For the fourth cycle: 𝑊𝑖 = 9,1119 ⋅ 𝑃100
𝐺2
𝐹80
.
5. CONCLUSION
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Comparison of the Different Ways of the Ball Bond Work Index Determining
The most effective alternative methods for determining the working index of Bond ball
grinding, three of which were modified, were reviewed and tested:
 For the Ahmadi method it was proposed to use a special number of revolutions in
the first grinding cycle depending on the genesis of the test sample, while the relative
error was reduced from 7.0 to 3.5%;
 For the Kapur method it was proposed to use special empirical coefficients
depending on the genesis of the test sample with the relative error reduced from 8.0
to 3.5%.
 For the Carr method it was proposed to use 3 or 4 grinding cycles with appropriate
empirical coefficients of the equation.
A flowchart was developed in order to select a method depending on the tasks facing the
researcher.
FUNDING STATEMENT
The work is carried out under financial support of the Ministry of Education and Science of the
Russian Federation, the project RFMEFI57417X0168
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