Minerals Engineering 20 (2007) 140–145
This article is also available online at:
www.elsevier.com/locate/mineng
The energy distribution theory of comminution specific surface
energy, mill efficiency and distribution mode
Elias Th. Stamboliadis
*
Department of Mineral Resources Engineering, Technical University of Crete, University Campus, 73100 Chania, Crete, Greece
Received 23 March 2006; accepted 29 July 2006
Available online 24 October 2006
Abstract
The present paper is a partial theoretical approach to the comminution process. A general theory of comminution should consist of
two parts, one that deals with the energy required to break mineral particles and another that examines how this energy is distributed to
the particles generated after breakage. The present approach deals with the second part that examines how the energy invested for comminution is distributed to the mill product. It uses the generally accepted concept, which assumes that the useful part of comminution
energy is consumed to create new surfaces and finds the relationship between a characteristic particle size of the mill product and the
energy consumed for grinding. The paper introduces the concept of potential energy and provides the means to give a value to the energy
state of a material produced by a specific type of equipment. The energy efficiency is also taken into consideration and is used to calculate
the energy actually invested for comminution. The main conclusion is that the specific surface energy is a physical property of materials
and can be used as a universal index characterizing their grindability, regardless of the mill type or the mill efficiency. The physical dimensions of this index are energy per unit surface area (J/m2) compared to energy per unit volume or unit mass, which are the dimensions of
the indices proposed so far.
2006 Elsevier Ltd. All rights reserved.
Keywords: Comminution; Crushing; Grinding; Mineral processing; Modeling
1. Introduction
The energy-particle size relationship is an important
aspect of the minerals comminution process and a lot of
experimental and theoretical efforts have been devoted to
it. Actually there are two different approaches to it that initially were confused and still are in almost all the relevant
textbooks and published research works. The first
approach is that of Kick (1885) and refers to the energy
required to break a particle of a given mass and hence of
a given size. It states that the energy required to break a
particle of a certain material is proportional to its mass
or in other words the specific energy (energy per unit mass)
required for breakage is constant. Some recent experimental work by Tavares and King (1998) has shown that this is
*
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doi:10.1016/j.mineng.2006.07.009
true for relatively large particles only and a theoretical
explanation is given by Stamboliadis (2005). According
to this at large particle sizes the surface energy required
to elastically deform a particle before breakage is negligible
compared to the internal energy required for breakage as
predicted by Kick. This explains why the specific energy
for breakage is practically constant. However at smaller
particle sizes the energy for elastic deformation before
breakage becomes appreciable and at some point it overpasses the internal energy for breakage. This explanation
does not exclude but at the same time it has not as a prerequisite Griffith’s hypothesis (1921) that at smaller sizes
solid materials become stronger due to the presence of
lesser structural defects. The above mentioned as Kick’s
approach is not related to the size distribution of the comminution products but deals only with the energy required
for breakage and it will be referred as ‘‘The Breakage Theory of Comminution’’.
E.Th. Stamboliadis / Minerals Engineering 20 (2007) 140–145
The second approach is that of Rittinger (1867) and
relates the energy consumed to the size distribution of the
products. According to this, the new surface area produced
is proportional to the energy consumed for breakage. It has
been shown by Stamboliadis (2004) that Kick’s and Rittinger’s statements can be true at the same time and are not
exclusive. The so called ‘‘Third Theory’’ of comminution,
proposed by Bond (1952) is only a partial case of a general
expression of the Rittinger’s statement as shown by Charles
(1957) and later by Stamboliadis (2003). These authors
have come to similar results using different breakage
models, although Charles still confuses the Kick’s and Rittinger’s approach. Although Bond’s statement is a partial
case of the energy-size relationship it has been proved very
useful especially in the design of ball mill circuits because in
this application most of the experimental data conform to
it. The theoretical approach to the energy-size relationship
will be referred as ‘‘The Energy Distribution Theory of
Comminution’’ and the present work will deal with this
part of the comminution theory.
The term ‘‘grindability index’’ is mainly used to provide
a measure of the difficulty or the energy required to comminute a certain material from an initial coarse size to a finer
one. The indices proposed by Hardgrove (1932) and Bond
(1961) are related to the equipment used and according to
Fuerstenau and Kapur (1994) do not satisfy the requirements for an ideal measure of the inherent grindability of
a solid that must be independent of feed size, product fineness, quantum of energy dissipation and the nature of the
comminution equipment employed. Using the experimental
methodology of Kerber and Schoenert, described by
Kerber (1984), consisting of an instrumented roll mill that
can crush single particles under compression, Fuerstenau
and Kapur have proposed a new grindability index that
still is dependant on feed size. One should note that
the dimensions of these indices are energy per unit mass
(J/kg) or (kwh/ton).
2. Theoretical background
2.1. Surface area, mass–size distribution and equivalent size
Mill products are usually tested for the distribution of
their mass over predetermined size classes. The cumulative
mass, finer than a certain size, is generally used to define
the mass distribution and is usually, but arbitrarily, called
‘‘size distribution’’ although we all understand the mass
distribution over the sizes. The mathematical equations
mainly used to describe such distributions are the Rosin–
Ramler (R–R) and the Gates–Gaudin–Schuhmann (G–
G–S). They are satisfactory to predict the mass distribution
and can calculate with acceptable accuracy the sizes d80 or
d50 that are commonly used to describe the fineness of mill
products. However both fail to describe the surface area
distribution of any sample and cannot be used to calculate
its specific surface area. Although they give negligible mass
141
at fine sizes the corresponding surface area is enormous
and tends to infinity as the size approaches to zero.
The reason is that there is no minimum to the size they
apply. However mill products have well defined and measurable surface areas. If the surface area of any sample is
known then one can calculate an equivalent size, ds that
has the same specific surface area as the sample itself and
attribute this size to the sample. In the following the size
ds will be referred as the equivalent surface size of the
sample and will be used to define it.
The problem is whether one can use a particular mass–
size distribution model in order to calculate the surface
area of a sample and hence its equivalent surface size. Such
a mass–size distribution model has been proposed by Stamboliadis (February 2004) according to which the surface
area of any particular sample is proportional to the energy
consumed for its creation and the distribution of energy
among the molecules follows the Maxwell–Boltzman one.
The key to this task is the definition of the ‘‘energy level’’
of the molecules that make a particle. This model, regardless of its theoretical merits, is difficult to use for every-day
practical work.
2.2. Mass–size distributions of mill products
Mill products follow a specific mass–size distribution
and the mathematical models used to describe them, R–R
or G–G–S, have two parameters that define the specific distribution, as it happens with most of the statistical mathematical distributions.
One of the parameters determines the size modulus,
which is a characteristic size of the distribution. G–G–S
have used the size d100 at which 100% of the material is finer,
while others have used the d80 or d50. The second parameter
determines the distribution modulus, which shows the deviation of the particles mass dispersion around the mean. It is
understood that during comminution the size modulus
moves to finer sizes and the experimental results show that
in most cases the distribution modulus remains more or less
constant. In these cases the ratio of any two characteristic
sizes, for instance d50/d80, remains constant for all mill
products at different energy inputs. If the distribution modulus is not constant then the ratio of the two characteristic
sizes is expected to change during comminution.
However, due to its definition the ratio of the equivalent
surface size ds to any other characteristic size is not
expected to be constant and must depend on the breakage
function of the material in the specific comminution equipment used.
2.3. Mill efficiency
Mill efficiency is an important parameter that affects the
energy utilization during comminution. It is affected by the
mechanical energy transition, the breakage mode, and
the energy lost in elastic and permanent deformation of
the solid particles before breakage. Obviously, energy
142
E.Th. Stamboliadis / Minerals Engineering 20 (2007) 140–145
efficiency does not depend only on the equipment used but
it is also dependant on the material itself especially at smaller sizes.
• The specific surface area s (m2/kg) of the mill product.
• The equivalent specific surface area size ds, of the sample
calculated according to Eq. (1) that applies to spherical
and cubic particles.
3. Experimental procedure
The experimental results used in this work were produced by Petrakis (2004), who describes the experimental
procedure in detail and are processed here in a new way.
Since the original reference is in Greek the experimental
procedure is repeated here in brief.
The materials used are coarse crystal marble from the
island of Naxos and quartz from the area of Assyros in
Greek Macedonia. Both samples were crushed to 4 mm,
in 20 · 10 cm laboratory jaw crusher and separated into
samples of 1 kg using a splitter.
The experimental device is a laboratory 20 · 30 cm
(D · L) rod mill loaded with 9 kg of rods and rotated on
controlled speed rollers at a predetermined frequency
N = 60.5 rpm (1.01 Hz) of the mill. According to Stamboltzis (1990), the net power P (W) invested for breakage is a
function of the mill diameter D (m), the load (rods plus
the material) W (kg), and the rotation frequency N (Hz).
It is calculated by the formula, P = 9.9 Æ D Æ W Æ N. For
the specific mill the power is 20 (W) and the energy E (J)
provided for grinding in each test is E = P Æ t where, t is
the time of the duration of the test in sec. If M is the mass
of the material in the mill the specific energy for grinding is,
e = E/M.
For each test the sample is 1 kg and the times used are
2.5–5–10–20 and 40 min. The initial feed for each mineral
and the individual mill products are wet screened at
38 lm (400 mesh). The +38 lm fraction is dried
pffiffiffi and
screened in a series of n screens of aperture ratio 2 and
the total 38 lm is dried and analyzed by a laser beam size
analyzer made by Malvern. The individual surface area of
the +38 lm fractions is calculated as follows. The average
size Di of a fraction passing a screen, with aperture ai1 and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
retained on a screen with aperture ai, is Di ¼ ai ai1 , its
weight is Dmi and its surface area assuming spherical parti6
cles is DS i ¼ Dmi qD
, where q is the density of the material
i
3
(2700 kg/m for marble and 2650 kg/m3 for quartz). The
surface area of the 38 lm fraction is DSi+1 = Dmi+1 Æ si+1
where sn+1 is the specific surface area automatically calculated by the laser analyzer software that assumes spherical
particles
P as well. The total surface
Pnþ1area of the sample is
S ¼ nþ1
DS
,
its
mass
is
M
¼
i
1
1 Dmi and the specific
surface area s = S/M.
ds ¼
6 1
;
q s
where q is the density of the material (kg/m3).
The data obtained are related as follows.
4.1. The relationship of d50 versus ds
The plot of d50 versus ds is presented in Fig. 1, both for
marble and quartz. There is some distinction between marble and quartz and it appears that for the same d50 quartz
gives a coarser ds.
Eq. (2) can describe the results:
d 50 ¼ b d ms :
ð2Þ
By the method of least squares (see Appendix) one finds:
marble: b = 1080, m = 1.542 and R2 = 0.996 and quartz:
b = 123, m = 1.353 and R2 = 0.998.
For decreasing sizes the ratio d50/ds decreases as well
indicating that the material tends to a closer distribution
as comminution proceeds.
4.2. Specific surface area versus specific energy
The specific surface area of the mill products versus the
specific energy consumed is plotted in Fig. 2 for marble and
quartz. It is reminded here that the energy is not the energy
drawn by the motor but the net energy to keep the mill load
running.
One can observe the following:
• At the beginning of each test the feed material has an
initial surface area.
• The specific surface increases as the specific energy
increases but it deviates from linearity indicating that
as comminution proceeds more energy is required for
the same surface production.
• The specific energy e0 that corresponds to the initial surface area of the feed can be found by extrapolation of
the curve until it crosses the horizontal axis at e0.
4. Analysis of the experimental results
During and after each test the parameters measured are:
• The net specific energy e (J/kg) consumed to comminute
the material.
• The size d50, derived from the mass–size distribution of
the product.
ð1Þ
Fig. 1. d50 versus ds for marble and quartz in rod mill.
E.Th. Stamboliadis / Minerals Engineering 20 (2007) 140–145
143
unit energy input as the residence time of the material in
the mill increase or the energy input increases. One can also
say that the surface production rate or the breakage rate
drops as the potential energy increases. At E = 0 the breakage rate has its maximum and if this rate was kept during
grinding the specific surface area produced smax would be
given by Eq. (5) that corresponds to Eq. (4) for n = 1:
smax ¼ a E:
The ratio Y = s/smax given by Eq. (6), derived from (4) and
(5), will be called breakage efficiency or breakage yield and
it is a dimensionless parameter:
Fig. 2. Specific surface area versus specific energy input.
Y ¼ s=smax ¼ En1 :
The addition of the quantity e0 to the energy e invested
for grinding gives the potential specific energy E of any mill
product produced by the particular mill and is independent
of the initial feed size.
E ¼ e þ e0 :
ð3Þ
The plot of the specific surface area s of the mill products
versus the corresponding specific potential energy E is presented in Fig. 3.
Eq. (4) can fit the experimental data and the parameters
a and n, together with e0 of Eq. (3), can be calculated mathematically by the method of least squares and are given in
Table 1 for marble and quartz:
s ¼ a En :
ð5Þ
ð4Þ
4.3. Surface production rate and energy efficiency
At any point of the curves of Fig. 3 the slope gives the
surface area produced per unit energy. The slope of the
curves is not constant and becomes smaller as grinding proceeds. This means that less surface area is produced per
ð6Þ
The plot of the breakage yield, calculated as the ratio of the
actual measurement of specific surface to the calculated
smax, versus the specific potential energy is plotted in
Fig. 4. The data for marble and quartz fit to the equations:
Y = E0.179 for marble and Y = E0.105 for quartz, according to Eq. (6) and the values of n from Table 1.
According to the above the breakage yield, or the usage
of the net energy available for comminution in producing
new surface, becomes unit (100% efficiency) as the specific
potential energy tends to zero. If the energy transmission
efficiency of the system is f0 and the breakage yield at each
time is Y then the overall energy efficiency f becomes:
f ¼ Y f0 :
ð6aÞ
For f0 = 1 (perfect grinding equipment) the energy efficiency is equal to the breakage efficiency or yield. For marble the yield is in the range 0.15–0.25 or 15%–25% and for
quartz it is in the range 0.35–0.45 or 35%–45%. For f0 < 1
the overall energy efficiency is even less. A general expression for the energy efficiency is:
f ¼ f0 En1 :
ð6bÞ
It is worth to mention that the efficiency for breaking
quartz, a hard material with high Young modulus, is higher
than breaking marble, a soft material with lower young
modulus. As it will be seen below the energy or breakage
efficiency is something different than the energy requirement for breakage.
Fig. 3. Specific surface versus specific potential energy.
Table 1
Parameters of the surface versus potential energy relationship
Marble
Quartz
a
n
e0
R2
0.0236
0.0058
0.821
0.895
5867
2587
0.989
0.990
Fig. 4. Breakage efficiency (yield) versus specific potential energy.
144
E.Th. Stamboliadis / Minerals Engineering 20 (2007) 140–145
5. Synthesis of the energy distribution model of comminution
The main assumption of this model is that the specific
surface area s (m2/kg) produced is proportional to the
net specific energy available for this task and the proportionality constant c (J/m2), called specific surface energy
(for liquids the term used is surface tension), is a physical
property of the comminuted material depending on its
structure and history and is approximately constant at certain size intervals. The value of c may change if the structure of the material changes at different particle sizes.
The specific energy required for surface formation is a part
of the available net specific energy and this is expressed by
Eq. (7):
E f ¼ c s;
ð7Þ
also written as s = E Æ f/c.
Combining Eqs. (4), (6a) and (7) one can calculate c:
c ¼ f0 =a:
ð8Þ
From the experimental data of Fig. 2, assuming that f0 = 1,
because the specific energy calculated is the net energy, one
can obtain the specific surface energy (surface tension) of
the minerals tested: Marble c = 1/0.0236 = 42.4 J/m2.
Quartz c = 1/0.0058 = 172.4 J/m2.
The energy required to create a unit of new surface for
quartz is higher than it is for marble but, as mentioned in
the previous paragraph, it is done more effectively for
quartz than for marble as grinding proceeds.
It is very customary for grinding calculations to relate
the energy to the particle size produced. In this case the
combination of Eqs. (1), (4), (6a) and (8) gives:
1=n
6c
1
E¼
1=n :
ð9Þ
q f0
ds
One should keep in mind that ds is the surface equivalent
diameter. For a potential energy E1 the corresponding size
is d1s. The energy E12 required to grind a sample of size d1s
to size d2s is calculated from Eq. (9) that takes the form
(10):
!
1=n
6c
1
1
E12 ¼
1=n :
ð10Þ
1=n
q f0
d 1s
d 2s
Dividing by d1s one can calculate the reduction ratio d1s/d2s
as a function of the energy E12, which is the energy usually
measured:
"
#n
1=n
d 1s
q f0 d 1s
¼ E12 þ1 :
ð11Þ
d 2s
6c
One can substitute the sizes ds by the corresponding d50
using the experimental data as predicted by Eq. (2) and
then (9)–(11) become (12)–(14):
!1=n
6 c b1=m
1
E¼
1=mn ;
ð12Þ
q f0
d 50
Fig. 5. The plot 1/d50 versus specific potential energy.
!1=n
!
6 c b1=m
1
1
1=mn ;
E12 ¼
1=mn
q f0
d 1;50
d 2;50
2
3mn
!
1=m 1=n
d 1;50 4
q f0 d 150
¼ E12 þ 15 :
d 2;50
6c
b
ð13Þ
ð14Þ
The value 1/d50 can be obtained from (12) as follows:
m
1
q f0
1
ð15Þ
¼ Emn :
d 50
b
6c
The plot of Eq. (15) is presented in Fig. 5 together with the
experimental data for marble and quartz.
In this particular case m Æ n = 1.27 for marble and
m Æ n = 1.21 for quartz. The inverse of the slope has the
units, J m/kg, and could be also used as an index of grindability but its physical interpretation is not so evident as that
of the specific surface energy c given in J/m2.
6. Discussion
The usual form of equations used to describe the relationship of energy and the size of the mill products is that
of (13). It is said that if 1/m Æ n = 1 it represents Rittinger’s
law while if 1/m Æ n = 0.5 it represents Bond’s law. Charles
and Stamboliadis have shown that 1/m Æ n can have any
intermediate value. In the present case 1/m Æ n = 0.79 for
marble and 1/m Æ n = 0.83 for quartz. However according
to the present approach although 1/m Æ n 5 1 Rittinger’s
law still holds in any case because the specific
surface
energy
1=m 1=n
c is always incorporated in the constant 6cb
. This
qf0
constant depends both on the material and also on the system used. The units of this constant, J m/kg, are not convenient to use as grindability index. In engineers mind such an
index should indicate energy per unit mass or volume, J/kg
or (kwh/ton), etc. Bond has bypassed the problem by defining the grindability index (work index) as the energy
required to grind a unit mass of a material from infinite size
down to 100 lm. It seems now that the exponent 1/m Æ n
cannot be the criterion used so far to define whether a
specific grinding relationship follows Rittinger’s, Bond’s
or any other intermediate law. The exponent n depends
on the fraction of the net energy provided, which is used
to produce new surface area and is not necessarily due to
E.Th. Stamboliadis / Minerals Engineering 20 (2007) 140–145
the inadequacy of the milling system but it can be also due
to the energy lost for elastic deformation of the material
during breakage as explained in a previous work, Stamboliadis (2005). On the other hand the other exponent m
depends on the mass–size distribution function of the material that determines the ratio d50/ds as explained above.
A new effort to derive a universal grindability index is that
of Fuerstenau and Kapur (1994), who have transformed Eqs.
(13) and (14). In the case that 1/m Æ n = 1 thereduction
ratio
d
1=n
0
is a linear function of energy and the slope is qf
1;50
.
6c
b
The units of the inverse of the slope, J/kg or (kwh/ton), seem
to fulfill the requirements of a grindability index. Fuerstenau
and Kapur (1994) have used a prototype mill, they assumed
that the energy efficiency is almost 100% and have produced
results that give the grindability of different minerals under
the condition that the mill feed belongs to a narrow size class,
namely 6 · 7 mesh (3.35 · 2.80 mm). It is clear now that the
values given as grindability indices, although they might
have been very accurately measured, cannot be used as a
characteristic constant of the material because they incorporate the size of the initial feed.
According to the theoretical approach followed in the
present work the parameter that depends only on the material and can be used as an index of its grindability is the specific surface energy or surface tension and can be calculated
from data that relate the net energy input and the specific
surface area of the feed and the products in a comminution
test. It can also be calculated from single particle breakage
tests for gradually diminishing particle size according to the
theory proposed in a previous paper, Stamboliadis (2005).
No matter how it is calculated the units of the specific surface energy (surface tension) c are energy per unit surface
area (J/m2) and not energy per unit mass (J/kg) or (kwh/
ton) as it was expected for an index of grindability.
7. Conclusions
The final conclusions of the present work are summarized as follows:
• The potential energy of a material, as applied to the specific comminution equipment used, is the sum of the
energy used to produce the material from a given feed
plus the energy that is calculated to produce the feed
material from an original one with practically zero specific surface area.
• The surface area of the mill products as a function of the
potential energy indicates how the energy provided is
used to create new surface. The deviation from linearity
is an index of the energy utilization efficiency.
• The new surface formed is assumed to be proportional
to the energy fraction actually invested for surface formation and the proportionality constant, called specific
surface energy or surface tension measured in J/m2, is a
physical property of materials and can be used to define
their grindability.
145
Appendix
The procedure applied to fit a curve of the form y = axn
to two sets of data x: [x1, x2, . . ., xn] and y: [y1, y2, . . ., yn] is
as follows:
Transform the formula y = axn . . . (Eq. (1)) to log(y) =
log(a) + nlog(x). . . (Eq. (2)).
Create two new sets of data log(x): [log(x1), log(x2),
. . ., log(xn)] and log(y): [log(y1), log(y2), . . ., log(yn)].
Use the method of least squares for the linear relationship of (Eq. (2)) and find the values of the parameters A
and B involved in log(y) = A + Blog(x). . . (Eq. (3)). The
values a and n of the parameters in (Eq. (1)) are found as
a = 10A and n = B.
The program Excel of Microsoft can automatically
calculate A and B once the data log(x): [log(x1), log(x2),
. . ., log(xn)] and log(y): [log(y1), log(y2), . . ., log(yn)] are
given and it also calculates the correlation coefficient R2.
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