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A cement Vertical Roller Mill modeling based on the number of
breakages
Rasoul Fatahi
a,
⇑
, Ali Pournazari
b
, Majid Parvez Shah
c
a
School of Mining Engineering, College of Engineering, University of Tehran, Tehran 16846-13114, Iran
b
CEO of Sarooj Cement International Corporation, Kangan-Bushehr, Iran
c
Process Engineer at A TEC production and service GmbH
article info
Article history:
Received 25 May 2022
Received in revised form 2 August 2022
Accepted 4 August 2022
Keywords:
Modeling
Matrix Model
VRMs
Comminution
MRT
abstract
This study investigated a mathematical model for an industrial-scale vertical roller mill(VRM) at the Ilam
Cement Plant in Iran. The model was calibrated using the initial survey’s data, and the breakage rates of
clinker were then back-calculated. The modeling and validation results demonstrated that according to
the bed-breakage mechanism in VRM, clinker particles only stay in the VRM for a short time. Particles
in the VRM haven’t a 1 to 3 times greater chance of breaking due to their brief time in the VRM.
Matrix model’s results model provides a more robust prediction based on the number of 2-times clinker
breakage in VRMs (R
2
= 0.9916, MSE = 5.3526, accuracy = 94.6474). Also shown by the results of the
matrix modeling, the S increased with decreasing the particle size. In contrast, the population balance
model increased with increased particle size.
Ó2022 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder
Technology Japan. All rights reserved.
1. Introduction
Vertical roller mills (VRM) are widely used to grind, dry, and
select powders from various materials in the cement, electric
power, metallurgical, chemical, and nonmetallic ore industries.
For the sectors above, the VRM is a powerful and energy-
intensive grinding field [1,2]. It is used to grind slag, nonmetallic
ore, and other block and granular raw materials into the fine pow-
ders necessary for production. In the mid-1990s, Loesche GmbH
developed the VRM technology, which was first used for grinding
clinker and slag [3].Fig. 1 shows the components of a Loesche mill
used for grinding. VRMs are the most popular choice for finished
cement grinding over other machines due to low power consump-
tion, higher capacity, process simplifications, and compactness.
However, VRMs are extremely sensitive to vibrations, and produc-
tivity can be negatively affected if process optimization is slightly
altered [4]. VRMs, which have achieved widespread adoption in the
cement industry and are used for crushing raw materials (mainly
limestone), represent an exciting alternative [5].
The VRM operates on the principle of interparticle comminu-
tion, considered the most efficient form of particle breaking [7].
The gap between the rotating grinding table and the stationary
grinding rollers is where interparticle comminution occurs [8].
Although there are various VRMs, the breakage principles are the
same. Loesche uses a master and supports the roller mechanism.
Support rollers(S-Rollers) are used to prepare and stabilize the
grinding bed, while the master rollers(M-Rollers) transmit the
grinding force for interparticle comminution [9]. This mechanism
decreases the dynamic load on each component during operation
and ensures that the mill runs smoothly [10].Fig. 2 illustrates
the operation principle of M-Rollers and S-Rollers in VRM. The
material layer generated between the roller and the revolving table
is the grinding bed. The grinding bed is the fundamental factor in
the proper operation of a VRM, as defined by the feed size, feed
material, moisture content, dam ring height, fineness of grind,
and nozzle ring airspeed [4]. VRMs have several advantages over
conventional grinding equipment regarding mining industry diffi-
culties. An energy-efficient breaking up of a particle bed results
in a smaller particle size distribution (PSD). Because they only
grind a small quantity of each particle before classification, VRMs
reduce the need for excessive over-grinding.
Simulation is a valuable tool in process technology if the pro-
cess models are accurate and can determine model parameters in
a laboratory or plant. It is now widely utilized for designing and
optimizing wet grinding circuits, resulting in significant cost sav-
ings [11].
Austin was the first to develop a full-scale cement mill mathe-
matical model [12,13]. His approach has two key concepts: the
https://doi.org/10.1016/j.apt.2022.103750
0921-8831/Ó2022 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
⇑
Corresponding author.
E-mail address: Rasoul.fatahi97@ut.ac.ir (R. Fatahi).
Advanced Powder Technology 33 (2022) 103750
Contents lists available at ScienceDirect
Advanced Powder Technology
journal homepage: www.elsevier.com/locate/apt
breakage rate and the mill residence time. The model considers the
mill equivalent to several grinding stages with internal classifica-
tion in series, assuming the cement mill model was equivalent to
a thoroughly mixed ball mill [14]. Only a few studies have been
conducted on the simulation of VRMs [15]. Wang, Chen et al.
2009 use a matrix model to replicate the grinding process in VRMs.
It was developed based on experimental data from cement clinker,
coal grinding lines, and laboratory experiments [15]. VRMs are
simulated using high-efficiency classifiers, matrix models, and
other methods. Faitli, J. and P. Czel developed a matrix model to
model a VRM with a high-efficiency slat classifier [16]. Esnault
et al. predicted the breakage particles in milling under pressure
using a population balance model [17]. There were no breakages
in the clinker cement, limestone, and quartz modeling tests. The
difference in particle weakening behaviors in pure and mixed feed
grinding by HPGR was effectively explored by [18].
In a study by [19], the compressed bed breakage test in a piston-
die cell device was used to establish the breakage distribution
Fig. 1. Schematic operation principle of a VRM [6].
Fig. 2. Working principle of the M and S rollers.
R. Fatahi, A. Pournazari and M.P. Shah Advanced Powder Technology 33 (2022) 103750
2
function of the material in the VRM, to simulate cement grinding,
the perfect mixing model was applied. The results demonstrated
that the perfect mixing model accurately predicted the grinding
process of a VRM (VRM). Fatahi and Barani used the population
balance model to model the VRM, and According to the findings,
the specific breakage rates increased as particle size increased
[20]. In the cement clinker grinding circuit, the residence time dis-
tribution in a VRM was measured by applying the dispersion
model, the tank-in-series model, and the perfect mixer with a
bypass [21]. In Research by [15], to the expression of the probabil-
ity of material breakages in the grinding circuit, by using the selec-
tion function, the breakage function, simulation of VRM has been
done in Research by [16]. Also, artificial intelligence techniques
have been used in the cement industry to simulate vertical roller
and ball mills [22,23]. A matrix model for a VRM with a
high-efficiency classifier has been used. The behavior of material
breakages, their mechanism, and their relationship with the mean
residence time (MRT) haven’t been investigated. In contrast, in this
Research, the relationship between the MRT of VRMs and the num-
ber of breakages has been studied, which provides a proper under-
standing of particle breakage behavior in VRMs.
So far, no research has been done to investigate the number of
particle breakages inside VRMs. Because the residence time is so
short in VRMs, the number of times the particles are broken is rel-
atively low, resulting in few particle breakages. Modeling the num-
ber of breakages in these mills can help us better understand the
impact of comminution behavior and mechanism as well as the
relationship between comminution efficiency and the mill’s capac-
ity to grind. The excellent efficiency of these mills is attributable to
the breakage of particles in the least amount of time; therefore, it is
nessacary to investigate the breaking of particles in these mills.
2. Experimental method
In this study, all the samples were obtained from the clinker
grinding line 2 of the Ilam cement plant in the Iranian province
of Ilam-karezan. Used a VRM (Loesche mill) with 160 t/h to grind
the clinker, fed into the mill during the grinding process (%90 pass-
ing 32 mm). Fig. 3 shows the grinding circuit for VRMs. The
Loesche VRMs at the Ilam cement plant are equipped with four
rollers, two of which are master rollers and small support rollers
performing grinding and layering of the material on the grinding
table, respectively.
2.1. Sampling
In the VRM, critical operational parameters, such as classifier
rotor speed, grinding pressure, and gas flow rate of the VRM, are
effective in the grinding process [24] and would be constant during
sampling Table 1. On the grinding circuit, we carried out two sam-
pling surveys. The first set of survey data was utilized for circuit
simulation and modeling, using the second set of survey data for
circuit validation and verification. Used the feed and product sam-
ples to measure the PSD with high accuracy in the Iran Mineral
Processing Research Center using the Laser Particle Size Analyzer
devices. The PSD of product results is needed to create the PSD
curve. Fig. 4 shows the PSD curve of feed and product.
2.2. Breakage function
The breakage function, B, represents the relative distribution of
each size fraction after the breakage. The function depends on the
property of the material and can be expressed as a matrix [15]:
B¼
B
ij
0... 00
... ... ... ... ...
B
i1
... B
ij
... ...
... ... ... ... 0
B
n1
... B
nj
... B
nn
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
ð1Þ
The breakage functions of VRMs are distinct compared to ball
mills because the breakage mechanism in VRMs is compression
and shear forced. The materials are placed between the rollers
and grinding tables in layers of particles and comminuted under
the pressure of compressive forces [25].
The breakage function utilized during this study was developed
by [19]. The breakage function was evaluated using the
well-known lab-scale compressed bed breakage test during a
piston-die cell system, which has been extensively investigated.
Furthermore, used the following equation to fit the data [26]:
B
i;j
¼x
i
x
j
c
þ1/ðÞ
x
i
x
j
b
or B
i;j
¼RðÞ
c
þ1/ðÞRðÞ
b
ð2Þ
Fig. 3. Grinding circuit of the VRMs.
R. Fatahi, A. Pournazari and M.P. Shah Advanced Powder Technology 33 (2022) 103750
3
B
i,j
is the fraction of particles from the jth interval broken and trans-
ferred to the ith interval of particles. X
i
and X
j
are the sizes of fric-
tion ith and jth fraction, respectively, u,
c
,and bare model
parameters, and Ris the relative size (x/x) ratio of i-th and first size
fraction. The values of u,
c
, and bwere determined using lab-scale
compressed bed breaking experiments as Shahgholi, Barani et al.
(2017). The fitted model parameters were employed since the sam-
ple feed of Shahgholi et al. and the sample feed (clinker) of the pre-
vious study (Fatahi and Barani 2020) was the same. By fitting Eq. (2)
to the model’s data, the following parameters were calculated:
u= 0.966,
c
= 0.31863 and b= 0.31862 [19].
Because clinker samples have the identical physical charactries
and chemical composition, the re-producibility of these coeffi-
cients for all PSDs of feed holds true. According to Eq. (1),Table 2
shows the breakage function as a lower triangular matrix, where:
B
1;1
¼B
2;2
¼B
3;3
......:...:B
n;n
¼1n1
B
2;1
¼B
3;2
¼B
4;3
......:...:B
n;n1
¼1n2
B
1;1
¼B
2;2
¼B
3;3
......:...:B
n;n2
¼1n3
2.3. Selection function
Let the breakage probability of size fraction iwill be S
i
. This
selection function can be expressed as a diagonal matrix [15]. Here
is what the selection function looks like, which can be represented
as a diagonal matrix:
S¼S
1
; :::; S
i
; :::; S
n
ðÞ ð3Þ
2.4. Matrix model
The relationship between the selection function S and feed anal-
ysis was illustrated using a matrix model [27]. The feed and product
size distributions were represented as N size ranges to represent
the feed and product size distributions. Created the matrix based
on the assumption that Si represents the proportion of particles
within a sieve fraction, i, that would break preferentially (the others
being too small). Now that the total number of broken and unbro-
ken particles has been calculated, the generic equation can use
Eq. (4) to express the entire breakage operation [28]:
P¼B:SþIS
ðÞ½
:Fð4Þ
This model, which provides a quantitative relationship between
feed and product, has been frequently used and presented with
acceptable results. Of course, this equation is only valid for one-
time breakage cycles. Eq. (4) can be stated in the following form
for further breakage:
In the grinding process of a VRM, three types of forces are
assigned: compression, shear, and centrifugal, with compression
forces attributed to breakage function (B) in the matrix model.
The grinding process (S) determines the mill’s selecting function.
When a compression force is given to the particle bed, the material
is moved and disseminated onto the rotating plate (table) by cen-
trifugal force. In contrast, the mill’s table rotates, and compression
breakage occurs.
2.5. Number of breakages
According to the previous study, the clinker particle spends
around 67 s in the VRM [21]. Because the residence time is short,
it is possible to estimate the number of particle breakages between
1 and 3 times the total number of particles [20].Table 3 compares
the breakages in cement VRM and cement ball mills [29]. In the
Table 1
Operational variables for the VRMs of the Ilam cement plant.
Description of variables Variables Value
The speed of the classifier rotation Classifier Speed
(rpm)
65
A load of mill feed Feed rate(ton/hr) 140
The required pressure of the master roller
for clinker grinding
Working Pressure
(bar)
84–85
The required pressure for lifting the
master roller
Counter Pressure
(bar)
18–20
The speed of mill fan rotation Mill fan Rotation
(rpm)
700–750
The required gas flow for cement
transportation
gas flow rate
(m
3
/hr)
510000–
520000
0
10
20
30
40
50
60
70
80
90
100
0.1 1 10 100 1000 10000
cumulative passing(%)
)μm(parcle size
Feed
Product
Fig. 4. Measured PSD of feed and product of the VRM.
Table 2
Lower triangular matrix of breakage function.
j
i12345678910...28
11000000000.....
2 0.93137 100000000.....
3 0.84979 0.93137 10000000.....
4 0.7468 0.84979 0.93137 1000000.....
5 0.68139 0.7468 0.84979 0.93137 100000.....
6 0.5973 0.68139 0.7468 0.84979 0.93137 10000.....
7 0.5549 0.5973 0.68139 0.7468 0.84979 0.93137 1000.....
8 0.51681 0.5549 0.5973 0.68139 0.7468 0.84979 0.93137 1 0 0 .....
9 0.47893 0.51681 0.5549 0.5973 0.68139 0.7468 0.84979 0.93137 1 0 .....
10 0.44493 0.47893 0.51681 0.5549 0.5973 0.68139 0.7468 0.84979 0.93137 1 .....
...28 ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 1
R. Fatahi, A. Pournazari and M.P. Shah Advanced Powder Technology 33 (2022) 103750
4
VRM, MRT indicates that the particles are broken just once and
maybe up to three times in the grinding process. Can use Eq. (5)
to calculate the Si-based on a 1 to 3-times breaking scenario
(V = 1 to 3).
P¼B:SþISðÞ½
V
:Fð5Þ
where P is product Column matrixes(N 1), F is feed Column
matrixes(N 1), and S is selection function diagonal matrixes
(N N). I is that the identity matrixes(N N), and B could be is a
lower triangular matrix(N N) where N: sieve number and V is
the number of breakage times, which is proportional to the (MRT).
2.6. S
i
calculations
To calculate S
i
by inserting the calculated values of the breakage
function(B) in Table 2 and identity matrix(I) and feed(F) as well as
product (P) matrixes in Equation (5), the number of times of break-
age(V) 1 to 3 times(Table 2). The selection function for the state
(V = 2) was well calculated using the inverted calculations in the
Excel spreadsheet. It used Eq. (5) to get the selection function val-
ues for 28 size fraction classes after applying the 2-times com-
minution(V = 2). Table 4 shows the results. The selection
function is zero for feed particles with a particle size of less than
79.43
l
m, indicating there’s no possibility of comminution. Only
a few particles have a particle size of less than 100
l
m. The feed’s
PSD must be considered while designing a VRM. The PSD of feed
and also the mill’s operational parameters are essential for the pro-
duct’s PSD.
The majority of the clinker particles (d80 = 25 cm) are coarse. In
VRMs, the bed-breakage mechanism is described Fig. 5. For break-
ing, coarse particles are chosen. Large feed particles were broken in
the 1-time, but within the 2- time, most of the particles were bro-
ken and therefore became fine particles. As a result, decreasing the
P
1
P
2
P
n
2
6
43
7
5¼
B
i;j
00
B
i;1
B
i;j
0
B
n;1
B
n;j
B
n;n
2
6
43
7
5
S
1
00
0S
2
0
00S
n
2
6
43
7
5
0
B
@1
C
Aþ
100
010
001
2
6
43
7
5
S
1
00
0S
2
0
00S
n
2
6
43
7
5
0
B
@1
C
A
#
v
F
1
F
2
F
n
2
6
43
7
5
2
6
4
Table 4
Model fitting results.
Sieve number (i) Sieve size (
l
m) Cumulative Passing (%) Selection function S
i
Feed Product
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
31,750
25,400
19,050
12,700
9525
6300
5000
4000
3150
2500
1190
1000
841
297
177
138.038
120.33
104.71
79.43
60.26
45.71
34.67
30.200
22.91
17.38
10.00
6.607
3.311
90.8
63.24
48.94
35.14
29.54
21.54
18.35
13.05
10.15
7.55
3.05
1.12
0.95
0.63
0.57
0.34
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
90.8000
100.1285
100.1048
100.0834
100.0819
99.9619
100.0011
99.8657
100.0356
99.6358
97.9242
97.8401
98.2963
98.6608
98.7733
98.7582
98.5721
98.2514
96.9090
92.9757
83.7794
75.6091
68.1534
61.4483
55.5822
49.7194
44.3520
39.1940
0.2180
0.1297
0.1701
0.1272
0.1864
0.1494
0.1968
0.1926
0.2045
0.2982
0.3207
0.3446
0.4622
0.4999
0.5399
0.5744
0.6244
0.6011
0.4720
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Table 3
Comparative performance parameters for the two systems when used for grinding cement.
Characteristics Ball mill (closed circuit) VRM
Comminution by Impact and attrition Pressure and shear forces
Residence time(miutue) 20–30 Less than 1 min
Crushing before separation(time) Infinite 1–3
Circulation factor(C.L/Feed) 2–3 6–20
Wear rate(g/ton) 50 3–6
R. Fatahi, A. Pournazari and M.P. Shah Advanced Powder Technology 33 (2022) 103750
5
particle size enhanced the values of the selection function for the
2-times. According to the matrix model in VRMs and the bed-
breakage mechanism, completed breakage in 2-times.
3. Results and discussion
3.1. Comparison between calculated S
i
matrix model and population
balance model
A previous study focused on the population balance model in
vertical roller [20]. This model used the (MRT), breakage function,
selection function, and feed and product PSD. For a perfect mixed
mill at a steady-state, the equation becomes [30]:
P
i
¼F
i
þ
s
X
i1
j¼1
i>1
b
ij
S
j
m
j
0
B
B
B
B
B
@
1
C
C
C
C
C
A
S
i
m
i
s
ð6Þ
for NPiPjP1P
i
¼F
i
þP
i1
j¼1
b
ij
s
j
s
p
j
1þs
i
s
where
s
= MRT; b
ij
= breakage function; S
j
= selection function;
F
i
= PSD of feed; and P
i
= PSD of product.
According to Eq. (6), to measure the size distribution of the final
product, the mill (P
i
), MRT (
s
), selection functions of different
classes (S
i
), breakage functions (b
ij
), and mass in the screen classes
(m
i
) must be computed. In the population balance model, based on
the modeling residence time distribution of mills, which predicts
the mixing pattern of particles, particles with different (S
i
), and
(b
ij
) have a different probability of breakages or different selection
functions. In the population balance model, the selection functions
of different classes (S
i
), indicates that particles with different mix-
ing patterns have different breakage probabilities, which are pre-
sented by units (1 / minute). At the same time, in the matrix
model, it is expressed as values between 0 to 1 or 0 % to 100 %,
which implies the probability of breakages using a number of
breakages.
To examine specific breakage rates, researchers [20] analyzed a
previous work that applied a population balance model to the
VRM. In step with the results, the specific breakage rate declined
as reduced the particle size from 25.4 cm to zero. Therefore the
specific breakage rate of particles under 100
l
m became zero
Fig. 6. Smaller than 100
l
m particles don’t have any possibility
of being ground. Breakage is proportional to particle size larger
than 79.43
l
m in the matrix model; hence, the likelihood of break-
age is zero to particle size under the 79.43
l
m. So, in the popula-
tion balance model, larger particles have a greater chance of
grinding, which is similar to the finding that larger particles are
more likely to be ground than smaller ones. As seen in the 1-
time matrix model, wherein the 2-times, smaller particles are more
likely to be ground because coarse particles do not exist between
the rollers and table. The common element in the matrix and pop-
ulation balance model is the selection function which is the break-
age possibility of particles with PSD bigger than 177
l
m.
In VRM, the particles don’t have much opportunity to be ground
because of the short residence time. Therefore, the number of
breakages is considerable, avoiding over-grinding. Concurrently,
the population balance model has predicated on the mixing pat-
tern inside the mill and also the hypothesis that the number of
times of breakage doesn’t amplify. The matrix model provides a
good insight into the number of particle breakages in the VRM.
In keeping with the results; it matches with the breakage number
of times equal to 2 in VRMs.
3.2. Validation of the matrix model
Used sampling data from the second survey to validate the
model’s parameters. Operating pressure, classifier speed, and feed
rate were all left unchanged. (Grinding pressure on the table or
working pressure equals 85 bar). Other parameters, like material
circulation rate, mill temperature, hot gas flow rate, material reject
rate, and grinding table speed, were constant between the primary
and second surveys. Because it didn’t change expected S values. In
Fig. 7, It can be deduced that the dimensions size distribution of
the simulated product within the range of 3175
l
m to 6.607
l
m
is approximately well fitted (R
2
= 0.9916, MSE = 5.3526, accu-
racy = 94.6474, Fig. 8) on the measured product for fraction classes
from 6.607
l
m to zero thanks to in 2-times of grinding. Before the
grinding by master rollers, particles leave the grinding table by the
ventilation of the mill fan. Depending on the ventilation rate of
mills, size fractions larger than 6.607
l
m may also leave the grind-
ing table before comminution.
3.3. Comparison between 1-time breakage and 2-times breakage
Fig. 9 shows the measured PSD of the VRM product supported
by the number of breakages(actual sample and calculated:1-time
Fig. 5. The grinding mechanism of VRMs.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 10 100 1000 10000 100000
Si
)μm(Parcle size
Matrix Model
-1
499
999
1499
1999
2499
2999
3499
3999
4499
1 10 100 1000 10000 100000
Si
Particle size(μm)
PB Model
Fig. 6. Comparison S
i
matrix model and population balance model.
R. Fatahi, A. Pournazari and M.P. Shah Advanced Powder Technology 33 (2022) 103750
6
and 2-times breakage). Consistent with the cumulative passing
percent of various fractions of product for two-state, for 1-time
breakages, most of the particle size is coarse. It means the particles
aren’t sufficiently comminuted for a 1-time breakage state and
must be comminuted 2- times. For two-times breakages, the parti-
cles produced are near the actual sample.
4. Conclusions
According to the MRT, clinker particles spent a short time
within the VRM during in this study, with an MRT of about 67 s.
Due to the short residence time, the number of particle breakages
within the VRMs is 1 to 3 times lower, avoiding unneeded
over-grinding. The matrix model’s results showed that this model
provides a more robust prediction that supported the quantity 2-
times clinker breakage in VRM. A comparison between the matrix
model and population balance model within the VRM; showed that
the matrix model provides adecent insight into the number of par-
ticle breakages, whereas the population balance model; in step
with the mixing pattern material inside the mill, the hypothesis
is that the number of times of breakage does not amplify. Also,
the comparison of the calculated product produced for the 1-
time and 2-time showed that the number of 2 breakages is far clo-
ser to the actual product size distribution. Confirmed the findings
of the matrix model by using 2-time grinding; however, because
the effective parameters on the number of particles breakage are
various, it’s possible to average the number of breakages for mul-
tiple states, equal 1 to 3 times.
Declaration of Competing Interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Acknowledgments
The author’s appreciation goes to Ilam cement Plant manage-
ment for providing access to production line 2 circuit the plant.
Also, A TEC production and service GmbH (A member of Loesche
family) is appreciated for guidance and advice.
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