Aplicación Práctica / Practical Issues
Recibido: 15-09-2023, Aprobado tras revisión: 15-12-2023
Forma sugerida de citación: Paredes, J.; Cepeda, J.; Lozada, J.; “Parameters for the Grinding Process in Vertical Mills Using
Optimization Methods”. Revista Técnica “energía”. No. 20, Issue II, Pp. 90-97
ISSN On-line: 2602-8492 - ISSN Impreso: 1390-5074
Doi: https://doi.org/10.37116/revistaenergia.v20.n2.2024.594
© 2024 Operador Nacional de Electricidad, CENACE
Esta publicación es de acceso abierto bajo una licencia Creative Commons
Parameters for the Grinding Process in Vertical Mills Using Optimization
Parámetros para el Proceso de Molienda en Molinos Verticales Usando
Métodos de Optimización
J. Paredes1 0009-0002-2256-3018 J. Cepeda1 0000-0002-2488-6796
J. Lozada2 0009-0009-0016-0913
1Escuela Politécnica Nacional, Quito, Ecuador
E-mail: jorge.paredes01@epn.edu.ec ; jaime.cepeda@epn.edu.ec
2Unión Cementera Nacional, Riobamba, Ecuador
E-mail: jlozada@ucem.com.ec
Vertical roller mills, (VRM), are widely used for
grinding raw materials in factories engaged in the
extraction and processing of minerals. Any machine
used for grinding or crushing consumes around 30 to
40% of the energy of a factory. The loading pressure,
table rotation speed, moisture content, outlet
temperature and pressure rollers are variables that can
be controlled to decrease the specific energy
consumption Ecs. This paper poses an optimization
problem in order to reduce the energy consumption of a
VRM used to produce cement and to find the optimal
parameters of the operating variables. Several packages
are used to solve the nonlinear programming problem,
with very good results in terms of accuracy and speed
of convergence, but those provided by the Pyomo
package are used because it obtains more accurate
results. Comparing the result of the objective function
with the energy consumption of a well-known cement
company in Ecuador, it is concluded that the optimized
parameters are able to reduce by 25% the energy
consumption guaranteeing a minimum production of
2200 tons of cement per day, so the model is correctly
Los molinos verticales de rodillos (VRM), son
máquinas muy utilizadas para moler materia prima en
fábricas dedicadas a la extracción y procesamiento de
minerales. Cualquier máquina ocupada para moler o
triturar consume alrededor del 30 o 40% de la energía
de una fábrica. La presión de carga, la velocidad de
rotación de la mesa, el contenido de humedad,
temperatura de salida del molino y presión de los
rodillos son variables que se pueden controlar para
disminuir el consumo de energía especifica (Ecs). Este
trabajo plantea un problema de optimización con el fin
de reducir el consumo energético de un VRM utilizado
para producir cemento y encontrar los parámetros
óptimos de las variables de operación. Se utilizan varios
paquetes para resolver el problema de programación no
lineal, obteniendo resultados muy buenos respecto a la
precisión y velocidad de convergencia, pero se usan
aquellos proporcionados por el paquete Pyomo ya que
obtienen resultados más exactos. Comparando el
resultado de la función objetivo con el consumo
energético de una empresa cementera en Ecuador se
concluye que los parámetros optimizados son capaces
de reducir en un 25% el consumo energético
garantizando una producción mínima de 2200 toneladas
de cemento diarias por lo que el modelo se valida
Index terms−−
efficiency, grinding process,
optimization, vertical mill.
Palabras clave −−
eficiencia, proceso de molienda,
optimización, molino vertical.
Edición No. 20, Issue II, Enero 2024
Minning is an economic activity that has been
growing since the beginning of the 20th century. From
1920 to 2018 the rate of exploitation of each mineral has
varied from 1.48% to 7.3% measured in megatons (Mt)
[1]. Since the end of the 1990s, mineral exploitation has
advanced at a much higher rate than in previous years,
which has caused an accelerated depletion of mineral
resources [2]. Among the largest industries that make use
of the exploited minerals, metallurgy, oil and cement can
be named. The cement industry complies with providing
the main material for construction, which is cement and
which has been key to the process of human civilization
For the cement grinding process, two machines can
be used, which are: a ball mill or a VRM vertical roller
mill. The ball mill makes use of grinding bodies and they
have been the main tool for more than 100 years,
although they have a low efficiency, while the vertical
mills are much more modern tools that are capable of
saving between 45 to 70% of the power consumed [4].
It is important to know that the cement classification
process in a VRM can be carried out by air sweep or by
overflow the use of the overflow model suggests more
energy savings than the sweep model, however it is
useless in certain conditions operation (VRM motor at
full load or rater load pressure) such as particle size,
moisture or hardness [5]. This directly influences energy
consumption and the characteristics and quality of the
Some methods to improve the grinding process vary
one parameter at a time such as the loading force, the rate
of revolution or the fractional filling [6]. Some studies
also try to predict energy consumption taking into
account the characteristics of the material to be ground in
the VRM [7]. An energy and exergy analysis are used to
compare the ball mill and the VRM, finding that the
VRM is more efficient and consumes less energy,
maintaining the quality parameters of the product, so its
use is recommended [8]. The grinding performance has
been improved by increasing the grinding surface, in
addition, it has been compared with other processes such
as jaw crushing and the ball mill, finding the VRM with
better characteristics in product quality and energy
efficiency [9]. A model of material failure can provide
different energy levels, which can help improve energy
efficiency [10]. With the above, it is observed that no
research makes use of optimization methods to find
adequate operating parameters in vertical mills.
Crushing processes consume about 3 to 4% of the
electricity generated worldwide and about 70% of the
energy required in an industry dedicated to mineral
processing [11]. That is why it is important to optimize
the cement grinding process, in order to obtain better
efficiency with the lowest consumption of electrical
energy, in addition, it is necessary to take into account
certain operating parameters to guarantee a quality
Most of the works related to energy consumption
vary a single parameter at a time, propose models, make
comparisons between grinding processes or perform
various energy analyses. The present work takes into
account five important parameters in the operation of any
VRM such as the load pressure, humidity, the speed of
rotation of the motor, outlet temperature and pressure
roller [12].
This document is structured by: section one presents
the introduction, in the second section is presented the
related work, the third section explains the milling
process in vertical mills, the methodology used is
explained in the fourth section, while in the fifth section
all the results obtained are presented and finally, in the
last section, the conclusions that were obtained.
The search for optimal operating parameters in
vertical milling processes is essential to guarantee a
product under all quality standards and that unplanned
shutdowns are considerably reduced. However, the
complexities of the models or the handling of proprietary
software make this task difficult.
There are several optimizations works related to
vertical roller mills. The one proposed by [13] proposes
the design of the lower rocker arm body that supports the
roller and take as optimization objective function the
mass of the rocker arm and as constraint conditions the
stresses and displacements generated by the roller in the
rocker arm in order to decrease the mass while
conserving the resistance and deformation. In this way,
the design is optimized and the cost is reduced by saving
A VRM in its normal operation undergoes cyclic
bending stresses due to the roller load which can cause
fractures in the mill table so [14] makes use of artificial
neural networks to solve the multi-objective optimization
problem by determining the maximum and minimum
stresses to which the vertical mill can be subjected.
High vibration in a VRM can considerably reduce
productivity, using the 7 quality tools and together with
a vibrational analysis, an optimization model is obtained
that achieves improved productivity and reduced
downtime [15].
To predict the vibrations in the upper case of the
VRM, use can be made of the fruit fly optimization
algorithm (FOA). It shows better precision in
simulations. As a result, a vibration pattern can be
predicted which will help to understand the state of the
VRM and take new safety precautions [16].
Paredes et al. / Parameters for the Grinding Process in Vertical Mills Using Optimization Methods
Another way to optimize the load and the quality of
the cement is by using expert PID controllers, increasing
the production of the cement in vertical roller mills [17].
Although there are already some optimization works
in the field of VRMs, there are still few or almost none
those works that aim to optimize key parameters in the
operation in order to reduce electricity consumption.
A vertical roller mill is suitable equipment for
grinding and drying wet materials, which can be carried
out in the same equipment [18]. Some of the materials
that can be processed in a VRM are:
1) Clinker for cement manufacturing
2) Raw materials for cement manufacturing
3) Pozzolan
4) Metal dross
The grinding process in a VRM is carried out by
placing a certain amount of material on a horizontal
surface that is in motion, at a pressure sufficient to
fracture the materials on the bed. The bed materials are
considerably smaller, so it is necessary to form a stable
grinding bed between the rollers and the grinding table to
withstand the pressure without the material being ejected
from the grinding pressure zone. [19].
A vertical roller mill can be divided into 3 sections.
1) Motor and gears
2) Grinding
3) Drying and separation.
In the grinding process, the material is fed to the
grinding table and due to its speed, the material is
directed towards the rollers where it is milled. This
process is one of the most efficient in the cement industry
The drying process consists of a stream of
recirculated gases, which can come from the clinker kiln
or from hot gas generators. Drying usually occurs on the
mill table and in the vertical sections towards the
separator [20].
Finally, the pressure exerted by the rollers on the table
causes the material to rise towards the separator which
together with fixed plates and the cage separates the
material to the desired size. The rejected coarse material
is recirculated back to the mill [20].
In Fig. 1, shows a vertical roller mill in which the
three sections mentioned above can be easily
distinguished. The motor, grinding table, rollers and
separator are clearly visible. The material and hot gas
inlet ducts and the material outlet can also be seen.
Figure 1: Vertical Roller Mill
4.1 Mathematical Model
In grinding processes, it is common to use the specific
energy [21]as observed in (1).
𝑬𝒄𝒔 =𝒌𝑾𝒉
𝒕𝒐𝒏 (1)
The ratio between the kilowatt hour, which is the unit
of measurement for accounting for electricity
consumption, and the kilogram, which is the unit of
measurement for mass, is also known as energy
efficiency and is widely used in the field of mineral
processing. This allows the performance of the grinding
process to be evaluated as well.
In vertical roller mills there are several factors that
influence the performance of the grinding process [22]
which are moisture content (mc), grinding table rotation
speed (s), load pressure (P), outlet temperature (T) and
the pressure rollers (Pr).
The specific energy model [23] can be seen in (2).
The resulting model is non-linear, so some variables,
such as roller pressure and speed, are of degree two and
other variables can be multiplied by others. There is only
one independent term.
The model was developed using the Box-Behnken
method, using historical data on the energy consumption
of the VRM and the number of tons of cement produced.
Edición No. 20, Issue II, Enero 2024
𝑬𝒄𝒔 = 𝟎. 𝟎𝟏𝟕𝟔𝟖 + 𝟎. 𝟎𝟏𝟕𝟔𝟓𝒔
−𝟎. 𝟎𝟎𝟑𝒎𝒄 − 𝟎. 𝟎𝟑𝑻 + 𝟎. 𝟎𝟏𝟕𝟑𝑷𝒓
−𝟎. 𝟎𝟎𝟒𝑷 ∗ 𝑷𝒓 + 𝟎. 𝟎𝟏𝟕𝒔 ∗ 𝑻
−𝟎. 𝟎𝟐 𝑻 ∗ 𝑷𝒓 + 𝟎. 𝟎𝟏𝟓 𝒔𝟐+ 𝟎. 𝟎𝟎𝟒 𝑷𝒓𝟐
Taking into account the above, the formulation of the
optimization problem is as follows:
𝑴𝒊𝒏𝒊𝒎𝒊𝒛𝒆 𝒁 = 𝟎. 𝟎𝟏𝟕𝟔𝟖 + 𝟎. 𝟎𝟏𝟕𝟔𝟓𝒔
−𝟎. 𝟎𝟎𝟑𝒎𝒄 − 𝟎. 𝟎𝟑𝑻 + 𝟎. 𝟎𝟏𝟕𝟑𝑷𝒓
−𝟎. 𝟎𝟎𝟒𝑷 ∗ 𝑷𝒓 + 𝟎. 𝟎𝟏𝟕𝒔 ∗ 𝑻
−𝟎. 𝟎𝟐 𝑻 ∗ 𝑷𝒓 + 𝟎. 𝟎𝟏𝟓 𝒔𝟐+ 𝟎. 𝟎𝟎𝟒 𝑷𝒓𝟐
Subject to:
𝟐𝟐 ≤ 𝑷 ≤ 𝟐𝟖 𝒎𝒃𝒂𝒓 (3a)
𝟖𝟖𝟔 ≤ 𝒔 ≤ 𝟏𝟏𝟗𝟖 𝒓𝒑𝒎 (3b)
𝟎 ≤ 𝒎𝒄 ≤ 𝟑 % (3c)
𝟎 ≤ 𝑻 ≤ 𝟗𝟕 ˚𝑪 (3d)
𝟏𝟎𝟎 ≤ 𝑷𝒓 ≤ 𝟏𝟗𝟎 𝒃𝒂𝒓 (3e)
The restriction data are obtained using a Pffeifer
vertical roller mill. Parameter intervals documented in
articles, manuals and INEN standards were taken into
account. To guarantee a desired fineness of less than 45
microns in the case of cement, a charge pressure between
22 and 28 mbar [24] as observed in restriction (3a), a
pressure of the rollers between 100 and 190 mbar [24]
defined in the restriction (3e), speed between 886 and
1198 rpm [25] as indicated in the restriction (3b) the
moisture content between 0 and 3% [26] as observed in
restriction (3c), and the outlet temperature between 0 and
97 ˚C [26] be used as indicated in the restriction (3d).
4.2 Model Optimization
The objective function of the proposed model clearly
presents non-linear terms, so it must be solved using non-
linear programming methods. To choose the appropriate
method it is necessary to check certain characteristics of
the objective function.
As a first step it is necessary to check the convexity
or concavity of the function. For this it is necessary to
obtain the gradient (4) and Hessian of the objective
function (5).
𝜵𝒇(𝑷, 𝒔, 𝒎𝒄, 𝑻, 𝑷𝒓
)= [−𝟎. 𝟎𝟎𝟒 ∗ 𝑷𝒓,
𝟎. 𝟎𝟑 ∗ 𝒔 + 𝟎. 𝟎𝟏𝟕 ∗ 𝑻 + 𝟎. 𝟎𝟏𝟕𝟔𝟓,
−𝟎. 𝟎𝟎𝟑,
𝟎. 𝟎𝟏𝟕 ∗ −𝟎. 𝟎𝟐 ∗ 𝑷𝒓 +𝟎. 𝟎𝟏𝟕𝟑
−𝟎. 𝟎𝟎𝟒 ∗ 𝑷 − 𝟎. 𝟎𝟐 ∗ 𝑻 + 𝟎. 𝟎𝟎𝟖 ∗ 𝑷𝒓]
𝑯𝒇(𝑷, 𝒔, 𝒎𝒄, 𝑷𝒓, 𝑻
𝟎 𝟎 𝟎 𝟎 𝟎
𝟎 𝟎. 𝟎𝟑 𝟎 𝟎 𝟎
𝟎 𝟎 𝟎 𝟎 𝟎
𝟎 𝟎 𝟎 𝟎 𝟎
𝟎 𝟎 𝟎 𝟎 𝟎. 𝟎𝟎𝟖]
From the Hessian matrix, its eigenvalues (6) are
obtained, which are equal to or greater than zero, so it is
a positive semi-definite matrix.
𝝀 = [𝟎, 𝟎. 𝟎𝟑, 𝟎, 𝟎. 𝟎𝟎𝟖, 𝟎
] (6)
Taking into account the considerations that the
gradient is greater than zero and the Hessian matrix is
positive semidefinite, the objective function presents
With this in mind, we proceed to solve the
optimization problem proposed in (3). In this case, in
addition to the Python programming language, nonlinear
programming solvers from the Pyomo and Scipy
packages are uses. Both solvers are used to ensure the
accuracy and speed of convergence of the model and to
select the model that yields parameter values that meet
the constraints of the optimization problem.
The Pyomo package uses the ipopt solver [27] to
solve nonlinear programming problems. Table 1 shows
the optimal values for each variable and the objective
Table 1: Optimized values obtained using Pyomo
Variables Optimized Value
P 28
s 886
mc 3
T 0
Pr 100
Z 11.82
Finally, the Scipy package makes use of sequential
least-squares programming [28], which is a widely used
solver for non-linear programming problems. In Table 2,
you can see the optimized values using Scipy.
Table 2: Optimized values obtained using Scipy
Variables Optimized Value
P 27.99
s 885.99
mc 2.99
T 0
Pr 99.99
Z 11.82
Paredes et al. / Parameters for the Grinding Process in Vertical Mills Using Optimization Methods
The benefits of optimizing any process or model are
many, such as better tuning the parameters of a particular
controller or reducing associated costs by making it more
efficient. The grinding process optimized in this work is
important because it provides the optimum operating
parameters. Optimized parameters avoid potential
damage to machine components, such as preventing
VRM rolls from breaking due to out-of-range operating
parameters or high energy consumption with low
Grinding in the cement industry is a critical process
as it is the last step before the finished product is
obtained. There are other sub-processes involved in
grinding, such as material transport or separation, but this
work focuses on the actual grinding process that takes
place within the VRM.
In Ecuador, there are several companies dedicated to
the production of cement. In the center of the country
there is a well-known cement industry from which the
energy consumption data for the grinding process could
be obtained. For reasons of confidentiality, the name of
the company is withheld.
The company's grinding process consists of a VRM,
which will replace a ball mill previously used for cement
production in 2021. The current VRM has a capacity of
100 tons per hour. This process also includes the raw
material feeding sub-process, which is carried out by
means of dosing tables, the finished product separation
sub-process and transport to the finished product silo.
The company also has two other VRMs, one for the
production of raw meal, which is heated to 1200 ˚C to
produce clinker, the main ingredient of cement, and the
other for the production of Petcoke powder, the main fuel
for the clinker kiln.
Taking this into account, the average consumption
data obtained is 78,408.5 kWh per day. Similarly, the
average cement production is 2228 tons. Dividing the
consumption value by the number of tons produced gives
a value of 34.39 kWh/tons, which means that it takes
34.39 kWh to produce one tons of cement. The Ecs value
obtained takes into account all the processes involved,
such as raw material dosing, separation of the final
product and transport. However, if only the VRM
grinding process, which accounts for 40%, is specifically
analyzed, the value obtained is 13.75 kWh/tons.
The packages used to optimize the problem proposed
in this work provide very similar data. The Pyomo
package provides optimized values of the operating
values within the given constraints, but Scipy packages
provide optimized values very close to the constraints,
which, by applying rounding methods, are equal to the
values provided by Pyomo. Taking this into account, for
further comparisons with the production data presented
above, the values optimized by the Pyomo package will
be used, which are considered to be the most accurate and
reliable values for this work.
The value of the objective function after the
optimization process is 11.82 kWh/tons, while that of the
normal process without optimization of the operating
parameters is 13.75 kWh/tons. This reduction
corresponds to 14.03% of the energy consumption of the
grinding process carried out by the VRM. To achieve
this, it is necessary to change the values of the current
mill parameters to the optimized values, which are: for a
feed pressure of 28 mbar, the table speed should be set to
886 rpm, the moisture content of the material should be
0% the outlet temperature should be 0 and the pressure
roller should be 100 bar. This ensures low energy
consumption and a minimum cement production of over
2200 tons per day. Table 4 summarizes the results
Table 4: Comparison of the results obtained
Variable Ecs (kWh/ton)
Actual 13.75
Optimized 11.82
Difference 1.93
It is important to know the savings in dollars of the
consumption of the milling process. Currently, in the year
2023, the cost per KWh in Ecuador is $0.105 for voltages
between 351V and 500V [29]. Considering this, before
the optimization process, the consumption was $3216 for
a production of 2228 tonnes of cement per day, and with
the optimized parameters, the consumption is $2686 for
the same production of cement. This represents a saving
of $530 per day.
5.1. Effect of the parameters in Ecs
When comparing the loading pressure P with the
pressure roller Pr, the specific energy Ecs increases if the
loading pressure and pressure roller also increases due to
the features of the raw material. Due to this increase in
ECS, it is necessary to carry out optimal quality control
of the raw material used in the cement manufacturing
process. By increasing the loading pressure and the
pressure roller to their maximum levels, the maximum
Ecs can be obtained as shown in Fig 2.
Figure 2: Change of Ecs by varying loading pressure and pressure