Artículo Científico / Scientific Paper 



https://doi.org/10.17163/ings.n28.2022.08 


pISSN: 1390650X / eISSN: 1390860X 

MATHEMATICAL MODEL OF A RESISTIVE OVEN FOR
THERMOFORMING POLYPROPYLENE SHEETS 

MODELO MATEMÁTICO DE UN HORNO RESISTIVO PARA TERMOFORMADO DE LÁMINAS DE POLIPROPILENO 
Received: 03052022,
Received after review: 06062022, Accepted: 13062022, Published:
01072022 
Abstract 
Resumen 
A
mathematical model of a resistive oven for the production of thermoformed sheets
is developed in this paper; such oven is located in a production plant in the
city of Riobamba. The objective of the research is to achieve temperature
stability and that the plates have a homogeneous dimension when going through
the thermoforming process, to guarantee customer satisfaction. For this
purpose, the physical variables that govern the heat transfer phenomena,
namely radiation, convection and conduction, are analyzed, to obtain a
mathematical model that predicts the temperature profile of the oven in the
thermoforming process, from which a controller is designed using various
control techniques that are efficiently coupled to the system. A theoretical
study of the physical phenomena and of the mathematical equations that
represent them is proposed in the first stage of the research. Then, they are
solved through computational techniques using Simulink to obtain the
temperature profile. Finally, this model is validated by comparing it with
those obtained in previous works through statistical techniques, and a new
controller that guarantees minimum temperature variability is proposed. As a
result of the simulation, a variation of ±1 mm in the width of the plate is
achieved. 
En el presente trabajo se desarrolla un modelo matemático de un horno resistivo para la producción de planchas termoformadas, localizado en una planta de producción de la ciudad de Riobamba. El objetivo de la investigación es conseguir la estabilidad de la temperatura y que, al pasar por el proceso de termoformado, las planchas tengan una dimensión homogénea que garantice la satisfacción del cliente. Para ello se analizan las variables físicas que rigen los fenómenos de transferencia de calor; radiación, convención y conducción, y así obtener un modelo matemático que prediga el perfil de temperatura del horno en el proceso de termoformado, a partir del cual se diseña un controlador utilizando varias técnicas de control que se acoplen al sistema de forma eficiente. En la primera etapa de la investigación se plantea un estudio teórico de los fenómenos físicos y las ecuaciones matemáticas que los representan. Luego son resueltas a través de técnicas computacionales usando Simulink para conseguir el perfil de temperatura. Por último, se valida este modelo comparándolo con aquellos ya obtenidos en trabajos anteriores a través de técnicas estadísticas; finalmente, se propone un nuevo controlador que garantice la variabilidad mínima de la temperatura. Como resultado de la simulación se consigue una variación de ±1 mm del ancho de la plancha 
Keywords: heat transfer, modelling, polypropylene,
temperature, thermoforming 
Palabras clave: modelado, polipropileno, temperatura, termoformado, transferencia de calor. 
^{1,*} Facultad de Mecánica, Escuela Superior Politécnica de Chimborazo  Ecuador. Corresponding author ✉: lidia.castro@espoch.edu.ec. ^{2} Jefe de Planta Prefabricados Riobamba, Unión Cementera Nacional UCEM S.A  Ecuador. Suggested citation: CastroCepeda, L. and CortésLlanganate, J. “Mathematical model of a resistive oven for thermoforming polypropylene sheets,” Ingenius, Revista de Ciencia y Tecnología, N.◦ 28, pp. 8091, 2022, doi: https://doi.org/10.17163/ings.n28.2022.08. 
1.
Introduction The
worldwide plastics industry has grown in recent years. Besides the virgin raw
material production data, this is also evidenced in the percentage of plastic
that is currently recycled in the countries of the world. Regarding
recycling, the plastic had an increase of 80 % in 10 years in European
countries [1]. In Ecuador, the
plastic is used in various production fields, such as: food industry (drinks
packages, snacks, etc.), construction industry (translucent roofs, plastic
tiles, curtains, etc.), kitchen utensils in general [2]. Specifically, there
is an industry in the city of Riobamba devoted to manufacturing translucent
polypropylene sheets that are used as a complement of fiber cement roofs. The plant has three
production lines that have been assembled with recycled technology, coming
from countries such as Spain; they are acquired at lower prices for reuse
after they have completed their useful life cycle. Then, it is repowered in
local industries making some adjustments and even mechanizing parts and
pieces that are coupled to the system so that they operate with the greatest
possible efficiency. The process for transforming the polypropylene is
summarized in
Figure 1. Process for transforming the polypropylene The main machine of
this line is a thermoforming machine that produces type P7 polypropylene
sheets. This research work has been carried out on the input oven of such
plastic thermoforming machine. Previous works were conducted to control and
improve the production of the thermoforming oven. 
The
first work consisted in setting up a SCADA system for temperature control. It
was based on a hysteresis control system, where the resistances of the
machine were turned on and then turned off when they reached the desired
value of temperature [3]. At present, the described system is being used in
the area of extrusion, distribution and output ovens. Figure 2 shows the
architecture of the network used in the SCADA system. Figure 2. Architecture of
the local area network of the machines [3] The second work
consisted in the design of a controller for the input oven, based on system
identification. This enabled obtaining the transfer function of the oven.
Once such function was identified, a PI controller was calculated for its
control [4]. This system is being currently used in the production plant to control
the input oven. Up to this moment,
the system has not been modeled mathematically, and thus the real behavior of
its variables is unknown. Therefore, it has been considered necessary to find
this mathematical model, which will be compared with the existing models.
Based on them, modifications that improve the product quality will be
suggested in the controllers. At present, the product meets the current
regulations of the country, but it is observed that its width is not
completely uniform. When it is stacked for distribution and sale, the
customer has the sensation and perception that the plates do not meet the
regulation, due to the variability of their dimensions. Therefore, the main
problem of this oven is that it produces plates with a very variable width
that causes that some clients do not accept the product. Among the research
works analyzed, [5–8], Neacă and others use
the heat transfer phenomena to quantify the temperature through the
mathematical model found, which is similar for this case study. However, each
publication is distinguished because its entire system is in accordance with
the particular features of each oven. In contrast, the study by Throne [9],
[10] is based on two variables, namely the molar absorptivity and the
emissivity, emphasizing that they are fundamental in the development of the
model. Following the recommendation of this author, the appropriate data will
be used in the radiation analysis, which are detailed in the solution
statement. Meanwhile, Khan
[11], Erdogan [12] and Chy [13] include in their
research work specific properties such as density, thermal diffusivity,
thickness, specific heat and thermal conductivity, data that are also
considered in the development of this research work, for the conduction and
convection analysis, considering the accessibility to the equipment and the
features detailed in the description of the oven structure. 
On the other hand,
Schmidt [14] and Ajersch [15] use infrared sensors
and thus their study is aimed at radiation analysis; these publications reach
results that fit the reality, but due to the experimental conditions of the
oven under study it is not possible to replicate this technique. However, it
will be useful to compare and discuss the results at the end of this
research. This is similar to Chy and Boulet [16], who divide in layers the material to be
heated, and then interpret the results as a single element through numerical
techniques; it is not possible to carry out this condition in this study, but
it will be useful to contrast such results when formulating the conclusions. Even though all
these researchers developed acceptable mathematical models of the oven,
especially for the heating phase, there are still some discrepancies between
the simulation, the experimental results and the variables used by each
author in his/her research work; in addition, such mathematical models fit
the particular features of each oven. Therefore, this
research is intended to develop an improved model for the oven under study that
fits its features. It is taken into account that the process is mainly
executed through trial and error, based on the intuition and experience of
the machine operator. Consequently, it arises the initiative of understanding
the oven heating process, specially emphasizing on the features typical of
its construction knowing that it is an oven manufactured empirically. With
this mathematical model it is sought to establish a more precise way to
predict the temperature profile inside the oven, in order to use it in the
future for a better control of the process. There is a study
carried out previously about the oven under analysis. In 2017, Cortés
implemented a temperature control system for the thermoforming oven of the
machine known as P7. It briefly describes the oven, stating that «it consists
of metallic plates, that constitute a chamber. Such chamber has a glass wool
insulation to reduce heat irradiation to the outside. Stype and Utype
resistances (known as such due to their shape) of different electric power
are located inside» [3]. The author obtains a model of the oven through
system identification graphical techniques, as a step prior to the design of
the corresponding controller. Various techniques are applied, and compare
with each other. At the end of the corresponding validation, it is chosen the
Analytical without Delay graphical method, which yielded the highest
percentage of 90.95%. Regarding the model
obtained by means of system identification, Cortés [4] performed the
associated calculations to obtain a system controller, which yielded
acceptable results with respect to the temperature variability. Nevertheless,
there is still the issue of the width of the plates. 
The purpose of obtaining the
mathematical model of the oven of interest seems simple, but putting it into
practice is much more complex, since it largely depends on the data
available. The aim is to provide a system model that shows the real dynamic
behavior of the oven, thus enabling to calculate different controllers that
finally optimize its operation. 2.
Materials and
methods The methodology to develop this
research work is summarized in the diagram of Figure 3.
Figure 3. Working methodology 2.1.
Description of
the oven structure The oven under study, which is
shown in Figure 4, was constructed locally, with ohmic
electric resistances that heat up by Joule effect and release their heat. The
oven has been constructed with recycled elements, machine pieces and
equipment, which are coupled to each other to fulfill the requirements of the
production process. For this specific case, the industrial production of
thermoformed plates is used nationally in different applications. 
The oven is
constituted by metallic plates. It is parallelepipedshaped, constituted by
two boxes, one inside and another outside made up of 2 mm galvanized steel
sheets. Such boxes are separated 5 cm with respect to each other, and
insulation is placed between plates. For this design it was used fiberglass
manufactured from a mixture of sands, borates and silicates, thus fulfilling
the parameters recommended by various authors mentioned in the state of the
art. For this reason, from this point on it will be supposed that the oven
has the insulation necessary to avoid significant losses that modify the
temperature profile of this oven.
Figure 4. Thermoforming oven of
machine P7 2.2.
Description
of the oven physical phenomena To take measurements
about the real operation of the oven under study, two J type thermocouples
were placed with the purpose of measuring the temperature at the center of
the oven and in a lateral wall every second, during the time necessary to
reach stability, approximately 5000 seconds. The oven has a builtin data
acquisition system, and these data were stored in a computer connected to the
equipment, and will be used to obtain the mathematical model. To make the
oven study simpler and more practical, the oven operation has been divided
into the three stages that are detailed in Figure 5.
Figure 5. Oven operation diagram 
Table 1 shows the nomenclatures
to be used in the different sections of the present paper. Table
1.
Nomeclature used in the different equations
2.2.1.
Calculation of
heat flow in the resistance It starts with the transformation
of electric energy into thermal energy in the resistance. It is fundamentally
based on the basic Ohm’s law given by equation (1).
The only element in
the oven that takes part in this energy transformation is the resistance, whose
calculation is fundamental because it is unknown when constructed manually. 
Equation (2) is used
for this purpose, which involves the resistivity, a function of temperature,
the length and the cross section of the resistance, data that can be easily
obtained through measurements, tables and knowing the material of which it is
made of.
Then, it is applied
the energy conservation law in thermodynamics, which states that the amount of
heat received by the system is transformed and performs work against external
forces, as shown in equation (3).
The thermal flow
stored in the resistance is expressed through equation (4).
The heating by radiation is verified in equation 5.
2.2.2.
Calculation of
thermal flow in the air After carrying out
the corresponding calculation of resistance, it is continued with the
equations for the air inside the oven, with the purpose of relating the heat
transfer phenomena and developed the desired mathematical model. The total
thermal flow, presented in equation (6), is obtained from the equation that
relates the thermal flow produced by the resistance and the thermal flow by
the air on the wall.
The
thermal flow produced by the air on the wall is expressed using again the
convection phenomenon. Afterwards it is calculated the thermal flow stored in
the air, which is expressed in (7).
2.2.3.
Calculation of
thermal flow in the insulator After the study of
heat transfer in air has been completed, it is carried out the insulation
analysis. Equation (8) is used for this purpose, which defines the heat flow
on the wall due to the air movement, which is equal to the sum of the thermal
flow transferred to the outside by conduction and the heat stored in the
insulator itself.

The oven has an insulating wall
made of fiberglass, and hence it is necessary to quantify it, and the thermal
conduction phenomenon is used for this purpose. The mathematical expressions
and definitions that will be used in the solution of the mathematical model
has been explained in a general manner, and the following section details the
most important calculations that will be fed to the system.
2.3.
Solution
statement After presenting in
detail the physical phenomena supported on classical theory that appear during
the heating of the oven for the thermoforming production process, it is
proceeded with the mathematical calculation with the purpose of obtaining the
differential equations that will enable finding the desired mathematical
model. The Simulink mathematical tool was used with the aim of implementing
computationally the solution to the problem, since it is visual programming
environment that operates on Matlab. The
data in the technical sheet of the material is used to start with the
calculations of resistance. These data were found in the company catalogs to
specifically verify that the correct information supplied by the provider is
used. The conductor used in this case is Nikrothal
70 [17]. Appropriate calculations are carried out that lead to equation (9).
This
equation is implemented in Simulink, and the scheme of inputs and outputs may
be visualized in
Figure
6.
Scheme of inputs and outputs for calculating the temperature of the
resistance Then,
the behavior of the resistance as a function of the temperature is obtained,
and with it the heat that transforms through the physical phenomena already
described. Such implementation may be observed in detail in Figure 7, which
corresponds to equation (9). 
Figure 7. Implementation of equation
(9) Figure 8 shows the
response of the resistance temperature as a function of time when it is
connected to a voltage of 440 VCA; such temperature profile will be used in
the model of the thermoforming oven.
Figure 8. Resistance temperature
along time Finally, the
temperature of the oven is calculated, resulting in equation (10) that is implemented
in Simulink. Such equation is the result of the mathematical calculations of
the fundamental equations of thermodynamics.
Figure
9 shows the basic scheme that describes the inputs and outputs of the
Simulink implementation of equation (10). 
Figure 9. Scheme of inputs
and outputs for calculating the temperature of the air inside the oven The calculation of the temperature
of the air inside the oven, equations and data obtained through tables and
calculations are implemented in Simulink. Such implementation may be observed
in detail in Figure 10, which corresponds to equation (10).
Figure 10. Implementation of equation (10) After the
calculation of the thermal flow of the resistance and the air of the oven is
available, the subsequent step is to calculate the temperature lost in the
insulation to obtain the temperature of the oven wall. The temperature of the
air can be obtained relating all these calculations, data that should be
known for its further processing in the calculation of the control systems.
The data obtained through the oven data acquisition system is used in the
case of the wall temperature. Figure 11 shows the total implementation for
calculating the temperature of the air inside the oven. The blocks defined in
previous steps have been used for this purpose. 
Figure 11. Total
implementation of temperatura calculation in the air
of the oven Figure 12 shows the
response of the mathematical model comparing it with the response of the oven
obtained through data acquisition. As it is observed, the model fits very
well the real performance curve. This enables to predict that the validation
of the mathematical model will be better than the validation of the model
obtained by means of graphical identification techniques published by Cortés
[4]. A step input has been
applied to both the real system and the mathematical model for obtaining
Figure 12. For the real system, a voltage of 440 VCA has been applied to the
resistances. A similar value of voltage has been applied to the mathematical
model. The data acquisition in the oven is carried out reading the type J
thermocouple, which is sampled at a period of 1 second. Figure 12. Comparison of
the response of the mathematical model vs the real response 3.
Results and
discussion 3.1.
Validation of
the model obtained The mathematical obtained is
validated using mathematical techniques that enable quantifying it. This
section considers various points of view, as explained below. 
3.1.1.
Comparison of
the root mean square error of the mathematical model vs the current
model Figure 13 is similar to Figure 12.
All the curves shown are the responses to a step input of 440 VCA. The
behavior of the real system is shown in blue, whose data were taken
experimentally through the data acquisition system and processed in MATLAB.
The response of the mathematical model obtained through the calculations
described in previous sections is shown in red. The difference is in the
green curve, which corresponds to the response of the model obtained by
graphical identification to a step input of 440 VCA; this response is
currently used in the plant for calculating the controller. It may be
visually inferred in this plot that the mathematical model is similar to the
profile of real temperatures, and so it is expected that this error is
smaller when it is quantified.
Figure 13. Comparison of
the responses of the mathematical model vs the model calculated by graphical
identification The root mean square
error (RMSE) [18], whose mathematical expression is represented by equation
(11), is calculated for the first validation.
The values of 2
RMSEs of the responses to a step input of 440 VCA were calculated: · RMSE between the model obtained by
graphical system identification, which is currently used for calculating the
controller, and the real system, whose result is represented in equation
(12). • RMSE between the mathematical model
found in this research work and the real system, whose result is represented
in equation (13). 
In the statement of
the solution, it has been obtained the curve the represents the real
mathematical model, but it is necessary to compare it with the experimental
data, and at the same time with the model currently used for controlling the
thermoforming machine.
From the comparison
of the errors calculated it is evident that the error of the mathematical model
is smaller than the error of the current model; therefore, the mathematical
model found in this research work fits the real curve of the thermoforming
oven with greater precision. 3.1.2.
Comparison of
control responses to a step input Starting from the original control
system shown in Figure 14, the mathematical model is inserted in the current
control system with the purpose of analyzing its response to a step value
(input signal) of 140 °C.
Figure 14. Temperature
control system using the identification model Figure 15 shows the
comparison of the response of the control system with the identification
model (blue line) versus the response of the control system with the
mathematical model (red line), and the step of 140 °C (black line). It is
observed that the responses are different despite using the same controller,
and thus it is interpreted that it is not the appropriate one for the
mathematical model and, therefore, it becomes necessary to find one that fits
better to the features typical of the mathematical model; in this way, it is
expected that there is a greater stability in the temperature and this
implicitly leads to improve the final product avoiding the mismatches in the
width of the thermoformed plates. 
Figure 15. Comparison of the responses of the control systems 3.1.3.
Calculation of
a new controller that fits to the conditions of the mathematical model, to
simulate and verify its response The classical ZieglerNichols
control method is applied to calculate the new controller and simulate the
response of the system. This is one of the most wellknown
methods for tuning the parameters of PID controllers, and its rules come from
an experimental response according to the dynamics of the process and without
assuming any previous knowledge of the plant to be controlled [19–21].
Figure 16. Calculation of the PID parameters by means of the
openloop ZN for the mathematical model 
Figure
16 shows the limits used to obtain the data required, and thus apply the
openloop ZieglerNichols method. As
it is observed in Figure 17, the response of the PID controller using the
ZieglerNichols method does not show improvements compared to the current
system. This is because the stabilizing time (time to reach the setpoint
value) is high and the temperature variation does not decrease. Due to this
it is necessary to calculate the PID controller through using method.
Figure
17. Comparison of the response of the PID controller applying
ZieglerNichols vs the current PI controller The
Simulink PID Tuner tool provides a singleloop PID tuning method of fast and
wide application for the PID controller blocks. The parameters of the PID
controller may be tuned with this method to achieve a robust design with the
desired response time. Figure
18 shows the responses of the control system, comparing the one obtained
through the Simulink PID Tuner tool (shown in blue), with the response of the
current controller (shown in red). It is verified that this controller generates
a better system response reaching temperature stability in the smallest time
possible, it adjusts much better resulting in a smaller temperature
variability. It should be taken into account that the data used for the
subsequent analysis are the data obtained after 4000 seconds, since it is
there when the system remains stable.
Figure
18. Detailed visualization of the responses of the control systems vs
controllers tuned 
The behavior of the data
is verified through statistical analysis techniques to compare them with the
current ones, to quantify the improvement that would be produced if this
controller is applied in the production process. The mean temperature of the
current controller is 140.62 ^{o}C, whereas
the mean temperature with the controller proposed in this work is 140.35 ^{o}C, so it is evidenced quantitatively that
there is a greater temperature stability applying this controller that uses
the mathematical model developed. In addition, this is reflected in the error
of the standard deviation, which is smaller for the new controller. The
72.81% of the data simulated with the new controller are normal with respect
to their mean, whereas only 65.55% of the data of the current controller are
normal with respect to their mean, but if the data of the current controller
are compared with respect to the standard deviation of the new controller,
only 54.41% of the data are normal. Hence, it is identified that the new
controller provides better results to the system, which guarantees a greater
stability of the temperature if it is implemented. It
is conducted a linear regression analysis with data provided by the quality
department, with measurements of the temperature and the width of the
thermoformed plates, with the purpose of obtaining an equation and verifying
the correlation between temperature and width, where A represents the width
and T the temperature of the oven, as shown in equation (14). With this
mathematical model it is calculated the width of the plates with the
temperatures obtained in the system with the current controller, and the
corresponding statistical analysis is conducted.
Figure
19 shows the histogram of the plates measured, and there is a variation of ±
2 mm; in addition, it is clearly seen that the data are quite disperse.
Hence, the plates are heterogeneous with respect to their width, and when
stacked it gives the sensation that the regulation is not met. Figure
19. Histogram of the measured width of the thermoformed plates On the other hand,
Figure 20 shows the histogram of the plates calculated with the linear
regression model using the system with the proposed controller, and it is
clearly observed that there is smaller dispersion of the data and there is a
variation of ± 1 mm; 
thus, it is assumed
that they will look more homogeneous when stacked, providing security to the
client regarding the product quality. Although it would be important to have
experimental data to verify the data obtained in this research work.
Figure 20. Histogram of the
calculated width of the thermoformed plates In this way, it is validated the
mathematical model and in turn the new proposed controller, which is expected
to be applied in the production plant to have access to these data; this
confirms the effectiveness of what has been detailed in this document. 4.
Conclusions It was obtained the mathematical
model of the resistive oven for producing thermoformed sheets, from which it
was calculated a new controller for the system. Its behavior was simulated
and the variability of the plate width was projected as ± 1 mm, as opposed to
the variation of ± 2 mm with the current controller. Such results show the
effectiveness of the new controller. It was found the
mathematical model of the oven. For this purpose, its operation was divided
in three parts. First, the equations that have influence on resistance
heating were formulated. Then, the equations that have incidence on oven
heating through radiation and convection were expressed. At last, the
equations that have influence on the heating of the oven wall temperature
were used. However, the last part was not calculated since the specific
information of the insulator on the oven walls is not available. Instead, the
data acquired by means of the equipment data acquisition system was used. It was found the
mathematical model that best fits the real behavior of the oven. This is
evidenced when calculating the root mean square error of the mathematical
model, which is smaller compared to the model obtained by system
identification. Therefore, it is concluded that the mathematical model
obtained in this work has a better validation than the one obtained with the
current control system. 
It
was simulated a system that uses the current oven controller coupled to the
mathematical model found. It was evident that the operation dynamics of the
oven is slower with respect to the simulation of the current control system.
This is due to the fact that the mathematical model takes into account
different phenomena that occur, such as the loss of heat in the oven walls.
For this reason, it was necessary to calculate a new controller that fits the
mathematical model in a better way. Another
controller was calculated that improved the final behavior of the system.
However, it was not obtained by means of traditional calculations of PID
controllers. Instead, automatic tuning techniques available in Simulink (PID
tuner tool) were used. The main difference is that the controller is not
calculated in terms of the initial behavior desired by the user (2%
overshoot, stabilization time of 3600 seconds). In
this case, the overshoot is approximately 3.5% and the stabilization time is
4000 seconds. The mathematical model obtained analyzing the data of the
polypropylene plates’ width was validated. For this purpose, it was necessary
to calculate a function that relates the temperature with the width of such
plates. After comparing the results of the two systems (current and new), it
is concluded that the new control system has a variation of ± 1 mm, whereas
the current control system has a variation of ± 2 mm. To
obtain a better validation of the systems with respect to the width of the
plates, it is necessary to design better methods for recording information,
that store the temperature together with the specific dimension features in
real time. In other words, knowing the width of the plate at the instant at
which the oven has a particular value of temperature. For this purpose, it
would be necessary to implement a more complex data acquisition system that
would have direct incidence on the production costs, which for the moment is
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