Multidisciplinary Teaching of Variable Complex Functions in Heat
Flow Systems
EDUARDO GAGO, PAOLA SZEKIETA, LUCAS D’ALESSANDRO
Computer and Multidisciplinary Laboratory of Basic Sciences,
Universidad Tecnológica Nacional, Facultad Regional Rosario,
Zeballos 1341,
ARGENTINA
Abstract: - In the context of new curriculum designs in engineering careers, new lines of work are being
developed in teaching to achieve the development of basic competencies, acquisition of skills, and the
replacement of old learning paradigms with new didactic techniques that are currently applicable. These
characteristics in teaching are consolidated when they are added to the integration of knowledge in the
teaching-learning process and the inclusion of computer resources. In the present work, the use of new
pedagogical methodologies and diverse didactic strategies has been implemented, aiming at multidisciplinary
work in the development of the topic of Complex Variable Functions corresponding to the Advanced Calculus
subject in the Mechanical Engineering program. The didactic proposal presented corresponds to a system
related to heat transfer in an incompressible fluid, where the concepts of complex field and fluid flow are
approached analytically and graphically, giving meaning to the curricular contents, and facilitating the
interpretation and conceptualization of the theory.
Key-Words: - Multidisciplinary Teaching, learning, complex variable, heat flow, laboratory, simulation,
models.
Received: August 29, 2023. Revised: April 15, 2024. Accepted: May 11, 2024. Published: June 28, 2024.
1 Introduction
The new challenges posed on Engineering
Education, supported by government organism-
established guidelines, for the accreditation of
degree programs, highlight the need for
multidisciplinary training of the student.
This multidisciplinary training is supported by
the implementation of learning proposals, that
involve the application of computer resources.
All universities appear to share a common goal,
that is, to ensure the appropriate education of
students which will allow them to play an active,
ethical, and responsible role in the complex society
of today, [1].
The recommendations made by accreditation
agencies urge students to engage in learning
activities using computational tools. This proposal
aims to move away from the traditional classroom
setting by proposing a methodological strategy
supported by a new perspective of competency-
based teaching.
University students must be prepared to make
decisions and effectively deal with the challenges
posed by a globalized, technological society,
respecting the rights of individuals and safeguarding
the legacy to be left to future generations, [1].
It is intended that the student acquires knowledge
through an experimentation process, using
simulation and visualization. To achieve this, the
student must formulate working hypotheses, induce
responses, question predictions, and validate them to
give meaning to the learning process.
It is possible to highlight a set of essential
minimum contents for the basic training of the
engineer, which are being addressed in the
Advanced Calculus subject.
Advanced Calculus subject is taught at the third
level of the Mechanical Engineering program.
This paper recounts several classroom
experiences undertaken on that subject.
In the last decades the voice of the student has
been rarely taken into consideration. Today’s
competitive market structure determines a shift in
higher education organisational focus towards the
needs and wants of its main actor/consumer. Student
satisfaction is increasingly associated with
institutional success. The antecendet of student
satisfaction is service quality. Thus, service quality
needs further attention, especially in new offers by
assessing as accurtaely as possible the features that
attract students or that are requested by them, [2].
The teaching-learning process is implemented
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Eduardo Gago,
Paola Szekieta, Lucas D’Alessandro
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through the integration of various disciplines within
the program, which requires the teacher to know
about the teaching programs of other parallel
subjects, not only those of the Integrator Core but
also of basic and specialty subjects.
Until not many years ago, it was common for
research on the teaching-learning of any area of
knowledge, including mathematics, to focus on
cognitive processes, or how a student can capture,
encode, store, and work with the information that is
normally transferred to him or her by the teacher.
But in this process, other factors also influence, such
as motivational, affective, metacognitive,
evolutionary, and social factors, which can have
their importance in the context of education, [3].
As a final goal in educational work, it is
necessary to organize coordinated practices to link
basic and simple problems of the subject with the
problematic situations that develop in Fluid
Mechanics and Heat Transfer.
2 Motivation and Objectives
The multidisciplinary teaching approach aims to
relate the content and concepts taught in class to real
Engineering projects.
In this way, the objective of the class is to
provide future engineering professional with the
necessary tools to detect relevant variables in a
problem, and interpret and propose solutions to
different alternatives, thus increasing their analytical
capacity.
The rational selection of teaching proposals and
perspectives is based on decision-making by
students based on the solutions found, [2].
Since the topics are related to current problems
actually present in the actuarial sector, undertaking
the projects is a very stimulating way to get familiar
with the sector context and required skills for a
professional, [4].
The didactic activities proposed consist of
emphasizing theoretical research by the students
with the teacher's guidelines, followed by making an
analogy of the physical parameters of the topic
under study. Finally, with the gathered information,
solve the proposed problematic situation using
technological resources.
The methodological proposal for presenting
mathematical content is based on the search for
models that simulate a simple situation in the
Engineering field, [5], [6].
Formulating the technical situation in
mathematical terms involves presenting a simplified
scenario translating that situation into mathematical
terminology, and working with that model.
After a critical review, a list of different aspects
to be changed/improved (as well as those to keep) is
proposed for each of the analyzed degrees in order
to incorporate the most appropriate evaluative and
methodological features to help in the achievement
of specific, generic and basic skills of each degree,
[7].
This methodology stimulates interest in
discovery and fosters confidence in the use of the
formative aspects of mathematics, related to other
areas of knowledge, such as in this case, the analysis
of the dynamics of a specific fluid.
. These characteristics in teaching are
consolidated when they are added to the integration
of knowledge in the teaching-learning process and
the inclusion of computer resources.
The pedagogical objectives are outlined in the
following phases:
Presentation of the system to be analyzed.
Theoretical research.
Parameter modeling and analysis
Interpretation of results in technical terms.
Mathematical competence formation in
university students is a pedagogical process that
takes place in several stages. According to the
formation logic in Mathematics academic discipline,
the formation of mathematical competence in
university students requires a stage that depends on
the motivational objective, [8].
Mathematical competence formation
effectiveness in university students depends on
certain pedagogical conditions. In our study, we
identified the following pedagogical conditions in
teaching mathematics: Formation of a stable
motivation for teaching mathematics; use of
personal developmental techniques; and designing
the content of the discipline.
The first condition for the formation of
mathematical competencies in university students is
the formation of a stable motivation for teaching
mathematics. Focusing on this condition is one of
the most important features enabling the future
specialist to realize the role of mathematics in his
professional activity, [8].
This problem is especially acute at present, when
students, are firstly guided by the study of subjects
measured by their professional significance and
increased competitiveness in the labor market. The
choice of the second condition is due to the personal
component of the methodological approach to
teaching.
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Paola Szekieta, Lucas D’Alessandro
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Developmental techniques used in education are
of personality-oriented education importance for
potential activities implementation, [8].
3 Methodology
Learning based on the development of simple,
integrative, and ideal projects is a methodology that
enables students to acquire key basic knowledge and
skills through the presentation of models that
address specialty-specific problems.
But without being necessary to arrive at concepts
that we can qualify as a high level of development,
Mathematics is present in all degrees in Engineering
in any university in the world.
Normally, the subjects of Mathematics are
placed in the first years of these degrees. And it is
here that from the consortium we have formed, we
see that Mathematics and its associated subjects are
fundamental elements of engineering education, and
proficiency in the area is expected.
Engineers are required to be analytical and be
able to utilize their mathematical toolkit to solve
problems that may be ill or well-defined depending
on the contextual situation of the engineer. Until
quite recently the determination of learning was
undertaken using face-to-face techniques such as
handwritten assessments, private and public
communication, and observation to name but a few.
Assessment and program delivery underwent a
seed change with the new millennium when
Educational Authorities and Professional bodies
adapted their validation and accreditation methods
to include learning outcomes within programs of
study.
The assessment techniques within programs
altered accordingly to address these requirements
and forces have evolved within higher education to
increase the online presence, [3].
The role of teachers in this teaching
implementation approach is to act as guides, and
facilitators of the teaching-learning process, and to
provide support throughout the experiences.
In addition to coordinating a series of
multidisciplinary activities that result in a
continuous collaboration task, strengthening the
project, [9].
The classroom experience takes place in the
Basic Sciences Multidisciplinary Computer
Laboratory, a learning space with 25 networked
computers and a small library.
The laboratory notebook is a common tool used
in research areas such as Biology, Biochemistry,
Chemistry, and Mathematics.
It is therefore interesting that students learn in
the laboratory how to work with it. Since the first-
year laboratory experiments, students use it and it is
also an item for the assessment.
All the measurements, computations, and results,
as well as the incidences that occurred during the
experiment, have to be written there.
We consider that the laboratory notebook helps
stud students to neatly save/report the results and the
experiences that happened during the laboratory
lessons [7].
The specific resources and methodology applied
in each case are different as the objectives pursued
are also different. Nevertheless, there are some
points in common. Special emphasis has been
placed on the use of the recommended textbooks,
because a worrying decline in using them has been
observed in most students in the last courses, [10].
4 Theoretical Foundations
The systems being analyzed correspond to heat flow
in a region of the complex plane of a fluid
First, it is proposed to analyze the heat flow
around a cylinder with a circular cross-sectional
area, followed by a second activity, where the heat
flow is analyzed knowing the stream function as
data.
4.1 Fourier's Law of Heat Conduction
(Molecular energy transport)
Heat conduction in fluids can be thought of as
molecular energy transport, inasmuch as the basic
mechanism is the motion of the constituent
molecules.
Energy can also be transported by the bulk
motion of a fluid, and this is referred to as
convective energy transport; this form of transport
depends on the density of the fluid. Another
mechanism is that of diffusive energy transport,
which occurs in mixtures that are interdiffusing,
[11].
A medium heat conductor is considered when a
temperature distribution may be variating.
The Fourier law of heat conduction allows us to
calculate the amount of heat conducted per unit area
in a unit of time through a surrounding medium.
This quantity is called the heat flow through the
surrounding medium, and is given by:
(1)
In equation (1), is heat flow; is the
complex temperature gradient, and is a
constant, known as thermal conductivity which
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depends on the material or medium through
which it is diffusing.
That is, the rate of heat flow per unit area is
proportional to the temperature decrease over the
distance.
Equation (1) is also valid if a liquid or gas is
placed between the two plates, provided that
suitable precautions are taken to eliminate
convection and radiation, [10], [11].
Actually Equation (1) is not a law of nature, but
rather a suggestion, which has proven to be a very
useful empiricism. However, it does have a
theoretical basis.
If the temperature varies in all three directions,
then we can write an equation like equation (1) for
each of the coordinate directions, [10], [11]:
(2)
If each of these equations (2) is multiplied by the
appropriate unit vector and the equations are then
added, we get the equation (1) which is the three-
dimensional form of Fourier's law.
This equation describes the molecular transport
of heat in isotropic media. By isotropic we mean
that the material has no preferred direction, so that
heat is conducted with the same thermal
conductivity K in all directions, [11].
The problems designed for students to solve in
class serve as a vehicle for conceptualizing the topic
of complex variable applications when the heat flow
meets the following conditions:
The fluid flow is two-dimensional: The basic
fluid model and the characteristics of fluid
motion in a plane are essentially the same in
every parallel plane.
The flow is steady or uniform: The fluid
velocity in a determinate point of the plane
depends only on the position and not
on time.
The fluid is incomprehensible: The density is
constant.
The fluid is non-viscous: It has no viscosity
or internal friction in the fluid layers, [12].
4.2 Modelling of the System under Study
After conducting theoretical research on the
fundamental concepts of heat transfer in fluids,
students will identify the variables involved in the
system under study.
With regards to the interest shown by students to
solve problematic situations, which will then be
described, enounces:
If temperature T is a complex variable function,
whose independent variable is a complex number Z,
such that:
(3)
where is the real part of , is the imaginary
part of , and is the imaginary unit.
Variable can be represented graphically in a
coordinate rectangular system called Argand
Diagram. However, the , and values
can not be plotted on a single set of axes, as can be
done with real functions of a real variable.
is known as analytic in a region of the
complex plane , if a derivative exists in
every point of [12], [13].
The complex temperature is composed of the
ordered pair whose real part is the temperature
potential and the imaginary part is the stream
function .
(4)
A necessary condition for to be analytic in a
region of the plane is that and in satisfy
the equations expressed in (5), called the Cauchy-
Riemann equations:
(5)
If these partial derivatives are continuous in ,
so the equations (5) are sufficient conditions for to
be analytic in . Functions that satisfy these
conditions are called conjugate and harmonic
functions.
The conjugate and harmonic functions fulfil the
following properties:
a) The family of the curves, for all and real
numbers, expressed in equations (6) are
orthogonal.
(6)
b) If, in addition, the second partial derivatives
of and exist and are continuous in , the
Laplace equation is satisfied. For and , the
Laplace equations are as shown in (7), [13],
[14].
(7)
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If the analysis of heat flow is limited to a two-
dimensional model, then the quantity of heat for
such a model is:
(8)
The magnitude of the heat quantity, also referred
to as the heat flow velocity, is calculated as the
modulus of the complex quantity expressed in
equation (8).
(9)
Where,
(10)
Whatever may be, represents a simple closed
curve in the complex plane , delineating the
boundary of the cross-sectional area of a cylinder,
and assuming and denote the normal and
tangential components of the heat flow,
respectively, with steady-state conditions ensuring
no net heat accumulation within , then the
following holds [12].
(11)
If is not any heat accumulation within that
means heat accumulation is null, this situation is
represented in (12).
(12)
Assuming there are no sources or sinks within .
The equation (8) can be expressed in the form of
education [11], [14].
(13)
5 Learning Space
A teaching sequence is designed with a theoretical-
practical-technological approach. Three classes of
three clock hours each are implemented to carry out
the proposed activity.
The class, which is called theoretical-practical-
technological, is a class that does not separate
theory from practical activities and takes place in a
computer-mediated mathematics laboratory.
In Laboratory Class, students work in groups of
no more than three members. For this purpose, two
cases are presented for students to solve with the
support of computer tools.
Each student has the freedom to choose their
group partners according to their preferences, but
the only established indispensable condition is
teamwork.
Each group has, at least, a desktop computer to
work but also, it is allowed for the students to bring
their personal computers for the task development.
In pedagogical science and practice, new
approaches and methods for solving the problem of
improving professionalism are constantly being
developed. Some of them are based on the use of
new information technologies in the training of
qualified specialists, others are focused on updating
the content of professional training, and several of
them are interested in strengthening practical
orientation, [15].
The characteristics of this class are to establish
the guidelines of a interactive task of Engineering
analysis, resulting in an activity that generates new
ideas.
In these cases, it is common for students to
replicate what they have seen in class, without a
clear idea of why or what for, and without knowing
very well what to do in the case of small variations
in the types of problems posed in class. We could
say that the students have learned the concepts, but
only to apply them in situations equal to those
created by the teacher. This is one of the reasons
why the contents may lack real meaning for these
students. Likewise, there are places in which results
are prioritized without concern for the mental
processes that the student develops when solving
mathematical exercises or problems, [3].
This teaching methodology enables to advance in
a learning system, where the student assumes an
active role and allow them to build concepts through
experimentation and the development of
conclusions.
For the implementation of project based learning
in the curricula of engineering degree programs
lecturers’ instructional abilities are critically
important as they take on increased responsibilities
in addition to the presentation of knowledge, [16].
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DOI: 10.37394/232010.2024.21.9
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Paola Szekieta, Lucas D’Alessandro
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5.1 Analysis of Heat Flow around a Circular
Cross-Sectional Cylinder
The first activity presented to students consists to
analyse the heat flow around a circular cylinder with
adiabatic walls. For this purpose, the complex
temperature function is provided to them as data,
with its law being:
.
(14)
where and are two positive constants.
As a guide for the development of the activity,
the following questions are posed to the students:
a) What relationships can be established
between the potential temperature and the
stream function ?
b) What are the equations that model the
isotherms and the heat flow lines of the
system?
c) What is the graphical representation of the
trajectories of the isotherms and the heat
flow lines?
d) What is the physical interpretation of the
parameters, plotted in the previous point?
e) What is the temperature profile at different
points along its trajectory?
f) What are the system stationary points?
g) What equation measures the heat amount at
all points of plane?
h) What happens if in the heat flow the
cylinder has an elliptical transversal cross?
i) What happens if the heat flow is not
confined to a adiabatic wall cylinder, and
there is a sink on point ?
The students answer the previous questions, with
the teacher collaboration, arriving to the following
conclusions:
The and functions, fulfil with te Cauchy-
Riemman equations, and consequentially with
Laplace equation, confirming that both functions are
conjugate and harmonics.
. The student determinates the line equations of the
isotherms and the heat flow lines through the
computational tools application.
The result obtained by MATHEMATICA
software in the calculus of and is presented:
Fig. 1: Student work with the software
The data provided by the software about the heat
flow and the isotherms are respectively showed in
the equation system (15).
(15)
In Figure 1 is observed that the isotherms are
been plotted with line points and are orthogonal to
the heat flow line, which are plotted with a
continuous line.
Far from point (coordinate origin),
observing the graphic from Figure 2, the heat flow
lines are parallel to axis , When we are
approximating to coordinate origin, the heat flow
lines presents a curve than reach near the axis,
surround the cylinder centred in the origin filed in
that region.
These curves indicate the trajectory the heat flow
follows in that region of plane. When , the
graph of the heat flow lines shows a tangent line
parallel to the axis, with a null slope.
Fig. 2: Heat flow and isotherm curves
In maths, graphics with the characteristics that
presents the isotherms and heat flow are called
contour lines. The contour lines have the
characteristic that they do not intersect each other.
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Fig. 3: Student computer work with the software
The software enables students to develop a
operation sequence to calculate the fluid velocity
circulation in each point of Complex Plane z, using
the versatility that the computational resources
provides.
Use of technology made it possible to make more
explicit the role of modes of representation. In
particular, the way in which the complementarity
between graphic, numerical and symbolic
representation, produce best comprehension using
technology and help develop coordination
processes.Thus, the descriptions of the construction
process followed by students have allowed relating
aspects of the particularization to the reflective
abstraction derived from the modes of
representation in construction knowledge. These
early educational experiences continued with
intervention and interaction activities with students,
inside and outside the laboratory, through virtual
platforms, always favoring the approach and
requests, [2].
Computers have become nowadays a valuable
tool for Education The animation of figures and
representations, achieved by using the proper
software, develop the students’ imagination and
enhance their problem-solving skills. In the
forthcoming era of the Fourth Industrial Revolution
computers will provide, through the advanced
Internet of Things (IoT), a wealth of information for
students and teachers, [17].
According to procedure observed in Figure 3 the
students determine that function that governs the
heat amount in a two dimensional fluid in the
Complex plane is:
(16)
Fig. 4: Heat flow and isotherm curves
If an observer at the origin moves away from the
obstacle the amount of heat has its maximum value,
, the amount of heat tends to a constant value
as the observer moves away from the obstacle.
The stationary system points are those where the
velocity is null, and in this case are given by the
values of and .
In Figure 4 and Figure 5, the simulation of two
alternative situations from the original is observed.
From Figure 4, it can be determined that when
the given heat flow encounters an elliptical cross-
sectional cylinder, it is observed that the flow lines
have a wider separation, while the isotherms are
closer to each other compared to the circular cross-
sectional cylinder.
Also, a new simulation is made, consists in
supress the cylinder and considers a sink in the
coordinate origin.
This change modifies the heat flow, determining
that if one is away from the origin, it behaves the
same as any cylinder, but at the coordinate origin,
the heat dissipates through the sink. Figure 5
illustrates this situation.
Fig. 5: Heat flow and isotherm curves
3
2
1
0
1
2
3
3
2
1
0
1
2
3
x
y
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DOI: 10.37394/232010.2024.21.9
Eduardo Gago,
Paola Szekieta, Lucas D’Alessandro
E-ISSN: 2224-3410
68
Volume 21, 2024
5.2 Analysis of Heat Flow Known Current
Function
In this opportunity, it is analyzed the function
stream current , in a heat transmitter system.
The announced data in equation (17) is the law
that determines the stream function:
(17)
Where and are two positive constant.
The following questions are posed to the students:
a) What is the potential temperature related
to ?
b) What are the heat flow lines and isotherms
equations?
c) What is the graphical representation of the
above trajectories and how can they be
physically interpreted?
d) What is the temperature profile at different
points along the fluid path?
e) What are the stationary points of the system
associated with equation (17)?
f) What is the equation that measures the
amount of heat at all points in the plane?
According to previous questions, the students
arriving to the following conclusions for this case:
The steps that were made with the software to
obtain the potential temperature are detailed in
Figure 6.
Fig. 6: Student work with computer tools
According to the works with computational
tools, the potential temperature obtained is:
(18)
Fig. 7: Heat flow and isotherms curves
The Equations (19), and (20) represents the
expressions of heat flow lines and isotherms, and
are respectively:
(19)
(20)
Figure 7 shows that the heat flow lines are
logarithmic spirals entering at the origin, and the
current lines are two families of orthogonal circles,
and have similarities to the field lines of a magnet
with the poles located at two points in the complex
plane of coordinates and .
Implementation of technology in teaching has
now become a trend of the modern world.
Specifically, in the field of mathematics, ICT is
bringing day by day a positive motivation in the
learning process by fostering and encouraging
interaction between students, by stimulating them
with quick feedback focusing on solution strategies
as well as interpretation of the final solution. This
implementation supports the constructive theory of
pedagogy to apply and deepen ideas, [18].
6 Conclusion
The approach to multidisciplinary activities in
Mathematics from the Basic Sciences in
Engineering careers, through the exploration of new
knowledge and pedagogical methods, fosters
creativity through the analysis and management of
simple mathematical models, resulting in an
innovative teaching-learning process.
To achieve the theoretical conceptualization of
the abstract contents of complex variable functions,
it is essential to be able to mathematically analyze
the heat transmission system presented to the
students. This proposal is a motivating approach that
allows the integration of theory, practice, and
3
2
1
0
1
2
3
3
2
1
0
1
2
3
x
y
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DOI: 10.37394/232010.2024.21.9
Eduardo Gago,
Paola Szekieta, Lucas D’Alessandro
E-ISSN: 2224-3410
69
Volume 21, 2024
technology, making this activity a challenge for
future research.
In the experience presented in this paper, the
creation of a dynamic and symbolic information
space is an important contribution to engineering
education, introducing a new paradigm in the
teaching of sciences.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Eduardo Gago: Implementation and development
of the classroom experience. Application of
mathematical, computational, or other formal
techniques to analyze or synthesize study data.
Conducting a research and investigation process,
specifically performing the experiments, or
data/evidence collection. Preparation, creation
and/or presentation of the published work,
specifically writing the initial draft (including
substantive translation). Management and
coordination responsibility for the research
activity planning and execution. Management and
coordination responsibility for the research
activity planning and execution.
- Paola Szekieta: Development of the classroom
experience. Development or design of
methodology; creation of models. Verification,
whether as a part of the activity or separate, of the
overall replication/reproducibility of
results/experiments and other research outputs.
- Lucas D’Alessandro: Development of the
classroom experience.Verification, whether as a
part of the activity or separate, of the overall
replication/reproducibility of results/experiments
and other research outputs.for the Statistics.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2024.21.9
Eduardo Gago,
Paola Szekieta, Lucas D’Alessandro
E-ISSN: 2224-3410
71
Volume 21, 2024
