Computational Engineering: Mathematical Models in Heat Flow
Dynamics Study
EDUARDO A. GAGO, LUCAS I. D’ALESSANDRO, MARCELO M. ZURBRIGGEN
Computer and Multidisciplinary Laboratory of Basic Sciences
Universidad Tecnológica Nacional – Facultad Regional Rosario
Zeballos 1341
ARGENTINA
Abstract: - According to the latest educational innovation trends in engineering, new teaching paradigms emerge
to achieve higher development in intellectual capacities, acquisition of skills, substitution of outdated techniques
for more efficient and fast means, and a better integration of knowledge in the teaching and learning processes.
In line with these paradigms, we have set up new pedagogic methodologies together with computer resources to
work with the development of functions with a complex variable and Laplace transform from a multidisciplinary
view in the subject Advanced Calculus of the Mechanical Engineering Programme. The didactic proposal
presented is about systems related with heat transfer in a fluid where the concepts used in the models are
approached analytically and graphically making the curricular content meaningful and facilitating the
interpretation and conceptualization of the theory.
Key-Words: - Modelling, multidisciplinary approach, simulation, complex variable, Laplace Transform, heat flow.
Received: July 21, 2021. Revised: April 13, 2022. Accepted: May 11, 2022. Published: June 1, 2022.
1 Introduction
The fast technology development and the use of
digital tools condition the academic work of faculty
in the university fields. In this context, professors
have to face continuous challenges, find new
teaching approached and agree on curricular
contents. The design of learning strategies should
comply with the opportunities which available and
new computational systems offer.
These new developments in technology have
potential to be used in the mathematic training of
engineering students. Professors have to distinguish
which students’ basic needs are to approach the
analysis and solution of simple models that can give
the opportunity to develop suitable strategies to
connect and integrate computational mathematics to
applied basic technologies in engineering.
Real systems modelling determines that the
mathematical objects dealt with in the classroom
must be adapted to students’ levels and learning
processes. This is why the tasks designed should be
adapted to tangible learning situations [1].
Thus, it must be considered that the topics
developed in Advanced Mathematics will be further
used in applied technologies and in the professional
career. These topics often present difficulties on the
complexities of mathematical structures meant to be
taught. Activities should then rely on didactic
resources from symbolic, numeric and graphic
Calculus tools.
This work presents two experiences carried out in
Advanced Calculus with modelling of the curricular
topics in connection to heat Flow systems dealing
with functions of complex variables and Laplace
Transform. The teaching approach gives relevance to
both the match contents meant to be taught and the
competencies which students should develop to
achieve meaningful learning functional applications
of specific software.
2 Methodology Criteria
The contents of the engineering programmes usually
have a high levels of abstraction and generalizations,
particularly in Advanced Math courses.
Teaching Math through modelling in simple
dynamic systems facilitates student’s engagement in
curriculum topics. This situation fosters learning
autonomy and the understanding of concepts in a
meaningful way, two important competences for
future professional development.
In fact, training suitable professionals requires the
articulation of an academic programme to achieve the
professional competencies, right contents and
activities to develop such competencies together with
political and curricular management [2].
Designing curricular activities to foster the
DESIGN, CONSTRUCTION, MAINTENANCE
DOI: 10.37394/232022.2022.2.23
Eduardo A. Gago, Lucas I. D’alessandro, Marcelo M. Zurbriggen
E-ISSN: 2732-9984
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construction of concepts aims at a student who can be
protagonist of the teaching and learning process
relying on a multidisciplinary approach to curricular
contents.
The selection of real mathematical models raises
students’ interest. When the work proposal results
engaging and convenient, students can analyse and
visualize complex systems and thus assimilate the
target topic.
These objects can be focused towards
collaborative learning and can be identified as an
exercise in determining the association scheme. This
tool is positive because it reinforces communication,
reflection, teamwork and helps strengthen the
educational role that is expected to give today [3].
Collective interdependence was observed in the
construction of learning supported by collaborative
instruments in which there was no evidence of
positions of competence but of group interaction. The
interests were shared in the vocational training
against their daily practice and work experience.
From the collaborative learning and the area of the
computer science was propitiated the construction of
significant knowledge in the subject of the mining of
texts [3].
The aim is to create a reflexive and critical
environment where the students can make decisions
and delimit mistakes made during the learning
process. This implies students’ need to know the
object of study and to create the internal conditions
for the assimilation of new knowledge in an active
and independent way.
In this type of experiences, the professor is a guide
and works as a facilitator of the learning process
towards the deep understanding of the target subject,
making use of questions which can activate a
continuing mechanism for further enhancement and
research of different lines of the proposed problem.
The task is carried out in the Informatic and
Multidisciplinary Lab of the Department of basic
Sciences at our College equipped with 25 computers
connected to Internet and to a specific software. This
modelling analysis interacts with Mathematics and
GeoGebra.
In order to carry out the didactic proposal, an
activity is designed with two topics from the subject
Advanced Calculus: Functions of complex variable
and Laplace Transform. The activity includes two
cases of heat transfer: the first one presents the
balance of heat flow in two concentric pipelines, and
the second one the analysis of thermometer when a
system is prompted by a sinusoidal perturbation
The methodological design has the following
stages: generation of a situation to train on symbolic,
numeric and graphic calculus; integration of
disciplines with a multidisciplinary perspective;
complementation of contents for analytic and
simulated methods in the reproduction and analysis
of the behaviours of two real simple systems.
3 Objectives
The teaching of Advanced Mathematics in
engineering should find the balance between the
formulation of mathematical models and the skills
which students have to use to solve the challenges
which they will have to face during applied
technologies and their future professional careers.
The general aim of this work is to show where
theoretical topics in the subjects Advanced Calculus
are used with special emphasis on the simulation and
visualization of target systems [4].
The professional competencies in the higher
education framework are achieved by mean of a
process which can allow the training of competent
professionals, not only of their knowledge and skills
to carry out their work as engineers, but also of
personal and social development.
It is important to educate engineers who can work
in a suitable way in the market labour place. In this
line, while academic competencies rely on theory and
reflection on the role of engineers, labour
competences focus on a more pragmatic perspective
which demands an efficient performance. Thus, an
integrated teaching and learning process is proposed
so as to try to give a different experience from
dialogue, convergence criteria and active students’
participation [5].
The right questions are sequenced in such a way
that can guide student’s thinking by means of
argumentation which accordingly can lead to
conclusions, convergent thinking in a dynamic and
collaborative experience.
Therefore, the training activities carried out with
systems engineering students have revealed new
problems that make it necessary to advance the
conceptualization of collaborative learning through
educational research. One of the first elements to take
into account as the basis of each learning, and
especially the collaborative learning, is the
communicative interaction [3].
The aim is also to foster scientific
experimentation to approach problems that require
mathematical modelling and application of different
methods to solve them and the identification of
different tools developed in each method.
All this implies selecting algorithms to solve
problems, a skill necessary for a professional who
will have to interpret and propose solutions when
facing alternatives together with decision making.
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Eduardo A. Gago, Lucas I. D’alessandro, Marcelo M. Zurbriggen
E-ISSN: 2732-9984
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4 Learning environment
The following two models show two learning
situations set up in the Advanced Calculus class. The
problems involve two topics: Functions of complex
variable and Laplace Transforms are based on the
analysis of the heat flow which undergo the fluid.
4.1 Heat transfer in pipelines
Students have to solve the heat transfer applying the
theory of functions of complex variable por a system
in stationary state [6].
It is given a cylindrical pipeline with an interior
cylindrical cavity which is off-centric through which
steam flows at . The exterior temperature of
the pipeline is . The radio of the interior
circumference is
from the radio of the exterior
circumference. Fig.1 shows a chart of the system.
Fig.1 Schematic diagram of the pipeline.
The law of the harmonic function which rules the
temperature for fluids in pipelines is
(1)
with and .
Knowing that the law of bilinear mapping which
is applied to observe the positions on the plan of the
pipeline display, and to transform the problems to
one which has axial symmetry is:
(2)
According to given data, student’s work is guided
by means of the following questions:
a) Verifying if the equation (1) is harmonic in
function of u and v.
b) Making the graphics which transforms the
enclosures in Fig. 1 applying the mapping of the
equation (2) so as to show the complex plane
from the domain (plane z) and from the image
(plane w).
c) Finding the equations of the graphics of the
enclosures determined in the item b, which
transform the circular regions of the domain into
other regions with axial symmetry.
d) What characteristics do the equations of item c
with its graphics have?
e) The value of the constants A and B for the
equation (1) which fits this case.
f) The law which rules the distribution of the
temperature in function of x-y.
g) The graphic of the function T is an orthogonal
cartesian system x-y-z.
h) Showing if T is harmonic in function of x and y.
i) If the potential in a heat transference process
follows the law f, analysing the graphics of the
isotherms and the flow lines.
The students carry out their work in groups, and
once the questions are answered, they agree on the
writing a document with the reached conclusions.
The answers which determine the following
conclusions are cited below:
One function of the variable is harmonic if
verifies
(3)
The function T has as second partial derivatives
(4)
(5)
The equations (4) and (5) verify the condition
expressed in the equation (3).
Fig.2 shows the graphic of the systems in Fig.1 in
a system of orthogonal axis in a complex plane of the
domain of a complex variable function.
Fig.2 Complex plane of the domain .
Fig.3 shows the application of the bilinear
mapping of the law f to the domain represented in
Fig.2.
As the students know the law of the function
expressed by the equation (2), and from previous
theoretical knowledge, they consider that it is
convenient to work with the inverse function
expressed in (2), resulting in:
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Eduardo A. Gago, Lucas I. D’alessandro, Marcelo M. Zurbriggen
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(6)
Fig.3 Complex plane of the image .
According to equation (6), students determine that
the equations of the enclosures of Fig.2 and 3 are the
ones which are shown on Table 1 and 2.
The same equations presented in Table 1 are
presented in Table 2, but the latter is made in function
of the variables z and w.
Table 1. Equations of the relation which exist
between the circular enclosures on the complex
planes and , with function of the variables -
and -.
Complex plane
Domain (
Image (
Table 2. Equations of the relation that there is
between the circular enclosures on the complex
planes and , function of the variables and
.
Complex plane
Domain (
Image (
The students conclude that on the complex plane
, the exterior area of the circumference of the unitary
radio of the equation is mapped on the
complex plane w in the interior area of the
circumference of the unitary radio of the equation
.
While on the complex plane , the interior area of
the circumference of the equation
is mapped on the complex plane w on the exterior
area of the circumference of the equation .
The required activity is turned into a problem of
axial symmetry on the plane w which consists on
finding a harmonic function so that
in and in .
The harmonic function T with axial symmetry has
the general formula of the equation (1).
The system:
(7)
(7) is a system of linear equations to true
coefficient, a unique solution, so that its solution is:
The equation (1) for this case is:
(8)
From the equation (2), and knowing that the
complex variable of the domain , and
the complex variable of the image , the
module of is the one indicated in equation (9).
(9)
Squaring both members of the equation (9), and
then substituting by , equation (10) is
(10)
Finally, from equation (10) it turns equation (1)
which determines the temperature in function of x-y
If now , and
(11)
The graphic of the function given by the equation
(11) can be observed in Fig.4.
Fig.4 Temperature in the function of -.
From Fig.4 students point out that the temperature
grows close to the origin of the coordinates, taking
DESIGN, CONSTRUCTION, MAINTENANCE
DOI: 10.37394/232022.2022.2.23
Eduardo A. Gago, Lucas I. D’alessandro, Marcelo M. Zurbriggen
E-ISSN: 2732-9984
176
Volume 2, 2022
significant values, while as the temperature is far
from the origin, the value gets lower, with values
which decrease smoothly.
A function of a complex variable is harmonic if
verifies
(12)
If and
, has a second partial
derivatives
(13)
(14)
Consequently, they verify the equations (13) and
(14) and conclude that the function T is harmonic
regardless the variable of the domain or the complex
image.
The mapping expressed in the equation (2),
presents two types of curves which are themselves
orthogonal: the isotherms and the flow curves [7].
The equations of the isotherms for the values
(constant) with and such that
are equation (15)
(15)
The equations of the curves of heat flow for the
values (constant) considering and such that
are equation (16)
(16)
The curves which represent the isotherms and the
curves of heat flow are shown in Fig.5.
Fig.5 Isotherms and Flow lines for the complex
potential of .
The curves in continuous lines represent the
isotherms and the dotted lines represent the curves of
heat flow.
4.2 Signal Response with sinusoidal variation
from a first order system
Next consider the case of a thermometer submerged
in a water bath which varies with a sinusoidal signal
between , with a period of And which
has a constant of time of
A guideline is presented to students in order to
look into the characteristics of the system:
a) Which is the law that rules the behaviour of the
system when a sinusoidal perturbation is present?
b) What does each term of the solution represent?
c) Which are the characteristics of the system?
d) Which is the highest output answer of the system
after applying the effect of the sinusoidal
function
e) Which is the function of the output signal of the
system?
f) Draw the curve which represents the system
output. Identify each component associated with
the graphic of the curve.
The students develop the given model from the
function of transference for a first order system which
has the equation (17)
(17)
Where, : function of transference, : time
constant and : transformed variable of the time
variable .
When the system is excited with a sinusoidal
signal, the prompting power of the input system
is:
(18)
Given, : the angular frequency, and if it’s
simbolized with to the period,
(19)
The Laplace Transform of the equation (18) is :
(20)
The transformed output signal of the system
(21)
(22)
In order to find the inverse Laplace Transform of
equation (22).
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Eduardo A. Gago, Lucas I. D’alessandro, Marcelo M. Zurbriggen
E-ISSN: 2732-9984
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They apply the convulsion, where the symbol *
doesn’t represent the product of algebraic
expressions but the product of the convulsion [8]
(23)
(24)
(25)
The solution of the integral (25) is the one
expressed in equation (26)
(26)
From where .
In equation (26) in the second member, the first
member represents the transitory answer from the
system because as grows, the value of that term
goes down until it finally disappears, the second term
represents the permanent answer of the system which
is also a sinusoidal function of narrower amplitude
and moved away angle as regards original value
[9].
The law of the function which represents the input
signal, according to the given data to students in the
original problem is
(27)
The output signal of the final temperature
informed by students is:
(28)
from where ; and
Fig.6 Transitory answer to the system
Students make the graphic of Fig.6 with the
transitory answer of the system, probably called this
way because its action extinguishes in a short period
of time, although its impact is never null, with a
negligible time value of about 70s.
The maximum amplitude of the temperature
reading is: . Students can observe
the value of the maximum temperature does not
match with the value of the maximum temperature of
the fluid as they are mismatched in .
Fig.7 Permanent answer to the system
Fig.7 shows the permanent answer to the system,
which results in a sinusoidal function of less
amplitude and mismatched by an angle as regards
the original value. It is called permanent because it is
the one which stands out in the final values of the
temperature.
Fig.8 Overall answer to the system
Fig.8 shows the graphic of the temperature in the
function of time and in both components (or the
graphics of both terms in equation (28). After ,
the total answer has the same value on images as the
permanent answer.
From the analysis of the equations (26) y (28)
students conclude:
The lower is the time constant of the termometer
(), more and get closer to the values of
maximum amplitude and less is the mismatch in
time.
If the period () decreases, the angular
frequency () grows, and a fluctuation between
minimum and maximum values in the graphic
can be observed.
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Eduardo A. Gago, Lucas I. D’alessandro, Marcelo M. Zurbriggen
E-ISSN: 2732-9984
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Volume 2, 2022
As a consequence, the thermometer has a very
low amplitude as an answer (tending to ) but
with the same frequency as the input signal.
On the other hand, the mismatch grows. It sould
be considered that the transitory period is not
significant, except for the overall answer in the
first cycle, and then that period is no longer
considered.
4 Conclusion
The most effect way to face the methodological
challenges when teaching Advanced Mathematics in
engineering is to design a curricular map that can
consider a multidisciplinary perspective in teaching.
The experience enriches as the traditional
classroom is turned into a workshop – laboratory
where the student can integrate the knowledge of
Functions of complex variables and the Laplace
Transform with dynamic models into real models.
The learning environment generated fosters
motivation when there is interaction between
computational mathematics and applied physics.
Applying this type of experiences and adopting it
as a teaching match approach has an impact on the
students’ acquisition of skills. The purpose is to
emulate the engineering labour environment.
With the use of virtual environments as a means
to facilitate knowledge acquisition and improve
education efficiency through the development of an
adequate language, collaborative tasks to solve wide
and complex problematic situations where research
work is central, respecting students’ learning time
and fostering students’ teamwork and active
participation.
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DESIGN, CONSTRUCTION, MAINTENANCE
DOI: 10.37394/232022.2022.2.23
Eduardo A. Gago, Lucas I. D’alessandro, Marcelo M. Zurbriggen
E-ISSN: 2732-9984
179
Volume 2, 2022
