The Influence of linear Heating on Free Convection in a Cylindrical
Enclosure
AKRAM MAZGAR1,2, BEN NEJMA FAYCAL2
1Ionized and Reactive Media Studies Laboratory,
Preparatory Institute for Engineering Studies of Monastir,
University of Monastir,
Ibn Eljazar Street 5019 Monastir,
TUNISIA
2Higher Institute of Applied Sciences and Technology,
University of Sousse,
Taher Ben Achour Street 4003 Sousse,
TUNISIA
Abstract: - The current study aims to numerically investigate free convection airflow within a horizontal
cylinder with a linearly heated side wall. The computation of heat transfer and fluid flow structure has been
carried out using the finite element software COMSOL Multiphysics. The influence of the heat source position
on fluid flow and heat transfer is inspected. Special attention is paid to the effect of Rayleigh number and the
heater position on energy efficiency within the cavity. The results indicate that the best heat transfer
performance is achieved for low Rayleigh numbers and when the active wall is centered in the vicinity of 90°.
Key-Words: - Free convection, linear heating, cylindrical enclosure, energy efficiency, COMSOL Multiphysics,
heat transfer.
Received: May 16, 2023. Revised: October 17, 2023. Accepted: December 8, 2023. Published: December 31, 2023.
1 Introduction
The rise in popularity of heat transfer induced by
free convection has been notable in recent times,
primarily because of its autonomy from external
energy sources, [1], [2], [3], [4]. Commonly referred
to as the buoyancy effect, natural convection is
employed to improve heat transfer in diverse
engineering fields. This includes applications in
building insulation, food storage processes, cooling
of metals and electronic components, and various
other domains.
The free convection challenge within an
enclosure encompasses the boundary layer flow near
the boundaries and the internal flow region beyond
the boundary layer. Consequently, the heat transfer
from the cavity wall is shaped by both the specified
boundary conditions and the thermophysical
characteristics of the working fluid within the
cavity. In many previous studies, a common
methodology involves uniformly heating one wall of
the cavity while uniformly cooling the opposite
wall. This scenario, known as the conventional
natural convection problem, typically employs air as
the working fluid. The interplay of temperature
distribution and fluid flow significantly affects heat
transfer induced by free convection in enclosures.
Consequently, the defined boundary conditions at
the control volume walls have a substantial impact
on the heat transfer from the cavity. In the majority
of instances, the cavity experiences uniform heating
or cooling from the wall, [5], [6], [7], [8], [9].
Employing such uniform heating can produce
reasonably accurate results when calculating the
thermal performance of numerous configurations of
interest. However, practical scenarios often arise
where the impact of non-uniform wall heating must
be considered, [10]. This type of process exists in
several industrial applications, such as addressing
specific heating needs, minimizing undesirable
temperature variations, and using heat transfer at
different temperatures in different areas of the
installation, etc. An example of specific burners can
be cited, used for heat treatment applications or for
non-uniform heating zones to meet particular
requirements. This phenomenon frequently
influences thermal performance in various industrial
applications. This is why many researchers have
directed their attention toward understanding the
influence of non-uniform heating on heat transfer
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phenomena and fluid flow properties. [11],
examined the impact of both uniform and non-
uniform heating on free convection within a
trapezoidal enclosure. Their findings indicated that
introducing non-uniform heating to the bottom wall
leads to an enhancement of the rate of heat transfer
at the midpoint of the bottom wall when compared
to the scenario with uniform heating, a trend
observed across all Rayleigh numbers. However, it's
noteworthy that the mean Nusselt number
consistently suggests a generally reduced heat
transfer rate for the case with non-uniform heating.
In their study [12], the authors utilized heatline
visualization techniques to examine the heat and
fluid flow in an inclined cavity subject to non-
uniform heating and filled with CuO nanofluids.
Their results suggested that non-uniform hot wall
temperatures induce the formation of small closed
cells at the top of the hot wall, particularly for
uninclined cavities. In a distinct investigation [13],
delved into the influence of both the height and
location of an obstacle on free convection within a
square cavity. In this setup, the cavity experienced
linear heating from the side walls, uniform cooling
from the bottom wall, and insulation on the top wall.
The study uncovered that when the obstacle is
located on the lower wall, this not only induces the
formation of a strong circulation cell but also a
weaker one within the cavity. Moreover, the
obstacle, characterized by high thermal
conductivity, led to a reduction in heat exchanges
with the lower wall and an increase in heat
exchanges with the side walls. In their study [14],
inspected heat transfer induced by free convection
in a square differentially-heated cavity, where
variations in the temperature of the left wall either
increased or decreased linearly along the wall, along
with similar variations in the right wall temperature.
Particularly noteworthy is the outcome observed
when the right wall's temperature rises linearly and
the left wall's temperature diminishes linearly,
resulting in the formation of two flow cells within
the enclosure, one at the bottom and the other at the
top. Additionally, [15], demonstrated the impact of
the linear position of heating on entropy creation
due to free convection in a square cavity. Their
findings highlighted that the position of the linear
heating center significantly influences both heat
transfer and entropy creation resulting from natural
convection. The study indicates that when the linear
heating center is elevated, the fluid flow shifts to the
right, accompanied by a concurrent reduction in its
intensity. However, there is a substantial increase in
heat transfer under these conditions. In [16], the
authors examined horizontal convection in a
rectangular cavity with linear temperature
distribution. Their findings revealed that the
examination of mean flows indicated a decrease in
the kinetic and thermal boundary layer thickness, as
well as the average temperature in the bulk region,
as the Rayleigh number increased. [17], conducted a
numerical investigation into the process of non-
uniform wall heating of a square cavity and the
resulting free convection. The study concluded that
the optimal heat transfer rate is achieved when the
heating occurs near the top wall of the heated
boundary. [18], numerically investigated free
convection in a porous cube under non-uniform
heating. They mentioned that the presence of the
porous layer causes the enhancement of heat
exchanges. [19], studied the influence of non-
uniform heating on velocity and temperature
distributions of free convection within a vertical
duct of a prismatic modular reactor core. They
concluded the temperature differences between the
wall and the fluid, along with axial variations in
velocity distributions reveal flow instabilities at the
top wall of the channel. [20], conducted
experimental research on the heat exchange
performance of high-temperature heat pipe exposed
to axial non-uniform heat flux. Their findings
indicated that the most promising performance of
heat transfer and temperature uniformity are
detected if the heating power is adjusted to
1133.7 W. Recently, [21], reported the 2D flow
micropolar nanofluid in mixed convection with
unsteady circumstances and non-uniform heat
source/sink. They found that the variables related to
unsteadiness and magnetism tend to increase the
motion of the fluid.
In summary, various boundary conditions such as
uniform, non-uniform (sinusoidal), or linear heating,
have been specified for control volume walls in the
literature. However, the effects of linear heating on
free convection have not been adequately studied,
even less for cylindrical geometry. The main goal of
the current research is to explore the influence of the
center of the heater location on free convection in a
cylindrical enclosure and determine the optimal
level of energy efficiency.
2 Problem Description
The present article model refers to a cylindrical wall
containing air (Pr = 0.71) and exposed to linear
heating. COMSOL Multiphysics software is
applied to model the comportment of flow field and
heat exchange processes.
In this study, the defined dimensionless parameters
are given as follows:
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A linearly heated boundary condition is
externally imposed along the wall:
where
Figure 1 shows the linear temperature
distributions on the walls for .The red line
indicates the boundary between the region where the
imposed dimensionless temperature is higher or
lower than 0.5. Note that heat exchange process
occurs from the fluid to the wall even in areas where
and conversely, the wall heats the fluid in
certain areas even for .
Fig. 1: Linear temperature distribution;
After the variable substitution, the governing
dimensionless equations are expressed as the
following:
(1)
(2)
(3)
(4)
Certainly, taking into account that the local
Nusselt number at the heated wall serves as a crucial
metric for characterizing heat exchange between the
gas and the walls. It is formulated in accordance
with equation (5).
(5)
Furthermore, the calculation of mean Nusselt
numbers for the heated wall is conducted to obtain a
comprehensive understanding of heat transfer
throughout the enclosure. This provides an
overarching perspective on the overall heat transfer
characteristics within the system. It is expressed as
follows:
(6)
or quite simply:
(7)
In addition, the efficiency of energy utilization
in the current process is defined to provide insight
into the minimum energy required to sustain a
constant mean temperature. This measurement acts
as a valuable indicator of the heating effectiveness
of the system in maintaining thermal stability while
minimizing energy. The corresponding expression is
given in equation (8).
(8)
The computation of the mean dimensionless
temperature and velocity, as outlined in Eq. (9), is
undertaken to enhance the understanding of fluid
flow and heat exchange processes. These averages
are expressed as follows.
(9)
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3 Numerical Procedure
Please note that solving the partial differential
equations in the COMSOL modeling environment is
accomplished by employing an implicit scheme,
utilizing the damping Newton technique within an
adaptive automatic triangular grid. Notably, a mesh
refinement strategy is implemented in proximity to
boundaries, as illustrated in Figure 2. The mesh
configuration comprises 21,014 domain elements
and 472 boundary elements.
As evident from Table 1, the use of an
automatic normal grid appears to be sufficient, but
in practice and for more precision, we opted for an
automatic extremely coarse grid and the error does
not exceed 8.10-3 %.
Fig. 2: COMSOL grid resolution
Table 1. Grid effect (0=0)
Solver automatic grid
Nu (Ra=104)
Nu (Ra=105)
Extremely coarse
2.0167
3.7982
Extra coarse
2.0166
3.7985
Coarse
2.0163
3.8006
Fine
2.0192
3.7990
Normal
2.0177
3.7892
4 Problem Solution
The calculations were conducted for a Prandtl
number (Pr) of 0.71 within the range of 104 ≤ Ra ≤
106. The obtained results are presented in Figure 3,
Figure 4, Figure 5, Figure 6, Figure 7, Figure 8,
Figure 9 and Figure 10, aiming to provide a
comprehensive analysis of the impact of linear
heating devices on fluid flow and heat exchange
processes. For instance, Figure 3, Figure 4, Figure 5,
Figure 6 and Figure 7 depict the local fields of
Nusselt number, velocity, and temperature for
various locations of the heat source center at Ra =
105. Upon initial observation, the Nusselt numbers
exhibit similar profiles regardless of the heater's
location but deviate from the expected flattened to
elongated profile. Notably, the specific profile of the
Nusselt number is of interest when the heater is
centered at 90°, prompting consideration of thermal
conduction between the heated and cooled walls.
Returning to Figure 3, which illustrates the local
distributions when the heater is centered at -90°, it
should be noted that reference must be made to the
fact that the flow velocity exhibits an asymmetric
profile despite the use of symmetry conditions. The
corresponding fields show a single rotating and
elongated convection cell spanning the entire
domain. Also, it should be mentioned that the
temperature profile given in Figure 3c displays a
zone of average temperature in the middle of the
cavity.
Figure 4 depicts the local distributions when the
center of the heater is located at -45°. At first blush,
we can signal that the Nusselt number profile is
more elongated, displaying a peak heat exchange
located where the temperature is extreme. The fluid
flow is visibly accelerated resulting in a
multicellular flow with a practically symmetrical
profile, following circular streamlines and
displaying higher velocities. The center of the cavity
seems to be in slow motion, featuring multiple cells.
What is noteworthy in Figure 5 when the heater is
centered at 0°, is that the Nusselt number
distributions are more developed and the flow
patterns display an asymmetric profile.
The local distributions of Figure 6 mention that
the Nusselt number profiles are less elongated and
the fluid flow exhibits a central zone containing a
single cell at the center of the cavity with two
discrete foci located at the extremities of the cell.
Additionally, it is essential to indicate the formation
of elliptical streamlines with the existence of a
bicellular flow in the middle of the cavity. It is
noteworthy also that the central multicellular zone
tends toward an elliptical shape as the center of the
source approaches 90°.
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(a)
(b)
(c)
Fig. 3: Local distributions; Ra = 105,
(a) Nusselt number; (b) dimensionless velocity; (c)
dimensionless temperature
(a)
(b)
(c)
Fig. 4: Local distributions; Ra = 105,
(a) Nusselt number; (b) dimensionless velocity; (c)
dimensionless temperature
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(a)
(b)
(c)
Fig. 5: Local distributions; Ra = 105,
(a) Nusselt number; (b) dimensionless velocity; (c)
dimensionless temperature
(a)
(b)
(c)
Fig. 6: Local distributions; Ra = 105,
(a) Nusselt number; (b) dimensionless velocity; (c)
dimensionless temperature
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(a)
(b)
(c)
Fig. 7: Local distributions; Ra = 105,
(a) Nusselt number; (b) dimensionless velocity; (c)
dimensionless temperature
Figure 7 depicts the local distributions when the
heater is centered at 90°. Firstly, it is essential to
acknowledge the existence of reduced profiles of the
Nusselt number indicating low heat exchanges
between the fluid and the wall. Moreover, it is
important to point out the appearance of a bi-zone
flow with the development of four symmetrical and
elliptical counter-rotating convection cells
exhibiting eccentric foci and displaying quasi-
stratified flow characteristics.
The impact of the heater center and Rayleigh
number on average distributions are exposed in
Figure 8, Figure 9 and Figure 10. At first blush and
according to Figure 8, the heat exchange process is
intensified in case the source is centered in the
vicinity of 0° and more precisely for high Rayleigh
numbers. We also observe a visible change in the
flow structure for a heater centered close to -50°. A
major transition appears from a single-cell flow to a
multicellular flow regime. Note also the visible
decreasing profiles of Nusselt numbers for heat
sources centered at values greater than 0°. It is also
worth mentioning that no significant variations
occur in the profile of average Nusselt numbers at
low Rayleigh numbers.
Let's not forget to mention that the
dimensionless average temperature remains uniform
regardless of the heater position and the Rayleigh
number at the active wall (Θa = 0.5).
The profiles of mean dimensionless velocity
shown in Figure 9 depict an initial increasing phase
for a source centered in [-90°,-70°], attaining an
absolute maximum close to -70°. This is
accompanied by a subsequent phase between -70°
and -40°, indicating a transition in the flow structure
from a single-cell flow to a multicellular flow and
leading to a strong decrease in average velocity. The
final phase between -40° and 90° displays
decreasing profiles, which becomes quasi-linear for
low Rayleigh numbers. It is interesting to mention
the emergence of quasi-stratified flow
characteristics in the vicinity of 90°.
Figure 10 depicts the energy efficiency of the
heat exchange process based on the Rayleigh
number and the center of the heater. One initial
point is that for a heater center localized between -
90° and 0°, the energy efficiency profiles of the heat
exchange process are practically constant.
Moreover, very strong increases are observed in the
energy efficiency profiles when the heater is
centered at [0°,90°]. It should be underlined that in
general, the profiles of energy efficiency are more
pronounced for low values of the Rayleigh number
regardless of the heating position center, except for
a source centered in the vicinity of 90°, where the
trends are reversed. Given this, the heating process
is recommended to be based on temperature step
heating along the duct.
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Fig. 8: Average Nusselt number
Fig. 9: Average Velocity
Fig. 10: Energy Efficiency
5 Conclusion
The present work describes a numerical
investigation of free convection airflow in a
horizontal heated cylinder with linear temperature
distributions on the walls. The effects of the heater's
angular position and the Rayleigh number on fluid
flow and heat transfer processes have been
numerically examined. Based on the achieved
results, it is evident to summarize that the airflow
structure is significantly influenced by the heater
location. It has been shown that convective heat
transfer is enhanced for high Rayleigh numbers and
more specifically when the active wall is centered at
0°. It is also worth mentioning that no significant
variations occur in the profile of average Nusselt
numbers at low Rayleigh numbers. Moreover, the
dimensionless average temperature remains
consistent irrespective of the Rayleigh number and
the placement of the active wall. Furthermore, the
optimum energy efficiency level is generally
attained for low Rayleigh numbers especially if the
heater center is localized at 90°. To conclude, it can
be said that the heating process is recommended to
be based on temperature step heating along the duct.
Subsequent research will focus on assessing
thermodynamic irreversibility and entropy
generation arising from heat transfer. This aims to
enhance both the performance and efficiency of our
heating process by employing entropy minimization
techniques. One can also consider the numerical
modeling of the same problem but in annular
configurations.
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0
1
2
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6
7
8
-90 -75 -60 -45 -30 -15 015 30 45 60 75 90
Nu
0 (°)
Ra=1E4
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0
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-90 -75 -60 -45 -30 -15 015 30 45 60 75 90
Ua
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ε
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Akram Mazgar prepared and wrote the
manuscript. He also contributed to the numerical
simulation of the physical problem on COMSOL
software and the verification of the overall
research outputs.
- Fayçal Ben Nejma used COMSO
MUTILPHYSICS to model the physical problem.
He also contributed to the verification of all the
results
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
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
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