Energy Efficiency Analysis of Coupled Thermal Radiation and Free
Convection within a Square Enclosure with Internal Heating
AKRAM MAZGAR, FADHILA HAJJI, FAYCAL BEN NEJMA
EMIR Laboratory, University of Monastir,
The Preparatory Institute for Engineering Studies of Monastir,
Ibn Eljazar Street, Monastir 5019,
TUNISIA
Abstract: - This study explores the simultaneous impact of thermal radiation and free convection within a
square cavity featuring internal heating. The walls are consistently held at a stable temperature through
isothermal cooling, while an internal heat source sustains a consistently higher temperature. The radiation
component is characterized by employing the FT40 discrete-ordinate approximation in conjunction with the
statistical narrow-band correlated-k method (SNBcK). The primary focus lies in discerning the influences of
radiation on both flow patterns and heat transfer. Particular emphasis is placed on investigating energy
efficiency and its correlations with key governing parameters, including the heat source temperature, wall
emissivity, and the size and placement of the heater. A noteworthy revelation from this analysis is the
substantial impact of radiation on the acceleration of vortices, leading to a homogenizing impact on temperature
distributions. Additionally, it is observed that the highest level of energy efficiency is realized by siting the
heater at the central lower section of the enclosure.
Key-Words: - Energy efficiency, Natural convection, Radiation, Heat source, SNBcK, square cavity.
Received: April 29, 2023. Revised: October 9, 2023. Accepted: November 28, 2023. Published: December 31, 2023.
1 Introduction
The effects of combined thermal radiation and
convection within enclosures have emerged as a
significant concern across various industries,
impacting applications such as heat exchangers,
boilers, thermal insulation systems, nuclear reactor
systems, burners, combustion chambers, and
metallurgical processes. Notably, enclosures
featuring internal heating have garnered substantial
attention due to their relevance in numerous heat
transfer processes and their implications in various
fields, including thermal engineering, culinary arts,
materials science, appliance safety, improvement of
industrial processes, heating device design such as
ovens, microwaves, water heaters, and other
practical applications. In-depth comprehension
enables the optimization of heat distribution, and
improvement of energy efficiency, and ensures
uniform cooking or heating. In such applications,
Heat transfer through thermal radiation and free
convection play pivotal roles in energy exchanges.
In these processes, maximizing energy efficiency is
fundamental, requiring the identification of ideal
configurations that provide the best heat transfer
performance, [1], [2]. Research in this area led to
several publications, highlighting the significance of
understanding the interaction between radiation and
convection to increase system performance. For
instance, [3] inspected the free convection flow in a
square enclosure having an internal heater. Their
findings underlined that the region where heat
transfer is focused is located in the lower region of
the heated surface area. In [4], the authors presented
a numerical simulation dealing with the interaction
of radiation with convection and conduction within
the confined volume of a square-shaped cavity
containing a heat generation source. Their results
specified that wall-to-wall heat transfer meets
resistance from the heated block. Additionally, [5],
performed a numerical investigation of combined
radiation-free convection heat transfer in a titled
enclosure that included a centrally positioned inner
source. Their findings indicated that the inclination
angle had a diminishing impact on heat transfer
within the enclosure, particularly when thermal
radiation was present. In a distinct investigation, [6],
delved into the characteristics of free convection
within a cooled square-shaped enclosure including
an isothermal cylindrical heat source. Their
comments highlighted that the development of
convection cells was significantly influenced by the
location of the heater and the Rayleigh number.
Furthermore, [7], presented the impact of the heater
size localized at the center of a square enclosure on
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fluid structure and heat transfer characteristics.
Their study reveals the specific importance of
understanding in what way the heater size marks
both heat transfer and the structure of the fluid flow.
[8], tested the influence of surface-to-surface
radiation on the disruption of natural convective
flows through a square cavity comprising a centrally
situated, heated inner body. Their results specified
that the radiative contribution raises the Rayleigh
number, principally during the initial transition to
the transitional regime. [9], used the lattice
Boltzmann approach to examine the free convection
of nanofluid around a horizontal circular cylinder
within a square cavity. They noted that, in contrast
to the Hartmann number, both mean Nusselt and
Rayleigh numbers exhibit a rise as the volume
fraction of the nanoparticle increases. [10],
investigated the impact of the heater location on
entropy creation resulting from mixed convection
within an open square enclosure. [11], studied heat
transfer, taking into account the interplay between
surface radiation and free convection in a square
cavity with a heated plate positioned at the center.
They highlighted that thermal radiation leads to
improved temperature homogenization within the
enclosure. Additionally, they demonstrated that the
contribution of convective heat transfer decreases
with emissivity when the plate is vertically located,
but increases when the plate is positioned
horizontally. [12], performed a numerical
investigation on free convection within a
differentially heated enclosure with volumetric heat
generation. Their findings indicated that a horizontal
magnetic field proved to be the most effective in
stabilizing the flow. In another numerical study,
[13], inspected the impact of thermal radiation and
internal heating on mass and heat transfer over an
upward-facing horizontal flat plate within a porous
medium containing a nanofluid. They concluded
that thermal radiation significantly enhances flow
velocity, as well as the profiles of temperature and
nanoparticle volume fraction. [14], investigated
laminar free convection in an inclined enclosure
featuring a centrally heated body. Their results
revealed that elevating the inclination angle of the
cavity leads to the suppression of heat transfer
throughout the cavity. In a numerical simulation,
[15], the unsteady characteristics of free convection
within a square enclosure with various heat source
configurations at the bottom wall were explored.
Notably, they observed that reducing the heat source
length resulted in a decrease in the heat generation
rate. In other studies, [16], [17] delved into the
analysis of generated irreversibility arising from
natural convection within a square enclosure
featuring an internal cylindrical heat source. Their
observations indicated a substantial increase in
entropy creation rates attributed to heat transfer and
fluid friction with an elevated Rayleigh number.
[18], conducted an extensive review of numerical
and experimental investigations related to free
convection phenomena in rectangular or square
cavities, both with and without an internal heat
source. Moreover, [19], explored convective-
radiative energy transport in a titled enclosure
featuring an energy-generating body. Their findings
highlighted that the influence of radiation on overall
thermal transmission becomes significant with an
increase in surface emissivity. [20], analyzed free
convection in a square enclosure containing a
circular heater subjected to nanoparticles and a
magnetic field. They concluded that the heating rate
primarily increases with a rise in the Rayleigh
number and inversely with a diminution in the
Hartmann number. In [21], the authors studied a
numerical investigation on free convection
combined with surface radiation within a square
enclosure filled with air and including two heat
generating elements. They showed the influence of
the conductivity ratio, the Rayleigh number, and the
emissivity on the heating process and fluid flow
structure. In a three-dimensional numerical
investigation, [22], explored entropy creation under
thermal convection in a cubic cavity comprising a
heater source. They revealed that their findings
carry significant implications for the optimization
and design of heat transfer processes. Recently,
[23], explored the impact of a heat source on hybrid
nanofluids using the heatline approach. Their
findings indicated direct proportionality between the
heater length and the Nusselt number. On a related
note, [24], conducted an analytical study on steady
free convection in a vertical rectangular enclosure
resulting from uniform volumetric heating. The
study incorporated two pairs of heat sources/sink
within a vertical rectangular enclosure. Their results
demonstrated a notable agreement in terms of
isotherms, streamlines, and temperature and velocity
distributions.
An examination of the chosen bibliography
highlights that there exist various industrial setups
where the impact of volumetric radiation on fluid
flow and heat transfer has not been sufficiently
explored. The primary objective of this paper is to
conduct a numerical investigation into the impact of
volumetric radiative heat transfer on laminar free
convection within an isothermally heated square
enclosure featuring an internal heat source.
Additionally, particular emphasis is placed on
analyzing energy efficiency to delineate the optimal
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configuration that yields the most effective heat
transfer performance.
2 Problem Formulation
This study focuses on a 2D square enclosure as the
physical model, incorporating an internal local heat
source and diffusive grey radiating walls. The
selected working fluid for the study is superheated
steam, identified as a non-grey medium with
emitting-absorbing properties and devoid of
scattering, as illustrated in Figure 1. Note that the
fluid motion is presumed to be steady, laminar, and
compressible. Moreover, the walls of the enclosure
are kept at a consistently low temperature (Tc),
while the internal heater operates at an elevated
temperature (Th). To scrutinize the characteristics of
fluid flow and heating process, we utilized a
numerical computation with COMSOL
Multiphysics, concurrently conducting an iterative
coupled simulation through Matlab software.
Fig. 1: Geometry of the enclosure with an internal
heater
Considering the specified assumptions detailed
above, the governing equations are provided as
follows:
(1)
(2)
where
denotes the force of buoyancy, it is given
by Archimedes' principle and can be expressed as:
(3)
(4)
Certainly, the radiative source term given in
Equation (4) symbolizes the radiation contribution
within the medium.
For the convection governing equations, the
pertinent boundary conditions are specified as the
following:
On cavity walls: T = Tc
On heater walls: T = Th
On all walls: ux = uy = 0
Assuming isothermal walls, the boundary conditions
for thermal radiation are specified as follows:
(5)
The radiation code is developed based on the
FT40 discrete-ordinate approximation. This
technique entails discretizing the angular space into
a finite set of ordinates or directions. The radiative
transfer equation (RTE) is resolved numerically for
each direction.
Furthermore, the SNBcK model allows
determining the radiative characteristics of
superheated steam. This method is based on the
extraction of the gas spectral absorption coefficients
from the corresponding transmissivities.
Please note that the radiation code is established
based on the FT40 discrete-ordinate approximation.
Therefore, the RTE can be expressed according to
equation (6):
(6)
It is worth emphasizing that the SNBcK
approach is associated with the 4-point Gauss-
Legendre quadrature to compute the radiative source
term which is expressed following Eq. (7), [25],
while the net radiative flux is given in Eq. (8), [26],
[27].
(7)
(8)
Note that COMSOL Multiphysics allows
delivering the thermo-physical properties of the
working fluid supposed to be an ideal gas. The
application of this model within the temperature
range of 380 to 850°C ensures that the simulated
physical properties align with the expected behavior
of the working fluid within this specific temperature
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interval. Moreover, a linear extrapolation was used
to estimate values outside the given range.
In order to enhance comprehension of the heat
exchange at different points on the walls, which is
essential for estimating the overall heating process
in the system, we give the formulas of the local
convection and radiation Nusselt numbers at walls,
respectively:
(9)
(10)
Certainly, computing the mean Nusselt number
for each of the eight surfaces involves averaging the
local Nusselt.
(11)
In addition, the average temperature and average
velocity can be calculated as follows:
(12)
To make the heating process more effective, we
defined the corresponding energy efficiency as the
ratio between the temperature increase resulting
from heat transfer and the necessary heat flux to
achieve this temperature rise. It is expressed as
follows:
(13)
3 Numerical Procedure and
Validation
Let's reiterate that we used a combination of Matlab
and COMSOL Multiphysics for simulating heat
transfer and fluid flow. In fact, COMSOL
Multiphysics® employs finite element analysis as a
general computational method for resolving the
governing equations with Matlab handling
parameter initialization and storage.
(a) (b)
Fig. 2: The computation mesh (a) inside the cavity;
(b) in vicinities of walls
Tc=400K; Th=800K; Ly=Lx=0.1m; Lyy= Lxx =0.025m;
εc=εh=1; P=1atm, Pyy=Pxx= 0.0375m
The use of an adaptive mesh in COMSOL
Multiphysics®, along with the information on the
total number of mesh elements and boundary
elements represents a dynamic feature that refines
the mesh in specific zones where higher resolution
is needed (Figure 2). The total number of mesh
elements is 25,386 whereas the number of boundary
elements is 852.
Note that the validation of the COMSOL code
was achieved by using the Rayleigh-Bénard
convection within a square cavity of an “L” edge
with two insulating vertical surfaces. The upper wall
is maintained at a consistently lower temperature
(Tc), while the bottom one is consistently held at a
higher temperature (Th). In the spirit of validating
our convection model, we established the local
distributions of the Nusselt number at the active
wall, as shown in Figure 3. In fact, the comparison
of our results and those of the article, [28], shows
good agreement with an error rate not exceeding
2%. This proves that our code is providing accurate
and reliable simulations for the Rayleigh-Bénard
convection case.
For the purpose of validating our radiative
model, we conducted some computations to
juxtapose outcomes derived from the SNBcK-based
band model with data accessible in the existing
literature. To achieve this, a physical model was
employed, featuring a rectangular enclosure with
dimensions of 1 m × 0.5 m, housing water vapor at a
temperature of 1000 K, and enclosed by black walls
maintained at a temperature of 0 K. The
distributions of the radiative source term given in
Figure 4, compare our current findings using the S8
quadrature and those documented by [29], using the
T7 quadrature. The findings reveal a significant
alignment between the two investigations and the
error perceived does not exceed 3%, indicating a
high level of concordance.
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Moreover, as indicated in Table 1, employing a
spaced grid with dimensions 40*40 demonstrates a
judicious compromise to ensure precise accuracy
calculations.
Furthermore, the local distributions of the
radiative heat flux kept at the upper enclosure wall
were established to determine the suitable
quadrature model, as illustrated in Figure 5. It is
noteworthy to mention the distortions present in the
respective trends due to the ray effect resulting from
the selected angular discretization approach. The
decision was made in favor of employing the FT40
with a substantial number of directions to mitigate
the influence of the ray effect, [30].
Fig. 3: Validation of the convective model
Th=800K; Tc=400; P=1atm; Ly = Lx = 0.1m
Fig. 4: Validation of the radiative model
T0=1000K; TW = 0; P0 =1atm; εh=εc=1; Lx = 1m; Ly
=0.5m
Table 1. Grid sensitivity test (case of pure radiation)
Th=800K; Tc=400K; P=1atm; εc=εh=1; Ly=Lx=0.1m;
Lyy=Lxx=0.025m; Pyy =Pxx=0.0375m
Fig. 5: Quadrature effect
Th=800K; Tc=400K; P=1atm; εc=εh=1; Ly=Lx=0.1m;
Lyy=Lxx=0.025m; Pyy =Pxx=0.0375m
4 Problem Solution
The primary characteristics of thermal radiation's
impact on the heating process and convective flow
within a square enclosure containing an internal heat
source are delineated in Figure 6, Figure 7, Figure 8,
Figure 9, Figure 10, Figure 11, Figure 12, Figure 13
and Figure 14. Examining the local velocity
distributions depicted in Figure 6(a), the flow
pattern reveals two symmetrical counter-rotating
convective cells distributed throughout the entire
domain. Notably, the buoyant thermal plume
extends over the upper boundary of the enclosure,
with the maximum flow velocity attained near this
zone and near the upper corners of the heater.
Moreover, the boundary layer beneath the heat
source experiences destabilization due to vortices
generated above the heater, leading to thermal
stratification in the region between the heater and
the bottom surface of the enclosure, resulting in low
fluid velocities. It is crucial to highlight that thermal
radiation significantly contributes to heat transfer
and flow characteristics, as illustrated in Figure
6(b). The presence of radiation amplifies
temperature profiles, resulting in more developed
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and deformed cells that occupy a larger portion of
the cavity space. Additionally, water vapor particles
exhibit noticeable acceleration near the upper wall
of the enclosure. Furthermore, local variations in the
source term resulting from the radiative contribution
are presented in Figure 6(c). It is crucial to
underscore that regions with a negative radiative
source term are located in proximity to the upper
wall of the enclosure and above the heat source,
where the hottest particles are concentrated.
(a)
(b)
(c)
Fig. 6: Local distributions without radiation (left);
with radiation (right)
(a) velocity fields (m/s); (b) temperature fields (K);
(c) radiative source term (kWm-3);
Th=800K; Tc=400K; P=1atm; εc=εh=1; Ly=Lx=0.1m;
Lyy=Lxx=0.025m; Pyy =Pxx=0.0375m
4.1 Impact of the Heater Temperature
The heater temperature effects on average
distributions are depicted in Figure 7. It is
noteworthy that, independently of the radiative
impact, the average temperature exhibits a nearly
linear profile. Furthermore, an increase in the heater
temperature significantly elevates the mean values
of the gas temperature and the flow rate.
Conversely, even at relatively lower heat source
temperatures, it can be inferred that volumetric
radiation contributes to achieving a more uniform
temperature, causing a substantial change in flow
despite the mean temperature remaining almost
constant. Figure 7(c) illustrates the distribution of
mean convection Nusselt numbers along the cavity
boundaries based on the heater temperature. It
should be observed that in the absence of radiation,
the convection Nusselt number along the bottom
surface of the cavity maintains a nearly constant
profile, approaching practically null values. This is
attributed in part to the retention and immobilization
of fluid particles beneath the heating element.
Moreover, the mean convective Nusselt numbers of
the upper and side walls of the enclosure exhibit
increasing trends, prominently visible in the
profile. The buoyancy force escalates with an
increase in the temperature of the heating element,
resulting in significant temperature gradients near
the upper surface. This, in turn, enhances convective
heat exchange between this wall and the gas.
Additionally, when considering the influence of
radiation, the average convective Nusselt number
along the upper surface remains nearly constant,
whereas those computed at the bottom and side
walls distinctly show substantial and amplified
profiles. In this scenario, fluid particles near the
bottom and side walls experience acceleration,
thereby improving heat transfer between these
surfaces. Figure 7(d) illustrates the influence of the
heater temperature on average convection Nusselt
numbers along the heat source walls. It is
noteworthy that an elevation in the heat source
temperature results in a reduction in the
corresponding Nusselt numbers, displaying a quasi-
linear profile for those computed at the upper wall.
Furthermore, the radiative contribution
demonstrates no significant impact on convection
heat exchange near the heater walls, presenting a
slight amplification in the profile of the Nusselt
number along the upper wall and a minor
attenuation in those computed at the other walls.
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(a)
(b)
(c)
(d)
(e)
(f)
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(g)
Fig. 7: The influence of the heater temperature on
average distributions
Tc=400K; P=1atm; εc=εh=1; Ly=Lx=0.1m;
Lyy=Lxx=0.025m; Pyy =Pxx=0.0375m
The data presented in Figure 7(e) and Figure
7(f) delineate the distribution of mean radiation
Nusselt numbers at each wall based on the
temperature of the heater. Notably, the
corresponding profiles exhibit increasing trends
characterized by quasi-exponential curves. In Figure
7(g), the impact of the heater temperature on energy
efficiency is demonstrated. It is evident that raising
the temperature of the heater diminishes energy
efficiency. In such conditions, a notable energy loss
is observed through surface-to-surface exchanges
that do not contribute to the temperature rise of the
gas.
4.2 Influence of the Heater Emissivity
Figure 8 depicts the emissivity impact of the
enclosure walls on average distributions. It is
noteworthy that an increase in emissivity results in
higher heat losses due to radiation at the enclosure
surfaces and reduces their reflectivity, consequently
decreasing fluid-fluid heat exchange in favor of
fluid-wall heat transfer. As depicted in Figure 8(a)
and Figure 8(b), lower wall emissivity leads to an
increased reflection of thermal radiation originating
from the heat source. This enhancement in reflection
creates additional chances for heating the gas. In the
case of a perfectly emissive wall, this translates to
the absorption of radiation from the heater upon first
contact with the surface. Conversely, when dealing
with a perfectly reflective wall, radiation reflects
until it reaches the heating element, facilitating heat
exchange and consequently elevating the average
temperature of the gas, while the mean velocity
remains relatively constant. Another noteworthy
point highlighted in Figure 8(c) and Figure 8(d) is
that elevating the emissivity of the cavity surfaces
increases the mean convection Nusselt numbers
along the heater boundaries compared to those
calculated along the cavity surfaces. Furthermore,
increasing the emissivity of the cold walls reduces
the radiation emitted from the heater, thus
enhancing heat transfer between the heater and the
gas. However, this results in reduced temperature
gradients at the cold surfaces, leading to a decrease
in mean convection Nusselt numbers along the
cavity surfaces. Conversely, increasing the
emissivity of the enclosure walls amplifies
temperature gradients near the heat source surfaces,
consequently raising the corresponding mean
convection Nusselt numbers. Figure 8(e) and Figure
8(f) show variations of the mean radiation Nusselt
numbers based on the emissivity of the enclosure
walls. As anticipated, the corresponding Nusselt
numbers increase significantly with higher
emissivity of the cavity walls, attributed to a
sustained increase in radiative heat flux. In Figure
8(g), an increasing profile of energy efficiency is
presented according to wall emissivity. Decreasing
the emissivity of the cavity walls reduces heat
transfer between surfaces, thereby enhancing gas-
gas heat transfer at the expense of heat transfer
between surfaces.
4.3 Effect of the Heater Size
The influence of the heater aspect ratio on the flow
structure is depicted in Figure 9, showcasing the
variation of the heater length (Lyy) while
maintaining the corresponding width (Lxx).
Reducing the heater size increases the flow rate
within the plume while preserving its structure.
Turning to the temperature distribution presented in
Figure 10, the plume appears thinner and more
confined above the heater, exhibiting a noticeable
increase in cold zones below the heater.
(a)
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(b)
(c)
(d)
(e)
(f)
(g)
Fig. 8: Impact of the wall emissivity on the mean
distributions
Th=800K; Tc=400K; P=1atm; εh=1; Ly=Lx=0.1m;
Lyy=Lxx=0.025m; Pyy =Pxx=0.0375m
It is noteworthy to mention a decrease in the
mean temperature of the fluid particles situated
above the heater due to the reduction in heat transfer
between surfaces resulting from the smaller heater
size. Additionally, a notable increase in the plume
width is observed under the influence of the
radiative contribution, particularly in areas below
the heater that are primarily involved in heat
exchange. This is anticipated given the downward
extension of the cells, but it also leads to an
augmentation in fluid particles trapped below the
heat source.
Figure 11(a) and Figure 11(b) demonstrate the
influence of the heater size on the mean
distributions of gas temperature and velocity,
respectively. It is essential to note that there is
practically no impact of the source size on average
velocity, whereas the mean temperature displays a
quasi-linear and increasing profile. Furthermore, it
should be observed that the radiation contribution
leads to significantly higher temperature and
velocity distributions. In Figure 11(c), the evolution
of mean convection Nusselt numbers at the cavity
surfaces is presented based on the heater aspect
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ratio. Notably, the corresponding profiles are
practically unaffected by the size of the heater,
while the radiative contribution results in a slight
amplification in the Nusselt numbers along the
lower and side walls of the cavity. The impacts of
the source aspect ratio on the average distributions
of convection Nusselt numbers along the heater
walls are depicted in Figure 11(d). It is evident that
the corresponding trends reveal decreasing profiles,
and more specifically, the
trends show
quasi-linear evolutions. It must be emphasized,
however, that the
values approach those of
the
as the upper wall of the enclosure is close
to the upper wall of the heater. Moreover, the
profiles of convection Nusselt numbers along the
horizontal walls of the heater are practically
unaffected by thermal radiation. Figure 11(e) and
Figure 11(f) display the variations in average
radiative Nusselt numbers with the heater aspect
ratio. It is noteworthy that increasing the aspect ratio
of the heater results in a rise in the average radiative
Nusselt number along the walls of the enclosure.
This can be attributed to the expansion of the heat
source volume, leading to an enlargement of the
heat-exchange surface area. Conversely, the
radiative Nusselt numbers at the heater walls exhibit
decreasing and quasi-linear trends. Additionally, it
should be mentioned that the radiative Nusselt
numbers at the heater sidewalls
approach
those along the bottom heater wall
as the
upper wall of the heater is positioned away from the
upper surface of the enclosure. Figure 11(g)
illustrates the evolution of the energy efficiency of
the heating process with the heater size. It is
noteworthy that despite the small variation in energy
efficiency, opting for a square-shaped heater
(AR=1) appears to be the most disadvantaged
geometry, yielding the lowest heat transfer
performance.
(a) AR=1.2
(b) AR=1
(c) AR=0.4
Fig. 9: Impact of the source size on the distributions
of local velocity (m/s); without radiation (left); with
radiation (right)
Tc=400K; Th=800K; P=1atm; εc= εh=1;
Lx=Ly=0.1m; Lxx=0.025m; Pxx =Pyy=0.0375m
4.4 Effect of the Source Location
The detailed analysis of the source location's effect
on the behavior of flow structure and convective-
radiative heating is presented in Figure 12 and
Figure 13, respectively. Notably, the flow structure
and the formation of fluid vortices are greatly
influenced by the location of the heating element. In
configuration "c", where the heater is positioned in
the center of the enclosure, one can observe the
formation of two symmetrical and counter-rotating
convection cells close to the upper cavity wall. This
change in the flow structure is accompanied by the
development of a thermal plume originating from
the heater.
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(a) AR=1.2
(b) AR=1
(c) AR=0.4
Fig. 10: Impact of the source size on the
distributions of local temperature (K); without
radiation (left); with radiation (right)
Tc=400K; Th=800K; P=1atm; εc= εh=1;
Lx=Ly=0.1m; Lxx=0.025m; Pxx =Pyy=0.0375m
Maximum velocities are localized at each top
corner of the heat source, while high velocities are
obtained above the heater and in the vicinity of the
upper surface of the enclosure. An interesting aspect
in cases where the source is situated close to the
lower-left corner of the cavity (configuration "a") is
the formation of a small recirculation cell above the
heat source and another cell encompassing the rest
of the cavity. When the source is situated close to
the upper-right corner of the cavity (configuration
"e"), the main finding is that the velocities of fluid
particles at the upper corners of the heater are lower
than those generated in the other configurations.
(a)
(b)
(c)
(d)
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(e)
(f)
(g)
Fig. 11: Impact of the heater size on average
distributions
Th=800K; Tc=400K; P=1atm; εc= εh=1;
Lx=Ly=0.1m; Lxx=0.025m; Pxx =Pyy=0.0375m
Furthermore, it should be pointed out that the
radiative contribution homogenizes the temperature
and increases fluid flow within the cavity.
Specifically, the analysis of the radiative effect
reveals significant local velocities when the heater is
situated close to the bottom wall of the enclosure.
Configuration "a"
Configuration "b"
Configuration "c"
Configuration "d"
Configuration "e"
Fig. 12: Impact of the source location on local
velocity profiles (m/s); without radiation (left); with
radiation (right)
Th=800K; Tc=400K; P=1atm; εc= εh=1;
Ly=Lx=0.1m; Lyy=Lxx=0.025m; Configuration "a"
Pxx=Pyy = 0.0125m; Configuration "b" Pxx=
0.0375m; Pyy=0.0125m; Configuration "c" Pxx=Pyy=
0.0375m; Configuration "d" Pxx=0.0625m,
Pyy=0.0375m; Configuration "e" Pxx=Pyy=0.0625m.
Figure 14(a) and Figure 14(b) depict the
variations in average temperature and velocity based
on the heat source location. It is noteworthy that,
independently of the radiative impact, the highest
values of the mean temperature and velocity are
attained when the source is situated at the bottom of
the cavity. Moreover, radiation serves to
homogenize the temperature fields within the cavity,
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thereby increasing both mean velocity and
temperature, with the exception of configuration "a"
where the mean temperature in the absence of
radiation is slightly higher than that computed with
radiation.
Following Figure 14(c) and Figure 14(d), the
higher values of
are attained when the
source is positioned close to the lower-left corner of
the cavity (configuration "a"). Furthermore, it is
noteworthy that the mean convection Nusselt
number along the upper wall of the cavity
reaches its maximum value when the heater is
situated in the center of the enclosure.
This is logical since the structure and location of
the fluid vortices generate large temperature
gradients in the vicinity of the upper wall of the
cavity. Furthermore, the higher values of
are
reached when the heater is situated at the center of
the cavity and close to the sidewalls (configuration
"d"). Regarding the radiative influence on the
average convection Nusselt numbers along the
heater walls shown in Figure 14(e) and Figure 14(f),
it is notable that the distributions of the Nusselt
numbers computed along the sidewalls of the heater
are attenuated, except for those in configuration "b".
Figure 14(g) indicates that there is hardly any
difference between the radiation Nusselt numbers of
the cavity walls when the heater is situated at the
center of the cavity, and therefore, the
corresponding profiles are practically comparable.
On the other hand, the radiation Nusselt numbers of
the enclosure walls display inverted trends for
configurations "a" and "e". This can be explained by
the physical boundary conditions imposed on the
boundary. It should be noted that the heater location
has practically no influence on the radiative Nusselt
numbers along the heater boundaries, as shown in
Figure 14(h). The corresponding trends are
practically comparable with a slight difference due
to the radiative contribution of the gas. It is evident
from Figure 14(i) that energy efficiency is most
pronounced when the source is positioned at the
center of the lower section of the cavity
(configuration "b"). The circulation flow rate
becomes more pronounced when the heater is
situated at the bottom of the cavity. Furthermore, if
the heater is placed at the center of the enclosure,
this allows better gas circulation and heat transfer.
4.5 Effect of the Pressure
According to Table 2, decreasing the gas pressure
leads to an increase in fluid density, subsequently
causing a rise in the average temperature and
velocity of the gas. Additionally, it should be noted
that increasing the gas pressure affects the optical
characteristics of the medium by enhancing its
optical thickness. This, in turn, reduces wall-to-wall
heat transfer, resulting in lower radiation Nusselt
numbers.
Configuration "a"
Configuration "b"
Configuration "c"
Configuration "d"
Configuration "e"
Fig. 13: Impact of the source location on local
temperature profiles (K); without radiation (left);
with radiation (right)
Th=800K; Tc=400K; P=1atm; εc= εh=1;
Ly=Lx=0.1m; Lyy=Lxx=0.025m; Configuration "a"
Pxx=Pyy = 0.0125m; Configuration "b" Pxx=
0.0375m; Pyy=0.0125m; Configuration "c" Pxx=Pyy=
0.0375m; Configuration "d" Pxx=0.0625m,
Pyy=0.0375m; Configuration "e" Pxx=Pyy=0.0625m.
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(a)
(b)
(c) without radiation
(d) with radiation
(e) without radiation
(f) with radiation
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Volume 18, 2023
(g)
(h)
(i)
Fig. 14: Impact of the source location on average
distributions
Th=800K; Tc=400K; P=1atm; εc= εh=1;
Ly=Lx=0.1m; Lyy=Lxx=0.025m; Configuration "a"
Pxx=Pyy = 0.0125m; Configuration "b" Pxx=
0.0375m; Pyy=0.0125m; Configuration "c" Pxx=Pyy=
0.0375m; Configuration "d" Pxx=0.0625m,
Pyy=0.0375m; Configuration "e" Pxx=Pyy=0.0625m
Table 2. Pressure effect (Th=800K; Tc=400K; εc=
εh=1; Ly=Lx=0.1m; Lyy=Lxx=0.025m; Pyy
=Pxx=0.0375m)
5 Conclusion
Maximizing energy efficiency in thermal processes
is crucial, necessitating the identification of the most
effective setup for realizing optimal heat transfer
performance. The present study delves into the
impact of a heat-generating body on coupled
thermal radiation and free convection in a square
enclosure. The FT40 discrete-ordinate
approximation, coupled with the (SNBcK) model, is
applied to assess thermal radiation heat transfer. The
findings indicate that thermal radiation significantly
influences fluid flow and heat transfer by
accelerating fluid vortices and homogenizing gas
temperature. An elevation in the source temperature
results in a considerable rise in gas temperature and
a robust acceleration in flow rate. Furthermore, with
an increase in the emissivity of the enclosure walls,
the average gas temperature is significantly reduced,
while the mean velocity profile remains almost
unchanged. The study also reveals that increasing
the heater aspect ratio enhances the mean
temperature of the medium. Additionally, the results
illustrate that maximum energy efficiency is attained
when the source is positioned at the center of the
lower section of the cavity.
Future investigations will be dedicated to the
evaluation of thermodynamic irreversibility and
entropy creation resulting from heat transfer,
thereby enhancing the efficiency and performance
of our heating process through the application of
entropy minimization techniques.
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Volume 18, 2023
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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.
- Fadhila Hajji realized the iterative simulation on
Matlab software and contributed to the numerical
simulation of the physical problem on COMSOL
software.
- Fayçal Ben Nejma used COMSOL Multiphysics®
to model the physical problem. He also verified
the overall research outputs.
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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