Identification of the Effective Radiative Properties of Cylindrical
Packed Bed Porous Media
CHAIMA BOURAOUI, FAYÇAL BEN NEJMA
Ionized and Reactive Media Studies Research Unit,
Preparatory Institute of Engineering Studies of Monastir,
Ibn Eljazzar Street, 5019
TUNISIA
Abstract: - Understanding radiative exchange in a porous medium is a crucial step that can provide significant
insights and improvements in its characteristics, enhancing its practical utility across various industrial
applications. In this paper, a numerical model, utilizing the finite element method (FEM), was developed to
predict the radiative transfer between a diffusely/specularly reflecting cylindrical packed bed porous medium
and a plane heating surface. Four different structures of the medium were suggested to examine the effect of the
particles ‘disposition on the radiative properties of the medium. The assessment of normalized flux distribution
enables the computation of effective radiative properties including reflectivity, transmissivity, and absorptivity
for particles exhibiting diffuse and specular reflection. The results underscore the significant influence of
particle arrangement on media properties. The structure of the second model allowed for the attainment of an
opaque surface from the first layer. Meaningful correlations can be established from the presented curves,
offering a streamlined and accurate method for determining effective radiative property coefficients based on
emissivity in future model applications.
Key-Words: - Porous media, cylindrical packed bed, particles disposition, numerical model, FEM, radiative
properties, diffuse and specular reflection.
Received: April 19, 2023. Revised: November 11, 2023. Accepted: December 19, 2023. Published: January 26, 2024.
Nomenclature:
Greek Symbols:
e
Porosity
Local effective absorptivity
I
Radiative intensity W.m-2
Mean effective absorptivity
L
Characteristic length, m
Local effective reflectivity
Normal vector
Mean effective reflectivity
Normalized radiant flux density
Local effective transmissivity
Radiant flux density, W.m-2
Mean effective transmissivity
R
Radius of the particle
Emissivity
S1
Radiant hot surface (m2)
eff
Mean effective emissivity
S2
Radiant cold surface (m2)
Solid angle, sr
Tp
Temperature of the particles, K
Subscripts
Trad
Temperature of the surface S1, K
b
Black body
Ts
Temperature of the surface S2, K
1 Introduction
Radiative heat transfer in porous media is an
important research topic as it is widely observed and
dominant in several high-temperature applications
such as catalytic combustion [1], heat exchangers
[2], solar thermochemical reactors [3], medical
applications [4]. The determination and
understanding of the radiative properties of the
porous medium are thus key steps for the efficient
design and operation of these applications.
A porous media is characterized by a complex
structure of pores with irregular sizes, shapes, and
connections. Therefore, the determination of its
radiative characteristics has remained a complicated
task. The most common assumption proposed in the
literature is to consider the porous medium as a
packed bed of cylinders, squares, spheres, and even
a mixture of them, etc … This hypothesis was
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adopted by a considerable amount of investigations
reviewed in, [5], [6], [7]. Cylindrical packed bed
configuration was frequently used in porous media
radiative transfer, as it has various advantages over
packed beds of other particle forms. Thanks to its
axisymmetric condition, it offers simplified
mathematical modeling, ease of manufacture, good
packing efficiency and enhanced mass transfer, etc...
To solve the problem of radiative transfer in such a
complex structure, porous media have been treated
as a continuous and homogenous system and the
standard Radiative Transfer Equation RTE has been
used with ‘‘effective radiative properties”, [8], [9].
Previous research showed that reliable
identifications of the radiative properties of porous
media have been obtained through experimental
measurements as reported in, [10], [11], [12].
Although its performance, this method can be an
expensive, onerous, and time-consuming task. As an
alternative method, a great number of studies have
proposed theoretical and analytical approaches. A
considerable body of literature examining their
methodologies and highlighting their assumptions,
findings and limits has been reported in, [13], [14].
Recently, along with the increase in computational
capacity and speed, some authors have suggested
numerical simulations for the determination of
radiative properties of porous media including
Monte Carlo methods, [15], [16], the Mie theory
[17], Finite Volume Method (FVM) [18], discrete
ordinate method (DOM) [19], Finite Element
Method (FEM) [20], discrete dipole approximation
[21]. More details about these methods can be found
in the literature [22], [23], [24].
The determination of radiative properties of
cylindrical packed bed porous media has been the
subject of long-term investigation. [25], investigated
the radiative heat transfer in participating media
composed of long cylindrical fibers with a diameter
within the geometrical optics limit. The suitability
of the Single and Multi-RTE Approaches was
examined by comparing the macroscopic optical
properties they yielded specifically, the radiative
flux, transmittance, and reflectance, with those
obtained through direct Monte Carlo simulation on
analogous morphologies. The results indicated that
the multi-RTE approach is better suited for fibrous
media with high porosity, whereas the single-RTE
approach is more appropriate for isotropic fibrous
media. [26], evaluated the radiative characteristics
of a set of cylindrical fibers with an arrangement
comparable to that of bird feathers using the Monte
Carlo method. The influence of different parameters
including the diameter, the volume fraction, the
color, and the arrangement of the fiber as well as the
rachis's angle relative to the skin, the angle between
the barb and the rachis, and the angle between the
barbon and the barbules on the radiative properties
of the fiber was examined. [27], suggested the
addition of spherical or cylindrical particles that
interact with IR radiation in the polymer matrix to
improve its thermal insulation performance. In their
study, a model was created to study the impact of
adding the particles on the conductive and radiative
properties of the opacified foam and to identify the
key parameters for effective particles. Experimental
tests on powders incorporated into polystyrene thin
films confirmed the efficiency of the proposed
solution. [28] and [29], concluded that utilizing
hollow cylindrical double-layer porous media with
suitable pore size and number led to improvements
in the combustion temperature and the efficiency of
hydrogen production.
For the past few years, there has been
considerable interest in numerical simulations using
commercial software as it can efficiently handle
complex geometries and material properties,
supporting cost-effective and time-efficient
simulations. This method enables accurate
characterization, optimization, and understanding of
industrial processes involving radiative heat transfer
in porous media, [30], [31]. The current paper
suggests employing COMSOL Multiphysics, a
commercial Finite Element Method (FEM)
software, to examine the radiative characteristics of
four regular configurations of cylindrical packed
bed porous media. The objective of this study is to
establish correlations between these properties and
emissivity, while also assessing the impact of
particle arrangement.
2 Mathematical Modelling
2.1 Problem Configuration
In this work, four models were considered, each
consisting of a packed bed of equally sized cylinders
with axes orthogonal to the radiating surface. These
models differ in their particles’ arrangement and
thus in their porosity, as resumed in Table 1 and
Table 2.
The symmetry character of these configurations
allows restricting the study to their cells whose
dimensions are related to the cylinder’s radius by
the relations listed in Table 1.
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Table 1. The porosity of the configurations and the dimensions of the studied cells
Model N°
Porosity
Dimensions of
the studied cells
1
Lx = 2R
Ly = 2R
2
Lx = 2R
Ly = 2R
3
Lx = 2R
Ly = 2R
4
Lx = 2R
Ly = 2R
Table 2. The details of the four studied configurations
Model N°
Arrangements of the particles
Representation of the studied cell
1
2
3
4
2.2 Hypotheses
To simplify our numerical calculations, the
following hypotheses are made:
The porous medium as well as the heating
surface are infinite.
The radiant hot surface (S1) is considered as a
black surface with the temperature Trad.
The output surface (S2), representing the
surrounding environment, is also modeled as a
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black surface at a temperature T = 0K,
neglecting its emission.
The emissivity of the particles is negligible.
Fig. 1: The adopted assumption
When the radiation flux incident from the
surface (S1), considered as the hot surface, strikes
up the porous medium surface, it undergoes three
primary interactions. First, a portion of the incident
radiation is reflected off the surface (S1). Second,
another fraction is absorbed by the porous medium,
signifying the conversion of radiant energy into
internal energy within the material. Lastly, the
remaining portion, if any, is transmitted through the
porous medium, reaching a second surface (S2), as
indicated in Figure 1 and Figure 2.
Fig. 2: The schematic presentation of the different
radiative transfer by the porous media
2.3 Boundary Conditions and Equations
The present study was conducted for both types of
reflection, namely diffuse and specular. The diffuse
boundary condition is written as:
1)
Concerning the specular boundary condition, it
can be defined as:
(2)
These equations present the outgoing radiance
which is composed of the emitted radiance by the
surface itself (the first term) and the reflected
radiance (the second term) in the case of a diffuse
reflection (Eq (1)) and of a specular reflection (Eq
(2)).
Where Ib is the radiative intensity of a black
body, which is given by:
(3)
The radiative flux density is generally expressed
as:
(4)
Given the different dispositions of the particles
in the studied models, a normalization of the flux
has been performed. It can be calculated by:
(5)
The local effective reflectivity is expressed as:
(6)
Therefore, the mean effective reflectivity can be
determined by:
(7)
For the local effective transmissivity, it is
expressed as:
(8)
The mean effective transmissivity is thus
calculated by:
(9)
Considering the following relation:
(10)
The effective absorptivity can be determined using
the relation:
(11)
2.4 Numerical Procedure
Three-dimensional geometries were generated and
simulated utilizing the software COMSOL
Multiphysics, employing the finite element method.
An adaptive triangular mesh was used with a
refinement near the contact points between the
black diffuse surface and the cylinder to get a better
resolution in this area.
I
.I
.I
α.I
Black diffuse
surface (S2)
Black diffuse
surface (S1)
Porous medium
Black diffuse
surface (S2)
TS=0K
Symmetry plans
Cylindrical
particule
; Tp=0K
Black diffuse
surface (S1)
Trad
L
D
D
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To ensure the independence of the solution on
the grid size, different grid densities were tested by
varying the boundary and edge elements. After
comparing the stability of the obtained results, a
grid size of 28610 boundary elements and 1812
edge elements was fixed, as it represents a good
compromise between the accuracy and the time
calculation. Figure 3 shows the grid structure
adopted for Model 1.
Fig. 3: The mesh grid adopted for Model 1 (L=2R
D=R/10)
3 Results and Discussions
This section entails an examination of the radiative
transfer within cylindrical packed bed porous
media, and more precisely the reflection,
transmission, and absorption phenomena. Both
diffuse and specular reflections are considered in
this analysis. Furthermore, the impact of the
emissivity of the cylinders and the number of layers
present is also explored.
The initial step in establishing our radiative
properties involves the computation and
presentation of the radiative flux. Due to the
different disposition of the particles in the proposed
models, a normalization of the flux has been
performed to ensure consistency. Figure 4, Figure
5, Figure 6 and Figure 7 depict the normalized flux
distribution at the inlet, outlet, and on the cylinders
within the four models. The assessments were
carried out while taking into account cylindrical
particles that exhibit both diffuse and specular
reflections, and possess an emissivity value of
ε=0.5. It can be seen that, at the inlet and the outlet
of the four models and as the rays enter and exit the
cylinder, a circular flux profile that gradually
expands moving away from the cylinder’s
perimeter is generated. Nevertheless, the intensity
of the flux distribution differs among the different
proposed models. Indeed, for models 1 and 2, the
flux intensity rises gradually as one moves away
from the perimeter of the cylinder’s face regarding
the radiation emitted from neighboring particles
through reflected rays.
Model 1
Diffuse Reflection
Specular Reflection
At the inlet
At the outlet
On the entire cylinder
Fig. 4: the normalized flux distribution at the inlet, outlet and on the cylinders obtained for: Model 1 with
L=2R, D=R/10, =0.5 and for diffuse and specular reflections
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Model 2
Diffuse Reflection
Specular Reflection
At the inlet
At the outlet
On the entire cylinder
Fig. 5: the normalized flux distribution at the inlet, outlet and on the cylinders obtained for: Model 2 with
L=2R, D=R/10, =0.5 and for diffuse and specular reflections
Model 3
Diffuse Reflection
Specular Reflection
At the inlet
At the outlet
On the entire cylinder
Fig. 6: the normalized flux distribution at the inlet, outlet and on the cylinders obtained for: Model 3 with
L=2R, D=R/10, =0.5 and for diffuse and specular reflections
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Model 4
Diffuse Reflection
Specular Reflection
At the inlet
At the outlet
On the entire cylinder
Fig. 7: the normalized flux distribution at the inlet, outlet and on the cylinders obtained for: Model 4 with
L=2R, D=R/10, =0.5 and for diffuse and specular reflections
However, it exhibits an inverse trend for models
3 and 4. The flux exhibits a lower intensity at the
regions where the particles are connected.
Conversely, the holes formed by the arrangement of
particles display a significantly higher flux intensity
due to the radiation emitted by adjacent particles in
all directions. These findings clearly show that the
cylinders ‘disposition has an impact on the flux
distribution.
The distribution of flux is comparable for both
particles exhibiting diffuse and specular reflection,
although there is a diminished intensity in the case
of diffuse reflection. This is because when light rays
hit diffusely reflecting particles, they scatter in
multiple directions instead of a single angle as with
specularly reflecting particles. As a result, fewer
rays are transmitted through diffusely reflecting
particles, leading to a decrease in the flux intensity.
The distribution of the flux over the entire
cylinder illustrates that the outer surface of the
particles in all four models receives a greater flux
compared to the inner surface. This can be attributed
to the radiation emitted by neighboring particles in
the second layer.
These findings hold importance in
comprehending the dynamics of radiative transfer
within the suggested configurations. This
knowledge bears relevance in various fields such as
radiative heat transfer, transport phenomena,
materials science, and engineering, as the ability to
regulate energy distribution is crucial for enhancing
system performance.
Based on these results, the effective reflectivity,
which encapsulates both diffuse and specular
reflections, was calculated and graphically depicted
in Figure 8, Figure 9, Figure 10 and Figure 11
against the corresponding emissivity values. The
trends observed in these figures reveal a notable
inverse relationship: as the emissivity of the porous
media increases, there is a discernible decrease in
the amount of incident radiation being reflected.
This behavior is consistent with the fundamental
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0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=12R
principles of emissivity and reflectivity, where
higher emissivity values signify an increased
capacity for a material to absorb and emit radiation,
consequently leading to reduced reflectivity.
Moreover, it is noteworthy that the observed
relationship between emissivity and reflectivity
aligns with fundamental principles governing
radiative heat transfer. Surfaces with emissivity
values approaching zero, as noted in the instances of
highest reflectivity, signify a diminished capacity to
absorb and emit radiation. This phenomenon can be
attributed to the fact that materials with low
emissivity tend to reflect a larger proportion of
incident radiation, resulting in heightened
reflectivity. Conversely, a surface characterized by
an emissivity value of 1, characteristic of a black
body, exhibited a complete absence of reflectivity.
This behavior adheres to the theoretical expectation
for a black body, which absorbs all incident
radiation without reflecting any.
Furthermore, the influence of the internal
disposition of the cylindrical particles within the
porous media was underscored. It was shown that,
in the case of diffusely reflected particles, the first
six layers of models 1, 3, and 4 contributed to the
reflection of the incident radiation. Indeed, the
arrangement of the particles in each layer creates
additional interfaces for scattering and reflection,
contributing to an increased likelihood of diffusely
reflected particles encountering surfaces conducive
to reflection. This phenomenon results in a higher
overall reflectivity compared to the scenario with a
single layer, where the opportunities for multiple
interactions and reflections are limited. In contrast,
for the specularly reflected particles in the four
models and diffusely reflected particles in model 2,
the first layer of the porous medium surface emerges
as the main reflecting surface. This can be attributed
to its smoothness and the medium structure.
(a) (b)
Fig. 8: Variation of effective reflectivity with emissivity for diffuse (a) and specular (b) reflecting cylindrical
particles in Model 1 across different Layers
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=12R
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(a) (b)
Fig. 9: Variation of effective reflectivity with emissivity for diffuse (a) and specular (b) reflecting cylindrical
particles in Model 2 across different Layers
(a) (b)
Fig. 10: Variation of effective reflectivity with emissivity for diffuse (a) and specular (b) reflecting cylindrical
particles in Model 3 across different Layers
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=4R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=4R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
0
0,05
0,1
0,15
0,2
0,25
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
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(a) (b)
Fig. 11: Variation of effective reflectivity with emissivity for diffuse (a) and specular (b) reflecting cylindrical
particles in Model 4 across different Layers
Using the obtained curves of the effective
reflectivity for the four models, correlations linking
the effective reflectivity coefficient and the
emissivity have been established for specular
reflection, as illustrated in Table 3. This insight is
crucial, particularly for engineers and scientists
working on energy-efficient building design. Such
relationships enable informed material selection,
optimizing the balance between reflectivity and
emissivity for enhanced control over heat transfer.
This has practical applications in diverse areas,
including thermal insulation, solar panels, building
materials, etc…
Table 3. The correlations of effective reflectivity in
the case of specular reflection
Model N°
Effective reflectivity
1
2
3
4
Under the assumption that the radiating surface
uniformly emits radiation to the porous medium, the
calculation and plotting of transmissivity for
different layers, corresponding to the four models,
were performed as a function of emissivity for both
reflection cases. The results, presented in Figure 12,
Figure 13, Figure 14 and Figure 15, indicated a
decrease in transmissivity with an increase in
emissivity, a trend particularly pronounced as the
number of layers escalates, leading to a faster
convergence of transmissivity curves towards zero.
In addition, a comparative analysis of the
obtained transmissivity curves for the four
suggested models revealed the significant impact of
model structure on transmissivity. Figure 13
highlights that Model 2, with its unique structural
configuration, exhibits an abrupt transition to an
almost complete opacity starting from the second
layer. This can be attributed to the closely packed
nature of the particles in each layer, hindering the
transmission of radiation through the medium.
However, as presented in Figure 12, Figure 14 and
Figure 15, opacity begins to manifest from the sixth
layer for Model 1 and the tenth layer for Models 3
and 4. This can be ascribed to the looser packing
arrangement of the particles and the existence of
numerous voids among them.
It can also be noted that a specular surface tends
to have higher transmissivity than a diffuse surface.
This is because specular reflection involves minimal
scattering, allowing more radiation to pass through
the material. In contrast, a diffuse surface scatters
radiation in multiple directions, which can result in
a portion of the radiation being absorbed or
redirected away from the original path, reducing
transmissivity.
In this study, the investigated media were treated
as opaque surfaces. Consequently, the utilization of
effective properties is deemed justified for this
purpose.
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0,09
0,1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
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(a) (b)
Fig. 12: Variation of effective transmissivity with emissivity for diffuse (a) and specular (b) reflecting
cylindrical particles in Model 1 across different Layers
(a) (b)
Fig. 13: Variation of effective transmissivity with emissivity for diffuse (a) and specular (b) reflecting
cylindrical particles in Model 2 across different Layers
0
0,02
0,04
0,06
0,08
0,1
0,12
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=12R
0
0,05
0,1
0,15
0,2
0,25
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=12R
0
0,005
0,01
0,015
0,02
0,025
0,03
0,035
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=4R
0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0,09
0,1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=4R
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(a) (b)
Fig. 14: Variation of effective transmissivity with emissivity for diffuse (a) and specular (b) reflecting
cylindrical particles in Model 3 across different Layers
(a) (b)
Fig. 15: Variation of effective transmissivity with emissivity for diffuse (a) and specular (b) reflecting
cylindrical particles in Model 4 across different Layers
The effective absorptivity is also computed and
graphically represented as a function of emissivity
for the four models and considering both reflection
cases, as depicted in Figure 16, Figure 17, Figure 18
and Figure 19. With an increase in emissivity,
coupled with the presence of voids in each model,
there is an augmentation in the likelihood of
radiation being absorbed by the particles.
The particles within the bed not only absorb
incident radiation but also have the potential to
receive radiation emitted by neighboring particles
and reflected within the voids. This intricate
interplay contributes to the overall increase in
absorptivity. The heightened emissivity, in
conjunction with the structural characteristics of the
bed, facilitates a more effective interaction with
thermal radiation, underscoring the nuanced
dynamics at play in the absorption process.
Additionally, it is important to underscore that a
substantial portion of radiation was absorbed by the
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
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cylindrical particles located in the initial layers of
the porous media. To be more precise, upon
comparing the absorptivity curves derived for the
four models under both reflection scenarios, it is
noteworthy that the first layer of Model 2, where the
cylindrical particles are strategically positioned,
exhibited the remarkable capability to absorb all
incoming radiation. Similarly, the first layer of
Model 1 absorbed nearly all the radiation, with a
minor portion being absorbed by the subsequent five
layers. As for Models 3 and 4, the initial ten layers
played a significant role in the absorption of
radiation, particularly in the case of specular
reflection. This variation in absorption behavior can
be attributed to the specific geometric configuration
and distribution of the cylindrical particles in each
model, emphasizing the nuanced impact of particle
arrangement on radiation absorption.
(a) (b)
Fig. 16: Variation of effective absorptivity with emissivity for diffuse (a) and specular (b) reflecting cylindrical
particles in Model 1 across different Layers
(a) (b)
Fig. 17: Variation of effective absorptivity with emissivity for diffuse (a) and specular (b) reflecting cylindrical
particles in Model 2 across different Layers
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=12R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=12R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=4R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=R L=2R
L=3R L=4R
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(a) (b)
Fig. 18: Variation of effective absorptivity with emissivity for diffuse (a) and specular (b) reflecting cylindrical
particles in Model 3 across different Layers
(a) (b)
Fig. 19: Variation of effective transmissivity with emissivity for diffuse (a) and specular (b) reflecting
cylindrical particles in Model 4 across different Layers
These findings provide valuable insights into the
impact of internal structural variations on the
radiative behavior of porous media. Such
considerations are crucial in the design and
optimization of materials for applications involving
radiative heat transfer, offering a nuanced
understanding of how internal configurations can
affect the overall thermal performance of porous
media. For instance, in the heating, ventilation and
air conditioning (HVAC) field, this can lead to the
development of advanced heat exchange systems
and HVAC components. This knowledge can be
applied to design more efficient thermal insulation
for buildings, improving temperature regulation and
reducing energy consumption. Additionally, it can
contribute to the enhancement of HVAC systems,
allowing for better control of indoor temperatures
and increased energy efficiency in both residential
and commercial settings. The implications extend to
the automotive industry, where optimized thermal
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R L=6R
L=8R L=20R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
eff
L=2R L=4R
L=6R L=8R
L=20R
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management systems can improve engine efficiency
and overall vehicle performance.
By comparing the obtained results to those
presented in another study on spherical packed bed
porous media [32], it can be noted that spherical
packed configurations facilitate the quicker
attainment of an opaque medium. They also appear
to be more reflective and absorptive in comparison
to a cylindrical packed bed porous media.
4 Conclusion
Numerical simulations were performed to assess the
radiative properties of four configurations of
cylindrical packed bed porous media, considering
both specular and diffusive reflection of particles.
Employing a COMSOL Multiphysics model, the
impact of the arrangement of particles was
conducted. The normalized flux calculations
revealed circular flux profiles with varying
intensities across the proposed models. The first
layer of Model 2 emerges as the main reflecting,
transmitting and absorbing surface, whereas the
other models required multiple layers to achieve
complete radiation reflection, resulting in an opaque
medium. These layers also contributed in the
absorption of radiation.
This approach, grounded in numerical simulation
calculations, provides an efficient and cost-effective
alternative to analytical and experimental methods,
offering insights into the complex nature of
radiative heat transfer through porous media.
References
[1] P. Qian, M. Liu, X. Li, F. Xie, Z. Huang, C.
Luo and X. Zhu « Combustion
characteristics and radiation performance of
premixed hydrogen/air combustion in a
mesoscale divergent porous media
combustor », International Journal of
Hydrogen Energy, vol. 45, no 7, p. 5002-
5013, févr. 2020, doi:
10.1016/j.ijhydene.2019.12.094.
[2] S. Rashidi, M. H. Kashefi, K. C. Kim, and O.
Samimi-Abianeh, « Potentials of porous
materials for energy management in heat
exchangers – A comprehensive review »,
Applied Energy, vol. 243, p. 206-232, juin
2019, doi: 10.1016/j.apenergy.2019.03.200.
[3] H. Zhang, B. Guene Lougou, R. Pan, Y.
Shuai, F. Wang, Z. Cheng and H. Tan
« Analysis of thermal transport and fluid
flow in high-temperature porous media solar
thermochemical reactor », Solar Energy, vol.
173, p. 814-824, oct. 2018, doi:
10.1016/j.solener.2018.08.015.
[4] L. Dombrovsky, J. H. Randrianalisoa, W.
Lipinski, and V. Timchenko, « Simplified
approaches to radiative transfer simulations
in laser-induced hyperthermia of superficial
tumors », CTS, vol. 5, no 6, 2013, doi:
10.1615/ComputThermalScien.2013008157.
[5] W. Fuqiang, Z. Xinping, D. Yan, Y.
Hongliang, X. Shi, L. Yang and C. Ziming.,
« Progress in radiative transfer in porous
medium: A review from macro scale to pore
scale with experimental test », Applied
Thermal Engineering, vol. 210, p. 118331,
juin 2022.
[6] S. Hajimirza and H. Sharadga, « Learning
thermal radiative properties of porous media
from engineered geometric features »,
International Journal of Heat and Mass
Transfer, vol. 179, p. 121668, nov. 2021.
[7] H. H. Kang, « Machine Learning
Implementation in Radiative Properties
Prediction for Porous Media », Thesis, 2019.
[8] M. Luo, C. Wang, J. Zhao, and L. Liu,
« Characteristics of effective thermal
conductivity of porous materials considering
thermal radiation: A pore-level analysis »,
International Journal of Heat and Mass
Transfer, vol. 188, p. 122597, juin 2022, doi:
10.1016/j.ijheatmasstransfer.2022.122597.
[9] B. Liu, J. Zhao, and L. Liu, « Continuum
approach based on radiation distribution
function for radiative heat transfer in densely
packed particulate system », Journal of
Quantitative Spectroscopy and Radiative
Transfer, vol. 253, p. 107028, sept. 2020.
[10] F. Retailleau, V. Allheily, L. Merlat, J.-F.
Henry, and J. H. Randrianalisoa,
« Experimental characterization of radiative
transfer in semi-transparent composite
materials with rough boundaries », Journal
of Quantitative Spectroscopy and Radiative
Transfer, vol. 256, p. 107300, nov. 2020,
doi: 10.1016/j.jqsrt.2020.107300.
[11] F. Retailleau, V. Allheily, L. Merlat, J.-F.
Henry, and J. Randrianalisoa, « Experimental
study of radiative transfer in semi-transparent
composite materials at different
temperatures », J. Phys.: Conf. Ser., vol.
2116, no 1, p. 012062, nov. 2021, doi:
10.1088/1742-6596/2116/1/012062.
[12] J. F. Sacadura and D. Baillis, « Experimental
characterization of thermal radiation
properties of dispersed media »,
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Volume 19, 2024
International Journal of Thermal Sciences,
vol. 41, no 7, p. 699-707, June 2002.
[13] S. Cunsolo, R. Coquard, D. Baillis, and N.
Bianco, « Radiative properties modeling of
open cell solid foam: Review and new
analytical law », International Journal of
Thermal Sciences, vol. 104, p. 122-134, June
2016.
[14] J. Randrianalisoa and D. Baillis, « Analytical
model of radiative properties of packed beds
and dispersed media », International Journal
of Heat and Mass Transfer, vol. 70, p. 264-
275, mars 2014.
[15] Y. Maanane, M. Roger, A. Delmas, M.
Galtier, and F. André, « Symbolic Monte
Carlo method applied to the identification of
radiative properties of a heterogeneous
material », Journal of Quantitative
Spectroscopy and Radiative Transfer, vol.
249, p. 107019, juill. 2020, doi:
10.1016/j.jqsrt.2020.107019.
[16] A. Sit, R. Wulf, T. Fieback, and P. Talukdar,
« Identification of spectral radiative
properties of closed cell polymeric foams
using coupled Monte Carlo-particle swarm
optimization », International Journal of
Thermal Sciences, vol. 189, p. 108263, juill.
2023, doi:
10.1016/j.ijthermalsci.2023.108263.
[17] Y. Wang, P. Hsu, and M. H. McCay, « The
Pore Size Dependence of the Radiative
Scattering Coefficient in Yttria-Stabilized
Zirconia Films », presented at ASME Turbo
Expo 2022: Turbomachinery Technical
Conference and Exposition, American
Society of Mechanical Engineers Digital
Collection, oct. 2022. doi: 10.1115/GT2022-
80853.
[18] P. Zhang, C. Sun, and X.-L. Xia, « Improved
Gold-SA algorithm for simultaneous
estimation of temperature-dependent thermal
conductivity and spectral radiative properties
of semitransparent medium », International
Journal of Heat and Mass Transfer, vol. 191,
p. 122836, August 2022, doi:
10.1016/j.ijheatmasstransfer.2022.122836.
[19] Z.-T. Niu, H. Qi, Y.-K. Ji, S. Wen, Y.-T.
Ren, and M.-J. He, « Real-time
reconstruction of thermal boundary condition
of porous media via temperature sequence »,
International Journal of Thermal Sciences,
vol. 177, p. 107570, July 2022, doi:
10.1016/j.ijthermalsci.2022.107570.
[20] H. Gonome, « Interference effect of localized
surface plasmon resonance on radiative
properties of plasmonic particle clusters in
3D assemblies », Journal of Quantitative
Spectroscopy and Radiative Transfer, vol.
230, p. 13-23, June 2019.
[21] Y. Tang, K. Zhu, and Y. Huang, « Radiative
properties of porous fly ash particles based
on the particle superposition model »,
Journal of Quantitative Spectroscopy and
Radiative Transfer, vol. 277, p. 107977, Jan.
2022, doi: 10.1016/j.jqsrt.2021.107977.
[22] O. Rozenbaum, C. Blanchard, and D. De
Sousa Meneses, « Determination of high-
temperature radiative properties of porous
silica by combined image analysis, infrared
spectroscopy and numerical simulation »,
International Journal of Thermal Sciences,
vol. 137, p. 552-559, march 2019.
[23] L. Ni, Z. Chen, P. Mukhopadhyaya, X.
Zhang, Q Wu, Q Yu and G. Miu,
« Numerical simulation on thermal
performance of vacuum insulation panels
with fiber /powder porous media based on
CFD method », International Journal of
Thermal Sciences, vol. 172, p. 107320, févr.
2022, doi:
10.1016/j.ijthermalsci.2021.107320.
[24] C. Chen, D. Ranjan, P. G. Loutzenhiser, and
Z. M. Zhang, « A numerical study of the
spectral radiative properties of packed bed
with mixed bauxite and silica spheres »,
International Journal of Heat and Mass
Transfer, vol. 207, p. 124025, june 2023, doi:
10.1016/j.ijheatmasstransfer.2023.124025.
[25] J. Randrianalisoa, S. Haussener, D. Baillis,
and W. Lipiński, « Radiative characterization
of random fibrous media with long
cylindrical fibers: Comparison of single- and
multi-RTE approaches », Journal of
Quantitative Spectroscopy and Radiative
Transfer, vol. 202, p. 220-232, nov. 2017,
doi: 10.1016/j.jqsrt.2017.08.002.
[26] M. Sedghi, A. Saboonchi, and M. Ghane,
« Numerical study of radiative properties of
birds’ feathers using Monte Carlo method »,
International Communications in Heat and
Mass Transfer, vol. 117, p. 104718, oct.
2020, doi:
10.1016/j.icheatmasstransfer.2020.104718.
[27] R. Coquard, D. Baillis, and D. Quenard,
« Numerical and experimental study of the
IR opacification of polystyrene foams for
thermal insulation enhancement », Energy
and Buildings, vol. 183, p. 54-63, janv. 2019,
doi: 10.1016/j.enbuild.2018.10.037.
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2024.19.1
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[28] A. Maznoy, A. Kirdyashkin, S. Minaev, A.
Markov, N. Pichugin, and E. Yakovlev, « A
study on the effects of porous structure on
the environmental and radiative
characteristics of cylindrical Ni-Al burners »,
Energy, vol. 160, p. 399-409, oct. 2018, doi:
10.1016/j.energy.2018.07.017.
[29] H. Zhu, H. Dai, Z. Song, X. Wang, Z. Wang,
H. Dai, S. He, « Improvement of hollow
cylinders on the conversion of coal mine
methane to hydrogen in packed bed burner »,
International Journal of Hydrogen Energy,
vol. 46, no 61, p. 31439-31451, sept. 2021,
doi: 10.1016/j.ijhydene.2021.07.036.
[30] E. Codau, T.-C. Codau, I.-G. Lupu, A. Raru,
and D. Farima, « Heat transfer simulation
through textile porous media », The Journal
of The Textile Institute, vol. 114, no 2, p. 257-
264, feb. 2023, doi:
10.1080/00405000.2022.2027608.
[31] M. Sans, O. Farges, V. Schick, and G.
Parent, « Solving transient coupled
conductive and radiative transfers in porous
media with a Monte Carlo Method:
Characterization of thermal conductivity of
foams using a numerical Flash Method »,
International Journal of Thermal Sciences,
vol. 179, p. 107656, sept. 2022, doi:
10.1016/j.ijthermalsci.2022.107656.
[32] C. Bouraoui and F. Ben Nejma, « Numerical
simulation for the determination of the
radiative properties of spherical packed bed
porous media: A COMSOL Multiphysics
Study », Advances in Mechanical
Engineering, vol. 15, no 10, p.
16878132231205724, oct. 2023, doi:
10.1177/16878132231205724.
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