Free Convection in Wavy Trapezoidal Porous Cavity with Hybrid Nanofluid Under MHD Effects

Case Studies in Thermal Engineering 56 (2024) 104243
Available online 12 March 2024
2214-157X/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Investigation of free convection in a wavy trapezoidal porous
cavity with MWCNT- Fe
3
O
4
/Water hybrid nanouid under MHD
effects: Galerkin nite element analysis
Kamel Guedri
a
, Abdel-Nour Zaim
b
, S. Mohammad Sajadi
c
, Dheyaa J. Jasim
d
,
Abderrahmane Aissa
b
, Soheil Salahshour
e , f , g
, Ahmad Almuhtady
h
, Obai Younis
i , **
,
Sh Baghaei
j , *
, Wael Al-Kouz
k
a
Mechanical Engineering Department, College of Engineering and Islamic Architecture, Umm Al-Qura University, P.O. Box 5555, Makkah, 21955,
Saudi Arabia
b
Laboratoire de Physique Quantique de la Mati` ere et Mod´ elisation Math´ ematique (LPQ3M), University of Mascara, Algeria
c
Department of Nutrition, Cihan University-Erbil, Kurdistan Region, Iraq
d
Department of Petroleum Engineering, Al-Amarah University College, Maysan, Iraq
e
Faculty of Engineering and Natural Sciences, Istanbul Okan University, Istanbul, Turkey
f
Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey
g
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
h
Mechanical & Maintenance Engineering Department, German Jordanian University, Madaba, Jordan
i
Department of Mechanical Engineering, College of Engineering in Wadi Alddawasir, Prince Sattam Bin Abdulaziz University, KSA
j
Department of Mechanical Engineering, Islamic Azad University, Iraq
k
Department of Engineering and Industrial Professions, University of North Alabama, Florence, Alabama, 35632, USA
ARTICLE INFO
Handling Editor: Huihe Qiu
Keywords:
Heat transfer
Natural convection
Magnetic eld
Trapezoid
Corrugated chamber
Porous medium
GFEM
ABSTRACT
Our study focuses on numerically investigating natural convection within a three-dimensional
trapezoidal cavity featuring a corrugated hot bottom wall, with a particular emphasis on
enhancing heat exchange using a water-based hybrid nanouid. Employing a steady-state regime
assumption, the study considers laminar and incompressible three-dimensional ow. The Darcy-
Forchheimer model is incorporated to account for inertial advection effects within the porous
layer. Predictions of the thermal and hydrodynamic behaviors of the system are achieved by
solving dimensionless equations utilizing the Galerkin Finite Element Method (GFEM). Notably, a
range of inuential parameters, including the volume fraction of nanoparticles (φ), Darcy number
(Da) spanning from 10
−5
to 10
−2
, Hartman number (Ha) across the range of 0–100, φ within the
range of 0–0.08, porosity (
ε ) ranging from 0.2 to 0.9, and Rayleigh numbers (Ra) varying from
10
3
to 10
6
was explored. The investigation also evaluates the geometrical impact of the enclosure
by considering undulation numbers (N) of the warm bottom ranging from 1 to 4. Our results
provide novel quantitative insights. Specically, we observe an inverse relationship between
undulation number (N), Hartman number (Ha), and porosity (
ε ) with large Rayleigh (Ra) ows,
signicantly inuencing effective heat transfer. Notably, this inuence diminishes at lower Ra
ows. At the highest Ra, we nd that increasing Da,
ε , and φ enhances the Nusselt number (Nu
avg
)
by 26 %, 17 %, and 23.5 %, respectively. Conversely, increasing Ha and N reduces Nu
avg
by 13 %
* Corresponding author.
** Corresponding author.
E-mail addresses: [email protected] (O. Younis), [email protected] (S. Baghaei).
Contents lists available at ScienceDirect
Case Studies in Thermal Engineering
journal homepage: www.elsevier.com/locate/csite
https://doi.org/10.1016/j.csite.2024.104243
Received 13 December 2023; Received in revised form 8 February 2024; Accepted 10 March 2024

Case Studies in Thermal Engineering 56 (2024) 104243
2
and 40 %, respectively. These ndings highlight the nuanced interplay of parameters and offer
valuable quantitative data for optimizing heat transfer processes in similar systems.
1. Introduction
In recent years, numerous techniques have been developed to enhance convective processes in thermal applications, including
variations in envelope shape and changes in the working uid. Towards the end of the 20th century, a new class of thermouids,
known as nanouids, emerged [1]. These nanouids exhibited remarkable thermal characteristics, sparking a multitude of research
efforts in a variety of engineering and technical applications [2–5], conducted in diverse operational settings [6,7]. Several nanouid
studies have highlighted the signicant inuence of nanoparticle type (φ) on ow pattern formation and system thermal performance
[8–10]. In this context, Selimefendigil et al. [11] conducted a study on the natural convection of two water-based nanouids (Al
2
O
3
and CuO) saturating different sides of a partitioned square cavity. Their results revealed distinct behaviors for the two nanouids. In
addition, Kasaeian et al. [12] carried out a comparative analysis of the heat transfer rates of three nanouids in a solar collector. They
concluded that the carbon nanotubes presented the best heat transfer rate with an optimum φ of 0.5%. Dat et al. [13] analyzed the
inuence of nanoparticles’ shape on the nanouid MHD heat transport mode within a porous medium. They reported that high values
of the shape factor contribute to an extended heat transfer, and when the magnetic force is increased, the conductive phenomena
become predominant over the one resulting from convection. Whilst extending their research on nanouids, scientists sought to create
hybrid nanouids by nanoparticle suspension in mixtures or composite materials [14,15]. A new category of hybrid nanomaterials is
being developed. They are mainly composed of CNT and metals, semiconductors or composites of non-conductive nanoparticles, and
carbon nanotubes. Wael et al. [16] conducted a study on a cavity with corrugated walls, investigating the thermal performance of
nanoliquid within a permeable medium and a magnetic eld. In another analysis, Slimani et al. [17] numerically examined the
performance of nanouid (Cu/Al
2
O
3
) in a differentially heated square enclosure, considering the impact of φ and buoyancy forces.
Their ndings indicated that thermal performance increases with the Rayleigh number and φ. Tayebi et al. [18] explored entropy
generation during free convection in a square cavity, incorporating a conductive cylinder with a wavy pattern. They observed that
higher Rayleigh numbers, coupled with the use of a hybrid nanouid, result in increased heat transfer rates along with a rise in thermal
entropy generation. Oztop et al. [19] investigated the inuence of φ and buoyancy forces on the enhancement of natural convection
within partially heated rectangles. Aghaei et al. [20] studied the effect of elliptic obstacle positions and orientations on the thermal and
hydrodynamic elds generated by natural convection inside a chamber lled with MWCNT/H2O, deducing that a horizontal location
of thermal obstacles induces higher heat transfer. In the broader context, various forms of enclosures containing nanoliquids have been
explored [21–25]. Researchers aiming to advance the thermal performance of systems have proposed the use of corrugated geometries
with metallic nanouid nanoparticles. Abdelmalek et al. [26] investigated various congurations of circular corrugated heating de-
vices, nding that the undulating heater’s conguration signicantly affects the heat transfer rate. Cimpean et al. [27] presented an
analysis of mixed convection in a trapezoidal cavity loaded with nanouid and a permeable medium. Another study by Said et al. [28]
emphasized the use of hybrid nanouids to improve the convection rate of a linear Fresnel reector prototype by using rGO--
Co
3
O
4
/H
2
O nanouid. Nguyen et al. [29], using the CVFEM method, examined a permeable enclosure lled with an ethylene gly-
col-Fe
3
O
4
nanouid, including the inuence of an electric force in the physical model. They found that convection is enhanced by
increasing the Da number and applying a high voltage. In a study on thermal ow within a microchannel with a corrugated wall,
Nguyen et al. [30] observed that increasing the slip coefcient from 0 to 0.1 allowed simultaneous improvement in convection and a 13
% reduction in entropy generation. Dutta et al. [31] researched free convective ow within a rhombic cavity with a corrugated upper
wall, revealing that a higher φ provides a higher average Nusselt number (Nu
avg
).
Trapezoidal shapes hold signicant appeal for a range of industrial and power applications due to their compelling advantages
[32–37]. In research conducted by Manikumar et al. [38], Reynolds et al. [39], and Dabiri et al. [40], the thermal characteristics and
heat loss in a trapezoidal cavity serving as an absorber for a linear Fresnel reector solar concentrator unit were thoroughly inves-
tigated. Examining various parameters, Sompong et al. [41] delved into the impact of free convection within a trapezoidal enclosure
featuring a wavy top wall. Building upon this study, Eshon et al. [42] extended the investigation by saturating the enclosure with
porous media. Selimefendigil et al. [43] conducted an optimization study on MHD mixed convection within a lid-driven trapezoidal
enclosure saturated with alumina–water nanouid. In a study by Hussein et al. [44], the focus was on the unsteady free convection of
air motion in a three-dimensional side-heated trapezoidal room. In the context of nanouid-based thermal transport and convective
heat transfer, our study introduces a unique research dimension by examining heat transfer in a trapezoidal corrugated cavity, a
geometric conguration relatively unexplored in the existing literature. While previous studies have addressed convective heat
transfer in various congurations, research involving trapezoidal corrugated chambers remains limited. This research departs from
convention by employing a hybrid nanouid containing multiwall carbon nanotubes (MWCNTs) and iron oxide nanoparticles (Fe
3
O
4
)
in this distinct geometry. The fusion of MWCNTs and Fe
3
O
4
nanoparticles in the unconventional trapezoidal cavity offers a promising
route to discover new heat transfer phenomena. We aim to gain new insights into the complex interaction between nanouids,
convective heat transfer and conductive heat transfer in this unconventional setting, thereby contributing to the advancement of
understanding of thermal transport in new congurations. The investigation into natural convection ow within a three-dimensional
trapezoidal cavity with a corrugated hot bottom wall using a water-based hybrid nanouid holds practical implications in various
domains. First, it is relevant to solar energy systems, offering insights for optimizing thermal performance in solar collectors and
absorbers. Second, the ndings could be applied to enhance electronic cooling, contributing to improved heat dissipation in electronic
K. Guedri et al.

Case Studies in Thermal Engineering 56 (2024) 104243
3
devices. Third, the study’s outcomes may nd application in industrial settings to optimize heat exchange processes within enclosed
spaces with complex geometries. Lastly, the results could be valuable in the design and optimization of thermal energy storage systems,
contributing to advancements in renewable energy storage.
1.1. Problem denition
As stated in this research, the problem statement is predicated on the magnetic force’s impact on the nanoparticle treatment process
within the porous media, with the effect of permeability taken into account. Specically, the goal is to determine how to alter ow
condition factors, such as the Darcy number, Rayleigh, and Hartmann numbers, and the number of corrugations on the bottom wall, to
optimize heat transfer inside the cavity. The physical domain, as shown in Fig. 1, is a 3-D trapezoidal enclosure with cold sidewalls (T
c
)
and a hot bottom wavy wall (T
h
). The cavity top wall is assumed to be adiabatic. The thermophysical properties of the hybrid nanouid
(MWCNT-Fe
3
O
4
/Water) are listed in Table 1. The bottom wall corrugation is given by x= − AsinN
π x
H.
2. Mathematical formulation
Our study makes several underlying assumptions to streamline its methodology. First, we assume laminar and incompressible ow
within the trapezoidal cavity, simplifying mathematical modeling and solution procedures. Second, a steady-state regime is consid-
ered, implying that the ow and thermal characteristics are time-independent. Third, the Darcy-Forchheimer model is employed to
describe the inertial advection effect within the porous layer. Fourth, a homogeneous distribution of nanoparticles within the nano-
uid is assumed, neglecting agglomeration or non-uniform distribution. Finally, the study employs a water-based hybrid nanouid,
assuming its composition enhances the heat exchange process within the cavity by leveraging its potential advantages in thermal
conductivity. These assumptions collectively frame the conditions under which the numerical study is conducted, providing clarity for
the interpretation of its ndings in specic real-world scenarios. The Trapezoidal enclosure is saturated with porous media and loaded
with Fe
3
O
4
-MWCNT-H
2
O nanouid. The Forchheimer-Brinkman extension of the Darcy model [46] and the Boussinesq approximation
are utilized [2–5],
∇·v=0,(1)
ρ
hnf
ε
2v·∇v= − ∇p+∇·
μ
hnf
ε
∇v−
μ
hnf
Kv−CF
K
√|v|v− (
ρ β)hnf (T−Tc)g
→+
σ
hnf
ε
v×B
→,(2)
ρ Cphnf v·∇T=∇·(km∇T),(3)
Where
|u| = 
u2+v2
√, K=
ε
3d2
m
150(1− ε )2 (4) The thermophysical characteristics of nanouid are [4–6]:
Fig. 1. (a) The Trapezoidal enclosure and boundary conditions, (b) computational domain meshing.
Table 1
Thermophysical characteristics of (MWCNT(50%)-Fe
3
O
4
(50%)/Water) [45].
Properties
ρ (kg/m
3
) C
p
(J/kg K) k (W/m K) σ (S/m) ×10
−4
β (K
−1
) ×10
−5
Pure water 997.1 4179 0.613 0.055 21
Fe
3
O
4
5810 670 6 2.5 1.3
MWCNT 1600 796 3000 1.9 4.2
K. Guedri et al.

Case Studies in Thermal Engineering 56 (2024) 104243
4
ρ
hnf = (1−φ)
ρ
f+φ
ρ
p
( ρ β)hnf = (1−φ)( ρ β)f+φ( ρ β)p
(6)
ρ Cphnf = (1−φ) ρ Cpf+φ ρ Cpp
α
hnf =khnf
ρ Cphnf
(7)
khnf
kf=knp + (n−1)kf− (n−1)kf−knpφ
knp + (n−1)kf+kf−knpφ(8)
μ
hnf =
μ
bf
(1−φ)2.5(9)
For nanoparticles Fe3O4 and MWCNT, the properties are obtained [18,45].
φnp =φFe2O4+φMwCNT
ρ
mp =φFe2O4
ρ
Fe2O4+φMWCNT
ρ
MwCNT
φ
cpnp =φFe2O4cpFe2O4+φMwCNTcpMWONT
φ
knp =φFe2O4kFe2O4+φMWCNTkMwCNT
φ
σ
np =φFe2O4
σ
Fe2O4+φMwCNI
σ
MwCNT
φ10)
2.1. Dimensionless equations
X=x
L,Y=y L,Z=z L(11)
U=uL
α
f
,V=vL
α
f
,W=wL
α
f
,θnf =Tnf −Tc
Th−Tc
,θs=Ts−Tc
Th−Tc
,Pr =
ν
f
α
f
(12)
Ra =gβf(Th−Tc)L3
ν
f
α
f
,P=pL2
ρ
f
α
2
f
,keff = ε knf +(1− ε )km,CF=1.75
150
√(13)
Da =λ
L2,Pr =vfl
α
fl
Ha =LB 
σ
nf
μ
nf
(14)
As a result, we have [2–5].
∂ U
∂ X+
∂ V
∂ Y+
∂ W
∂ Z=0(15)
1
ε
2
ρ
hnf
ρ
fluid U
∂ U
∂ X+V
∂ U
∂ Y+W
∂ U
∂ Z= −
∂ P
∂ X+1 ε
υ
hnf
υ
fluid
Pr
∂
2U
∂ X2+
∂
2U
∂ Y2+
∂
2U
∂ Z2−
υ
hnf
υ
fluid
Pr
Da U−Fc
Da
√U
U2+V2+W2
−
σ
hnf
σ
fluid
Pr Ha2
ε
U
(16)
1
ε
2
ρ
hnf
ρ
fluid U
∂ V
∂ X+V
∂ V
∂ Y+W
∂ V
∂ Z= −
∂ P
∂ Y+1 ε
υ
hnf
υ
fluid
Pr
∂
2V
∂ X2+
∂
2V
∂ Y2+
∂
2V
∂ Z2−
υ
hnf
υ
fluid
Pr
Da V−Fc
Da
√V
U2+V2+W2
−
σ
hnf
σ
fluid
Pr Ha2
ε
V
(17)
1
ε
2
ρ
hnf
ρ
fluid U
∂ W
∂ X+V
∂ W
∂ Y+W
∂ W
∂ Z= −
∂ P
∂ Z+1 ε
υ
hnf
υ
fluid
Pr
∂
2W
∂ X2+
∂
2W
∂ Y2+
∂
2W
∂ Z2−
υ
hnf
υ
fluid
Pr
Da W−Fc
Da
√W
U2+V2+W2
+βhnf
βhnf
Pr Raθ
(18)
K. Guedri et al.

Case Studies in Thermal Engineering 56 (2024) 104243
5
U
∂ θ
∂ X+V
∂ θ
∂ Y+W
∂ θ
∂ Z=
α
hnf
α
fluid 
∂
2θ
∂ X2+
∂
2θ
∂ Y2+
∂
2θ
∂ Z2(19)
The boundary conditions are as follows:
On the upper wall:
U=V=W=0,
∂ θnf
∂ Y=0,(20)
The Heated Part of the Bottom Wall
U=V=W=0,θnf =1,
On the lower hot wall:
U=V=W=0,θnf =0,(21)
The Nu
avg
at the bottom horizontal wall:
Nuloc = − keff
kfl 
∂ θpo
∂ nwall
,Nuavg =1
S2
s
0
s
0
Nulocdydz (22)
3. Methodology
The non-dimensional equations that governed the system understudy Eqs. (14)–(18) and the associated boundary conditions of Eqs.
(19)–(21) are resolved using Galerkin weighted and nite element methods [47]. The Galerkin weighted residuals approach is used to
convert the system equations into a set of integral equations. Firstly, the computational zone is discretized into tiny triangular
components, as shown in Fig. 1(b). Lagrange triangular nite elements of different forms are introduced to each ow parameter that is
being computed inside of the computational zone. When the approximations of the dimensionless governing equations are replaced
with their exact values, the residuals for any conservation equation may be calculated. An application of the Newton-Raphson iteration
process is required in order to provide an explanation for the nonlinear expressions seen in momentum equations. When it comes to the
obtained numerical solution, convergence is only considered sufcient when the following convergence requirement of the relative
errors is met:
Γi+1−Γi
Γi+1≤
η
Table 2
Mesh sensitivity (Ha =0, and Ra =10
5
, Da =10
−2
and N =4).
Mesh size 36430 56164 122162 273994 531828 1889299
Nu
avg
8.8197 9.1230 9.3964 10.299 11.207 11.200
Fig. 2. Dimensionless temperature for Ra =10
5
and Pr =0.7.
K. Guedri et al.