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Citation: Madhukesh, J.K.; Sarris,
I.E.; Prasannakumara, B.C.;
Abdulrahman, A. Investigation of
Thermal Performance of Ternary
Hybrid Nanofluid Flow in a
Permeable Inclined Cylinder/Plate.
Energies 2023,16, 2630. https://
doi.org/10.3390/en16062630
Academic Editor: Gabriela Huminic
Received: 13 February 2023
Revised: 1 March 2023
Accepted: 7 March 2023
Published: 10 March 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
energies
Article
Investigation of Thermal Performance of Ternary Hybrid
Nanofluid Flow in a Permeable Inclined Cylinder/Plate
Javali Kotresh Madhukesh 1, Ioannis E. Sarris 2, * , Ballajja Chandrappa Prasannakumara 1
and Amal Abdulrahman 3
1Department of Studies in Mathematics, Davangere University, Davangere 577007, India
2Department of Mechanical Engineering, University of West Attica, 12244 Athens, Greece
3Department of Chemistry, College of Science, King Khalid University, Abha 61421, Saudi Arabia
*Correspondence: sarris@uniwa.gr
Abstract:
This article comprehensively investigates the thermal performance of a ternary hybrid
nanofluid flowing in a permeable inclined cylinder/plate system. The study focuses on the effects of
key constraints such as the inclined geometry, permeable medium, and heat source/sink on the thermal
distribution features of the ternary nanofluid. The present work is motivated by the growing demand for
energy-efficient cooling systems in various industrial and energy-related applications. A mathematical
model is developed to describe the system’s fluid flow and heat-transfer processes. The PDEs (partial
differential equations) are transformed into ODEs (ordinary differential equations) with the aid of
suitable similarity constraints and solved numerically using a combination of the RKF45 method and
shooting technique. The study’s findings give useful insights into the behavior of ternary nanofluids
in permeable inclined cylinder/plate systems. Further, important engineering coefficients such as skin
friction and Nusselt numbers are discussed. The results show that porous constraint will improve
thermal distribution but declines velocity. The heat-source sink will improve the temperature profile.
Plate geometry shows a dominant performance over cylinder geometry in the presence of solid volume
fraction. The rate of heat distribution in the cylinder will increase from 2.08% to 2.32%, whereas in the
plate it is about 5.19% to 10.83% as the porous medium rises from 0.1 to 0.5.
Keywords:
cylinder and plate; porous medium; ternary nanofluid; heat source/sink; inclined geometry
1. Introduction
Nanofluids are a form of a fluid composed of a base fluid (such as water, oil or
ethylene glycol) with nanoscale particles (usually with sizes less than 100 nm) scattered
within it. The incorporation of nanoparticles into the base liquid can increase its thermal
characteristics, such as thermal conductivity and heat capacity, resulting in enhanced
thermal efficiency. Nanofluids have been proven to have better thermal characteristics
than conventional coolants, making them appealing for application in cooling systems for
electronic devices, power plants, and cars. Nanofluids can be utilized as heat-transfer fluids
in solar thermal systems, which utilize solar energy to generate heat. The increased thermal
characteristics of nanofluids can aid in the efficiency of these systems, resulting in more
energy production and reduced costs. Nanofluids can be utilized as drug-delivery systems,
allowing medications to be administered directly to specific tissues or cells. Moreover,
nanofluids have several potential uses in a variety of sectors, including lubrication systems
and heat exchangers. Their distinct features, including increased thermal conductivity,
physical qualities, and targeting capabilities, make them appealing for various industrial
and biological applications. Experiments have demonstrated that nanofluids can have
much greater thermal conductivities than the base fluid alone, with up to several-fold
improvements recorded in some circumstances. The precise improvement in thermal
performance will be determined by a number of parameters, including the kind and
Energies 2023,16, 2630. https://doi.org/10.3390/en16062630 https://www.mdpi.com/journal/energies
Energies 2023,16, 2630 2 of 18
concentration of nanoparticles, the characteristics of the base fluid, and the operating
circumstances. Many reports are made in this view, some of them listed as [
1
–
5
]. These
works conclude that nanofluids exhibit better thermal performance than base fluids.
A hybrid nanofluid is a type of fluid that mixes nanoparticles with additional additives,
such as agents or polymers, to improve its thermal efficiency beyond that of a normal
nanofluid. Babu et al. [
6
] reported that hybrid nanofluid shows better thermal conductivity
than nanofluid and viscous fluids. Yang et al. [
7
] and Esfahani et al. [
8
] also reviewed hybrid
nanofluid and agree with the results of [
6
]. Atashafrooz et al. [
9
] showed an improvement in
the thermal performance of hybrid nanofluid over viscous fluid. Recently, Adun et al. [
10
]
reviewed the synthesis, stability, thermophysical properties, and thermal distribution of
ternary nanofluids. The study revealed that the overall performance of ternary nanofluids
is better than hybrid and nanofluids.
Ternary nanofluids (TNFs) are fluids composed of a base liquid and three types of
nanoparticles. These fluids have distinct characteristics that can be used for a wide range
of possible applications. Ternary nanofluids can have higher thermal conductivity than
their base fluid, making them helpful for cooling electronic devices and converting solar
thermal energy. Ternary nanofluids, for example, can be utilized in electronic cooling to
move heat away from sensitive components, reducing overheating and damage. Ternary
nanofluids may be utilized to capture and transport heat from the sun in solar thermal
energy conversion, enhancing system efficiency. Ternary nanofluids have several poten-
tial uses, including heat transmission, power storage, increased oil extraction, medicinal
applications, and environmental services. Adun et al. [
10
] in 2021 conducted a review of
the preparation, thermophysical properties, and stability of TNFs. The study provided
a thorough examination of the latest advancements in the field and offered insights into
potential future research directions. Alharbi et al. [
11
] investigated the flow of a TNF over
an expanding cylinder while accounting for induction effects using computational ap-
proaches. Using a non-Fourier heat flux concept, Sarada et al. [
12
] studied the consequence
of exponential internal heat production on the movement of a TNF. Sharma et al. [
13
]
inspected the movement of TNFs through parallel plates with Nield boundary conditions
using numerical and LMBNN. Yogeesha et al. [
14
] explored the TNF circulation around an
unsteady permeable stretched sheet with Dufour and Soret impacts.
Geometries embedded in a porous material, such as in this topic, are of interest for
scientific and topographical purposes such as geothermal reservoirs, thermal insulation,
nuclear reactor cooling, treatment of water, carbon capture, sensors, actuators, and in-
creased oil recovery. Ullah et al. [
15
] utilized the Keller box tactic to investigate the impact
of MHD and temperature slip on viscous liquid movement across a symmetrically verti-
cally hot plate in a permeable material. Rekha et al. [
16
] investigated the effect of TPD on
temperature transmission and nanofluid circulation in the context of permeable media.
Using local thermal non-equilibrium conditions for non-Newtonian liquid circulation in-
tegrating hybrid nanoparticles, Alsulami et al. [
17
] examined heat transfer in permeable
media. Yu et al. [
18
] explored the movement of a nanofluid through a saturated porous
surface positioned on a horizontal plane while accounting for the effects of generalized
slip in a three-dimensional stagnation-point stream. Rawat et al. [
19
] investigated HNF
movement in a Darcy–Forchheimer porous medium among two parallel spinning discs
using a non-uniform HS–S and the Cattaneo–Christov model.
Because of their unique thermal characteristics, nanofluids have been investigated as
a possible coolant for heat sources and sinks (HS–S). The nanoparticles in a nanofluid can
improve the fluid’s thermal conductivity and heat convection, making it more efficient in
removing heat. Interior heat generation or absorption is difficult to accurately calculate;
nonetheless, certain simple numerical models may represent its regular behavior for utmost
physical situations. As a result, the external heat generation or absorption factor (or
heat source/sink) must be considered. This heat source/sink is widely used in cooling
electronic equipment, in many industrial processes, in the conversion of solar thermal
energy, and in automotive engines. Ahmad et al. [
20
] explored the motion of mixed
Energies 2023,16, 2630 3 of 18
convection in a radiative Oldroyd B nano liquid in the attendance of HS–S. Khan et al. [
21
]
examined the stability of the buoyancy magneto circulation of a hybrid nanofluid over a
shrinkable/stretchable perpendicular surface generated by a micropolar liquid and subject
to a nonlinear HS–S. Kumar et al. [
22
] looked into the movement of carbon nanotubes
floating in dusty nanofluid through a stretched permeable rotating disc with a non-uniform
HS–S. Ramesh et al. [
23
] inspected the impacts of ternary nano liquid with HS–S and
permeable media in a stretched divergent/convergent path. Waqas et al. [
24
] conducted the
quantitative examination for 3-dimensional bioconvection circulation of Carreau nanofluid
with HS–S and motile microorganisms.
The cylinder/plate geometry provides a large surface area for heat transfer to oc-
cur, which can lead to an improvement in the overall thermal transfer performance. The
addition of nanofluid particles to the fluid further enhances heat transfer by providing
additional thermal energy pathways and by growing the effective thermal conductivity of
the fluid. This combination provides a promising approach for improving the temperature-
transfer performance of a wide range of systems, including those used in energy generation
and cooling processes. There are several real-world applications in various sectors. Re-
frigeration and air conditioning, electricity production, solar energy systems, thermal
energy storage, and industrial operations all rely on these systems. In a Darcy–Forchheimer
movement of NF containing gyrotactic organisms under the influence of Wu’s slip over
a stretched cylinder/plate, Waqas et al. [
25
] investigated bio-convection heat radiation.
Ali et al. [
26
] made a study to discover the effects of the C-C double diffusions theory on
the bioconvective slip movement of a magneto-cross-nanomaterial on a stretching cylin-
der/plate. Waqas et al. [
27
] achieved a numerical model of the bio-convection circulation
of a non-Newtonian NF related to a stretched cylinder/plate containing floating motile
microorganisms. Selimefendigil and Öztop [
28
] examined the mixed convection nanoliquid
circulation inside a cubic container that was separated by an inner revolving cylinder
and a plate. Waqas et al. [
29
] explored the behavior of NF of type magneto-Burgers flow-
ing along with swimming motile microorganisms, which was organized by a stretching
cylinder/plate and characterized by dual variables’ conductivity.
The present study provides a novel contribution to the field of thermal manage-
ment by investigating the thermal performance of a TNF flowing in a permeable inclined
cylinder/plate system. The study focuses on the effects of the inclined geometry, porous
medium, and heat source/sink on the thermal distribution characteristics of the ternary
nanofluid, which has not been well-studied in the literature. This study represents a
significant advancement in the field of thermal management and contributes to a better
understanding of the thermal performance of ternary nanofluids in porous media over an
inclined cylinder/plate with a heat source/sink.
The aim of this investigation is to answer the following research insight questions
regarding a fluid system:
1.
How does the solid volume fraction affect the velocity and temperature profile of
the system?
2. How does the fluid profile change as the porous constraint value is increased?
3. What is the influence of the heat source/sink on the thermal profile of the system?
To answer these questions, the investigation will analyze the behavior of the fluid system
under varying conditions and parameters. The findings of this investigation will provide
valuable insights into the impact of different factors on the behavior of the system, which
can be useful in optimizing the performance and efficiency of the system in various practical
applications. By understanding the complex dynamics of the fluid system, we can develop
more effective solutions for a wide range of industrial, scientific, and engineering applications.
2. Mathematical Formulation
The current study examines a steady, laminar flow of a ternary nanofluid in two di-
mensions over an inclined cylinder/plate system that includes a porous medium and a heat
source/sink. The coordinates of the physical model are presented in Figure 1.
Vz1w
denotes
Energies 2023,16, 2630 4 of 18
the reference velocity;
r1
and
z1
represent the radial and axial coordinates of the cylinder.
The temperature of the system is denoted by
Tw1
and far-field temperature is represented by
T∞(Tw1>T∞)
. Further, external forces and pressure gradients are assumed to have no influ-
ence on the system. Under these assumptions, the mathematical expressions for continuity,
momentum, and thermal transfer for the ternary nanofluid flow in the presence of a porous
medium and a heat source/sink are stated as follows (see [26,27,29–32]).
∂(r1Vz1)
∂z1+∂(r1Vr1)
∂r1=0 (1)
Vz1∂(Vz1)
∂z1+Vr1∂(Vz1)
∂r1=νmn f ∂2V z1
∂r2
1+1
r1
∂Vz1
∂r1+(ρβ)mn f g(T1−T∞)cos ς
ρmn f
−νmn f
K∗
1Vz1
(2)
Vz1∂(T1)
∂z1+Vr1∂(T1)
∂r1=αmn f ∂2T1
∂r2
1
+1
r1
∂T1
∂r1!+Q1
(ρCp)mn f
(T1−T∞)(3)
Figure 1. Pictorial representation of the problem.
With boundary conditions (see [25]).
Vz1=Vz1w=U∗
0z1
l
Vr1=0
T1=Tw1
at r1=R1(4)
Vz1→0, T1→T∞r1→∞(5)
In the above expressions,
Vz1
,
Vr1
are the velocity component along
z1
and
r1
direction;
ν
is the kinematic viscosity defined by
ν=µ
ρ
(
µ
- Dynamic viscosity and
ρ
is density);
K∗
1
denotes porous medium permeability;
β
is the thermal expansion factor;
g
acceleration due to
gravity;
ς
is the inclination angle;
α=k
ρCp
is the thermal diffusivity of the fluid; (
k
is thermal
conductivity and Cp is the specific heat); Q1is the heat generation/absorption coefficient.
Energies 2023,16, 2630 5 of 18
For similarity variables (see [33]).
ψ=qVz1wνfz1R1f(η),Vz1=1
r1
∂ψ
∂r1,Vr1=−1
r1
∂ψ
∂z1
η=qVz1w
νfz1r2
1−R2
1
2R1,θ=T1−T∞
Tw1−T∞
(6)
By introducing Equation (6) into the Equations (1)–(3) and boundary conditions (4)–(5).
It is reduced into the following form:
h(1+(2δ1)η)f000+(2δ1)f00 i
B1B2−f02+f f 00 −Pm
B1B2f0+B3
B2γ1θcos ζ=0 (7)
kmn f [(1+2δ1η)θ00 +2δ1θ0]
kfPrB4+fθ0+HSS
B4θ=0 (8)
And reduced boundary conditions:
f(η),f0(η),θ(η)=0, 1, 1atη=0 (9)
f0(∞)=0, θ(∞)=0asη→∞(10)
From the Equations (7)–(10), the controlling parameter
δ1=rνfl
U∗
0R2
1
is the Curva-
ture parameter (
δ1=
0 denotes Plate geometry and
δ1>
0 denotes Cylinder geometry);
Pm=νfl
U∗
0K∗
1
is the porosity constraint;
γ1=Gr
Re2=gβ(Tw1−T∞)l
Vz1wU∗
0
is the Buoyancy param-
eter/Mixed convection parameter;
ζ
is the inclined angle;
Pr =νf
αf
is the Prandtl
number;
HSS =Q1l
U∗
0(ρCp)f
is the heat source/sink parameter;
Gr =gβ(Tw1−T∞)z13
ν2
f
is the
local Grashof number; and Re =Vz1wz1
νfis the local Reynolds number. Further,
B1=(1−φ1)2.5(1−φ2)2.5(1−φ3)2.5,B2=(1−φ3)(1−φ2)h(1−φ1)+ρS1φ1
ρfi+ρS2φ2
ρf
+ρS3φ3
ρf
,
B3=(1−φ3)(1−φ2)h(1−φ1)+ρS1φ1βS1
βfρfi+ρS2φ2βS2
βfρf+ρS3φ3βS3
βfρf
, and
B4=(1−φ3)
(1−φ2)h(1−φ1)+ρS1CpS1φ1
ρfCp fi+ρS2C pS2φ2
ρfCp f+ρS3C pS3φ3
ρfCp f.
For engineering coefficients and their reduced form:
C f =∂Vz1
∂r1r1=R1
µmn f
ρfVz1w2andNu =∂T1
∂r1r1=R1
−z1kmn f
kf(Tw1−T∞)(11)
C f =f00 (0)
B1(Re)0.5 and Nu =−kmn f θ0(0)(Re)0.5
kf
(12)
The thermophysical properties of ternary nanofluid are provided below (see [34]).
µmn f =µf/(1−φ1)2.5(1−φ2)2.5(1−φ3)2.5 (13)
ρmn f = (1−φ3) (1−φ2)"(1−φ1)+ρS1φ1
ρf#+ρS2φ2
ρf!+ρS3φ3
ρf!ρf(14)
(ρβ)m n f = (1−φ3) (1−φ2)"(1−φ1)+ρS1βS1φ1
ρfβf#+ρS2βS2φ2
βfρf!+ρS3βS3φ3
βfρf!βfρf(15)
(ρCp)mn f = (1−φ3) (1−φ2)"(1−φ1)+ρS1CpS1φ1
ρfCpf#+ρS2CpS2φ2
ρfCpf!+ρS3CpS3φ3
ρfCpf!ρfCpf(16)
Energies 2023,16, 2630 6 of 18
kn f =1
kf−1ks1+2kf−φ12[kf−ks1]
ks1+2kf+φ1[kf−ks1]
khn f =1
kn f −1ks2+2kn f −φ22[kn f −ks2]
ks2+2kn f +φ2[kn f −ks2]
kmn f =khn f ks3+2khn f −2φ3[khn f −ks3]
ks3+2khn f +φ3[kn f −ks3]
(17)
3. Numerical Procedure and Validation
The governing Equations (1)–(3) along with the boundary conditions (4) and (5) can be
very challenging to solve. To simplify the problem, similarity variables (6) are often used to
reduce the governing equations into a set of ordinary differential equations (ODEs) without
compromising the originality of the equations. However, the reduced ODEs (7) and (8)
and boundary conditions (9) and (10) remain highly nonlinear and two-point in nature,
making them difficult to solve directly. One effective approach to overcome this challenge
is to transform the reduced ODEs into a first-order system. This technique simplifies
the problem and provides a more straightforward method to solve the equations. This
approach has been widely used in various fields of science and engineering and has been
proven to be effective in solving complex problems. By converting the reduced ODEs into
a first-order system, we can also analyze the stability and behavior of the system more
effectively. This is particularly useful in understanding the dynamics of complex systems,
where the behavior can be affected by various parameters and boundary conditions. For
this consider, [f,f0,f00 ]=[p,q,r]
[θ,θ0]=[p1,q1].
By substituting these terms in the resultant equations, the equation becomes
f00 0=−B1B2
(1+2δ1η)2δ1r
B1B2+rp −(q)2−Pm
B1B2q+B3
B2γ1p1cos ζ(18)
θ00 =−kfPrB4
kmn f (1+2δ1η) kmn f 2δ1q1
kfPrB4+pq1+HSS
B4p1!(19)
and the boundary conditions become
p(0)=0
q(0)=1
r(0)=ϑ1
p1(0)=1
q1(0)=ϑ2
(20)
The terms p,q,r,p1&q1represent f,f0,f00 ,θ&θ0respectively; δ1,γ1,ζand HSS denote
parameters.
The Runge–Kutta–Fehlberg (RKF-45) technique is implemented to crack the system of
equations represented by (18) and (19), as well as the boundary conditions stated in Equa-
tion (20). A shooting strategy is used to identify the unknown variables in Equation (20):
the thermophysical properties of the nanofluid as indicated in Equations (13) to (17), ther-
mophysical characteristics stated in Table 1, along with the values of the thermophysical
properties of the nanoparticles mentioned in Table 1and setting the initial values of the
parameters to
Pm=γ1=
0.1,
ζ=
30
0
,
φ1=φ2=φ3=
0.01&
HSS =
0.5. The step size is
set to 0.01, and the error tolerance is set to 10
−6
, ensuring that the solution is reliable and
well-resolved.
Energies 2023,16, 2630 7 of 18
Table 1. Thermophysical characteristics of nanoparticles and base liquid are provided by (see [35]).
Properties H2O Al2O3TiO2CuO
ρkg/m3997.1 3970 4250 6320
Cp (J/kgK)4179 765 686.2 531.8
k(W/mK)0.613 40 8.9538 76.5
β×10−5(1/K)21 0.85 0.9 1.8
The algorithm of the RKF-45 strategy is given below.
The 6 step sizes of the RKF-45 scheme:
h1f1(xi,yi)=k1(21)
h1f1xi+1
4h1,yi+1
4k1=k2(22)
h1f1xi+3
8h1,yi+3
32k1+9
32k2=k3(23)
h1f1xi+12
13h1,yi+1932
2147k1−7200
2147k2+7296
2147k3=k4(24)
h1f1xi+h1,yi+439
216k1−8k2+3680
513 k3−845
4104k4=k5(25)
h1f1xi+1
2h1,yi+2k2−8
27k1−11
40k5−3544
2565k3−1859
4104k4=k6(26)
yi+1=−1
5k5+25
216k1+1408
2565k3+2197
4104k4+yi(27)
zi+1=2
55k6+16
135k1+6656
12825k3+28561
56430k4−9
50k5+yi(28)
The present numerical scheme is validated with the existing available literature and
found to be the best match for each other (see Table 2).
Table 2.
Comparison of present numerical scheme with the work of [
36
] in the absence of
δ1
,
γ1
,
B1
,
and B2.
Parameter [36] Present Study
PmAnalytical
(−f00 (0))
SRM
(−f00 (0))
RKF-45
(−f00 (0))
1 1.41421356 1.41421356 1.4142375
2 1.73205081 1.73205081 1.7320517
5 2.44948974 2.44948974 2.4494897
10 3.31662479 3.31662479 3.3166247
4. Results and Discussion
The current inquiry focuses on the numerical solution of reduced Ordinary Differential
Equations (ODEs) and Boundary Conditions (BCs), as well as the graphical display of the
findings. The effects of numerous dimensionless factors on thermophysical characteristic
profiles are carefully investigated and assessed. The findings are provided for two different
geometries: cylinder
(δ1=0.3)
and plate
(δ1=0)
. The use of efficient mathematical
software allows for accurate and precise answers, allowing for the investigation of the
impact of the many dimensionless factors on the system’s thermophysical properties. This
Energies 2023,16, 2630 8 of 18
gives a thorough knowledge of the nanofluid’s behavior under various situations, offering
significant insights into its performance and behavior.
Figures 2and 3show the variation of velocity and temperature profiles in the presence
of porous constraint
Pm
. As seen in Figure 2, increasing the value of the porosity constraint
Pm
causes the fluid velocity profile to drop. The porous medium acts as a barrier to fluid
flow, reducing fluid velocity and increasing the thickness of the Momentum Boundary Layer
(MBL). This decrease in velocity emphasizes the porous constraint’s inhibitory influence on
fluid flow. Figure 3shows the fluctuation of the thermal profile in the presence of a porous
constraint
Pm
. The inclusion of a porous medium improves system thermal performance
by increasing the thickness of the Thermal Boundary Layer (TBL), which advances thermal
dispersion. It is detected from the figure that velocity is more in the cylinder case than the
plate in the case of Pm, but the reverse trend is observed in the thermal profile.
Figure 2. The role of porosity parameter on velocity profile (keeping γ1=0.1, ζ=300,φ1=φ2
=φ3=0.01&HSS =0.5).
Figures 4and 5are plotted to show the influence of a mixed convection constraint
γ1
on velocity and thermal profiles, respectively. The rise in the values of
γ1
will improve
the velocity profile (see Figure 4) and decline the thermal profile (see Figure 5). When
the magnitude of the mixed convection coefficient is greater, it implies that the thermal
buoyancy force in the fluid system is more dominating. This superiority of the thermal
buoyancy force results in an increase in the fluid’s velocity profile. It operates perpendicular
to the direction of flow. When the thermal buoyancy force is greater, it pushes the fluid
movement, resulting in an increase in fluid velocity that decreases the temperature. From
the diagram it is observed that velocity is higher in the cylinder than the plate, and
temperature is higher in the plate than the cylinder in the presence of γ1.
Energies 2023,16, 2630 9 of 18
Figure 3. The role of porosity parameter on temperature profile (keeping γ1=0.1, ζ=300,
φ1=φ2=φ3=0.01&HSS =0.5).
Figure 4. The role of mixed convection parameter on velocity profile (keeping Pm=0.1, ζ=300,
φ1=φ2=φ3=0.01&HSS =0.5).
Energies 2023,16, 2630 10 of 18
Figure 5. The role of mixed convection parameter on temperature profile (keeping Pm=0.1,
ζ=300,φ1=φ2=φ3=0.01&HSS =0.5).
The influence of
ζ
on velocity and thermal profiles is displayed in Figures 6and 7,
respectively. At
ζ=
0, the system experiences maximum buoyancy impact. This is due to
the presence of
γ1
with
ζ
and
cos(ζ=0)→1
. As the value of inclination angle reaches to
π
2
, the inclined angle term tends to zero. This leads the minimum buoyancy impact on the
flow system. When the inclination angle increases, the influence of the buoyancy force on
fluid flow decreases, leading to a decrease in fluid velocity. This decrease in fluid velocity
enhances the thermal distribution and leads to improved thermal performance. This effect
is particularly relevant in applications where thermal management is critical, such as in
heat exchangers, cooling systems, and electronic devices. By controlling the inclination
angle of the fluid system, it is possible to optimize the thermal performance and enhance
the overall efficiency of the system. From the diagrams it is further observed that velocity
is less in the plate than the cylinder, and thermal distribution is greater in the case of the
plate than the cylinder in the presence of ζ.
Figure 8represents the nature of thermal profile in the presence of the HS–S constraint
HSS
. The escalation in
HSS
will enhance the thermal profile. The heat sink
(HSS <0)
will
remove the temperature from the fluid and as a result, temperature decreases. Heat source
(HSS >0)
, which generates the temperature from the surface of the geometry, leads to
improvement in the temperature. From the illustration, it is further observed that thermal
distribution is more in the case of the plate than the cylinder. Figures 9and 10 show the
alteration of velocity and thermal profiles in the presence of solid volume fraction
φ3
. The
momentum and thermal boundary layers are gradually improving by escalating the
φ3
value. This leads the fluid flow slowly, which leads velocity decreases. The velocity of
the plate is very lower than the velocity of the cylinder (see Figure 9). The enhancement
in the TML makes the system deliver more temperature with the improved values of
φ3
.
Distribution of temperature is more in the case of the plate than the cylinder over
φ3
(see
Figure 10).
Energies 2023,16, 2630 11 of 18
Figure 6. The role of inclined angle on velocity profile (keeping Pm=γ1=0.1, φ1=φ2=φ3
=0.01 & HSS =0.5).
Figure 7. The role of inclined angle on temperature profile (keeping Pm=γ1=0.1, φ1=φ2=φ3
=0.01 & HSS =0.5).
Energies 2023,16, 2630 12 of 18
Figure 8. The role of heat source/sink parameter on temperature profile (keeping Pm=γ1=0.1,
ζ=300&φ1=φ2=φ3=0.01).
Figure 9. The role of solid volume fraction on velocity profile (keeping Pm=γ1=0.1, ζ=300,
φ1=φ2=0.01 & HSS =0.5).
Figure 11 shows the important engineering coefficient
C f
on
Pm
for various values of
φ3
. The improved values of
φ3
will enhance the MBL thickness, and the presence of
Pm
will
make the fluid flow slowly. The presence of these two factors will enhance the surface drag
force in the system. Further, it is observed that C f is more in the cylinder than the plate.
Energies 2023,16, 2630 13 of 18
Figure 10. The role of solid volume fraction on temperature profile (keeping Pm=γ1=0.1,
ζ=300,φ1=φ2=0.01 & HSS =0.5).
Figure 11.
Variation of skin friction on porosity parameter for increment in solid volume fraction
(keeping γ1=0.1, ζ=300,φ1=φ2=0.01 & HSS =0.5).
Figure 12 displays the variation of
Nu
on
HSS
for different values of
φ3
. The improve-
ment in the
φ3
will thicken the TBL, and the improvement in
HSS
enhances the thermal
distribution rate. Therefore, in the presence of these two factors, the thermal distribution
rate enhances.
Energies 2023,16, 2630 14 of 18
Figure 12.
Variation of Nusselt number on heat source/sink parameter for increment in solid volume
fraction (keeping Pm=γ1=0.1, φ1=φ2=0.01 & ζ=300).
Table 3displays the improvement in the Nusselt number percentage in the presence
and absence of a solid volume fraction for various dimensionless constraints. From the
table it is observed that, as the porous constraint rises from 0.1 to 0.5, the rate of heat
distribution in the cylinder will increase from 2.08% to 2.32%, whereas in the plate it is
about 5.19 % to 10.83%. In the presence of a mixed convection constraint (0.5 to 1.0),
Nu
will
decline by 2.08% to 1.98% in the cylinder but in the plate, it will decline by 5.19% to 2.74%.
As the inclination angle varies from 0
◦
to 60
◦
,
Nu
will increase slightly from 2.08% to 2.09%
in the cylinder, but in the plate it improves from 5.13% to 5.38%. In the presence of a heat
source/sink, the rate of heat distribution augments by 0.14% to 0.81% in the cylinder, and
in the plate it decreases from 3.37% to 2.05 %. In all the cases, the addition of nanoparticles
shows gradual improvement in the process of the rate of thermal distribution than in the
absence of nanoparticles.
Table 3. Computational values of Nu % for various dimensionless constraints.
Parameters Cylinder (δ1= 0.3)Plate (δ1= 0.0)
Pmγ1ζHSS
Nu|φ1=φ2=φ3=0.01 −Nu|φ1=φ2=φ3=0
Nu|φ1=φ2=φ3=0
×
100
Nu|φ1=φ2=φ3=0.01 −Nu|φ1=φ2=φ3=0
Nu|φ1=φ2=φ3=0
×
100
0.1
0.1 30◦0.5 2.08% 5.19%
0.3
2.20% 7.54%
0.5
2.32% 10.83%
0.1
0.1 30◦0.5 2.08% 5.19%
0.5 2.03% 3.84%
1.0 1.98% 2.74%
0.1
0.1 0◦0.5 2.08% 5.13%
30◦2.08% 5.19%
60◦2.09% 5.38%
0.1
0.1 30◦−0.2 0.14% 3.37%
0 0.27% 2.96%
0.2 0.81% 2.05%
Energies 2023,16, 2630 15 of 18
5. Final Remarks
By exploring the thermal performance of a ternary hybrid nanofluid flowing in a
permeable inclined cylinder/plate system, the current work makes a unique addition to
the field of thermal management. The research looks at the impacts of inclined geometry,
porous media, and heat source/sink.
The major findings in the study disclose that in the presence of ternary nanofluid
(contains
Al2O3
,
TiO2
,
CuO
and base fluid as
H2O
), the velocity of the fluid decreases with
improved values of porous factor while the temperature distribution increases. A change
in the angle of inclination and heat source/sink will improve the thermal profile. The
addition of nanoparticles in the fluid will decrease the velocity but improve the thermal
distribution. The rate of thermal distribution percentage will always be greater in the
presence of a ternary nanofluid than a base fluid (water) over all the constraints. Further, it
is concluded that the rate of thermal distribution percentage is higher in a ternary nanofluid
with water as the base fluid compared to pure water in a plate geometry than cylinder
geometry. The present work finds its importance in the field of heat exchangers, cooling
systems, renewable energy systems, and heating ventilation and air conditioning.
Author Contributions:
Conceptualization, J.K.M. and B.C.P.; methodology, I.E.S.; software, A.A and
J.K.M.; validation, B.C.P., I.E.S. and A.A.; formal analysis, J.K.M.; investigation, B.C.P. and I.E.S.;
resources, A.A.; data curation, J.K.M.; writing—original draft preparation, B.C.P. and A.A.; writing—
review and editing, I.E.S., J.K.M. and B.C.P.; visualization, A.A.; supervision, I.E.S. and B.C.P.; project
administration, I.E.S.; funding acquisition, A.A. All authors have read and agreed to the published
version of the manuscript.
Funding:
The authors would like to extend his appreciation to the Deanship of Scientific Research at
King Khalid University, Saudi Arabia, for funding this work through the Research Group Program
under grant No. RGP.2/218/44.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are available on reasonable request.
Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature
Symbols:
Vz1wReferencevelocity ms−1
r1Radialcoordinate (m)
R1Cylinderradius (m)
z1Axialcoordinate (m)
Tw1Walltemperature (K)
T∞Far −fieldtemperature (K)
Vz1,Vr1Velocitycomponents ms−1
Cp Specificheat m2s−2K−1
kThermalconductivity kgms−3K−1
Q1Heatgeneration/absorptioncoefficient kgm−1s−3K−1
PmPorosityconstraint (−)
K∗
1Porousmediumpermeability m2
Pr Prandtlnumber (−)
HSS Heatsource/ sin kparameter (−)
Gr LocalGrashofnumber (−)
Re LocalReynoldsnumber (−)
C f Skinfriction (−)
Nu Nusseltnumber (−)
Energies 2023,16, 2630 16 of 18
Greek symbols
νKinematicviscosity m2s−1
µDynamicviscosity kgm−1s−1
ρDensity kgm−3
βThermalexpansionfactor K−1
gAccelerationduetogravity ms−2
ςInclinationangle (−)
αThermaldiffusivity m2s−1
θDimensionlesstemperature (−)
ψStreamfunction (−)
ηSimilarityvariable (−)
δ1Curvatureparameter (−)
γ1Buoyancyparameter/Mixedconvectionparameter (−)
φSolidvolumefraction (−)
Subscripts
S1, S2, S3 Solid particles
mn f Modified nanofluid
hn f Hybrid nanofluid
n f Nanofluid
fFluid
Abbreviations
ODEs Ordinary differential equations
PDEs Partial differential equations
TNF Ternary nanofluids
HS–S Heat source and sink
NF Nanofluid
C–C Cattaneo–Christov
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