Thermal energy analysis of multi-walled carbon nanotubes-Fe 3 O 4 /H 2 O flow over non-uniformed surface with Darcy

Abstract

In this study, a new cavity shape was filled with an extension multi-walled carbon nanotubes-Fe₂O₃/H₂O nanofluid under a constant magnetic field. The Darcy–Forchheimer model is used to account for the inertial impact of advection in the porous layer while maintaining the laminar and incompressible nature of the nanofluid flow. The dimensionless version of the governing equations is used to describe the issue and the finite element approach is used to resolve it. Through this complex geometry, various thermophysical factors such as Rayleigh number $(10^{2} \leq Ra \leq 10^{5})$ , Hartmann number $(0 \leq Ha \leq 100)$ , and nanoparticle concentration are considered $(0.02 \leq ϕ \leq 0.08)$ . The porous layer's numerous characteristics are also explored. For example, its porosity $(0.2 \leq ε \leq 0.8)$ and Darcy number $(10^{- 5} \leq Da \leq 10^{- 2})$ , which indicates the permeability of the porous medium. The content of the hybrid nanofluid is considered to be Newtonian, stable, incompressible, and following a constant Prandtl number for the base fluid $(\Pr = 6.2)$ . Calculations are made according to the finite element method. The results of this work are presented in terms of rheology, isotherms, entropy generation, and mean Nusselt numbers. They have demonstrated that increasing the Rayleigh and Darcy numbers improve heat transfer in the enclosure.

Keywords

Hybridity nanofluid cavity carbon nanotubes Darcy–Forchheimer heat transfer

Introduction

In recent years, many academics have worked on improving heat transmission by altering the thermo-physical parameters of the working fluid. Because of its superior thermal conductance and convection heat transference coefficient compared to the conventional liquid, nanofluid, an engineered dispersion suspension of nanoparticles in a conventional liquid, has proven its remarkable potential applicability in many industrial implementations. Energy management is one of the most important issues in our society. It is, therefore, necessary to improve our energy consumption to protect our natural resources. Convective evaporation is involved in energy storage, generation, and cooling. This motion causes a mixture of fluid particles. The process of heat transfer is called natural convection. In their investigation of the convection of aluminum and water hybrid nanofluid (HNF) in a cavity, Bordaniani et al.¹ observed the impact of magnetic fluidity on the effectiveness of heat transfer. A water nanofluid Al₂O₃ was explored by Pure et al.² in an inclined square cavity. Using a revolving cylinder, Mebarek-Oudina et al.³ examined the behavior of convective heat in a hollow filled with pores. They considered the magnetic field of aqueous Al₂O₃ HNFs in a motivated square cavity.

Carbon nanotubes (CNTs) were discovered and have since developed into one of the most intensely investigated nanostructures, perhaps contributing to the nanotechnology revolution due to their superior thermal, physical, optical, and electrical capabilities. A Japanese scientist discovered CNTs. Because of CNTs’ extraordinarily high thermal conductivity, many scientists have changed their emphasis to better comprehend the basic properties of CNTs and their applicability in heat transference systems. Furthermore, Selimefendigil and Chamkha⁴ studied the properties of convection and entropy generation in a magnetic field container. A vertical beam was used to divide the room into two parts. The higher and lower lips were fixed, whereas the left and right lips were warm and icy, respectively. On the other hand, Selimefendigil and Öztop⁵ and Hamzah et al.⁶ used the finite element method to study the convection properties and entropy generation in a split cavity filled with CNTs/aqueous HNFs under magnetic flux. A Boltzmann grid technique was used to investigate the impact of cooling fan size on MHD natural convection in a partly differentiated box in a work by Gangawan and Bharti.⁷ Furthermore, results of thermophysical characteristics on spontaneous laminar convection in containers where HNF experiments are monitored using the nuance approach are now accessible, according to research by Ismail.⁸

Numerous scholars have examined and reviewed the usage of mono nanofluids in heat transfer applications. The reports regarding the nanofluid have been published, subjected to the Cattaneo–Christov magnetized nanofluid model,⁹ the radiating nanofluid with a magnetic dipole,¹⁰ and convective heat and mass transfer in a nanofluid flow with the activation energy.¹¹ The use of HNF is a relatively new notion that has lately attracted greater study interest. The use of HNFs in heat transference implementations such as plate heat exchangers, passive heatsinks, heat pumps, photo-voltaic modules, free convection housings, cooling and air-conditioning systems, vaporization implementations, impeller coolers, and thermal energy storage systems is thoroughly discussed. Abstract analysis of a porous crack occupied by a copper/water HNF with LBM was performed by Hoseinpour et al.¹² The HNF containing Ti6Al4 V and AA7075 nanoparticles with a base fluid sodium alginate over a stretching sheet in a porous media with magnetic effect is studied by Alsulami et al.¹³ Heat transfer analysis in the three-dimensional unsteady magnetic fluid flow of water-based ternary HNF is reported by Sarada et al.¹⁴ Subsequently, Sarada et al.¹⁵ reported the convective boundary layer flow of ternary HNF (water-based graphene-CNT-Silver) with activation energy. Recently, Varun Kumar et al.¹⁶ analyzed the Arrhenius activation energy on kerosine oil HNF with manganese zinc ferrite and nickel zinc ferrite as nanoparticles over a curved stretchable surface.

The higher the porosity, the better the overall entropy generation. Abu Abdulsahib et al.¹⁷ considered the effect of different thermophysical factors on hydrothermal generation and entropy within a new porous cavity. Kasaypur et al.¹⁸ performed an analysis of heat transfer by convection and entropy generation in a container with multi-walled carbon nanotubes (MWCNT)–MgO/water HNF-charged solid-cooling element. More Rahimi et al.¹⁹ completed a mathematical analysis of natural convection and entropy generation in a container, which is to be shipped with double dividers of CNTs and liquid nanotubes equipped with cold and hot baffles, while the scheme has succumbed to LBM. Ikram et al.²⁰ deliberated the optimization of entropy generation contained by a square crack overloaded with an Ag/water HNF. A square cavity filled with HNF that contains multiple configurations of a solid corrugated conductor. Brahimi and co-workers²¹ studied the thermal behavior of a porous container filled with a water-based Ag–MgO HNF in the attendance of a fixed magnetic field. A parametric study by Hua et al.²² was carried out on a triangular cavity of a porous nature with an undulating inclined side. The parametric influence can be classified into two categories for this work. One on the structural aspects such as shape and size and the other hand the parameters influencing the physical nature.

CNTs and magnetic nanocomposites (Fe₃O₄) have attracted great interest in recent years due to their interesting properties, especially at the nanoscale.^23,24 Sajadi et al.²⁵ used MWCNT–Fe₃O₄/water hybrid nanotube Hydrothermal assay for MHD natural convection in a porous environment using Boltzmann's double-clamp MRT technique inside a square bore. The working fluid composed of MWCNT, Fe₃O₄ nano-solids, and water is considered Newtonian and incompressible. Small et al.²⁶ conducted an experimental study to compare the heat-enhancing properties of different types of HNFs. And MWCNT-Fe₃O₄ HNF_. Shen et al.²⁷ studied the magnetic effect on improving the photothermal energy conversion performance of the MWCNT/Fe₃O₄ HNF. Their study aims to experimentally determine the effect of external magnetic forces on the photothermal energy conversion efficiency of the MWCNT/Fe₃O₄ HNF with variable magnetic forces. Zafar et al.²⁸ studied the thermophysical properties and stability of water-based magnetite (Fe₃O₄₎ HNFs coated with MWCNT using a multi-level sensor coupled with an artificial neural network. Al-Kulaibi and co-workers²⁹ conducted an experimental investigation to evaluate the efficacy of water-based MWCNT/Fe₃O₄ HNFs for cooling plate heat exchangers.

Several parameters showed that MWCNT + Fe₃O₄/water nanofluids improved plate heat exchanger performance. Increases in coolant flow velocity and nanoparticle concentration both contribute to amplification. The main objective of our research is to examine rectangular cavities. In addition, this effort aims to discuss the natural convective heat transfer of nanofluids (MWCNT-Fe₃O₄/H₂O). The effect of different control variables such as $Ra$ , $Da$ , and $Ha$ and the pore numbers on the evaporation and heat transfer properties were studied. The application area of this work is a new system of some exterior finned condensers which are designed to contain barriers and porous media. This study is about the performance of geometry and the influence of porous media with specific shapes. In the future, we try to introduce some additional statistical and reliability properties of mixing models of Lindley that have been studied by References,^30–34 such as the probability-generating function, and factorial moment-generating function to study the transfer of heat and matter in porous media. The selected numerical method, namely the Galerkin finite element method (GFEM) is implemented to provide the numerical finding.^35,36 However, the high-accuracy meshless numerical methods^37,38 can become the range of selections as the future works for the Darcy-Forchheimer model.

Problem explanation

Figure 1 displays the analyzed geometry. It displays a 2D rectangle. The walls that are entirely insulated are shaded. Additionally, while maintaining temperatures at Th and Tc, respectively, the sides are provided with hot and cold sections, and the other components are similarly segregated. B applies a constant magnetic field in the x direction. HNF (MWCNT-Fe₃O₄/H₂O), a Newtonian, incompressible substance with laminar flow and no viscous dissipation, fills the hollow. Table 1 is a list of the thermophysical characteristics of HNF.

Figure 1.

(a) The somatic design and (b) the network problem.

Table 1.

Thermophysical properties of basic liquid and MWCNT-Fe₃O₄ /H₂O nanoparticles (50%–50%).^39,40

Material	$C_{p} (J / kg \times K)$	$k (W / m \times k)$	$ρ (kg / m^{3})$	$β \times 10^{- 5} (K^{- 1})$	$σ (s / m)$
Water	4179	0.613	997.1	21	5.5 × 10⁻⁶
Fe₃O₄	670	6	5810	1.3	2.5 × 10⁴
MWCNT	711	3000	2100	4.2	1.9 × 10⁻⁴

The main thermo physics of the improper liquid (water) and the auxiliary nanoparticles are in Table 1.

Mathematical model

For numerical modeling of porous media, the Darcy–Brinkman–Forchheimer model is used. For predicting high-velocity flow in porous media, especially in the vicinity of gas wells,⁴¹ the governing equations given in Cartesian coordinates can be written in the dimension model.^42,43

In the HNF:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,

(1)

u \frac{\partial u}{\partial x} + v \frac{\partial v}{\partial y} = - \frac{1}{ρ_{h n f}} \frac{\partial P}{\partial x} + ν_{h n f} (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}),

(2)

u \frac{\partial u}{\partial x} + v \frac{\partial v}{\partial y} = - \frac{1}{ρ_{h n f}} \frac{\partial P}{\partial y} + ν_{h n f} (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}) + β_{h n f} g (T - T_{a v g}) + \frac{σ_{h n f}}{ρ_{h n f}} B_{0}^{2} v,

(3)

(ρ c_{p})_{h n f} (u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y}) = k_{h n f} (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}})

(4)

In a porous area:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,

(5)

\frac{1}{ε^{2}} (u \frac{\partial u}{\partial x} + v \frac{\partial v}{\partial y}) = - \frac{1}{ρ_{h n f}} \frac{\partial P}{\partial x} + \frac{ν_{h n f}}{ε} (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}) - ν_{h n f} \frac{u}{K} - \frac{1.75}{\sqrt{150} ε^{3 / 2}} \frac{1}{\sqrt{K}} u \sqrt{u^{2} + v^{2}},

(6)

\frac{1}{ε^{2}} (u \frac{\partial u}{\partial x} + v \frac{\partial v}{\partial y}) = - \frac{1}{ρ_{h n f}} \frac{\partial P}{\partial y} + \frac{ν_{h n f}}{ε} (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}) - ν_{h n f} \frac{v}{K} - \frac{1.75}{\sqrt{150} ε^{3 / 2}} \frac{1}{\sqrt{K}} v \sqrt{u^{2} + v^{2}} + β_{h n f} g (T - T_{a v g}) + \frac{σ_{h n f}}{ρ_{h n f}} B_{0}^{2} v,

(7)

(ρ c_{p})_{h n f} (u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y}) = k_{h n f} (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}}),

(8)

where u is the velocity component in the horizontal vector, v is the velocity component in the vertical vector,

ε

is its porosity,

P

is pressure,

B_{0}

is the strength of the magnetic field,

ρ_{h n f}

the density of HNF,

v_{h n f}

is the kinematic viscosity of HNF,

β_{h n f}

is the thermal expansion coefficient of HNF,

σ_{h n f}

is the electrical conductivity of HNF, and

(ρ c_{p})_{h n f}

is the nanofluid heat capacitance. Meanwhile, K is the permeability. The above-governing equations are transformed into non-dimensional equations using the following variables^44,45:

X = \frac{x}{L}, Y = \frac{y}{L}, U = \frac{u L}{α_{b f}}, V = \frac{v L}{α_{b f}}, θ = \frac{T - T_{f}}{T_{h} - T_{f}}, P = \frac{(p + ρ_{b f} g_{y}) L^{2}}{ρ_{b f} α_{b f}^{2}},

(9)

Ra = \frac{β_{b f} g (T_{h} - T_{f}) L^{3}}{α_{b f} v_{b f}}, Ha = L B_{0} \sqrt{\frac{σ_{b f}}{μ_{b f}}}, Da = \frac{K}{L^{2}}, \Pr = \frac{v_{b f}}{α_{b f}} .

(10)

where

Ra

is the Rayleigh number,

Ha

is the Hartmann number,

Da

is the Darcy number, and

\Pr

is the Prandtl number.

The non-dimensional equations are as given in the following structures:

\frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} = 0,

(11)

U \frac{\partial U}{\partial X} + V \frac{\partial V}{\partial Y} = - \frac{\partial P}{\partial X} + Pr \frac{ρ_{f}}{ρ_{h n f}} \frac{μ_{h n f}}{μ_{f}} (\frac{\partial^{2} U}{\partial X^{2}} + \frac{\partial^{2} U}{\partial Y^{2}}),

(12)

U \frac{\partial U}{\partial X} + V \frac{\partial V}{\partial Y} = - \frac{\partial P}{\partial Y} + Pr \frac{ρ_{f}}{ρ_{h n f}} \frac{μ_{h n f}}{μ_{f}} (\frac{\partial^{2} V}{\partial X^{2}} + \frac{\partial^{2} V}{\partial Y^{2}}) - Ra \Pr \frac{ρ_{f}}{ρ_{h n f}} [1 - ϕ + ϕ \frac{{(ρ β)}_{h n f}}{{(ρ β)}_{f}}] θ,

(13)

U \frac{\partial θ}{\partial X} + V \frac{\partial θ}{\partial Y} = \frac{α_{h n f}}{α_{b f}} (\frac{\partial^{2} θ}{\partial X^{2}} + \frac{\partial^{2} θ}{\partial Y^{2}}) .

(14)

A dimensional equation in the porous region can be formulated in the following system:

\frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} = 0,

(15)

\frac{1}{ε^{2}} \frac{ρ_{h n f}}{ρ_{b f}} (U \frac{\partial U}{\partial X} + V \frac{\partial V}{\partial Y}) = - \frac{\partial P}{\partial X} + \frac{1}{ε} \frac{ν_{h n f}}{ν_{b f}} Pr (\frac{\partial^{2} U}{\partial X^{2}} + \frac{\partial^{2} U}{\partial Y^{2}}) - \frac{ν_{h n f}}{ν_{b f}} \frac{\Pr}{Da} - \frac{1.75}{\sqrt{150} ε^{\frac{3}{2}}} \frac{1}{\sqrt{Da}} \sqrt{U^{2} + V^{2}} U,

(16)

\begin{aligned} \frac{1}{ε^{2}} \frac{ρ_{h n f}}{ρ_{f}} (U \frac{\partial U}{\partial X} + V \frac{\partial V}{\partial Y}) = & - \frac{\partial P}{\partial Y} + \frac{1}{ε} \frac{ν_{h n f}}{ν_{f}} \Pr (\frac{\partial^{2} V}{\partial X^{2}} + \frac{\partial^{2} V}{\partial Y^{2}}) - \frac{ν_{h n f}}{ν_{f}} \frac{\Pr}{Da} V \\ - \frac{1.75}{\sqrt{150} ε^{3 / 2}} \frac{1}{\sqrt{Da}} \sqrt{U^{2} + V^{2}} V + \frac{β_{h n f}}{β_{f}} \Pr . Ra θ + \frac{σ_{f}}{ρ_{h n f}} \frac{ρ_{f}}{ρ_{h n f}} \frac{PrH a^{2}}{ε \sqrt{Ra}} V, \end{aligned}

(17)

U \frac{\partial θ}{\partial X} + V \frac{\partial θ}{\partial Y} = \frac{α_{h n f}}{α_{b f}} (\frac{\partial^{2} θ}{\partial X^{2}} + \frac{\partial^{2} θ}{\partial Y^{2}}) .

(18)

Speed and function are related to⁴⁵

\begin{matrix} U = \frac{\partial ψ}{\partial Y}, \\ V = - \frac{\partial ψ}{\partial X} \end{matrix}}

(19)

Then the odd equation can be written by,

\frac{\partial^{2} ψ}{\partial X^{2}} + \frac{\partial^{2} ψ}{\partial y^{2}} = \frac{\partial U}{\partial Y} - \frac{\partial V}{\partial X} .

(20)

Based on the Brinkmann mode, the operative dynamic viscosity and conductivity of the HNFs are as follows:

μ_{h n f} = \frac{μ_{b f}}{{(1 - ϕ)}^{2.5}},

(21)

\frac{σ_{h n f}}{σ_{b f}} = 1 + \frac{3 (σ_{n p} - σ_{b f}) ϕ}{(σ_{n p} + 2 σ_{b f}) - (σ_{n p} - σ_{b f}) ϕ} .

(22)

where

ϕ

is the solid volume fraction of nanoparticles.

Boundary conditions for cavity walls are explained as follows:

hot wall:

u = v = 0, T = T_{h} .

cold wall:

u = v = 0, T = T_{c} .

insulated walls:

u = v = 0, \frac{\partial T}{\partial n} = 0.

N u_{l o c} = - \frac{k_{h n f}}{k_{b f}} \frac{\partial θ}{\partial X},

(23)

N u_{a v g} = \int_{0}^{L} N u_{l o c a l} d Y .

(24)

The multidimensional local entropy given by Woods⁴⁶ through the process of convection and the magnetic field is given as:

\begin{aligned} S_{G e n} = & \frac{k_{m}}{{(T_{a v g})}^{2}} [{(\frac{\partial θ}{\partial X})}^{2} + {(\frac{\partial θ}{\partial Y})}^{2}] + \frac{μ_{n f}}{T_{a v g}} [\frac{ε_{p}}{K} (U^{2} + V^{2}) + 2 {(\frac{\partial U}{\partial X})}^{2} + 2 {(\frac{\partial V}{\partial Y})}^{2} + {(\frac{\partial U}{\partial Y} + \frac{\partial V}{\partial X})}^{2}] \\ + \frac{σ_{n f}}{T_{a v g}} B^{2} V, \end{aligned}

(25)

when

T_{a v g} = \frac{T_{H} + T_{C}}{2}

Authentication and grid independence analysis

The effect of the network on the numerical solution was examined before moving on to the calculations in our study. For this, we have studied the state of flow in mixed convection and the Prandtl number equal to 6.2. The current problem is solved using the GFEM, and the GFEM described by Redouane et al.³⁵ has become our main reference in this model. GFEM transforms the nonlinear differential equations (15)-(25) into a system of integral equations. In addition, the non-uniform triangular grids are utilized to decertify the field solution. Furthermore, the mesh assessment process is accomplished to ensure that the current technique is independently grid. Eight alternative networks were tabulated in Table 2. This table gives results of average Nusselt number $Nu$ values of maximum velocities, as well as entropy generation functions $S_{a v g}$ . The accuracy of the outcomes is established by a comparison of the current effort with the obtainable works. Figure 2 showed a good arrangement with those established by Corcione et al.⁴⁷

Figure 2.

Comparison of streamlines (left) and isotherms (right) (a) current work and (b) Corcione et al.⁴⁷ for $Ra = 10^{5}$ and $ϕ = 0.04$ .

Table 2.

Contrast $Ψ_{m a x}$ , $N u_{a v g}$ , and $S_{a v g}$ as single functions of mesh items for $ϕ = 0.02$ , $Ra = 10^{5}$ , $Da = 0.01$ , and $ε = 0.2$ .

Number	250	400	1020	1500	2362	6534	17,562	25,064
$Ψ_{m a x}$	1.2205	1.1085	1.07845	1.0743	1.0711	1.061	1.0570	1.055
$N u_{a v g}$	5.3757	5.5230	5.6991	5.7427	5.7642	5.876	5.9423	5.9419
$S_{a v g}$	0.54578	0.5820	0.6335	0.6438	0.6496	0.6512	0.6535	0.6549

Results and discussion

Five parameters related to simplification, isotherm, and general entropy are presented, these parameters refer to Rayleigh amount ( $10^{2} \leq Ra \leq 10^{5}$ ), Hartmann numeral (0≤ Ha 100), nanoparticle size fraction ( $0.02 \leq ϕ \leq 0.08$ ), Darcy quantity ( $10^{- 5} \leq Da \leq 10^{- 2}$ ), and porosity ( $0.2 \leq ε \leq 0.8$ ). The thermophysical properties, basic liquid (water), and solids (MWCNT and Fe₃O₄) are shown in Table 1. The results are categorized into the following four subsections.

Effect of Rayleigh’s number

Figure 3 shows the smooth and isometric lines in the considered cavity filled with a Fe₃O₄–MWCNT HNF of different Rayleigh records. It is clearly observed that rationalizes designed two circumferences near the heated vertical wall and one circumference next to the cooled vertical wall of the cavity. For lower Rayleigh number values, the heated aerodynamic contours dominate over the cooled particular contour. Growing the Rayleigh amount ( $Ra$ ) moves the dominance in the direction of the conclusion of the quenching. This might be in line with conventional trans-cavity transfer.

Figure 3.

Streamlines and isothermal for different Rayleigh numbers ( $Ra$ ) when $Da = 0.01$ , $Ha = 0$ , $ϕ = 0.02$ , and $ε = 0.4$ .

From both ends of the thermal points, isotherms with a reduced Rayleigh number ( $Ra$ ) were dispersed vertically. The isotherms were drawn closer to one another to raise the values of $Ra$ . It is interesting to note that although the cooled isotherms flow through the bottom of the hollow, the heated isotherms spread to the coolant wall that fills the highest. This demonstrates how the heat in the cavitation fluid causes a drop in density.

Figure 4 illustrates the example of the mean Nusselt number across the cavity for various physical parameters. After 10², all the physical factors involved in the problem, the Rayleigh number ( $Ra$ ) increases and so does the rate of heat transmission. It is discovered that while Hartmann number ( $Ha$ ) tends to decrease, physical characteristics like solid volume fraction ( $ϕ$ ), porosity ( $ε$ ), and Darcy number ( $Da$ ) should enhance apparent heat transfer rates by raising the average $N u_{a v g}$ number.

Figure 4.

Average Nusselt presentation with different Rayleigh numbers $Ra$ .

Hartmann number effect

The confrontation of the forces caused by electromagnetism with viscous changes to changes in temperature can be designed via the Hartmann number ( $Ha$ ).

Figure 5 shows the streamlines in which the transverse magnetic field B was applied to isotherms inside the investigated cavity that had temperature variations on each side. When the magnetic field was absent ( $Ha = 0$ ), the streamlines designed a solitary contour close to the cooler adjacent and two faint streaks on the hot adjacent. The intensity of these lines changed further toward the bottom of the cavity with higher Hartmann numbers due to the increased flow resistance. Temperature is slowly dissipated from both ends toward each other thanks to the cavity's magnetically restricted fluid flow. The isotherms start to diverge from the surface with larger Hartmann number ( $Ha$ ) levels. Coolant isotherms cover the bottom portions of the cavity, as heated isotherms assert.

Figure 5.

Differences in simplifications and equivalencies as a function of Hartmann number $Da = 0.01$ , $Ra = 10^{5}$ , $ϕ = 0.02$ , and $ε = 0.$ .

Influence of Darcy's number

Around the cavity's porous structure, the result of Darcy's number becomes important. The porousness of the medium rises with Darcy's number, enabling flow across it. Higher values of Darcy's number can be obtained by dregs building up on each side of a porous medium and flowing toward it ( $Da$ ). The cooler side has a condensed perimeter in comparison to the hotter side, which may be caused by the coolant's sluggish permeability.

Figure 6 shows the flow line variation of the flow lines (top) by the variation of the Darcy number, and the obtained results show that the distribution of the flow lines decreases in the porous medium and increases in the non-porous medium by an amount. Thus, the increase in the number of Darcy varies relatively someone opined that the opposition to isotherms such as the increase in temperature increases with the increase in the number of Darcy.

Figure 6.

Simplifying differences and equals by the difference of Darcy number $Ra = 10^{5}$ , $Ha = 0$ , $ϕ = 0.02$ , and $ε = 0$ .

Influence of porosity

In nanofluid flows including a porous media, porosity becomes a crucial characteristic in contrast to issues with regular fluid flow. The nature of the dispersed nanoparticles may be the cause of this. Porosity ( $ε$ ) also has streamlines grouped around the porous media, albeit to a lesser extent than the effects of Darcy's number ( $Da$ ). The streamlines are only slightly altered because the increased porosity ( $ε$ ) makes the flow through the cavity easier.

Regarding Figure 7, the isotherms of the porosity effect appear to be negligible. It is clearly noted from the graphs that no such significant changes have occurred.

Figure 7.

Rheological and isometric differences as a function of porosity $Ra = 10^{5}$ , $Da = 0.01$ , $Ha = 0$ , and $ϕ = 0.02$ .

Entropy

Figure 8 displays entropy changes as a function of Rayleigh; we note that the heat transfer rate increases in the non-porous medium and decreases in the porous medium in addition to increasing the Rayleigh number values.

Figure 8.

Entropy variations with Rayleigh number ( $Ra$ ) when $Da = 0.01$ , $Ha = 0$ , $ϕ = 0.02$ , and $ε = 0.4$ .

Conclusion

Based on the finite element method, the parametric organization was studied on the MWCNT–Fe₃O₄ hybrid Nano-fluid/water packed in the middle of the porous cavity in layers and under the influence of a magnetic field. We extracted the following.

Activation of the dual streamlines for the hot side and individual cooling for the lower values of Darcy's number, Rayleigh's number, and higher Hartmann values.

At the top of the cavity, a cryogenic temperature is required below at the changing values of physical parameters.

Low Darcy and Hartmann index, Rayleigh index, high porosity, and low Fracture Volume controlled by high $N u_{a v g}$ in all porous media.

The higher rate of heat transfer within the cavity is due to an increase in the Darcy number with $Ra$ and a solid volume fraction. These results are reversed when Hartmann is high.

The components of entropy are determined by the production of entropy.

Footnotes

Acknowledgments

This work is funded by the Deputyship of Research & Innovation, Ministry of Education in Saudi Arabia, through project number 804 (Group Research Program 1). In addition, the authors would like to express their appreciation for the support provided by the Islamic University of Madinah.

Data availability

All data generated or analyzed during this study are included in this published article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Wasim Jamshed

Chibani Lotfi is an active researcher in the LGIDD, University of Relizane, 48000, Relizane, Algeria. His research focuses on heat transfer, computational fluid dynamics; renewable energy and nanofluids.

Fares Redouane is an assistant professor in the LGIDD, University of Relizane, 48000, Relizane, Algeria. His research focuses on experimental physics, heat transfer, fluid mechanics, materials science, and catalysis atmospheric.

Chikr Djaoutsi Zineb is an active researcher in the LGIDD, University of Relizane, 48000, Relizane, Algeria. Her research focuses on effect of nanoparticle, heat and mass transfer in fluid mechanics via porous media. Airfoil Numerical analysis of hydrofoil cavitation flows heat transfer enhancement through annulus using nanofluids CFD of Airships Analytical analysis and CFD of IC engines theoretical and numerical analysis of rocket engines.

Wasim Jamshed is an active researcher in the Department of Mathematics at Capital University of Science & Technology (CUST), Islamabad, Pakistan. His research focuses on effect of nanoparticle, heat and mass transfer in fluid mechanics via porous media. In particular he is studying the following topics: nanofluids; hybrid nanofluids, non-Newtonian fluids, chemical reactions effect, numerical treatments of nanofluids flow, porous media, magnetic field and solar energy applications. He has been ranked in the top 2% of the world's most cited scientists for research conducted by Stanford University in collaboration with Elsevier Publishing and Scopus in Mechanical Engineering and Transport field.

Mohamed R. Eid is an Associate Professor in the Department of Mathematics at the University of New Valley, Al-Kharga, Al-Wadi Al-Gadid, Egypt, and in the Department of Mathematics at Northern Border University, Arar, Northern Borders, KSA. His research focuses on the effects of nanoparticles, heat, and mass transfer in fluid mechanics via porous media. He is studying the following topics: nanofluids, non-Newtonian fluids, chemical reactions effects, numerical treatments of nanofluid flow, porous media, and magnetic fields. He has published more than 175 scientific papers in highly impactful journals. For the third year in a row, He has been ranked in the top 2% of the world's most cited scientists for research conducted by Stanford University in collaboration with Elsevier Publishing and Scopus in Mechanical Engineering and Transport field.

Rabha W. Ibrahim is a full Professor in the Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon. Her research focuses on convolution, edge detection, image enhancement, image resolution, object recognition, text detection, traffic engineering computing.

Siti Suzilliana Putri Mohamed Isa is an active researcher in the institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia. Her research focuses on mathematical sciences, convection heat transfer, boundary layer, nanofluids mechanics & numerical modeling flow.

Haifa Alqahtani is an active researcher in the Department of Statistics and Business Analytics, United Arab Emirates University, UAE Her research focuses on Computation of 2D and 3D shock/turbulent boundary-layer interactions Soot formation and MILD combustion Scramjet combustion CFD of Laden flows Numerical micro convection heat transfer Sustainable and Renewable Energy technology CFD of Variable Camber Continuous Trailing Edge (VCCTE).

Syed M Hussain is working as an Associate Professor in the Department of Mathematics, Faculty of Science, Islamic University of Madinah, Saudi Arabia. His area of research is Magnetohydrodynamics, Mono and Hybrid nanofluids, etc.

Nomenclature $a$

Length of porous aria ( $m$ )

x,y

Coordinates ( $m$ )

Ha

Hartmann number

K

Permeability ( $m^{2}$ )

Nu

Nusselt number

\Pr

Prandtl number

Da

Darcy number

U, V

Dimensional velocity components

B_{0}

The intensity of the magnetic field

Fc

Forchheimer coefficient

k

Thermal conductivity ( $W . m^{- 1} . K^{- 1}$ )

L

Height of cavity ( $m$ )

P

Pressure ( $Pa$ )

Ra

Rayleigh number

u, v

Velocity components ( $m . s^{- 1}$ )

T

Temperature ( $K$ )

S

Entropy

Greek symbols

α

Thermal diffusivity ( $m^{2} . s^{- 1}$ )

ε

Porosity

μ

Dynamic viscosity ( $W . m^{- 1} . K^{- 1}$ )

θ

Adimentional temperature

σ

Electrical conductivity ( $Ω . M$ )

β

Thermal expansion coefficient ( $K^{- 1}$ ₎

λ

The length of the Baffle ( $m$ )

ν

Kinematic diffusivity ( $m^{2} . s^{- 1}$ )

ρ

Density ( $kg . m^{- 3}$ )

ϕ

Solid volume fraction

Subscripts

a v g

Average

f

Fluid

l o c

Local

t o t

Total

c

Cold

h

Hot

h n f

Hybrid nanofluid

Base fluid

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Thermal energy analysis of multi-walled carbon nanotubes-Fe 3 O 4 /H 2 O flow over non-uniformed surface with Darcy–Forchheimer model

Abstract

Keywords

Introduction

Problem explanation

Mathematical model

Authentication and grid independence analysis

Results and discussion

Effect of Rayleigh’s number

Hartmann number effect

Influence of Darcy's number

Influence of porosity

Entropy

Conclusion

Footnotes

Acknowledgments

Data availability

Declaration of conflicting interests

Funding

ORCID iD

References