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Cu and Al2O3-based hybrid nanofluid flow through a porous cavity

  • Ebrahem A. Algehyne , Zehba Raizah , Taza Gul , Anwar Saeed , Sayed M. Eldin EMAIL logo and Ahmed M. Galal
Published/Copyright: March 25, 2023
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Abstract

In this study, the (Cu and Al2O3/water) hybrid nanofluid flow is carried out in a porous cavity. The thermophysical structures of solid materials are used from the available literature to improve the thermal performance of the base fluid. The mathematical model as a porous cavity is mainly used in the distillation process and is vital for the storage of thermal energy. The magnetic field is also employed perpendicular to the flow field and the impact of the magnetic parameter examined versus fluid motion. Similarity variables are used to transform governing equations as simplified partial differential equations. The model is solved using the control volume-based finite element method. Boussinesq–Darcy force is employed for the motion of the fluid flow, and the Koo–Kleinstreuer–Li model is used to assess the characteristics of the hybrid nanofluids. The roles of the Hartmann number, Rayleigh number, porosity factor in the porous medium, and drag fin improve traditional fluids’ thermal distribution presentation. Recent results predict that the two different kinds of nanoparticles speed up the heat transfer through the porous cavity. The percentage analysis shows that the hybrid nanofluids (Cu and Al2O3/water) are prominent in improving traditional fluids’ thermal distribution. Finally, the grid sensitivity test is also carried out for hybrid nanoparticles to demonstrate that the results are asymptotically coherent.

Nomenclature

List of symbols

u , v

components of velocity ( m s 1 )

( c p ) s

specific heat ( J kg 1 K 1 )

B 0

strength of magnetic field ( Nm A 1 )

Ha

Hartmann number

T

temperature of fluid ( K )

T l , T u

lower, upper wall temperature ( K )

Ra

Rayleigh number

Da

Darcy parameter

Nu

Nusselt number

C f

skin friction

Greek symbols

μ hnf

dynamic viscosity ( mPa )

ψ

stream function

μ f

dynamic viscosity of the base fluid ( mPa )

ρ hnf

hybrid nanofluid density ( kg m 3 )

ρ f

base fluid density ( kg m 3 )

σ nf

electrical conductivity

ϕ

nanoparticle volume fraction

Subscripts

f

base fluid

hnf

hybrid nanofluids

1 Introduction

This research aims to investigate the improvement of heat transfer via a hybrid nanofluid. The hybrid is the combination of two different nanoparticles with different chemical and thermophysical properties and distributed stably in the same base liquid.

The nanoparticles under consideration have several cooling applications in a variety of industrial processes. Based on those applications, the current analysis is performed to examine the effect of two different nanoparticles on the fluid flow through a porous cavity. A hybrid nanofluid is the combination of base fluid (water) and two different types of nanoparticles.

Nanofluids are potential heat transfer fluids containing suspensions of nanoparticles that significantly improve their thermophysical properties [1,2]. The area of nanofluids (NFs) has acquired great courtesy since the idea of dissolving nanoparticles in a fluid was introduced in the late 20th century [3,4]. When compared to conventional colloids, nanofluids have significant advantages over conventional solid–liquid suspensions including advanced explicit surface area, more prominent for the heat transfer surface among fluid and particles, the higher immovability of the colloidal suspension precise by Brownian motion of the particles, and reduced particle clogging [5,6,7]. Due to such advantages, nanofluids can be incorporated into a myriad of potential applications including heat exchangers [8,9], pharmaceutical processes [10], nuclear reactors [11,12], engine cooling [13,14], electronics [15,16], and food and cosmetics [17,18].

The scientific vision of nanomaterials is more suitable for various industrial and engineering devices to enhance their thermal performance. Most researchers used new and advanced techniques to evaluate the analysis of flows across these new strategies [19]. Shafiq et al. [20,21,22] used statistical analysis to predict the flow of nanofluids across the expansion surface for temperature distribution. They also carried out sensitivity and stability analyses on the proposed nanofluids using the artificial neural network. The performances of the individual model parameters are analyzed and deliberated.

Nanomaterials are also used in electronic devices, including DC-powered in combination with electro-osmosis in the micro-pump as mentioned by Alsharif et al. [23]. Hybrid nanofluids are used as a drug supplement using the coronary artery model as described by Kot and Elmaboud [24]. El-Masry et al. [25] investigated the idea of inserting the magnetic field into the flow phenomena of the hybrid nanofluid using the micropump to study heat transfer.

The cooling and heating strategy is necessary for many appliances such as heaters. Some porous structures also control long-term cooling and heating in industries, matching the demand for thermal devices.

The saturation of the nanofluid stream on the stretching and permeable surface can be seen in the work of Al-Srayyih et al. [26], Agarwal [27], and Yadav and Wang [28]. They used various types of nanofluid flows in model varieties to investigate thermal analysis.

The researchers focused on the hybrid nanofluid which is more advanced in the augmentation of heat transmission for different industrial instruments and heat exchangers. The U, V, circular, square, and elliptic porous cavities are mostly used to consider heat transport analysis using nanofluids. Chandra et al. [29] analyzed heat transfer across nanofluids taking into account the porous cavity. Chandra Sekhar et al. [30] investigated thermal expansion with the help of the square cavity filled with nanofluids. Alluguvelli et al. [31] examined the flow of the Fe3O4 nanofluid through a porous cavity and analyzed the thermal distribution. Ali et al. [32] have studied the flow of Fe3O4 nanofluid in a Couette channel in the presence of thermal radiation.

Extensive experimental and theoretical research has been undertaken on the energy transport features of NFs, most of which demonstrated that nanofluids can boost the rate of heat transmission [33,34,35]. However, studies have shown that pre-eminent heat transfer has been replaced by the use of different models [36,37,38]. A porous medium is a solid matrix containing interconnected voids or solid particles commonly filled with liquid. Its tortuous form and large contact surface are advantageous for increasing the rate of heat transfer [39,40]. Thus, nanofluids and porous media can be mixed in one to augment heat transport. The combination of nanofluids and porous media, as special functional materials, has major uses in improving heat transfer [41,42]. Recently, the use of nanoliquid across a permeable surface has got considerable attention and research in this area. Porous media enhance the link zone between a liquid and a solid surface, while nanoparticles dispersed within a nanofluid effectively increase thermal conductivity. As a result, it seems that the use of nanoliquid and porous media can significantly increase the efficiency of conventional thermal systems [43,44].

The researchers focused on the various cheap energy resources, including renewable energy to meet the requirements of industry and daily life. Nanofluids are one of the active and passive strategies used to enhance the heat transmission rate in heat exchangers, medications, and the solar sector. In general, researchers have used both active and passive strategies to improve energy propagation. Active techniques require external forces, like porous medium, and magnetic field, while passive procedures need unique surface models like a coarse surface and stretched surface liquids [45,46,47]. The improvement of thermal transfer by an applied electric field is due to the electrical force exerted on the nanoparticles, which may affect the microstructure of the nanoliquid and the motion of nanoparticles [48,49,50]. The benefits of this upgrade include easy control, simple design, and low energy usage [51,52].

Kang et al. [53] developed an experimental approach to guess the presence of thermal waves considering lag ratio measurement. They used Laplace’s processing method to solve a unidirectional flux by considering a cylindrical heater to heat a half-infinite space. The technique has been tested by experiments with lean pork and sand, which showed that the ratio in the sand is lesser than that in lean pork. In the interim, the time lag ratios for both were less than 1 within adequate dimension ambiguity, signifying that no thermal waves were produced. Sheikholeslami group [54] simulated electrohydrodynamic nanofluid in a porous cavity. The simulation was supported by the use of CVFEM in the presence of heat radiation and an electrical field. Moreover, the number of Nusselt increased as buoyancy forces and radiation parameters increased. The Galerkin finite element method (GFEM) was employed by Hamida et al. [55] for demonstrating the heat transport in a channel occupied by hybrid nanoliquids (HNFs) under the electric field. Four types of HNFs, i.e., TiO2–CuO, Al2O3–CuO, TiO2–Al2O3, and Al2O3–Cu have been exploited to progress the rate of heat transfer.

The novelty of the present work is highlighted as follows:

  • TiO2–Al2O3 hybrid nanofluid has been used to enhance the heat transmission rate.

  • The published work [38] is extended using the half-porous enclosure.

  • The effect of the magnetic field is investigated with the combination of the porous medium.

For us to remain motivated, we must have a purpose and be pushed to achieve it.

In this article, we aim to develop new ideas by introducing new models and strategies to improve the heat transfer ratio. Now, our main objective is to work toward the end of our goals to define cheap and easily accessible energy sources.

2 Description of the problem

The Cu and Al2O3 are used in the water to study the hybrid nanoliquids. The HNF flow is considered in a porous enclosure in the existence of a magnetic field. The magnetic field has been applied in the upright direction. The wall design of sinusoidal is defined as

(1) b = a ( 1 ε ) 2 .

The magnetic field is applied to the outflow field. Hybrid Al2O3 and CuO nanofluid flux are taken into account in a porous chamber. All the assumptions are used as per refs. [37,38].

3 Formulation and simulation of the Problem

3.1 Governing equation

Employed the Boussinesq–Darcy force and the non-equilibrium thermal model to the temperature model, the governing partial differential equations are as follows [37,38] (Tables 1 and 2).

(2) V = 0 ,

(3) ρ hnf β hnf g ( T ˜ hnf T ˜ c ) ˆ + μ hnf K + p + σ hnf ( V × B ) ˆ = 0 ,

(4) h hnfs ρ s ( c p ) s ( 1 ε ) ( T ˜ hnf T ˜ s ) + k s ρ s ( c p ) s 2 T ˜ s = 0 ,

where

(5) ϕ = ϕ Cu + ϕ Al 2 O 3 , ρ hnf = ( 1 ϕ ) ρ f + ρ Al 2 O 3 ϕ Al 2 O 3 + ρ Cu ϕ Cu , ( ρ c p ) hnf = ( 1 ϕ ) ( ρ c p ) f + ( ρ c p ) Cu ϕ Al 2 O 3 + ( ρ c p ) Cu ϕ Cu , ( ρ β ) hnf = ( 1 ϕ ) ( ρ β ) f + ( ρ β ) Al 2 O 3 ϕ Al 2 O 3 + ( ρ β ) Cu ϕ Cu ,

(6) σ hnf = σ f + 3 σ np σ f 1 ϕ 1 σ np σ f ϕ + σ np σ f + 2 σ f ,

Table 1

Thermophysical properties of Cu and Al2O3 nanoparticles [36]

Properties Water Cu Al2O3
Density [ ρ ] ( kg/m 3 ) 997.10 8,933 3,970
Heat capacity [ C p ] ( J / kg K ) 4,179 385 765
Thermal conductivity [ k ] ( W/m K ) 0.6130 401 40
Thermal expansion [ β × 10 5 ] ( K 1 ) 21 1.67 × 10 5 0.85 × 10 5
Electrical conductivity [ σ ] ( S / M ) 5.5 × 10 6 5.96 × 10 7 1 × 10 10
Table 2

Validation of current and published findings with common parameters

Ha Nu ave [37] Nu ave [38] Nu ave (present)
0 2.4634 2.4849 2.49327
6 2.28743 2.29532 2.29843
8 2.1458 2.16143 2.17327
10 1.6675 1.69758 1.72885

The k nf , and μ nf is defined by Koo–Kleinstreuer–Li model as follows:

(7) μ hnf = μ f 1 ( 1 ϕ Ag ϕ MgO ) 5 / 2 + k Brownian Pr k f ,

(8) k hnf = k f 3 k f ( k f k np ) ϕ ( k f k np ) ϕ + ( k np + 2 k f ) + 5 × 10 4 ρ f ϕ k b T ˜ ( ρ d ) np 1 / 2 c p,f g ( T ˜ , d p , ϕ ) ,

where for CuO/water nanofluid the function g ( T ˜ , d p , ϕ ) is defined as:

(9) g ( T ˜ , d p , ϕ ) = ln ( T ˜ ) b 1 + b 2 ln ( d p ) + b 3 ln ( ϕ ) + b 4 ln ( d p ) + b 5 ln ( d p ) 2 + b 6 + b 7 ln ( d p ) + b 8 ln ( ϕ ) + b 9 ln ( ϕ ) ln ( d p ) + b 10 ln ( d p ) 2 , R f = d p k p,eff d p k p = 4 × 10 8 km 2 / W .

The coefficients b i , i = [ 0 , 10 ] depend on nanofluid types.

The Non-dimensional variations are as follows:

(10) v = ψ x , u = ψ y , Ψ = ψ α nf , ( X , Y ) = ( x , y ) l , θ s = ( T ˜ s T ˜ c ) ( T ˜ h T ˜ c ) , θ nf = ( T ˜ nf T ˜ c ) ( T ˜ h T ˜ c ) .

Equation (12) in equations (2)–(4) yields the following non-dimensional differential system:

(11) 2 Ψ X 2 + 2 Ψ Y 2 = L 6 L 5 Ha 2 Ψ X 2 cos 2 γ + 2 2 Ψ X Y cos γ sin γ + 2 Ψ Y 2 sin 2 γ L 3 L 4 L 2 L 5 Ra θ nf X L 5 L 1 Pr Da Ψ X ,

(12) 2 θ nf X 2 + 2 θ nf Y 2 = θ nf ε X Ψ Y Nhs ( θ s θ nf ) ε θ nf ε Y Ψ X ,

(13) 2 θ s X 2 + 2 θ s Y 2 = Nhs ( θ nf θ s ) ε ,

where

(14) L 1 = ρ hnf ρ f , L 2 = ρ hnf ( c p ) hnf ρ f ( c p ) f , L 3 = ρ hnf ( β ) hnf ρ f ( β ) f , L 4 = k hnf k f , L 5 = μ hnf μ f , L 6 = σ hnf σ f , Ra = g K ρ f ( β ) f Δ T ˜ μ f α f , Nhs = h hnfs l 2 k hnf , δ s = k hnf k f ( 1 ε ) , Ha = K B 0 2 σ f μ f , Da = k l 2 .

For the boundary conditions, as the inner wall is considered hot, we have:

(15) Ψ = 0 , on all walls, θ s = 0 , θ nf = 0 , on the outer wall, θ s = 1 , θ nf = 1 , on the inner wall .

When the wall is cold, we have:

(16) Nu loc = k nf k f θ nf r , Nu ave = 0.5 π 0 2 π Nu loc d r .

3.2 Simulation through control volume-based finite element method

For the numerical solution of the proposed flow model presented in the equations, an advanced Control-based finite element technique has been employed in the spatial domain. In the finite element method, it is standard practice to display the discrete form of differential equations in a space using a locally defined coordinate system. To discretize the physical domain, the suggested technique employs hexahedral components. In the new space, components are divided up into smaller control volumes.

4 Grid independence test and verification

The optimum grid design should be taken into account for the best results. Most notably, the overall computing cost and precision of simulation analysis findings are affected by the number of grids. Inaccurate analysis findings are produced by using coarse grids that cause a large spatial discretization error. However, if the grid is too fine, the round-off error may become much larger than the truncation error, leading to less reliable findings [6]. Accordingly, selecting the right number of grids is critical [7]. Many CFD research used grid independence testing to determine the best grid size. By comparing the numerical results obtained with different grid sizes and densities, the grid independence test may determine which grid condition produces the best overall numerical performance with the fewest number of grids.

In CVFEM code, the solution set is not only observed on grid size but the appropriate meshed has been used in each case. More importantly, a more powerful machine has been used to find the solution in case of the high grids for accurate resolution.

5 Results and discussion

The influence of a magnetic field on the motion of hybrid nanofluids within a permeable container has been shown using a non-equilibrium model. Computational volume finite element method (CVFEM) has been used to represent high grids. The predictions show the effect of varying the Hartmann number (Ha = 05, 10, 15, and 20), Rayleigh number (Ra = 50, 100, 150, and 200), and the porosity factor (Da = 5, 10, 15 to 20). Figure 1 represents the proposed model for the number of smaller and larger grids, respectively. The stability of the problem has improved as the grids have multiplied. Figures 2 and 3 show the effects of various Rayleigh and Hartmann numbers for the velocity profile in the axial and angular directions. Figures 2 and 3 show the effects of different Rayleigh and Hartmann numbers on the velocity profile in the axial and angular directions. When the Rayleigh number is low, the convective flow is low and the conductive phase is stable. When the Rayleigh number increases (50, 100, 150, and 200) as shown in Figure 2(a)–(d), the thermal boundary layer on the inner wall’s surface thins down, indicating that convection plays a higher role in heat transmission at these high values. In addition, a plume begins to appear at the top of the inner circular wall. The flow is pushed hard against the top of the box by a powerful plume. Moreover, as the speed of convection increases, the center of the main vortices continues to increase. As expected, increasing the value of the Rayleigh number involves improving the buoyancy strength in entire cases. Consequently, we have noticed an improvement in the flow force and thermal convection in the field with the improvement of the Rayleigh number.

Figure 1 
               Grid presentation of the proposed model.
Figure 1

Grid presentation of the proposed model.

Figure 2 
               Different Rayleigh numbers for velocity profile.
Figure 2

Different Rayleigh numbers for velocity profile.

Figure 3 
               Different Hartmann numbers for velocity profile.
Figure 3

Different Hartmann numbers for velocity profile.

Thus, the rise with an increase in Rayleigh number but a decrease in Hartmann number as augmented the Lorentz force restricts the nanofluid velocity as shown in Figure 3(a)–(d). The isotherms on the porous medium region become more contorted as Ra increases because of increasing quantities of natural convection in the free flow. Consequently, the transfer of heat in the porous medium is changed from controlled by thermal conduction to controlled by conduction and natural convection. While the increasing amount of (Ha = 5, 10, 15, and 20) improves the strength more efficiently to stop the fluid movement as prescribed in Figure 3(a)–(d). That’s why when Ra rises, resulting in a decrease in the average temperature at the contact of the porous medium. Furthermore, the control of natural convection and hence the maximum absolute value of the flow function decreases as Ha increases. In addition, as Ha rises, the central line extends, and the fundamental temperature stratification decreases. Thus, the thermal layers of the two walls will disappear. Actually, in this case, the isotherms run parallel. As an increasing Ha leads to a reduction in the buoyancy force.

The thermal plume shrinks as the (Da = 5, 10, 15, and 20) rises as shown in Figure 4(a)–(d).

Figure 4 
               Porous parameter for the porosity factor.
Figure 4

Porous parameter for the porosity factor.

Increasing values of the permeability parameter increase the strength to reduce liquid motion. The Rayleigh number increases the skin friction for its larger values, and this effect is more prominent in the case of the HNF as displayed in Figure 5. Physically, the drag effect near the wall surface improved with a greater value of the Rayleigh number and consequently improved skin friction. Similarly, the increasing values of the Hartmann number increase skin friction as shown in Figure 6. The strength of the Lorentz effect has improved in terms of resistance, raising skin friction. The heat transfer rate increases with the rise in the nanoparticle volume fractions as displayed in Figure 7. The rise in the volumetric fraction of nanoparticulates shows that HNFs are more effective in improving heat transfer, as shown in Figure 8. The % wise enhancement shows that hybrid nanofluids are more effective to improve the heat transfer rate for the ϕ 1 + ϕ 2 = 0.01 , 0.03 , 0.05 . The combined volumetric fraction of the Al2O3 and Cu nanoparticles is essential to improve heat transfer compared to conventional fluid.

Figure 5 
               Skin friction versus Ra.
Figure 5

Skin friction versus Ra.

Figure 6 
               Skin friction versus Ha.
Figure 6

Skin friction versus Ha.

Figure 7 
               Nusselt number versus volume fraction.
Figure 7

Nusselt number versus volume fraction.

Figure 8 
               Enhancement in heat transfer (in percentage).
Figure 8

Enhancement in heat transfer (in percentage).

6 Conclusion

The flows of hybrid nanofluids considering the nanoparticles of Cu and Al2O3 are used for increasing heat transfer in a permeable enclosure driven by the cover exposed via CVFEM. The implications are transmitted for various values of ϕ, Da, Ra, Rd, and Ha. The results validate the improvement in drag force due to the increasing extent of Rayleigh numbers. Hybrid nanofluids indicate that with a small increment in the volume fraction, the rate of heat transfer increases significantly. The results are offered in 3D graphical form to show the maximum and minimum points. The number of extrema in the thermal profile is compatible with the presence of the number of ripples and thermal plume. Permeability affects heat transfer, and the buoyancy force is more efficient in thermal expansion.

  • According to the observations the motion of fluids Cu + Al2O3/water, significantly reduces near the center of the channel due to an intensification in Ra, Ha and ϕ .

  • The thermal profile of hybrid nanofluids seems ever higher in traditional fluids.

  • The value of Skin friction is heightened for M .

  • The extent of the Nusselt number is a growing function of ϕ .

  • Increased thermal radiation parameter makes Nusselt number to be improved.

  • This idea can also be extensible for the upper part of the enclosure. It is suggested that the entropy scheme and the use of other hybrid nanofluids for the same proposed model are possible for future research.

  1. Funding information: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP.2/300/44).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2022-09-05
Revised: 2023-02-09
Accepted: 2023-02-23
Published Online: 2023-03-25

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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