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Research Papers

# Numerical Analysis of Unsteady Conjugate Natural Convection of Hybrid Water-Based Nanofluid in a Semicircular CavityOPEN ACCESS

[+] Author and Article Information
Ali J. Chamkha

Mechanical Engineering Department,
Al-Khobar 31952, Saudi Arabia;
Prince Sultan Endowment for Energy
and Environment,
Al-Khobar 31952, Saudi Arabia

Igor V. Miroshnichenko

Laboratory on Convective
Heat and Mass Transfer,
Tomsk State University,
Tomsk 634050, Russia

Mikhail A. Sheremet

Laboratory on Convective
Heat and Mass Transfer,
Tomsk State University,
Tomsk 634050, Russia;
Department of Nuclear and
Thermal Power Plants,
Tomsk Polytechnic University,
Tomsk 634050, Russia

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF THERMAL SCIENCE AND ENGINEERING APPLICATIONS. Manuscript received May 15, 2016; final manuscript received February 22, 2017; published online April 19, 2017. Assoc. Editor: Giulio Lorenzini.

J. Thermal Sci. Eng. Appl 9(4), 041004 (Apr 19, 2017) (9 pages) Paper No: TSEA-16-1127; doi: 10.1115/1.4036203 History: Received May 15, 2016; Revised February 22, 2017

## Abstract

Unsteady conjugate natural convection in a semicircular cavity with a solid shell of finite thickness filled with a hybrid water-based suspension of Al2O3 and Cu nanoparticles (hybrid nanofluid) has been analyzed numerically. The governing equations for this investigation are formulated in terms of the dimensionless stream function, vorticity, and temperature and have been solved by the finite difference method of the second-order accuracy. The effects of the dimensionless time, Rayleigh number, thermal conductivity ratio, and the nanoparticles volume fraction on the flow patterns and heat transfer have been studied. The obtained results have revealed essential heat transfer enhancement at solid–fluid interface with addition of nanoparticles. In addition, a comparison of the heat transfer enhancement level due to the suspension of various nanoparticles materials (Al2O3 and Cu) in water as regular nanofluids (Al2O3/water and Cu/water) and as a hybrid Al2O3–Cu/water nanofluid is reported.

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## Introduction

The study of natural convection in enclosures has received attracted attention in a wide range of engineering applications, such as a nuclear reactor design, solar energy, electronic cooling, geophysics and underground storage of nuclear rubbish, energy stockpiling systems, and others. Most of these applications are simulated as natural convection inside enclosures of different geometries. There are many excellent theoretical and experimental studies for natural convective heat transfer of fluids in enclosures [113]. Conjugate natural convective heat transfer for a regular fluid has very important practical engineering applications in frosting practicalities and refrigeration of the hot obtrusion in a geological framing. For example, modernistic construction of thermal insulators which are formed of two diverse thermal conductivities (solid and fibrous) materials can be modeled by the partition length and conductivity model. There are some excellent studies considering the impact of partition length and conductivity on the heat transfer rate. Some previous theoretical and experimental works on conjugate natural convective heat transfer have been exhibited in Refs. [1418]. Ben-Nakhi and Chamkha [19] have presented a numerical analysis of conjugate natural convection in a square enclosure with an inclined thin fin of arbitrary longitude. Saeid [20] examined the problem of conjugate natural convection in a heated porous enclosure with a finite wall thickness. Varol et al. [21] have analyzed conjugate heat transfer by natural convection and conduction in a triangular enclosure with thick lower dike. Chamkha and Ismael [22] have focused on the investigation of conjugate natural convection heat transfer in a square domain composed of a cavity heated by a triangular solid wall. Martyushev and Sheremet have studied numerically natural convection combined with surface thermal radiation in an enclosure bounded by solid walls of finite thickness and conductivity with a heat source of constant temperature using two-dimensional approach [23] and three-dimensional model [24].

The analysis of nanofluids flow has been the subject of comprehensive exploration in the past decade as an innovative method for improving the thermal conductivity and convective heat transfer rendering of base liquids. A nanofluid is envisaged to refer to a fluid in which nanometer-sized particles are suspended in classical heat transfer primary fluids [25,26]. Classical heat transfer fluids encompassing oil, water, and ethylene glycol blend are pauper heat transfer fluids. Nanofluids are utilized in numerous engineering applications like microelectronics, microfluidics, heating or cooling in buildings, heat interchangers, lubrication systems, and refrigeration of electronic apparatuses. Hence, in the analysis of such systems, considering conjugate heat transfer is very important. Kuznetsov and Sheremet [27] have investigated natural convection of nanofluid in a square cavity bounded by heat-conducting solid walls. Chamkha and Ismael [28] studied the steady conjugate natural convection heat transfer in a square porous cavity heated diagonally and filled with a nanofluid. Sheremet and Pop [29] have carried out a numerical investigation of the problem of steady natural convection flow and heat transfer in a square porous cavity saturated by a nanofluid with solid walls of finite thickness. Ismael and Chamkha [30] have also reported a numerical investigation of conjugate conductive-convective heat transfer in a square enclosure composed of a heated solid wall and a nanofluid-saturated porous layer. Ismael et al. [31] have recently numerically investigated the conjugate natural convection-conduction heat transfer in a square domain filled with a nanofluid. Interesting analysis of convective heat transfer in nanofluids can be found in Refs. [3236].

Recently, there is a new type of nanofluids (hybrid nanofluid) which is prepared by suspending various kinds of nanoparticles in a base fluid or composite nanoparticles in a base fluid. A hybrid substance is a material which consolidates chemical and physical properties of various materials together and supplies these properties in a homogeneous phase. Synthetic hybrid nanomaterials demonstrate noteworthy physicochemical properties that cannot be found in the individual components. A significant number of studies have been accomplished concerning the properties of these composites [37] and electrochemical-sensors, bio-sensors, nanocatalysts, etc. [38], but the use of these hybrid nanomaterials in nanofluids has not advanced as such. Presently, studies on hybrid nanofluids are extremely confined and a lot of experimental work is still being completed. Due to their scope of applications and progressively improving published studies interest with hybrid nanofluids, some investigators [3943] published some results using such fluids.

The main objective of the current study is to investigate unsteady conjugate natural convection flow of Al2O3–Cu/water hybrid nanofluid in a semicircular cavity with a solid shell of finite thickness. The obtained results will reveal essential heat transfer enhancement at solid–fluid interface with the addition of nanoparticles.

## Problem Formulation

Figure 1 presents a schematic plan of the considered enclosure. This cavity consists of semicircular domain of radius R filled with a hybrid water-based suspension of Al2O3 and Cu nanoparticles and solid wall of finite thickness h and conductivity. A constant low temperature (Tc) is maintained at the upper wall of the cavity, while a constant high temperature (Th) is kept at external surface of solid wall. The nanofluid is considered to be Newtonian and heat-conducting fluid. The dimensional polar coordinates $r¯$ and γ define the polar radius and polar angle, respectively. The upper ends of the solid wall are assumed to be adiabatic. The water-based hybrid nanofluid contains solid spherical nanoparticles of copper and alumina and their physical properties are presented in Table 1. The flow and heat transfer inside the cavity is considered to be two-dimensional, time-dependent, and laminar. Thermal equilibrium between the fluid phase and nanoparticles without slip effects are supposed. The fluid properties are constant except for the density in the buoyancy term for momentum equation where the Boussinesq approximation is valid.

Taking into account the above-mentioned assumptions, the governing equations can be written in dimensional polar coordinates as follows for the nanofluid domain [46]: Display Formula

(1)$r¯∂u¯∂r¯+u¯+∂v¯∂γ=0$
Display Formula
(2)$ρhnf(∂u¯∂t+u¯∂u¯∂r¯+v¯r¯∂u¯∂γ−v¯2r¯)=−∂p¯∂r¯+μhnf(∂2u¯∂r¯2+1r¯∂u¯∂r¯+1r¯2∂2u¯∂γ2−u¯r¯2−2r¯2∂v¯∂γ)−(ρβ)hnfg(T−Tc)sin(γ)$
Display Formula
(3)$ρhnf(∂v¯∂t+u¯∂v¯∂r¯+v¯r¯∂v¯∂γ+u¯⋅v¯r¯)=−1r¯∂p¯∂γ+μhnf(∂2v¯∂r¯2+1r¯∂v¯∂r¯+1r¯2∂2v¯∂γ2−v¯r¯2+2r¯2∂u¯∂γ)−(ρβ)hnfg(T−Tc)cos(γ)$
Display Formula
(4)$∂T∂t+u¯∂T∂r¯+v¯r¯∂T∂γ=αhnf(∂2T∂r¯2+1r¯∂T∂r¯+1r¯2∂2T∂γ2)$

Energy equation for the solid wall can be written as follows: Display Formula

(5)$∂T∂t=αsw(∂2T∂r¯2+1r¯∂T∂r¯+1r¯2∂2T∂γ2)$

Here, we utilize the following expressions for the hybrid nanofluid properties such as

• density $ρhnf=ϕAl2O3ρAl2O3+ϕCuρCu+(1−ϕCu−ϕAl2O3)ρf$,

• thermal expansion coefficient $(ρβ)hnf=ϕAl2O3(ρβ)Al2O3+ϕCu(ρβ)Cu+(1−ϕCu−ϕAl2O3)(ρβ)f$,

• thermal diffusivity $αhnf=(khnf)/(ρcp)hnf$,

• heat capacitance $(ρcp)hnf=ϕAl2O3(ρcp)Al2O3+ϕCu(ρcp)Cu+(1−ϕCu−ϕAl2O3)(ρcp)f$,

• thermal conductivity taking into account the Maxwell–Garnett model

$khnfkf={ϕAl2O3kAl2O3+ϕCukCuϕAl2O3+ϕCu+2kf+2(ϕAl2O3kAl2O3+ϕCukCu)−2(ϕAl2O3+ϕCu)kf}×{ϕAl2O3kAl2O3+ϕCukCuϕAl2O3+ϕCu+2kf−(ϕAl2O3kAl2O3+ϕCukCu)+(ϕAl2O3+ϕCu)kf}−1$

• dynamic viscosity taking into account the Brinkman model $μhnf=μf(1−ϕAl2O3−ϕCu)−2.5$.

The formulated dimensional partial differential equations (1)(5) have been written in nondimensional form using the following dimensionless variables: Display Formula

(6)$r=r¯/R, u=u¯/gβf(Th−Tc)R, v=v¯/gβf(Th−Tc)Rτ=tgβf(Th−Tc)/R, θ=(T−Tc)/(Th−Tc)$

and new dependent dimensionless functions such as stream function ψ$(u=(1/r)(∂ψ/∂γ), v=−(∂ψ/∂r))$ and vorticity $ω=(∂u/∂γ)−v−r(∂v/∂r)$. Therefore, the governing equations (1)(5) using the above-mentioned variables can be written as follows: Display Formula

(7)$r∂2ψ∂r2+∂ψ∂r+1r∂2ψ∂γ2=ω$
Display Formula
(8)$∂ω∂τ+∂(u⋅ω)∂r+1r∂(v⋅ω)∂γ+u∂v∂r=H1(ϕ)PrRa(∂2ω∂r2−1r∂ω∂r+1r2∂2ω∂γ2+ωr2)+H2(ϕ)(r∂θ∂rcos(γ)−∂θ∂γsin(γ))$
Display Formula
(9)$∂θ∂τ+u¯∂θ∂r+v¯r¯∂θ∂γ=H3(ϕ)Ra⋅Pr(∂2θ∂r2+1r∂θ∂r+1r2∂2θ∂γ2)$
Display Formula
(10)$∂θ∂τ=ARa⋅Pr(∂2θ∂r2+1r∂θ∂r+1r2∂2θ∂γ2)$

The governing equations are subject to the following initial and boundary conditions: Display Formula

(11)$τ=0: ψ=ω=θ=0τ>0: ψ=0, ω=∂2ψ∂r2, {θhnf=θsw,∂θhnf∂r=K⋅H4(ϕ)∂θsw∂r at internal solid-fluid interface (r=1)θ=1 at external surface of solid wall (r=1+h/R)ψ=0,ω=1r∂2ψ∂γ2,θ=0 at top wall (0

Here, $Pr=μf(ρcp)f/(ρfkf)$ is the Prandtl number, $Ra=g(ρβ)f(Th−Tc)(ρcp)fR3/(μfkf)$ is the Rayleigh number, $A=αsw/αf$ is the thermal diffusivity ratio, $K=ksw/kf$ is the thermal conductivity ratio, and the functions $H1(ϕ)$, $H2(ϕ)$, $H3(ϕ)$, and $H4(ϕ)$ are given by Display Formula

(12)$H1(ϕ)=1(1−ϕCu−ϕAl2O3)2.5[1−ϕCu−ϕAl2O3+ϕAl2O3ρAl2O3/ρf+ϕCuρCu/ρf]H2(ϕ)=1−ϕCu−ϕAl2O3+ϕAl2O3(ρβ)Al2O3/(ρβ)f+ϕCu(ρβ)Cu/(ρβ)f1−ϕCu−ϕAl2O3+ϕAl2O3ρAl2O3/ρf+ϕCuρCu/ρfH3(ϕ)={ϕAl2O3kAl2O3+ϕCukCuϕAl2O3+ϕCu+2kf+2(ϕAl2O3kAl2O3+ϕCukCu)−2(ϕAl2O3+ϕCu)kf}×{ϕAl2O3kAl2O3+ϕCukCuϕAl2O3+ϕCu+2kf−(ϕAl2O3kAl2O3+ϕCukCu)+(ϕAl2O3+ϕCu)kf}−1×{1−ϕCu−ϕAl2O3+ϕAl2O3(ρcp)Al2O3/(ρcp)f+ϕCu(ρcp)Cu/(ρcp)f}−1H4(ϕ)={ϕAl2O3kAl2O3+ϕCukCuϕAl2O3+ϕCu+2kf−(ϕAl2O3kAl2O3+ϕCukCu)+(ϕAl2O3+ϕCu)kf}×{ϕAl2O3kAl2O3+ϕCukCuϕAl2O3+ϕCu+2kf+2(ϕAl2O3kAl2O3+ϕCukCu)−2(ϕAl2O3+ϕCu)kf}−1$

The physical quantities of interest are the local and average Nusselt numbers along the solid–fluid interface and along the upper wall that are defined as Display Formula

(13)$Nutop=−khnfkf1r∂θ∂γ, Nuinterface=−khnfkf∂θ∂r|r=1Nutop¯=∫01Nutop dr, Nuinterface¯=1π∫0πNuinterface dγ$

## Numerical Method

The governing equations (7)(10) with the corresponding initial and boundary conditions (11) are solved using the finite difference method [23,27,29,44,46,47]. The diffusive terms have been approximated by central differences. The convective terms have been discretized applying the second-order Samarksii monotonic difference scheme. The parabolic equations have been solved on the basis of Samarksii locally one-dimensional scheme. The obtained systems of algebraic equations have been solved by Thomas algorithm. The partial differential equation for the stream function (7) has been discretized by means of the five-point difference scheme on the basis of central differences for the partial derivatives. The obtained linear discretized equation has been solved by the successive over relaxation method.

The present model, in the form of an in-house computational fluid dynamics code, has been validated successfully against the works of Shi et al. [48], Kuehn and Goldstein [49], and Shahraki [50] for the steady-state natural convection between two concentric cylinders. It should be noted that Shi et al. [48] have used polar coordinates in their numerical analysis and a finite difference-based lattice Boltzmann method, Kuehn and Goldstein [49] have conducted experimental analysis, while Shahraki [50] have utilized Cartesian coordinates and the penalty finite element method. The results are presented in Fig. 2; the isotherms have a good agreement with numerical data of Shi et al. [48]. Table 2 shows a good comparison of the average Nusselt number at the hot wall of the present study with published results of other authors.

We have conducted also a grid-independent test, analyzing natural convection in a semicircular cavity presented in Fig. 1 for Ra = 105, Pr = 6.82, K = 1.0, $ϕAl2O3=ϕCu=0.03$. Four cases of the uniform grid are tested: a grid of 50 × 50 points, a grid of 80 × 80 points, a grid of 100 × 100 points, and a grid of 150 × 150 points. Figure 3(a) shows the effect of mesh parameters on average Nusselt number at solid–fluids interface.

Taking into account the conducted verifications, the uniform grid of 100 × 100 points has been selected for the further investigation. The utilized uniform grid of 100 × 100 points for the analyzed domain of interest is presented in Fig. 3(b).

## Results and Discussion

A numerical analysis of the problem under consideration has been conducted at the following values of the governing parameters: Rayleigh number (Ra = 104, 105, 106), thermal conductivity ratio (K = 1, 5, 20, ∞), nanoparticles volume fraction 0 ≤ $ϕ=ϕAl2O3+ϕCu$  ≤ 0.05, and dimensionless time (0 ≤ τ ≤ 200). It should be noted that in the present analysis, we have used equal volume fractions of alumina and copper $ϕAl2O3=ϕCu=0.5ϕ$. Particular efforts have been focused on the effects of these parameters on the nanofluid flow and heat transfer inside the cavity. Isolines of stream function and temperature as well as the average Nusselt number for different values of the governing parameters mentioned above are illustrated in Figs. 411.

Figure 4 shows the evolution of streamlines and isotherms for Ra = 105, K = 5, ϕ = 0.03. At initial time level (τ = 3), two convective cells are formed inside the cavity due to the effect of the temperature difference between the cooled upper wall and the heated bottom wall. These vortices illustrate an appearance of ascending flows along the solid–fluid interface and descending flow in the central part of the cavity. For τ = 3, the dominating heat transfer mechanism is a heat conduction. Therefore, the isotherms are parallel to the bottom heated wall. An increase in time (τ = 10 in Fig. 4(b)) produces an intensification of circulation inside the cavity with a formation of thermal plumes in the right and left upper corners of the domain of interest. One can find that convective cells cores displace from the solid–fluid interface to the central part of the cavity. Further increase in time leads to a collision of these thermal plumes in the center with a generation of one thermal plume of low temperature from the upper cold wall. At the same time, the convective cells cores drift to the vertical symmetry line of the cavity. The solid wall is heated fully except for the bottom part where cold descending flow impacts to this wall and the left and right upper parts where we have a cooling effect from the cold upper wall. A formation of a thin thermal boundary layer along the solid–fluid interface characterizes an interaction of the high- and low-temperature waves and the high Rayleigh number value. The latter also defines a thickness of the thermal boundary layer. Symmetrical fluid flow and temperature patterns for the steady-state regime (τ = 200) are defined by the symmetry of the boundary conditions and the laminar regime of convective heat transfer for Ra = 105.

The effect of the dimensionless time on the average Nusselt number at the solid–fluid interface is presented in Fig. 6. These obtained data illustrate an increase in $Nuinterface¯$ at the initial time level when the heat conduction is a dominated heat transfer mechanism, and after that, an evolution of side thermal plumes leads to a reduction of the temperature gradient, and as a result, a decrease in the average Nusselt number with a reaching of steady state is predicted.

It is well known that the Rayleigh number is a governing dimensionless parameter that defines different convective heat transfer modes. Figure 5 demonstrates the effect of Ra on the streamlines and isotherms at K = 5, ϕ = 0.03, τ = 200. A low Rayleigh number value characterizes less intensive circulation inside the cavity where the heat conduction is a major heat transfer mechanism. For Ra = 104, one can find inside the cavity a thick thermal boundary layers near the solid–fluid interface and the top cold wall with a weak penetration of hot and cold thermal plumes from one side to another one. An increase in Ra leads to an intensification of the convective flow and heat transfer with some modification of convective cells cores and thinning of the thermal boundary layers. It is worth noting that Ra = 106 (Fig. 5(c)) defines a beginning of the fluid flow and heat transfer chaotization where one can find a two-side low-temperature thermal plume near the top wall. Reduction of the thermal and velocity boundary layers close to the solid–fluid interface is illustrated by an increase in the streamlines and isotherms density near this wall. Also, high values of Ra characterize a less intensive heating of the solid wall, namely, for Ra = 104, the bottom part of the solid wall is heated and isotherm θ = 0.95 is located inside the fluid cavity near the interface; for Ra = 105, this isotherm is located inside the bottom part of the solid wall; and for Ra = 106, one can find an essential less intensive heating of this wall. Such behavior can be explained by intensive circulation of the nanofluid and cooling of the cavity. It is interesting to note a displacement of the convective cells cores from Ra = 105 close to the descending flow, but for Ra = 106, close to the high corners of the cavity.

An effect of the Rayleigh number and the dimensionless time on the average Nusselt number at the solid–fluid interface is pictured in Fig. 6. As has been mentioned above, a growth of Ra leads to the heat transfer enhancement. High values of the Rayleigh number illustrate a protraction of reaching of both the maximum value of $Nuinterface¯$ during the initial heat conduction regime and the steady-state regime. Moreover, for Ra = 106, we have a formation of heat oscillating regime due to a chaotization of the fluid flow and heat transfer inside the cavity.

Taking into account the conjugate heat transfer considered in the present paper, further on, we will analyze the effect of the thermal conductivity ratio that defines an intensity of heat conduction inside the solid wall. An increase in K is due to an increase in the thermal conductivity coefficient of solid wall. It should be noted that K = ∞ defines the nonconjugate problem with a solid wall having an infinitely thin wall. An increase in the thermal conductivity ratio leads to a more intensive heating of this solid wall. For example, for K = 1 (Fig. 7(a)), we have a weak heating of the solid wall that is also promoted by an effect of descending flow from the upper cold wall. Further increase in K leads to more intensive heating of the heat-conducting solid wall, where only the isotherm θ = 0.95 can be found inside this wall, and for K = 20 (Fig. 7(c)), this wall is heated fully. Therefore, K = ∞ is an extreme case when the solid–fluid interface is kept at a high temperature θ = 1. Such changes of K lead to less intensive cooling of the cavity, while the flow and temperature patterns are similar. At the same time, one can find a weak intensification of convective flow with K$|ψ|maxK=1=0.026<|ψ|maxK=5=0.029<|ψ|maxK=20=0.03<$$|ψ|maxK=∞=0.031$.

The effect of the thermal conductivity ratio on the evolution of the average Nusselt number at the solid–fluid interface is presented in Fig. 8. An increase in K leads to a growth of the temperature difference between the solid–fluid interface and surrounding nanofluid that reflects an increase in the value of $Nuinterface¯$.

An insertion of nanoparticles does not lead to modification of thermohydrodynamic behavior presented in Fig. 9, but the heat transfer rate increases with nanoparticles volume fraction (see Fig. 10). It should be noted that an addition of only 5% of Al2O3–Cu nanoparticles illustrates a growth of $Nuinterface¯$ from 4.9 to 5.4. At the same time, an addition of 5% of Al2O3 nanoparticles leads to a growth of $Nuinterface¯$ from 4.9 to 5.36. Moreover, the effect of hybrid nanofluid becomes more essential for high values of the Rayleigh number and the thermal conductivity ratio (see Fig. 11).

## Conclusions

A numerical simulation of unsteady conjugate natural convection in a semicircular cavity filled with a hybrid nanofluid has been carried out for a wide range of the Rayleigh number, thermal conductivity ratio, nanoparticle volume fraction, and the dimensionless time. The obtained results illustrate an essential effect of these governing parameters on the velocity and temperature patterns as well as the average Nusselt number at the solid–fluid interface. It has been found that the variation of the average Nusselt number with the dimensionless time can be divided into two parts: the first one illustrates an increase in $Nuinterface¯$, when the heat conduction is a dominated heat transfer mechanism, and the second part reflects an evolution of side thermal plumes that leads to a decrease in the average Nusselt number with the approach of the steady-state mode. A growth of the Rayleigh number characterizes the heat transfer enhancement. High values of Ra illustrate a protraction of the reach to the steady-state regime. An increase in K leads to a rise in the value of $Nuinterface¯$ and weak intensification of the convective flow. An addition of low-nanoparticles volume fraction inside the cavity leads to the heat transfer enhancement. Moreover, the effect of a hybrid nanofluid becomes more essential for high values of the Rayleigh number and the thermal conductivity ratio.

## Acknowledgements

This work of Igor V. Miroshnichenko and Mikhail A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project No. 13.9724.2017. The authors also wish to express their sincere thanks to the very competent Reviewers for the valuable comments and suggestions.

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Kuznetsov, G. V. , and Sheremet, M. A. , 2011, “ Unsteady Natural Convection of Nanofluids in an Enclosure Having Finite Thickness Walls,” Comput. Therm. Sci., 3(5), pp. 427–443.
Chamkha, A. J. , and Ismael, M. A. , 2013, “ Conjugate Heat Transfer in a Porous Cavity Filled With Nanofluids and Heated by a Triangular Thick Wall,” Int. J. Therm. Sci., 67, pp. 135–151.
Sheremet, M. A. , and Pop, I. , 2014, “ Conjugate Natural Convection in a Square Porous Cavity Filled by a Nanofluid Using Buongiorno's Mathematical Model,” Int. J. Heat Mass Transfer, 79, pp. 137–145.
Ismael, M. A. , and Chamkha, A. J. , 2015, “ Conjugate Natural Convection in a Differentially Heated Composite Enclosure Filled With a Nanofluid,” J. Porous Media, 18(7), pp. 699–716.
Ismael, M. A. , Armaghani, T. , and Chamkha, A. J. , 2016, “ Conjugate Heat Transfer and Entropy Generation in a Cavity Filled With a Nanofluid-Saturated Porous Media and Heated by a Triangular Solid,” J. Taiwan Inst. Chem. Eng., 59, pp. 138–151.
Noghrehabadi, A. , Pourrajab, R. , and Ghalambaz, M. , 2012, “ Effect of Partial Slip Boundary Condition on the Flow and Heat Transfer of Nanofluids Past Stretching Sheet Prescribed Constant Wall Temperature,” Int. J. Therm. Sci., 54, pp. 253–261.
Noghrehabadi, A. , Ghalambaz, M. , Ghalambaz, M. , and Ghanbarzadeh, A. , 2012, “ Comparing Thermal Enhancement of Ag-Water and SiO2-Water Nanofluids Over an Isothermal Stretching Sheet With Suction or Injection,” J. Comput. Appl. Res. Mech. Eng., 2(1), pp. 37–49.
Noghrehabadi, A. , Saffarian, M. R. , Pourrajab, R. , and Ghalambaz, M. , 2013, “ Entropy Analysis for Nanofluid Flow Over a Stretching Sheet in the Presence of Heat Generation/Absorption and Partial Slip,” J. Mech. Sci. Technol., 27(3), pp. 927–937.
Zaraki, A. , Ghalambaz, M. , Chamkha, A. J. , Ghalambaz, M. , and De Rossi, D. , 2015, “ Theoretical Analysis of Natural Convection Boundary Layer Heat and Mass Transfer of Nanofluids: Effects of Size, Shape and Type of Nanoparticles, Type of Base Fluid and Working Temperature,” Adv. Powder Technol., 26(3), pp. 935–946.
Pop, I. , Ghalambaz, M. , and Sheremet, M. , 2016, “ Free Convection in a Square Porous Cavity Filled With a Nanofluid Using Thermal Non Equilibrium and Buongiorno Models,” Int. J. Numer. Methods Heat Fluid Flow, 26(3–4), pp. 671–693.
Li, H. , Ha, C. S. , and Kim, I. , 2009, “ Fabrication of Carbon Nanotube/SiO2 and Carbon Nanotube/SiO2/Ag Nanoparticles Hybrids by Using Plasma Treatment,” Nanoscale Res. Lett., 4(11), pp. 1384–1388. [PubMed]
Guo, S. , Dong, S. , and Wang, E. , 2008, “ Gold/Platinum Hybrid Nanoparticles Supported on Multi Walled Carbon Nanotube/Silica Coaxial Nanocables: Preparation and Application as Electrocatalysts for Oxygen Reduction,” J. Phys. Chem. C, 112(7), pp. 2389–2393.
Sarkar, J. , Ghosh, P. , and Adil, A. , 2015, “ A Review on Hybrid Nanofluids: Recent Research, Development and Applications,” Renewable Sustainable Energy Rev., 43, pp. 164–177.
Suresh, S. , Venkitaraj, K. P. , Selvakumar, P. , and Chandrasekar, M. , 2011, “ Synthesis of Al2O3–Cu/Water Hybrid Nanofluids Using Two Step Method and Its Thermo Physical Properties,” Colloids Surf., A, 388(1–3), pp. 41–48.
Ho, C. J. , Huang, J. B. , Tsai, P. S. , and Yang, Y. M. , 2011, “ On Laminar Convective Cooling Performance of Hybrid Water-Based Suspensions of Al2O3 Nanoparticles and MEPCM Particles in a Circular Tube,” Int. J. Heat Mass Transfer, 54(11–12), pp. 2397–2407.
Ho, C. J. , Huang, J. B. , Tsai, P. S. , and Yang, Y. M. , 2011, “ Water-Based Suspensions of Al2O3 Nanoparticles and MEPCM Particles on Convection Effectiveness in a Circular Tube,” Int. J. Therm. Sci., 50(5), pp. 736–748.
Takabi, B. , and Shokouhmand, H. , 2015, “ Effects of Al2O3–Cu/Water Hybrid Nanofluid on Heat Transfer and Flow Characteristics in Turbulent Regime,” Int. J. Mod. Phys. C, 26(04), p. 1550047.
Sheremet, M. A. , Grosan, T. , and Pop, I. , 2015, “ Free Convection in a Square Cavity Filled With a Porous Medium Saturated by Nanofluid Using Tiwari and Das' Nanofluid Model,” Transp. Porous Media, 106(3), pp. 595–610.
Cho, C. C. , Chiu, C. H. , and Lai, C. Y. , 2016, “ Natural Convection and Entropy Generation of Al2O3–Water Nanofluid in an Inclined Wavy-Wall Cavity,” Int. J. Heat Mass Transfer, 97, pp. 511–520.
Sheremet, M. A. , 2012, “ Interaction of Two-Dimensional Thermal “Plumes” From Local Sources of Energy Under Conditions of Conjugate Natural Convection in a Horizontal Cylinder,” J. Appl. Mech. Tech. Phys., 53(4), pp. 566–576.
Sheremet, M. A. , and Pop, I. , 2014, “ Thermo-Bioconvection in a Square Porous Cavity Filled by Oxytactic Microorganisms,” Transp. Porous Media, 103(2), pp. 191–205.
Shi, Y. , Zhao, T. S. , and Guo, Z. L. , 2006, “ Finite Difference-Based Lattice Boltzmann Simulation of Natural Convection Heat Transfer in a Horizontal Concentric Annulus,” Comput. Fluids, 35(1), pp. 1–15.
Kuehn, T. H. , and Goldstein, R. J. , 1978, “ An Experimental Study of Natural Convection Heat Transfer in Concentric and Eccentric Horizontal Cylindrical Annuli,” ASME J. Heat Transfer, 100(4), pp. 635–640.
Shahraki, F. , 2002, “ Modeling of Buoyancy-Driven Flow and Heat Transfer for Air in a Horizontal Annulus: Effects of Vertical Eccentricity and Temperature-Dependent Properties,” Numer. Heat Transfer, Part A, 42(6), pp. 603–621.
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Chamkha, A. J. , and Ismael, M. A. , 2013, “ Conjugate Heat Transfer in a Porous Cavity Heated by a Triangular Thick Wall,” Numer. Heat Transfer A, 63(2), pp. 144–158.
Martyushev, S. G. , and Sheremet, M. A. , 2014, “ Conjugate Natural Convection Combined With Surface Thermal Radiation in an Air Filled Cavity With Internal Heat Source,” Int. J. Therm. Sci., 76, pp. 51–67.
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Das, S. K. , Choi, S. U. S. , Yu, W. , and Pradeep, T. , 2007, Nanofluids-Science and Technology, Wiley, Hoboken, NJ.
Buongiorno, J. , 2006, “ Convective Transport in Nanofluids,” ASME J. Heat Transfer, 128(3), pp. 240–250.
Kuznetsov, G. V. , and Sheremet, M. A. , 2011, “ Unsteady Natural Convection of Nanofluids in an Enclosure Having Finite Thickness Walls,” Comput. Therm. Sci., 3(5), pp. 427–443.
Chamkha, A. J. , and Ismael, M. A. , 2013, “ Conjugate Heat Transfer in a Porous Cavity Filled With Nanofluids and Heated by a Triangular Thick Wall,” Int. J. Therm. Sci., 67, pp. 135–151.
Sheremet, M. A. , and Pop, I. , 2014, “ Conjugate Natural Convection in a Square Porous Cavity Filled by a Nanofluid Using Buongiorno's Mathematical Model,” Int. J. Heat Mass Transfer, 79, pp. 137–145.
Ismael, M. A. , and Chamkha, A. J. , 2015, “ Conjugate Natural Convection in a Differentially Heated Composite Enclosure Filled With a Nanofluid,” J. Porous Media, 18(7), pp. 699–716.
Ismael, M. A. , Armaghani, T. , and Chamkha, A. J. , 2016, “ Conjugate Heat Transfer and Entropy Generation in a Cavity Filled With a Nanofluid-Saturated Porous Media and Heated by a Triangular Solid,” J. Taiwan Inst. Chem. Eng., 59, pp. 138–151.
Noghrehabadi, A. , Pourrajab, R. , and Ghalambaz, M. , 2012, “ Effect of Partial Slip Boundary Condition on the Flow and Heat Transfer of Nanofluids Past Stretching Sheet Prescribed Constant Wall Temperature,” Int. J. Therm. Sci., 54, pp. 253–261.
Noghrehabadi, A. , Ghalambaz, M. , Ghalambaz, M. , and Ghanbarzadeh, A. , 2012, “ Comparing Thermal Enhancement of Ag-Water and SiO2-Water Nanofluids Over an Isothermal Stretching Sheet With Suction or Injection,” J. Comput. Appl. Res. Mech. Eng., 2(1), pp. 37–49.
Noghrehabadi, A. , Saffarian, M. R. , Pourrajab, R. , and Ghalambaz, M. , 2013, “ Entropy Analysis for Nanofluid Flow Over a Stretching Sheet in the Presence of Heat Generation/Absorption and Partial Slip,” J. Mech. Sci. Technol., 27(3), pp. 927–937.
Zaraki, A. , Ghalambaz, M. , Chamkha, A. J. , Ghalambaz, M. , and De Rossi, D. , 2015, “ Theoretical Analysis of Natural Convection Boundary Layer Heat and Mass Transfer of Nanofluids: Effects of Size, Shape and Type of Nanoparticles, Type of Base Fluid and Working Temperature,” Adv. Powder Technol., 26(3), pp. 935–946.
Pop, I. , Ghalambaz, M. , and Sheremet, M. , 2016, “ Free Convection in a Square Porous Cavity Filled With a Nanofluid Using Thermal Non Equilibrium and Buongiorno Models,” Int. J. Numer. Methods Heat Fluid Flow, 26(3–4), pp. 671–693.
Li, H. , Ha, C. S. , and Kim, I. , 2009, “ Fabrication of Carbon Nanotube/SiO2 and Carbon Nanotube/SiO2/Ag Nanoparticles Hybrids by Using Plasma Treatment,” Nanoscale Res. Lett., 4(11), pp. 1384–1388. [PubMed]
Guo, S. , Dong, S. , and Wang, E. , 2008, “ Gold/Platinum Hybrid Nanoparticles Supported on Multi Walled Carbon Nanotube/Silica Coaxial Nanocables: Preparation and Application as Electrocatalysts for Oxygen Reduction,” J. Phys. Chem. C, 112(7), pp. 2389–2393.
Sarkar, J. , Ghosh, P. , and Adil, A. , 2015, “ A Review on Hybrid Nanofluids: Recent Research, Development and Applications,” Renewable Sustainable Energy Rev., 43, pp. 164–177.
Suresh, S. , Venkitaraj, K. P. , Selvakumar, P. , and Chandrasekar, M. , 2011, “ Synthesis of Al2O3–Cu/Water Hybrid Nanofluids Using Two Step Method and Its Thermo Physical Properties,” Colloids Surf., A, 388(1–3), pp. 41–48.
Ho, C. J. , Huang, J. B. , Tsai, P. S. , and Yang, Y. M. , 2011, “ On Laminar Convective Cooling Performance of Hybrid Water-Based Suspensions of Al2O3 Nanoparticles and MEPCM Particles in a Circular Tube,” Int. J. Heat Mass Transfer, 54(11–12), pp. 2397–2407.
Ho, C. J. , Huang, J. B. , Tsai, P. S. , and Yang, Y. M. , 2011, “ Water-Based Suspensions of Al2O3 Nanoparticles and MEPCM Particles on Convection Effectiveness in a Circular Tube,” Int. J. Therm. Sci., 50(5), pp. 736–748.
Takabi, B. , and Shokouhmand, H. , 2015, “ Effects of Al2O3–Cu/Water Hybrid Nanofluid on Heat Transfer and Flow Characteristics in Turbulent Regime,” Int. J. Mod. Phys. C, 26(04), p. 1550047.
Sheremet, M. A. , Grosan, T. , and Pop, I. , 2015, “ Free Convection in a Square Cavity Filled With a Porous Medium Saturated by Nanofluid Using Tiwari and Das' Nanofluid Model,” Transp. Porous Media, 106(3), pp. 595–610.
Cho, C. C. , Chiu, C. H. , and Lai, C. Y. , 2016, “ Natural Convection and Entropy Generation of Al2O3–Water Nanofluid in an Inclined Wavy-Wall Cavity,” Int. J. Heat Mass Transfer, 97, pp. 511–520.
Sheremet, M. A. , 2012, “ Interaction of Two-Dimensional Thermal “Plumes” From Local Sources of Energy Under Conditions of Conjugate Natural Convection in a Horizontal Cylinder,” J. Appl. Mech. Tech. Phys., 53(4), pp. 566–576.
Sheremet, M. A. , and Pop, I. , 2014, “ Thermo-Bioconvection in a Square Porous Cavity Filled by Oxytactic Microorganisms,” Transp. Porous Media, 103(2), pp. 191–205.
Shi, Y. , Zhao, T. S. , and Guo, Z. L. , 2006, “ Finite Difference-Based Lattice Boltzmann Simulation of Natural Convection Heat Transfer in a Horizontal Concentric Annulus,” Comput. Fluids, 35(1), pp. 1–15.
Kuehn, T. H. , and Goldstein, R. J. , 1978, “ An Experimental Study of Natural Convection Heat Transfer in Concentric and Eccentric Horizontal Cylindrical Annuli,” ASME J. Heat Transfer, 100(4), pp. 635–640.
Shahraki, F. , 2002, “ Modeling of Buoyancy-Driven Flow and Heat Transfer for Air in a Horizontal Annulus: Effects of Vertical Eccentricity and Temperature-Dependent Properties,” Numer. Heat Transfer, Part A, 42(6), pp. 603–621.

## Figures

Fig. 1

Physical model and coordinate system

Fig. 2

Comparison of isotherms θ for different values of Rayleigh and Prandtl numbers: (a) present results and (b) numerical data of Shi et al. [48]

Fig. 3

Variations of average Nusselt number at solid–fluid interface for Ra = 105, Pr = 6.82, K = 1.0, ϕAl2O3=ϕCu=0.03 and different mesh parameters (a) and the utilized uniform grid of 100 × 100 points (b)

Fig. 4

Streamlines ψ and isotherms θ at Ra = 105, K = 5, ϕ = 0.03: τ = 3 (a), τ = 10 (b), τ = 20 (c), τ = 50 (d), and τ = 200 (e)

Fig. 5

Streamlines ψ and isotherms θ at K = 5, ϕ = 0.03, τ = 200: Ra = 104 (a), Ra = 105 (b), and Ra = 106 (c)

Fig. 6

Variation of the average Nusselt number at solid–fluid interface versus the dimensionless time and Rayleigh number for K = 5, ϕ = 0.03

Fig. 7

Streamlines ψ and isotherms θ at Ra = 105, ϕ = 0.03, τ = 200: K = 1 (a), K = 5 (b), K = 20 (c), and K = ∞ (d)

Fig. 8

Variation of the average Nusselt number at solid–fluid interface versus the dimensionless time and thermal conductivity ratio for Ra = 105, ϕ = 0.03

Fig. 10

Variation of the average Nusselt number at solid–fluid interface versus the dimensionless time and nanoparticles volume fraction for Ra = 105, K = 5

Fig. 11

Variations of average Nusselt number at solid–fluid interface versus the nanoparticles volume fraction and Rayleigh number for K = 5, τ = 200 (a), versus the nanoparticles volume fraction and thermal conductivity ratio for Ra = 105, τ = 200 (b)

Fig. 9

Streamlines ψ and isotherms θ at Ra = 105, K = 5, τ = 200: ϕ = 0.0 (a), ϕ = 0.03 (b), and ϕ = 0.05 (c)

## Tables

Table 1 Physical properties of base fluid, copper (Cu), and alumina (Al2O3) nanoparticles [44,45]
Table 2 Comparison of the average Nusselt number at the hot wall

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