**1. Introduction**

Fluid flow and heat transfer due to natural convection in porous annulus is one of the most considerable research issues due to its wide applications in science and engineering [1–3], such as thermal insulators, chemical catalytic convectors, thermal storage systems, geothermal energy utilization, electronic cooling, and nuclear reactor systems. Therefore, natural convection in porous annulus has been extensively investigated during the past decades. Caltagirone [4] was the first to study natural convection in a saturated porous medium bounded by two concentric, horizontal, isothermal cylinders experimentally and numerically. The author used the Christiansen effect in order to visualize a fluctuating three-dimensional thermal field for Rayleigh number exceeding some critical value. Rao et al. [5,6] performed steady and transient investigations on natural convection in a horizontal porous annulus heated from the inner face using Galerkin method. The effects of Rayleigh number and Darcy number on heat transfer characteristics were studied. In addition, the bifurcation point was obtained numerically, which compared very well with that from experimental observation.

Himasekhar [7] examined the two-dimensional bifurcation phenomena in thermal convection in horizontal, concentric annuli containing saturated porous media. The fluid motion is described by the Darcy–Oberbeck–Boussinesq equations, which were solved using regular perturbation expansion. The flow structure was obtained under different parameters, such as the radius ratio and Rayleigh–Darcy number. A parametric study was performed by Leong et al. [8] to investigate the effects of Rayleigh number, Darcy number, porous sleeve thickness, and relative thermal conductivity on heat transfer characteristics. Braga et al. [9] presented numerical computations for laminar and turbulent natural

**Citation:** Zhang, L.; Hu, Y.; Li, M. Numerical Study of Natural Convection Heat Transfer in a Porous Annulus Filled with a Cu-Nanofluid. *Nanomaterials* **2021**, *11*, 990. https:// doi.org/10.3390/nano11040990

Academic Editor: S M Sohel Murshed

Received: 26 March 2021 Accepted: 8 April 2021 Published: 12 April 2021

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convection within a horizontal cylindrical annulus filled with a fluid saturated porous medium. Computations covered the range 25 < *Ra*<sup>m</sup> < 500 and 3.2 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>&</sup>gt; *Da* > 3.2 <sup>×</sup> <sup>10</sup>−<sup>6</sup> and made use of the finite volume method. Khanafer et al. [10] carried out a numerical simulation in order to examine the parametric effects of Rayleigh number and radius ratio on the role played by natural convection heat transfer in the porous annuli. The model was governed by Darcy–Oberbeck–Boussinesq equations and solved using the Galerkin method. In order to investigate the buoyancy-induced flow as affected by the presence of the porous layer, Alloui and Vasseur [11] studied natural convection in a horizontal annular porous layer filled with a binary fluid under the influence of the Soret effect using the Darcy model with the Boussinesq approximation. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters, such as Rayleigh number, Lewis number, buoyancy ratio, radius ratio of the cavity, and normalized porosity.

Belabid and Cheddadi [12] solved natural convection heat transfer within a twodimensional horizontal annulus filled with a saturated porous medium using ADI (Alternating Direction Implicit) finite difference method. This work placed emphasis on the mesh effect on the determination of the bifurcation point between monocellular and bicellular flows for different values of the aspect ratio. In a recent work, Belabid and Allali [13,14] studied the effects of a periodic gravitational and temperature modulation on the convective instability in a horizontal porous annulus. Results showed that the convective instability is influenced by the amplitude and the frequency of the modulation. Rostami et al. [15] provided a review of recent natural convection studies, including experimental and numerical studies. The effects of the parameters, such as nanoparticle addition, magnetic fields, and porous medium on the natural convection were examined.

The traditional fluids in engineering, such as water and mineral oils, have a primary limitation in the enhancement of heat transfer due to a rather low thermal conductivity. The term nanofluids, which was first put forward by Choi [16], is used to describe the mixture of nanoparticles and base fluid. Due to the relatively higher thermal conductivities, nanofluids are considered as an effective approach to meet some challenges associated with the traditional fluids [17]. In the last few decades, studies on the natural convection of nanofluids in enclosure were conducted by a number of researchers [18]. For instance, Jou and Tzeng [19] investigated the numerically natural convection heat transfer enhancement utilizing nanofluids in a two-dimensional enclosure. Results showed that increasing the buoyancy parameter and volume fraction of nanofluids caused an increase in the average heat transfer coefficient. Ghasemi et al. [20] simulated the natural convection heat transfer in an inclined enclosure filled with a CuO–water nanofluid. The effects of pertinent parameters such as Rayleigh number, inclination angle, and solid volume fraction on the heat transfer characteristics were studied. The results indicated that the heat transfer rate is maximized at a specific inclination angle depending on Rayleigh number and solid volume fraction.

In a related work, Abu-Nada and Oztop [21] found that the effect of nanoparticles concentration on Nusselt number was more pronounced at low volume fraction than at high volume fraction and the inclination angle could be a control parameter for nanofluid filled enclosure. Soleimani et al. [22] studied the natural convection heat transfer in a semiannulus enclosure filled with a Cu–water nanofluid using the Control Volume based Finite Element Method. The numerical investigation was carried out for different governing parameters, such as Rayleigh number, nanoparticle volume fraction, and angle of turn for the enclosure. The results revealed that there was an optimal angle of turn in which the average Nusselt number was maximum for each Rayleigh number. Seyyedi et al. [23] simulated the natural convection heat transfer of Cu–water nanofluid in an annulus enclosure using the Control Volume-based Finite Element Method. The Maxwell–Garnetts and Brinkman models were employed to estimate the effect of thermal conductivity and viscosity of nanofluid. The results showed the effects of the governing parameters on the local Nusselt number, average Nusselt number, streamlines, and isotherms. Boualit et al. [24] and Liao [25] studied respectively natural convection heat transfer of Cu- and

Al2O3-water nanofluids in a square enclosure under the horizontal temperature gradient. Both the flow structure and the corresponding heat transfer characteristics at different Rayleigh numbers and nanoparticle volume fractions were obtained. Wang et al. [26] investigated numerically the natural convection in a partially heated enclosure filled with Al2O<sup>3</sup> nanofluids. The results indicated that at low Rayleigh numbers, the heat transfer performance increased with nanoparticle volume fraction, while at high Rayleigh numbers, there existed an optimal volume fraction at which the heat transfer performance had a peak. In a recent work, Mi et al. [27] examined the effects of graphene nano-sheets (GNs) nanoparticles by comparing the thermal conductivity of graphene nano-sheets (GNs)/ethylene glycol (EG) nanofluid with EG thermal conductivity. Results showed that the presence of nanoparticles improved the thermal conductivity, and with increasing temperature, the effect of adding GNs was strengthened. water nanofluids in a square enclosure under the horizontal temperature gradient. Both the flow structure and the corresponding heat transfer characteristics at different Rayleigh numbers and nanoparticle volume fractions were obtained. Wang et al. [26] investigated numerically the natural convection in a partially heated enclosure filled with Al2O3 nanofluids. The results indicated that at low Rayleigh numbers, the heat transfer performance increased with nanoparticle volume fraction, while at high Rayleigh numbers, there existed an optimal volume fraction at which the heat transfer performance had a peak. In a recent work, Mi et al. [27] examined the effects of graphene nano-sheets (GNs) nanoparticles by comparing the thermal conductivity of graphene nano-sheets (GNs)/ethylene glycol (EG) nanofluid with EG thermal conductivity. Results showed that the presence of nanoparticles improved the thermal conductivity, and with increasing temperature, the effect of adding GNs was strengthened.

[24] and Liao [25] studied respectively natural convection heat transfer of Cu- and Al2O3-

*Nanomaterials* **2021**, *11*, x FOR PEER REVIEW 3 of 24

Although several numerical and experimental studies on the natural convection heat transfer were published, most of them concentrated on traditional fluids in cavities, and only a few of them consider a nanofluid in a porous annulus [28–30]. In the present work, steady natural convection heat transfer in a porous annulus filled with a Cu-nanofluid has been investigated, and the governing equations, including the Darcy–Brinkman equation, were solved using the Galerkin method. This paper presented a systematical examination on the effects of Brownian motion, solid volume fraction, nanoparticle diameter, Rayleigh number, Darcy number, porosity on the flow pattern, temperature distribution, and heat transfer characteristics. To the best of our knowledge, no study on this problem has been considered before, and accordingly, the current paper will address this topic. Although several numerical and experimental studies on the natural convection heat transfer were published, most of them concentrated on traditional fluids in cavities, and only a few of them consider a nanofluid in a porous annulus [28–30]. In the present work, steady natural convection heat transfer in a porous annulus filled with a Cu-nanofluid has been investigated, and the governing equations, including the Darcy–Brinkman equation, were solved using the Galerkin method. This paper presented a systematical examination on the effects of Brownian motion, solid volume fraction, nanoparticle diameter, Rayleigh number, Darcy number, porosity on the flow pattern, temperature distribution, and heat transfer characteristics. To the best of our knowledge, no study on this problem has been considered before, and accordingly, the current paper will address this topic.

#### **2. Problem Formulation 2. Problem Formulation**

#### *2.1. Physical Description 2.1. Physical Description*

We consider porous annulus filled with a Cu–water nanofluid between a horizontal inner and outer cylinder of radius *r*<sup>i</sup> and *r*o, respectively, as shown in Figure 1. The inner and outer cylinders are kept at uniform high temperature *T*<sup>i</sup> and low temperature *T*o, respectively. It is taken into consideration that the flow is two-dimensional, steady, and laminar due to the low velocity. The porous medium is considered as isotropic, homogeneous, and filled with a nanofluid, which is thermal equilibrium with the solid matrix, and the Darcy–Brinkman equation without inertia item is adopted. For the nanofluid, the effect of Brownian motion [31–33] is considered. The thermophysical properties of the base water, copper nanoparticles, and solid structure of the porous medium used in this study are presented in Tables 1 and 2 [34,35]. We consider porous annulus filled with a Cu–water nanofluid between a horizontal inner and outer cylinder of radius *r*i and *r*o, respectively, as shown in Figure 1. The inner and outer cylinders are kept at uniform high temperature *T*i and low temperature *T*o, respectively. It is taken into consideration that the flow is two-dimensional, steady, and laminar due to the low velocity. The porous medium is considered as isotropic, homogeneous, and filled with a nanofluid, which is thermal equilibrium with the solid matrix, and the Darcy–Brinkman equation without inertia item is adopted. For the nanofluid, the effect of Brownian motion [31–33] is considered. The thermophysical properties of the base water, copper nanoparticles, and solid structure of the porous medium used in this study are presented in Tables 1 and 2 [34,35].

**Figure 1. Figure 1.**  Sketch of problem geometry. Sketch of problem geometry.


**Table 1.** Thermal physical properties of the base fluid (water), nanoparticle (Cu), and solid of the porous medium (glass balls).


*<sup>C</sup>* = 3.6 <sup>×</sup> <sup>10</sup><sup>4</sup> ; *<sup>k</sup>*<sup>b</sup> = 1.38065 <sup>×</sup> <sup>10</sup>−<sup>23</sup> J K−<sup>1</sup> , *d*bf = 0.1[6*M*/(*Nπρ*m)]1/3: *M* = 0.018 kg mol−<sup>1</sup> , *<sup>N</sup>* = 6.022 <sup>×</sup> <sup>10</sup><sup>23</sup> mol−<sup>1</sup> , *ρ*, (*ρc*p) and *µ* denote the density, heat capacitance, and dynamic viscosity, respectively, *k* is the thermal conductivity, *φ* is the nanoparticle volume fraction, and the subscripts bf, sp, and nf designate the base fluid, nanoparticle, and nanofluid.
