Numerical Study of Natural Convection Heat Transfer in a Porous Annulus Filled with a Cu-Nanoﬂuid

nanomaterials

Article Numerical Study of Natural Convection Heat Transfer in a Porous Annulus Filled with a Cu-Nanoﬂuid

Lingyun Zhang 1,2, Yupeng Hu 2,* and Minghai Li 2,*

1 School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100000, China; [email protected] 2 Institute of System Engineering, China Academy of Engineering Physics, Mianyang 621999, China * Correspondence: [email protected] (Y.H.); [email protected] (M.L.); Tel.: +86-0816-2494908 (Y.H.); +86-0816-2484335 (M.L.)

Abstract: Natural convection heat transfer in a porous annulus filled with a Cu nanofluid has been investigated numerically. The Darcy–Brinkman and the energy transport equations are employed to describe the nanofluid motion and the heat transfer in the porous medium. Numerical results including the isotherms, streamlines, and heat transfer rate are obtained under the following parameters: Brownian motion, Rayleigh number (103–105), Darcy number (10−4–10−2), nanoparticle volume fraction (0.01–0.09), nanoparticle diameter (10–90 nm), porosity (0.1–0.9), and radius ratio (1.1–10). Results show that Brownian motion should be considered. The nanoparticle volume fraction has a positive effect on the heat transfer rate, especially with high Rayleigh number and Darcy number, while the nanoparticle diameter has an inverse influence. The heat transfer rate is enhanced with the increase of porosity. The radius ratio has a significant influence on the isotherms, streamlines, and heat transfer rate, and the rate is greatly enhanced with the increase of radius ratio. Keywords: natural convection; heat transfer; water-based nanofluid; Brownian motion; porous

Citation: Zhang, L.; Hu, Y.; Li, M. medium; Darcy–Brinkman equation; numerical simulation Numerical Study of Natural Convection Heat Transfer in a Porous Annulus Filled with a Cu-Nanofluid. Nanomaterials 2021, 11, 990. https:// 1. Introduction doi.org/10.3390/nano11040990 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 engi- Academic Editor: S M Sohel Murshed neering [1–3], such as thermal insulators, chemical catalytic convectors, thermal storage systems, geothermal energy utilization, electronic cooling, and nuclear reactor systems. Received: 26 March 2021 Therefore, natural convection in porous annulus has been extensively investigated during Accepted: 8 April 2021 the past decades. Caltagirone [4] was the first to study natural convection in a saturated Published: 12 April 2021 porous medium bounded by two concentric, horizontal, isothermal cylinders experimen- tally and numerically. The author used the Christiansen effect in order to visualize a Publisher’s Note: MDPI stays neutral fluctuating three-dimensional thermal field for Rayleigh number exceeding some critical with regard to jurisdictional claims in published maps and institutional affil- value. Rao et al. [5,6] performed steady and transient investigations on natural convection iations. 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 Copyright: © 2021 by the authors. convection in horizontal, concentric annuli containing saturated porous media. The fluid Licensee MDPI, Basel, Switzerland. motion is described by the Darcy–Oberbeck–Boussinesq equations, which were solved This article is an open access article distributed under the terms and using regular perturbation expansion. The flow structure was obtained under different conditions of the Creative Commons parameters, such as the radius ratio and Rayleigh–Darcy number. A parametric study was Attribution (CC BY) license (https:// performed by Leong et al. [8] to investigate the effects of Rayleigh number, Darcy number, creativecommons.org/licenses/by/ porous sleeve thickness, and relative thermal conductivity on heat transfer characteris- 4.0/). tics. Braga et al. [9] presented numerical computations for laminar and turbulent natural

Nanomaterials 2021, 11, 990. https://doi.org/10.3390/nano11040990 https://www.mdpi.com/journal/nanomaterials Nanomaterials 2021, 11, 990 2 of 24

convection within a horizontal cylindrical annulus filled with a fluid saturated porous −4 −6 medium. Computations covered the range 25 < Ram < 500 and 3.2 × 10 > Da > 3.2 × 10 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 two- dimensional horizontal annulus filled with a saturated porous medium using ADI (Al- ternating 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 semi- annulus 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 Nanomaterials 2021, 11, x FOR PEER REVIEW 3 of 24

Nanomaterials 2021, 11, 990 3 of 24 [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 Alflow2O 3structure-water nanofluids and the correspondi in a squareng enclosure heat transfer under characteristics the horizontal at different temperature Rayleigh gradient. num- Bothbers and the nanoparticle flow structure volume and the fractions corresponding were obtained. heat transferWang et characteristicsal. [26] investigated at different numer- Rayleigh numbers and nanoparticle volume fractions were obtained. Wang et al. [26] ically the natural convection in a partially heated enclosure filled with Al2O3 nanofluids. The investigatedresults indicated numerically that at low the naturalRayleigh convection numbers, in the a partially heat transfer heated performance enclosure filled increased with Alwith2O 3nanoparticlenanofluids. volume The results fraction, indicated while thatat hi atgh low Rayleigh Rayleigh numbers, numbers, there the existed heat transferan opti- performancemal volume increasedfraction at with which nanoparticle the heat transf volumeer performance fraction, while had at a high peak. Rayleigh In a recent numbers, work, thereMi et existedal. [27] anexamined optimal the volume effects fraction of graphe at whichne nano-sheets the heat transfer (GNs) nanoparticles performance hadby compar- a peak. Ining a recentthe thermal work, conductivity Mi et al. [27] of examined graphene the nano effects-sheets of graphene (GNs)/ethylene nano-sheets glycol (GNs) (EG) nanofluid nanoparticles by comparing the thermal conductivity of graphene nano-sheets (GNs)/ethylene with EG thermal conductivity. Results showed that the presence of nanoparticles improved glycol (EG) nanofluid with EG thermal conductivity. Results showed that the presence the thermal conductivity, and with increasing temperature, the effect of adding GNs was of nanoparticles improved the thermal conductivity, and with increasing temperature, strengthened. the effect of adding GNs was strengthened. Although several numerical and experimental studies on the natural convection heat Although several numerical and experimental studies on the natural convection transfer were published, most of them concentrated on traditional fluids in cavities, and heat transfer were published, most of them concentrated on traditional fluids in cavities, only a few of them consider a nanofluid in a porous annulus [28–30]. In the present work, and only a few of them consider a nanofluid in a porous annulus [28–30]. In the present steady natural convection heat transfer in a porous annulus filled with a Cu-nanofluid has work, steady natural convection heat transfer in a porous annulus filled with a Cu-nanofluid been investigated, and the governing equations, including the Darcy–Brinkman equation, has been investigated, and the governing equations, including the Darcy–Brinkman equa- were solved using the Galerkin method. This paper presented a systematical examination tion, were solved using the Galerkin method. This paper presented a systematical exam- inationon the effects on the of effects Brownian of Brownian motion, motion,solid volu solidme fraction, volume fraction,nanoparticle nanoparticle diameter, diameter, Rayleigh Rayleighnumber, number,Darcy number, Darcy number,porosity porosityon the flow on thepattern, flow temperature pattern, temperature distribution, distribution, and heat andtransfer heat characteristics. transfer characteristics. To the best To of the our best knowledge, of our knowledge, no studyno onstudy this problem on this problemhas been hasconsidered been considered before, and before, accordingly, and accordingly, the current the paper current will paper address will addressthis topic. this topic.

2.2. ProblemProblem FormulationFormulation 2.1.2.1. PhysicalPhysical DescriptionDescription WeWe consider consider porous porous annulus annulus filled filled with with a Cu–watera Cu–water nanofluid nanofluid between between a horizontal a horizontal in- nerinner and and outer outer cylinder cylinder of radiusof radiusri and ri andro, respectively,ro, respectively, as shown as shown in Figure in Figure1. The 1. inner The inner and outerand outer cylinders cylinders are kept are atkept uniform at uniform high high temperature temperatureTi and Ti lowand temperaturelow temperatureTo, respec- To, re- tively.spectively. It is taken It is taken into consideration into consideration that the th flowat the is two-dimensional,flow is two-dimensional, steady, andsteady, laminar and duelaminar to the due low to velocity.the low velocity. The porous The porous medium me isdium considered is considered as isotropic, as isotropic, homogeneous, homoge- andneous, filled and with filled a nanofluid, with a nanofluid, which is which thermal is equilibriumthermal equilibrium with the with solid the matrix, solid and matrix, the Darcy–Brinkmanand the Darcy–Brinkman equation equation without inertiawithout item inerti is adopted.a item is adopted. For the nanofluid, For the nanofluid, the effect the of Brownianeffect of Brownian motion [31 motion–33] is considered.[31–33] is consid The thermophysicalered. The thermophysical properties ofproperties the base water,of the copperbase water, nanoparticles, copper nanoparticles, and solid structure and solid of structure the porous of the medium porous used medium in this used study in arethis presentedstudy are inpresented Tables1 andin Tables2[34,35 1]. and 2 [34,35].

FigureFigure 1.1. SketchSketch ofof problemproblem geometry.geometry.

Nanomaterials 2021, 11, 990 4 of 24

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

Physical Properties Base Fluid (Water) Nanoparticle (Cu) Porous (Glass Balls) ρ [kg/m3] 997.1 8933 2700 cp [J/(kg·K)] 4179 385 840 k [W/(m·K)] 0.613 76.5 1.05 µ [kg/(m·s)] 0.001003 - - β × 105 [1/K] 21 1.67 0.9

Table 2. Applied models for thermophysical properties of the nanoﬂuid with or without Brownian motion.

Applied Model Physical Properties Without Brownian Motion With Brownian Motion

knf ksp+2kbf−2ϕ(kbf−ksp) ksp+2kbf−2ϕ(kbf−ksp) 2kb To k [W/(m·K)] = k = k + C ρcp 2 ϕdsp kbf ksp+2kbf+ϕ(kbf−ksp) nf bf ksp+2kbf+ϕ(kbf−ksp) nf πµnfdsp 2.5 −0.3 1.03 µ [kg/(m·s)] µnf = µbf/(1 − φ) µnf = µbf/[1 − 34.87(dsp/dbf) φ ]; 3 ρ [kg/m ] (ρcp)nf = (1 − φ)(ρcp)bf + φ((ρ cp)sp

cp [J/(kg·K)] ρnf(T) = (1 − φ)ρbf(T) + φρsp; 4 −23 −1 1/3 −1 23 −1 C = 3.6 × 10 ; kb = 1.38065 × 10 JK , dbf = 0.1[6M/(Nπρm)] : M = 0.018 kg mol , N = 6.022 × 10 mol , ρ,(ρcp) 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.

2.2. Governing Equations and Boundary Conditions By introducing the Boussinesq approximation, the governing equations for the heat transfer and ﬂuid ﬂow can be written as follows: ∂u ∂v + = 0 (1) ∂x ∂y

∂u ∂u 1 ∂p ∂2u ∂2u ν + = − + + − 2 nf u v ε ενnf 2 2 ε u (2) ∂x ∂y ρnf ∂x ∂x ∂y K ∂v ∂v 1 ∂p ∂2v ∂2v ν + = − + + − 2 nf + 2 ( − ) u v ε ενnf 2 2 ε v ε gβ T To (3) ∂x ∂y ρnf ∂y ∂x ∂y K ∂T ∂T ∂2T ∂2T ρc u + v = k + (4) p nf ∂x ∂y mnf ∂x2 ∂y2 where (x, y) are the Cartesian coordinates of the geometry, (u, v) are the velocity components, T and p are the temperature and pressure, respectively, ρ,(ρcp), ν, β and α denote the density, heat capacitance, kinematic viscosity, thermal expansion coefficient, and thermal diffusion, respectively, k is the thermal conductivity, ε is the porosity, and K is the medium permeability. The subscripts nf and mnf designate nanofluid and porous medium filled with nanofluid. The effective thermal conductivity of the porous medium filled with nanofluid can be as modeled as: kmnf = εknf + (1 − ε)ks. (5) Using the following dimensional variables, 2 (x,y) (u,v)(ro−ri) p(ro−ri) T−To kmnf (X, Y) = , (U, V) = , P = 2 , θ = , αmnf = , (ro−ri) αmnf ρ α Ti−To ρc nf mnf ( p)nf ( − )( − )3 Ra = gβ Ti To ro ri , Pr = vnf , Da = k . vnfαmnf αmnf L2 The governing Equations (1)–(4) reduce to a dimensionless form:

∂U ∂U + = 0 (6) ∂X ∂Y Nanomaterials 2021, 11, 990 5 of 24

∂U ∂U ∂P ∂2U ∂2U Pr U + V = −ε + εPr + − ε2 U (7) ∂X ∂Y ∂X ∂X2 ∂Y2 Da ∂V ∂V ∂P ∂2V ∂2V Pr U + V = −ε + εPr + − ε2 U + ε2RaPrθ (8) ∂X ∂Y ∂X ∂X2 ∂Y2 Da ∂θ ∂θ 1 ∂2θ ∂2θ U + V = + (9) ∂X ∂Y Pr ∂X2 ∂Y2 where (X, Y) are the dimensionless Cartesian coordinates of the geometry, (U, V) are the dimensionless velocity components, and θ and P are the temperature and pressure, respectively. Ra, Pr, and Da denote respectively the Raleigh number, Prandtl number, and Darcy number. The boundary conditions for this problem are as follows: On the outer cylinder surface: θ = 0. (10) On the inner cylinder surface: θ = 1. (11) On the outer and inner cylinder surfaces:

U = V = 0. (12)

The local and overall heat transfer rate along the inner cylinder surface are estimated using the local Nusselt number and average Nusselt number, respectively: s 2 2 ∂θ ∂θ ∂θ Nuloc = = + (13) ∂N S ∂X ∂Y 1 Z s Nuavg = Nuloc (14) S 0 where S is the non-dimensional length along the inner cylinder surface.

3. Numerical Procedure The dimensionless governing Equations (6)–(9) together with the boundary conditions (10) and (11) have been solved numerically using the commercial software tool, which is known as COMSOL Multiphysics (Version 5.5, COMSOL Inc., Stockholm, Sweden). The software employs the Galerkin ﬁnite element method, which enforces the orthogonality of residuals to all basis functions in a basis. In Galerkin formulation, weighting functions are chosen to become identical to basis functions [36]. In this paper, we have employed a segregated and parallel direct (Pardiso) solver to solve those equations. As convergence criteria, 10−6 has been chosen for all dependent variables.

3.1. Grid Generation and Independence Test In the finite element method, grid generation is the technique to discretize the compu- tational domain into subdomains. In this study, we have adopted unstructured triangular elements in the interior and structured quadrilateral elements on the boundary. Figure2 is the grid generation of the structure with a legend of quality measure. A quality of 1 represents the best possible grid quality, and a quality of 0 represents the worst possible grid quality. For purpose of ensuring the grid independence of the numerical solution, different grid levels in the Comsol Multiphysics are examined. As shown in Table3, the average Nusselt number of on the inner cylinder at different grids is presented for Cu nanofluid when the nanoparticle volume fraction (φ) is 0.5, nanoparticle diameter (dsp) is 50 nm, Rayleigh number (Ra) is 105, porosity (ε) is 0.5, and Darcy number (Da) is 10−2. The difference between normal and extremely fine is within 1.35%. By comprehensive considering the calculation accuracy and cost, the extra fine level is chosen in this study. Nanomaterials 2021, 11, x FOR PEER REVIEW 6 of 24

of residuals to all basis functions in a basis. In Galerkin formulation, weighting functions are chosen to become identical to basis functions [36]. In this paper, we have employed a segregated and parallel direct (Pardiso) solver to solve those equations. As convergence criteria, 10−6 has been chosen for all dependent variables.

3.1. Grid Generation and Independence Test In the finite element method, grid generation is the technique to discretize the com- putational domain into subdomains. In this study, we have adopted unstructured triangular elements in the interior and structured quadrilateral elements on the boundary. Fig- ure 2 is the grid generation of the structure with a legend of quality measure. A quality of 1 represents the best possible grid quality, and a quality of 0 represents the worst possible grid quality. For purpose of ensuring the grid independence of the numerical solution, different grid levels in the Comsol Multiphysics are examined. As shown in Table 3, the average Nusselt number of on the inner cylinder at different grids is presented for Cu nanofluid when the nanoparticle volume fraction (ϕ) is 0.5, nanoparticle diameter (dsp) is 50 nm, Rayleigh number (Ra) is 105, porosity (ε) is 0.5, and Darcy number (Da) is 10−2. The difference between normal and extremely fine is within 1.35%. By comprehensive consid- Nanomaterials 2021, 11, 990 6 of 24 ering the calculation accuracy and cost, the extra fine level is chosen in this study.

Figure 2. Grid generation of the structure with a legend of quality measure. Figure 2. Grid generation of the structure with a legend of quality measure. Table 3. Comparison of the average Nusselt number for Cu nanofluid at different levels when φ = 0.5, 5 −2 Tabledsp = 3. 50 Comparison nm, Ra = 10 of, ε the= 0.5, average and Da Nusselt= 10 number. for Cu nanofluid at different levels when ϕ = 0.5, dsp = 50 nm, Ra = 105, ε = 0.5, and Da = 10−2. Level Number of Elements Minimum Quality Average Quality Nuavg Level Number of Elements Minimum Quality Average Quality Nuavg normalnormal 894 894 0.4632 0.4632 0.7779 0.7779 2.2466 2.2466 fine 1344 0.4938 0.8284 2.2632 fine 1344 0.4938 0.8284 2.2632 finer 1882 0.5084 0.8305 2.2668 finer 1882 0.5084 0.8305 2.2668 extra fine 6394 0.5276 0.8424 2.2766 extremelyextra fine fine 6394 17858 0.5276 0.5053 0.8424 0.8466 2.2766 2.2769 extremely fine 17858 0.5053 0.8466 2.2769

3.2. Code Validation 3.2. Code Validation Due to the lack of experimental data for conjugate heat transfer of nanofluid in an Due to the lack of experimental data for conjugate heat transfer of nanofluid in an an- annulus filled with porous medium, we have compared our results with the numerical nulus filled with porous medium, we have compared our results with the numerical results results of Abhishek Kumar Singh and Tanmay Basak et al. [37] for a square cavity filled of Abhishek Kumar Singh and Tanmay Basak et al. [37] for a square cavity filled with base with base fluid (φ = 0). The comparisons are presented in Table4, when ε = 0.4, and the fluid (ϕ = 0). The comparisons are presented in Table 4, when ε = 0.4, and the deviations are deviations are 0%, 0%, and 0.84% with Ra = 103, 104, and 105, respectively. When ε = 0.9, 0%, 0%, and 0.84% with Ra = 103, 104, and 105, respectively. When ε = 0.9, the deviations are the deviations are 0.5%, 1.2%, and 1.3% with Ra = 103, 104, and 105, respectively. It is clear that the current results are in good agreement with the earlier work, and the maximum deviation is 1.3%. In addition, it can be seen that the deviation is increased with the increase of Rayleigh number and porosity. The validation work has enhanced the confidence in the numerical solution of the current study.

Table 4. Comparison of the average Nusselt number with those of Singh et al. [37] for different Rayleigh number when ϕ = 0, ε = 0.4, 0.9, Da = 10−2, and Pr = 1.

Average Nusselt Number (Nu ) Darcy Reyleigh avg Number Number ε = 0.4 ε = 0.9 (Da) (Ra) Ref [37] Present Work Diff(%) Ref [37] Present Work Diff(%) 105 2.983 3.008 0.84 3.91 3.95 1.3 10−2 104 1.408 1.408 0 1.64 1.66 1.2 103 1.01 1.01 0 1.023 1.028 0.5

4. Results and Discussion In this section, numerical simulations are carried out to investigate the flow and heat transfer characteristics of the nanofluid filled in the porous annulus. The results present the effects of several parameters, such as Brownian motion, nanoparticle diameter dsp Nanomaterials 2021, 11, 990 7 of 24

(10–90 nm), nanoparticle volume fraction φ (0.01–0.09), Rayleigh number Ra (103–105), Darcy number Da (10−4–10−2), porosity ε (0.1–0.9), and radius ratio RR (1.1–10) on the isotherms and streamlines, the local Nusselt number (Nuloc), and the average Nusselt number (Nuavg).

4.1. Effects of Brownian Motion In this section, we have investigated the effect of Brownian motion on the heat transfer characteristics using two applied models in Table2. One ignores the Brownian motion of the nanoparticles. The other takes Brownian motion into consideration and the modified effective thermal conductivity and effective dynamic viscosity are adopted. Figure3 presents the effect of Brownian motion on the average Nusselt number along the inner wall under different parameters. From Figure3a, the average Nusselt number increases generally as Brownian motion is considered. With the increase of nanoparticle volume fraction, the influence of Brownian motion becomes more noticeable. In addition, with the decrease of nanoparticle diameter, the influence of Brownian motion becomes more remarkable. 3 −2 For instance, when Ra = 5 × 10 , Da = 10 , dsp = 90 nm, Nuavg with Brownian motion increased by 0.21% compared with that without Brownian motion at φ = 0.01, while the 3 −2 growth rate is 2.7% at φ = 0.09. When Ra = 5 × 10 , Da = 10 , dsp = 10 nm, Nuavg with Brownian motion increased by 1.98% compared with that without Brownian motion at φ = 0.01, while the growth rate is 23.76% at φ = 0.09. Comparing Figure3a,b and Figure3b with Figure3c, the effect of Brownian motion becomes more remarkable with the increase of Rayleigh number and the decrease of Darcy number. For example, when Da = 10−2, φ = 0.09, dsp = 10 nm, Nuavg with Brownian motion increased by 23.76% compared with that without Brownian motion at Ra = 5 × 103, while the growth rate is 35.92% at Ra = 5 × 104. 4 When Ra = 5 × 10 , φ = 0.09, dsp = 10 nm, Nuavg with Brownian motion increased by 35.92% compared with that without Brownian motion at Da = 10−2, while the growth rate is 46.29% at Da = 10−3. Therefore, the effect of Brownian motion on the natural convection heat transfer of the nanofluid should be considered. Furthermore, the positive effect of Brownian motion on the overall heat transfer rate is different at different parameters.

4.2. Effects of Nanoparticle Volume Fraction Figure4 shows the isotherms and streamlines for different nanoparticle volume −2 fraction and Rayleigh number at Da = 10 , dsp = 50 nm, ε = 0.5, and RR = 2. The color scales on the left represent the dimensionless temperature and those on the right represent the dimensionless velocity, as in other sections of the article. From Figure4, for Ra = 103, the isotherms have a uniform distribution; this is because the buoyancy force is weak compared with the viscous force, and it indicates that the heat transfer in the annulus is dominated by thermal conduction. The effect of volume fraction on the isotherms is weak. For Ra = 104, a slight thermal disturbance appeared, which indicates that the flow is enhanced and the transition from conduction to natural convection takes place. In this case, the effect of volume fraction becomes more important. For Ra = 105, the isotherms are almost horizontally distributed, especially when φ = 0.9, which means that the natural convection heat transfer turns out to be more significant and the effect of volume fraction is more pronounced. With the increase of Rayleigh number, the streamlines become denser near the walls and the cell becomes bigger and has a tendency to move upward due to the enhanced buoyance force. In addition, for Ra = 103 and Ra = 104, the effect of volume fraction on the streamlines is very weak, while for Ra = 105, the effect is more pronounced. Figure5 displays the effect of volume fraction on the overall heat transfer rate along the inner wall at different Rayleigh numbers. The figure shows that an increase in volume fraction leads to heat transfer enhancement for all considered Rayleigh numbers, and the effect of volume fraction is more pronounced when the Rayleigh number is high. Nanomaterials 2021, 11, x FOR PEER REVIEW 8 of 24

Nanomaterials 2021, 11, 990 8 of 24

2.5 5 without brownian motion without brownian motion

with brownian motion dsp = 90nm with brownian motion dsp = 10nm 2.0 4 with brownian motion dsp = 10nm with brownian motion dsp = 90nm

1.5 3 ave ave

Nu 1.0 Nu 2

0.5 1

0.0 0 0.01 0.03 0.05 0.07 0.09 0.01 0.03 0.05 0.07 0.09 ϕ ϕ

(a) Ra = 5 × 103, Da = 10−2 (b) Ra = 5 × 104, Da = 10−2

3 without brownian motion

with brownian motion dsp = 90nm

with brownian motion dsp = 10nm

2 ave

Nu 1

0 0.01 0.03 0.05 0.07 0.09 ϕ

avg FigureFigure 3. 3.Effects Effects of of Brownian Brownian motion motion on on the the average average Nusselt Nusselt number number (Nu (Nuavg) along) along the the inner inner wall wall for for different different nanoparticle nanoparticle volume fractions (ϕ) and nanoparticle diameters (dsp) at ε = 0.5, RR = 2, and (a) Ra = 5 × 1033, Da = 10−2−, (2b) Ra = 5 × 104, Da4 = volume fractions (φ) and nanoparticle diameters (dsp) at ε = 0.5, RR = 2, and (a) Ra = 5 × 10 , Da = 10 ,(b) Ra = 5 × 10 , 10−2, (c) Ra = 5 × 104, Da = 10−3. Da = 10−2,(c) Ra = 5 × 104, Da = 10−3. 4.2. Effects of Nanoparticle Volume Fraction Figure 4 shows the isotherms and streamlines for different nanoparticle volume fraction and Rayleigh number at Da = 10−2, dsp = 50 nm, ε = 0.5, and RR = 2. The color scales on the left represent the dimensionless temperature and those on the right represent the dimensionless velocity, as in other sections of the article. From Figure 4, for Ra = 103, the isotherms have a uniform distribution; this is because the buoyancy force is weak compared with the viscous force, and it indicates that the heat transfer in the annulus is dominated by thermal conduction. The effect of volume fraction on the isotherms is weak. For Ra = 104, a slight thermal disturbance appeared, which indicates that the flow is enhanced and the transition from conduction to natural convection takes place. In this case, the effect of volume fraction becomes more important. For Ra = 105, the isotherms are almost horizontally distributed, especially when ϕ = 0.9, which means that the natural convection heat transfer turns out to be more significant and the effect of volume fraction is more pronounced. With the increase of Rayleigh number, the streamlines become denser near the walls and the cell becomes bigger and has a tendency to move upward due to the enhanced buoyance force. In addition, for Ra = 103 and Ra = 104, the effect of volume fraction on the streamlines is very weak, while for Ra = 105, the effect is more pronounced.

Nanomaterials 2021, 11, x FOR PEER REVIEW 9 of 24

Figure 5 displays the effect of volume fraction on the overall heat transfer rate along the inner wall at different Rayleigh numbers. The figure shows that an increase in volume Nanomaterials 2021, 11, 990 fraction leads to heat transfer enhancement for all considered Rayleigh numbers, and9 of the 24 effect of volume fraction is more pronounced when the Rayleigh number is high.

ϕ = 0.01 ϕ = 0.09

Figure 4.4. IsothermsIsotherms (left)(left) and and streamlines streamlines (right) (right) for for Cu–water Cu–water nanoﬂuid nanofluid for for different different nanoparticle nanoparticle volume volume fractions fractions (φ) and (ϕ) Rayleighand Rayleigh numbers numbers (Ra) at(Rad)sp at= d 50sp = nm, 50 nm,Da = Da 0.01, = 0.01,ε = 0.5,ε = 0.5, and andRR =RR 2. = 2.

Nanomaterials 2021, 11, x FOR PEER REVIEW 10 of 24

Nanomaterials 2021, 11, 990 10 of 24

5 Ra = 10 3 4 Ra = 10 3 Ra = 10 avg

Nu 2

1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ϕ

FigureFigure 5. Average 5. Average Nusselt Nusselt number number (Nuavg) of (Nu theavg inner) of thewall inner for differe wallnt for nanoparticle different nanoparticle volume frac- volume −2 tions (fractionsϕ) and Rayleigh (φ) and Rayleighnumbers numbers(Ra) at Da ( Ra= 10) at−2,Da dsp == 1050 nm,, d spε == 0.5, 50 nm,and εRR=0.5, = 2.and RR = 2.

FigureFigure 6 illustrates6 illustrates the isotherms the isotherms and andstreamlines streamlines for different for different nanoparticle nanoparticle volume volume 5 fractionfraction and Darcy and Darcy number number at dsp = at 50dsp nm,= 50Ra nm, = 10Ra5, ε= = 100.5,, andε = 0.5,RR and= 2. FromRR = Figure 2. From 6,Figure for 6 , −4 Da = for10−4Da, the= isotherms 10 , the have isotherms a uniform have distribution a uniform distribution due to the low due permeability, to the low permeability, and it indicatesand itthat indicates the heat that transfer the heat in transferthe annulus in the is annulusdominated is dominated by thermal by conduction. thermal conduction. The −3 effectThe of volume effect of fraction volume fractionon the isotherms on the isotherms is slight. is slight.For Da For = 10Da−3=, due 10 to, due the toenhanced the enhanced permeability,permeability, the flow the flowin the in annulus the annulus is stre isngthened, strengthened, and the and transition the transition from fromconduction conduction to naturalto natural convection convection takes takesplace. place. In this In case, this case,the effect the effect of volume of volume fraction fraction becomes becomes more more −2 important.important. For Da For = Da10−=2, 10the isotherms, the isotherms are almost are almost horizontally horizontally distributed, distributed, especially especially whenwhen ϕ = 0.9,φ = which 0.9, which means means the natural the natural convection convection heat heattransfer transfer plays plays a significant a significant role role and the effect of volume fraction is more pronounced. With the increase of Darcy number, and the effect of volume fraction is more pronounced. With the increase of Darcy number, the streamlines become denser near the walls and the cells become bigger and have the streamlines become denser near the walls and the cells become bigger and have a ten- a tendency to move upward due to the enhanced flow. In addition, for Da = 10−4 and dency to move upward due to the enhanced flow. In addition, for Da = 10−4 and Da = 10−3, Da = 10−3, the effect of volume fraction on the streamlines is very weak, while for Da = 10−2, the effect of volume fraction on the streamlines is very weak, while for Da = 10−2, the effect the effect is more pronounced. Figure7 displays the effect of volume fraction on the overall is more pronounced. Figure 7 displays the effect of volume fraction on the overall heat heat transfer rate along the inner wall at different Darcy numbers. The figure shows that transfer rate along the inner wall at different Darcy numbers. The figure shows that an an increase in volume fraction leads to heat transfer enhancement for all considered Darcy increase in volume fraction leads to heat transfer enhancement for all considered Darcy numbers and the effect of volume fraction is increased with the increase of Darcy number. numbers and the effect of volume fraction is increased with the increase of Darcy number. Figure8 presents the evolution of the local Nusselt number along the inner wall Figure 8 presents the evolution of the local Nusselt number5 along− 2the inner wall for for different nanoparticle volume fractions at Ra = 10 , Da = 10 , dsp = 50 nm, ε = 0.5, different nanoparticle volume fractions at Ra = 105, Da = 10−2, dsp = 50 nm, ε = 0.5, and RR = and RR = 2. The increase of Nuloc in the whole region means that the local heat transfer rate 2. The increase of Nuloc in the whole region means that the local heat transfer ◦rate is en- is enhanced. It can be found that the maximum Nuloc occurred at γ = 180 , which means hanced.the It natural can be convection found that heat the maximum transfer is moreNuloc occurred intense in at the γ = bottom 180°, which half of means the inner the wall. naturalIn addition,convection the heat heat transfer transfer is is more enhanced intense with in thethe increasebottom half of volume of the fraction.inner wall. In addition, the heat transfer is enhanced with the increase of volume fraction.

Nanomaterials 2021, 11, x FOR PEER REVIEW 11 of 24 Nanomaterials 2021, 11, 990 11 of 24

ϕ = 0.01 ϕ = 0.09

FigureFigure 6.6.Isotherms Isotherms (left) (left) and and streamlines streamlines (right) (right) for for Cu–water Cu–water nanoﬂuid nanofluid for for different different nanoparticle nanoparticle volume volume fractions fractions (φ) and (ϕ) 5 Darcyand Darcy numbers numbers (Da) at(DaRa) =at 10 Ra,=d 10sp5=, d 50sp = nm, 50 nm,ε =0.5, ε = and0.5, andRR =RR 2. = 2.

Nanomaterials 2021, 11, x FOR PEER REVIEW 12 of 24

Nanomaterials 2021, 11, 990 12 of 24

4 −2 Da = 10 −3 3 Da = 10 −2 −4 Da Da = = 10 10 3 −3 avg Da = 10

−4 Nu Da = 10 2 avg

Nu 2

1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1 ϕ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Figure 7. Average Nusselt numberϕ (Nuavg) of the inner wall for different nanoparticle volume frac-

tions (ϕ) and Darcy numbers (Da) at Ra= 105, dsp = 50 nm, ε = 0.5, and RR = 2. FigureFigure 7. Average 7. Average Nusselt Nusselt number number (Nuavg) of (Nu theavg inner) of thewall inner for differe wall fornt nanoparticle different nanoparticle volume frac- volume 5 5 tions (fractionsϕ) and Darcy (φ) and numbers Darcy numbers(Da) at Ra (=Da 10) at, dRasp = 50 10 nm,, dsp ε= = 500.5, nm, andε =RR 0.5, = 2. and RR = 2.

54 ϕ= 0.01 loc 43 ϕ = 0.03 Nu ϕ = 0.05 ϕ= 0.01 loc 2 ϕ = 0.07 3 ϕ = 0.03 Nu ϕ = 0.09 ϕ = 0.05 21 ϕ = 0.07 ϕ = 0.09 10 0° 60° 120° 180° 240° 300° 360° 0 γ 0° 60° 120° 180° 240° 300° 360°

FigureFigure 8. Local 8. LocalNusselt Nusselt number numberγ (Nuloc) (alongNuloc )the along inner the wall inner for differe wall fornt differentnanoparticle nanoparticle volume frac- volume 5 −2 tionsfractions (ϕ) at Ra (=φ )10 at5,Ra Da= = 1010−,2,Da dsp= = 1050 nm,, dsp ε = 0.5, 50 nm, andε RR= 0.5, = 2. and RR = 2. Figure 8. Local Nusselt number (Nuloc) along the inner wall for different nanoparticle volume frac- 4.3. Effects of5 Nanoparticle−2 Diameter tions4.3. Effects(ϕ) at Ra of = Nanoparticle 10 , Da = 10 Diameter, dsp = 50 nm, ε = 0.5, and RR = 2. Figure9 shows the isotherms and streamlines for different nanoparticle diameters and Figure 9 shows the isotherms and streamlines for different nanoparticle diameters 4.3. EffectsRayleigh of Nanoparticle numbers at DiameterDa = 10 −2, φ = 0.05, ε = 0.5 and RR = 2. From Figure9, for Ra = 103, and Rayleigh numbers at Da = 10−2, ϕ = 0.05, ε = 0.5 and RR = 2. From Figure 9, for Ra = 103, the isotherms have a uniform distribution due to the weak buoyancy force, and it indicates the isothermsFigure 9 shows have a theuniform isotherms distribution and streamli due tones the forweak different buoyancy nanoparticle force, and diametersit indicates that the heat transfer in the annulus−2 is dominated by thermal conduction. The nanoparticle3 andthat Rayleigh the heat numberstransfer in at the Da annulus= 10 , ϕ =is 0.05, dominated ε = 0.5 andby thermal RR = 2. Fromconduction. Figure The9, for nanoparti- Ra = 10 , diameter has a weak effect on the isotherms. For Ra = 104, the natural convection heat thecle isotherms diameter hashave a aweak uniform effe ctdistribution on the isotherms. due to the For weak Ra = buoyancy 104, the natural force, andconvection it indicates heat transfer is strengthened due to the enhanced buoyancy force. The isotherms near the thattransfer the heat is strengthened transfer in the due annulus to the enhancedis dominated buoyancy by thermal force. conduction. The isotherms The near nanoparti- the top cle diametertop half has of thea weak inner effe wallct on are the disturbed, isotherms. and For the Ra change = 104, the of thenatural isotherms convection distribution heat is half of the inner wall are disturbed, and the change of the isotherms distribution4 is more transfermore is strengthened remarkable at duedsp to= 10the compared enhanced with buoyancydsp = 90.force. For TheRa =isotherms 5 × 10 , thenear isotherms the top are remarkable at dsp = 10 compared with dsp = 90. For Ra = 5 × 104, the isotherms are almost half ofalmost the inner horizontally wall are disturbed, distributed, and especially the change when of thedsp isotherms= 10, whichdistribution means that is the more natural horizontally distributed, especially when dsp = 10, which means that the natural convection convection dominates the heat transfer and the effect of the4 nanoparticle diameter is more remarkable at dsp = 10 compared with dsp = 90. For Ra = 5 × 10 , the isotherms are almost dominatespronounced. the heat In addition,transfer forandRa the= 10effect3 and ofRa the= 10nanoparticle4, the effect ofdiameter nanoparticle is more diameter pro- on horizontally distributed, especially3 when dsp = 10,4 which means that the natural convection nounced.the streamlines In addition, is for very Ra weak, = 10 and while Ra for = 10Ra, =the 10 effect5, the of effect nanoparticle is more pronounced, diameter on andthe the dominatesstreamlines the is veryheat weak,transfer while and for the Ra effect = 10 5,of the the effect nanoparticle is more pronounced, diameter is andmore the pro- cell cell becomes bigger and has3 a tendency to4 move upward. Figure 10 displays the effect of nounced.becomes Inbigger addition, and has for a Ra tendency = 10 and to Ramove = 10 upwa, therd. effect Figure of nanoparticle 10 displays the diameter effect of on nano- the nanoparticle diameter on the overall heat5 transfer rate along the inner wall at different streamlinesparticle diameter is very on weak, the overa whilell heatfor Ra transfer = 10 , ratethe alongeffect theis more inner pronounced, wall at different and Rayleighthe cell becomesRayleigh bigger numbers. and has a The tendency ﬁgure to shows move that upwa anrd. increase Figure in10the displays nanoparticle the effect diameter of nano- leads numbers.to reduced The figure heat transfershows that for an all increase considered in the Rayleigh nanoparticle numbers. diameter For Ra leads= 10 to3, reduced the effect of particle diameter on the overall heat transfer rate along the inner3 wall at different Rayleigh heatnanoparticle transfer for all diameter considered is less Rayleigh pronounced. numbers. For ForRa Ra= 5= 10× 10, the4, theeffect effect of nanoparticle of nanoparticle numbers. The figure shows that an increase in the nanoparticle diameter leads to reduced diameterdiameter is less is pronounced. remarkable, especiallyFor Ra = 5when × 104, itthe is effect at a low of nanoparticle value. For instance, diameterNu is remark-decreased heat transfer for all considered Rayleigh numbers. For Ra = 103, the effect of nanoparticleavg able, especially when it is at a low value. For instance, Nuavg decreased by 11.79% from dsp = by 11.79% from dsp = 10 to dsp = 30 and4 decreased by 5.81% from dsp = 30 to dsp = 90 at diameter is less pronounced.4 For Ra = 5 × 10 , the effect of nanoparticle diameter is remark- Ra = 5 × 10 , while Nuavg decreased by 0.48% from dsp = 10 to dsp = 30 and decreased by able, especially when it is at a low value. For instance,4 Nuavg decreased by 11.79% from dsp = 0.12% from dsp = 30 to dsp = 90 at Ra = 10 .

Nanomaterials 2021, 11, x FOR PEER REVIEW 13 of 24

10 to dsp = 30 and decreased by 5.81% from dsp = 30 to dsp = 90 at Ra = 5 × 104, while Nuavg Nanomaterials 2021, 11, 990 decreased by 0.48% from dsp = 10 to dsp = 30 and decreased by 0.12% from dsp = 30 to d13sp of= 90 24 at Ra = 104.

FigureFigure 9.9. IsothermsIsotherms (left)(left) andand streamlinesstreamlines (right)(right) forfor Cu–waterCu–water nanoﬂuidnanofluid forfor differentdifferent nanoparticlenanoparticle diametersdiameters ((ddspsp)) andand RayleighRayleigh numbersnumbers ((RaRa)) atat DaDa== 10−22, ,ϕφ == 0.05, 0.05, εε == 0.5, 0.5, and and RRRR == 2. 2.

Nanomaterials 2021, 11, x FOR PEER REVIEW 14 of 24

Nanomaterials 2021, 11, 990 14 of 24

3.8 3.6 3.4 3.2 3.0

2.8 4 Ra = 5×10 2.6 4 avg 2.4 Ra = 10 3 Nu 2.2 Ra = 10 2.0 1.8 1.6 1.4 1.2 10 20 30 40 50 60 70 80 90 d(nm)

FigureFigure 10. Average 10. Average Nusselt Nusselt number number (Nuavg (Nu) ofavg the) of inner the innerwall for wall different for different nanoparticle nanoparticle diameters diameters −2 (dsp)( anddsp) Rayleigh and Rayleigh numbers numbers (Ra) (atRa Da) at =Da 10−=2, 10ϕ = 0.05,, φ = ε 0.05, = 0.5,ε =and 0.5, RR and = 2.RR = 2.

FigureFigure 11 illustrates 11 illustrates the isotherms the isotherms and andstreamlines streamlines for different for different nanoparticle nanoparticle diame- diam- 4 ter andeter Darcy and Darcy numbers numbers at Ra = at 5Ra × 10=4, 5 ϕ× = 100.05,, φ ε == 0.05,0.5, andε = RR 0.5, = and2. FromRR =Figure 2. From 11, forFigure Da 11, −4 = 10for−4, theDa isotherms= 10 , the have isotherms a uniform have distribution a uniform due distribution to the low duepermeability, to the low which permeability, indi- cateswhich that the indicates flow is that weak the and flow thermal is weak conduction and thermal plays conduction a leading plays role. a The leading effect role. of na- The ef- −3 noparticlefect of diameter nanoparticle on the diameter heat transfer on the is heat weak. transfer For Da is = weak.10−3, due For toDa the= enhanced 10 , due per- to the −2 meability,enhanced the permeability,natural convection the natural heat transfer convection is strengthened. heat transfer For is strengthened. Da = 10−2, theFor isothermsDa = 10 , the isotherms are almost horizontally distributed, especially when d = 10, which means are almost horizontally distributed, especially when dsp = 10, which meanssp that the natural convectionthat the heat natural transfer convection plays heata significant transfer role plays and a significant the effect roleof nanoparticle and the effect diameter of nanoparti- is Da −4 Da −3 morecle pronounced. diameter is moreIn addition, pronounced. for Da = In 10 addition,−4 and Da for= 10−3, =the 10 effectand of nanoparticle= 10 , the diam- effect of nanoparticle diameter on the streamlines is very weak, while for Da = 10−2, the effect is eter on the streamlines is very weak, while for Da = 10−2, the effect is more pronounced; more pronounced; the cell becomes bigger and has a tendency to move upward. Figure 12 the cell becomes bigger and has a tendency to move upward. Figure 12 displays the effect displays the effect of the nanoparticle diameter on the overall heat transfer rate along the of the nanoparticle diameter on the overall heat transfer rate along the inner wall at dif- inner wall at different Darcy numbers. The figure shows that an increase in the nanoparticle ferent Darcy numbers. The figure shows that an increase in the nanoparticle diameter diameter leads to heat transfer enhancement for all considered Darcy numbers, and it can leads to heat transfer enhancement for all considered Darcy numbers, and it can be found be found that the effect of nanoparticle diameter is increased with the increase of the that the effect of nanoparticle diameter is increased with the increase of the Darcy number. Darcy number. Figure 13 presents the evolution of the local Nusselt number along the inner wall for Figure 13 presents the evolution of the local Nusselt number along the inner wall different nanoparticle diameters at Ra = 105, Da = 10−2, ϕ = 0.05, ε = 0.5, and RR = 2. It can for different nanoparticle diameters at Ra = 105, Da = 10−2, φ = 0.05, ε = 0.5, and RR = 2. be found that the local Nusselt number has a symmetrical distribution due to the symmet- It can be found that the local Nusselt number has a symmetrical distribution due to the rical geometry and boundary conditions, and the maximum Nuloc occurred at γ= 180°, symmetrical geometry and boundary conditions, and the maximum Nuloc occurred at whichγ= means 180◦, which that the means natural that convection the natural heat convection transfer is heat more transfer intense is in more the bottom intense half in the of thebottom inner halfwall. of Furthermore, the inner wall. the Furthermore, local Nussel thet number local Nusselt is increased number with is the increased decrease with of the nanoparticledecrease ofdiameter, nanoparticle which diameter, indicates which that the indicates nanoparticle that the diameter nanoparticle at high diameter value will at high weakenvalue the will natural weaken convection the natural heat convection transfer. heat transfer.

Nanomaterials 2021, 11, x FOR PEER REVIEW 15 of 24 Nanomaterials 2021, 11, 990 15 of 24

FigureFigure 11. 11.Isotherms Isotherms (left) (left) and and streamlines streamlines (right) (right) for for Cu–water Cu–water nanoﬂuid nanofluid for for different different nanoparticle nanoparticle diameters diameters ( d(spdsp)) and and DarcyDarcy numbers numbers ( Da(Da)) at atRa Ra= = 5 5× × 104,, ϕφ = 0.05, 0.05, εε = 0.5, 0.5, and and RRRR == 2. 2.

Nanomaterials 2021, 11, x FOR PEER REVIEW 16 of 24

Nanomaterials 2021, 11, 990 16 of 24 3.8 3.6 3.43.8 3.23.6 3.03.4 2.8 −2 3.2 Da = 10 2.63.0 −3

avg Da = 10 2.42.8 −2 Da = 10−4 Nu 2.2 Da = 10 2.6 −3

avg 2.02.4 Da = 10 −4 Nu 1.82.2 Da = 10 1.62.0 1.41.8 1.21.6 10 20 30 40 50 60 70 80 90 1.4 1.2 d(nm) 10 20 30 40 50 60 70 80 90

Figure 12. Average Nusselt numberd(nm) (Nuloc) of the inner wall for different nanoparticle diameters

(dsp) and Darcy numbers (Da) at Ra = 5 × 104, ϕ = 0.05, ε = 0.5, and RR = 2. Figure 12. Average Nusselt number (Nu ) of the inner wall for different nanoparticle diameters Figure 12. Average Nusselt number (Nuloc) of locthe inner wall for different nanoparticle diameters 4 (dsp) (anddsp) Darcy and Darcy numbers numbers (Da) (atDa Ra) at= Ra5 × =10 54,× ϕ 10= 0.05,, φ = ε 0.05,= 0.5,ε and= 0.5, RR and = 2.RR = 2.

loc 34 d =10 nm

Nu d =30 nm d =50 nm

loc 23 d =10 nm d =70 nm

Nu d =30 nm d =90 nm 12 d =50 nm d =70 nm d =90 nm 01 0° 60° 120° 180° 240° 300° 360° 0 γ 0° 60° 120° 180° 240° 300° 360°

FigureFigure 13. Local 13. Local Nusselt Nusselt number numberγ (Nuloc ()Nu alongloc) alongthe inner the wall inner for wall diffe forrent different nanoparticle nanoparticle diameters diameters 5 −2 (dsp) (atdsp Ra) at= 10Ra5,= Da 10 =, 10Da−2,= ϕ 10 = 0.05,, φ =ε = 0.05, 0.5, εand= 0.5, RR and = 2.RR = 2. Figure 13. Local Nusselt number (Nuloc) along the inner wall for different nanoparticle diameters 4.4. Effects5 of Porosity−2 4.4.(dsp) Effectsat Ra = of10 Porosity, Da = 10 , ϕ = 0.05, ε = 0.5, and RR = 2. Figure 14 shows the isotherms and streamlines for different porosity and Rayleigh Figure 14 shows the isotherms− and2 streamlines for different porosity and3 Rayleigh 4.4. Effectsnumbers of Porosity at dsp = 50 nm, Da = 10 , and RR = 2. From Figure 14, for Ra = 10 , the isotherms numbers at dsp = 50 nm, Da = 10−2, and RR = 2. From Figure 14, for Ra = 103, the isotherms have a uniform distribution, and the isotherms are almost unchanged when the porosity have Figurea uniform 14 shows distribution, the isotherms and the andisotherms streamli arenes almost for different unchanged porosity when and the Rayleighporosity increased from ε = 0.1 to ε =−2 0.9. This is because at low Rayleigh numbers,3 the buoyancy increasednumbers atfrom dsp =ε 50= 0.1nm, to Da ε == 0.9.10 This, and is RR because = 2. From at low Figure Rayleigh 14, for numbers, Ra = 10 , the buoyancyisotherms haveforce a uniform is very distribution, weak, and the and heat the transfer isotherms in the are annulus almost is unchanged dominated bywhen thermal the porosity conduction. force is very weak, and the heat transfer in the annulus is dominated4 by thermal conduc- increasedThe porosity from ε has= 0.1 a slightto ε = effect0.9. This on theis because isotherms. at low For RayleighRa = 10 , thenumbers, isotherms the havebuoyancy a similar tion. The porosity has a slight effect3 on the isotherms. For Ra = 104, the isotherms have a forcetrend is very with weak, that and for Rathe= heat 10 transferat ε = 0.1, in whilethe annulus the isotherms is dominated have by an thermal obvious conduc- change at ε similar trend with that for 4Ra = 103 at ε = 0.1, while the isotherms have an obvious change = 0.9. For Ra = 5 × 10 , the natural convection heat transfer plays4 a leading role due to attion. ε = The 0.9. porosityFor Ra = 5has × 10 a 4slight, the natural effect on convection the isotherms. heat transfer For Ra =plays 10 , thea leading isotherms role havedue to a similarthe trend enhanced with ﬂow,that for and Ra the = 10 heat3 at transferε = 0.1, while is more the intense isotherms at ε =have 0.9 comparedan obvious with changeε = 0.1. the enhanced flow, and the heat transfer is more intense at ε = 0.9 compared with ε = 0.1. 3 The streamlines have4 a similar distribution, but some details are different. For Ra = 10 , at ε = 0.9. For Ra = 5 × 10 , the natural convection heat transfer plays a leading role due to3 The thestreamlines streamlines have have a similar little change distributi at ε =on, 0.1 but and someε = 0.9. details For Ra are= different. 104 and Ra For= 5 Ra× 10= 104, it, can the streamlinesenhanced flow, have and little the change heat transfer at ε = 0.1 is andmore ε =intense 0.9. For at Ra ε = = 0.9 10 4compared and Ra = 5 with × 10 4ε, it= can0.1. be found the cells become bigger and have a tendency to move upward when the porosity3 beThe found streamlines the cells have become a similar bigger distributi and haveon, a tendency but some to details move upwardare different. when For the Raporosity = 10 , increases. Figure 15 displays the effect of porosity on the overall4 heat transfer rate4 along the increases.the streamlines Figure have 15 displays little change the effect at ε = of0.1 poro andsity ε = 0.9.on the For overall Ra = 10 heat and transfer Ra = 5 × rate 10 , alongit can be foundinner the wall cells at different become bigger Rayleigh and numbers. have a tendency It can be to found move that upward a continuous when the increase porosity of the the inneroverall wall heat at transferdifferent rate Rayleigh occurs numbers. with the It increase can be found of porosity that a for continuous all considered increase Rayleigh of increases. Figure 15 displays the effect of porosity on the overall heat transfer rate along the overallnumbers heat and transfer the effect rate of occurs porosity with is morethe increase pronounced of porosity at high for Rayleigh all considered numbers. Ray- leighthe inner numbers wall at and different the effect Rayleigh of porosity numbers. is more It can pronounced be found that at high a continuous Rayleigh numbers.increase of the overall heat transfer rate occurs with the increase of porosity for all considered Ray- leigh numbers and the effect of porosity is more pronounced at high Rayleigh numbers.

Nanomaterials 2021, 11, x FOR PEER REVIEW 17 of 24 Nanomaterials 2021, 11, 990 17 of 24

FigureFigure 14. 14. IsothermsIsotherms (left) (left) and and streamlines streamlines (right) (right) for for Cu–water Cu–water nanofluid nanoﬂuid for for different different porosity porosity (ε (ε)) and and Rayleigh Rayleigh numbers numbers −2 ((RaRa)) at at DaDa == 10 10−2, ϕ, φ= =0.05, 0.05, dspd sp= 50= 50nm, nm, and and RRRR = 2.= 2.

Nanomaterials 2021, 11, x FOR PEER REVIEW 18 of 24

Nanomaterials 2021, 11, 990 18 of 24

3.6 3.4 3.2 3.0 2.8 Ra = 5×104 2.6 4 Ra = 10 2.4 3 avg Ra = 10

Nu 2.2 2.0 1.8 1.6 1.4 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ε

Figure 15. Average Nusselt number (Nu ) of the inner wall for different porosity (ε) and Rayleigh Figure 15. Average Nusselt number (Nuavg) ofavg the inner wall for different porosity (ε) and Rayleigh −2 numbersnumbers (Ra) (atRa Da) at =Da 10−=2, 10ϕ = 0.05,, φ = d 0.05,sp = 50dsp nm,=50 and nm, RR and = 2.RR = 2.

Figure 16 displays the isotherms and streamlines for different porosity and Darcy Figure 16 displays the4 isotherms and streamlines for different porosity and Darcy −4 numbers at Ra = 5 × 10 , φ = 0.05, dsp = 50 nm, and RR = 2. From Figure 16, for Da = 10 , numbers at Ra = 5 × 104, ϕ = 0.05, dsp = 50 nm, and RR = 2. From Figure 16, for Da = 10−4, the the isotherms have a uniform distribution at ε = 0.1 and ε = 0.9; the reason is that the flow is isotherms have a uniform distribution at ε = 0.1 and ε = 0.9; the reason is that the flow is constrained by the low permeability. In this case, porosity has little effect on the isotherms. constrained by the low permeability. In this case, porosity has little effect on the isotherms. For Da = 10−3, due to the enhanced permeability, the flow in the annulus is strengthened, For Da = 10−3, due to the enhanced permeability, the flow in the annulus is strengthened, and a disturbance occurs in the isotherms. For Da = 10−2, the isotherms at ε = 0.1 and and a disturbance occurs in the isotherms. For Da = 10−2, the isotherms at ε = 0.1 and ε = ε = 0.9 have a remarkable difference. The isotherms are almost horizontally distributed 0.9 have a remarkable difference. The isotherms are almost horizontally distributed at ε = at ε = 0.9, which means that the overall heat transfer is dominated by natural convection. 0.9, which means that the overall heat transfer is dominated by natural convection. For Da For Da = 10−4, the streamlines are almost the same at ε = 0.1 and ε = 0.9. For Da = 10−3 and = 10−4, the streamlines are almost the same at ε = 0.1 and ε = 0.9. For Da = 10−3 and Da = 10−2, Da = 10−2, the cell becomes bigger and has a tendency to move upward when porosity the cell becomes bigger and has a tendency to move upward when porosity increases from increases from ε = 0.1 to ε = 0.9. Figure 17 displays the effect of porosity on the overall ε = 0.1 to ε = 0.9. Figure 17 displays the effect of porosity on the overall heat transfer rate heat transfer rate along the inner wall at different Darcy number. The figure shows that an alongincrease the inner in porositywall at different leads to heatDarcy transfer number. enhancement The figure forshows all considered that an increase Darcy in numbers, po- rosityand leads the to effect heat of transfer porosity enhancement is more pronounced for all considered at high Darcy Darcy numbers. numbers, and the effect of porosityFigure is more 18 presents pronounc theed evolution at high Darcy of the numbers. local Nusselt number along the inner wall for Figure 18 presents the evolution 4of the loca−l 2Nusselt number along the inner wall for different porosity at Ra = 5 × 10 , Da = 10 , φ = 0.05, ε = 0.5, dsp = 50 nm, and RR = 2. 4 −2 differentThe local porosity Nusselt at Ra number = 5 × 10 has, Da a symmetrical = 10 , ϕ = 0.05, distribution ε = 0.5, duedsp = to50 the nm, symmetrical and RR = 2. geometry The local Nusselt number has a symmetrical distribution due to the symmetrical◦ geometry and boundary conditions, and the maximum Nuloc occurred at γ = 180 . Furthermore, and theboundary local Nusselt conditions, number and is the increased maximum with Nu theloc increaseoccurred of at porosity, γ = 180°. so Furthermore, it can be concluded the localthat Nusselt the porosity number hasis increased a positive with effect the on increase the overall of porosity, heat transfer so it can rate. be concluded that the porosity has a positive effect on the overall heat transfer rate.

Nanomaterials 2021, 11, x FOR PEER REVIEW 19 of 24

Nanomaterials 2021, 11, 990 19 of 24

FigureFigure 16. 16.Isotherms Isotherms (left) (left) and and streamlines streamlines (right) (right) for for Cu–water Cu–water nanoﬂuid nanofluid for differentfor different porosity porosity (ε) and (ε) Darcyand Darcy numbers numbers (Da) 4 4 at(DaRa) =at 5Ra× =10 5 ×, φ10=, 0.05,ϕ = 0.05,dsp = dsp 50 =nm, 50 nm, and andRR =RR 2. = 2.

Nanomaterials 2021, 11, x FOR PEER REVIEW 20 of 24

Nanomaterials 2021, 11, 990 20 of 24 3.4 3.2 3.03.4 2.83.2 −2 2.63.0 Da = 10 −3 2.8 Da = 10 2.4 −2 avg −4 2.22.6 Da = 10

Nu −3 2.02.4 Da = 10 avg −4 1.82.2 Da = 10 Nu 1.62.0 1.41.8 1.21.6 1.4 0.2 0.4 0.6 0.8 1.2 ε 0.2 0.4 0.6 0.8

Figure 17. Average Nusselt numberε (Nuavg) of the inner wall for different porosity (ε) and Darcy

numbers (Da) at Ra = 5 × 104, ϕ = 0.05, dsp = 50 nm, and RR = 2. Figure 17. Average Nusselt number (Nu ) of the inner wall for different porosity (ε) and Darcy Figure 17. Average Nusselt number (Nuavg) ofavg the inner wall for different porosity (ε) and Darcy 4 numbersnumbers (Da) (atDa Ra) at = Ra5 ×= 10 54,× ϕ10 = 0.05,, φ = d 0.05,sp = 50dsp nm,=50 and nm, RR and = 2.RR = 2.

loc 34

Nu ε = 0.1 ε = 0.3 loc 23 ε = 0.5

Nu ε = 0.1 ε = 0.7 12 ε = 0.3 ε = 0.9 ε = 0.5 01 ε = 0.7 0° 60° 120° 180° ε = 0.9 240° 300° 360° 0 γ 0° 60° 120° 180° 240° 300° 360° × 4 FigureFigure 18. Local 18. Local Nusselt Nusselt number numberγ (Nuloc (Nu) alongloc) along the inner the inner wall wall for different for different porosity porosity (ε) (atε) Ra at Ra= 5= × 5 10 , −2 104, DaDa == 10 10−2, ϕ, φ= =0.05, 0.05, dspd sp= 50= 50nm, nm, and and RRRR = 2.= 2. Figure 18. Local Nusselt number (Nuloc) along the inner wall for different porosity (ε) at Ra = 5 × 4 4.5. Effects−2 of Radius Ratio 4.5.10 , DaEffects = 10 of, ϕRadius = 0.05, Ratio dsp = 50 nm, and RR = 2. Figure 19 illustrates the isotherms and streamlines for different radius ratio at Ra = 5 × 104, Figure −192 illustrates the isotherms and streamlines for different radius ratio at Ra = 5 4.5. EffectsDa = 10 of Radius, φ = 0.05,Ratiod sp = 50 nm, and ε = 0.5. The radius ratio is an important parameter × 104, Da = 10−2, ϕ = 0.05, dsp = 50 nm, and ε = 0.5. The radius ratio is an important parameter that affects the fluid flows and natural convection heat transfer according the previous re- that affectsFigure the19 illustrates fluid flows the and isotherms natural and convec streamlinestion heat for transfer different according radius ratiothe previous at Ra = 5 4search [10−–2 12]. From Figure 19, when the radius ratio increased from RR = 1.2 to RR= 6, the flow research× 10 , Da [10–12].= 10 , ϕ From= 0.05, Figure dsp = 50 19, nm, when and theε = 0.5.radius The ratio radius increased ratio is anfrom important RR = 1.2 parameter to RR= 6, is strengthened, a main cell is formed in the middle of the annulus, and the cell becomes bigger thethat flow affects is strengthened,the fluid flows a mainand naturalcell is formed convec tionin the heat middle transfer of the according annulus, the and previous the cell researchand has[10–12]. a tendency From Figure to move 19, upward.when theIt radius is worth ratio noting increased that thefrom color RR = scales 1.2 to on RR the= 6, right becomeshave bigger a downward and has trend a tendency with the to move increase upward. of RR ,It which is worth is inverse noting that to the the above color conclusion.scales onthe the flow right is strengthened, have a downward a main trend cell with is formed the increase in the ofmiddle RR, which of the is annulus, inverse toand the the above cell In fact, the reason is that the change of RR leads to a change of the characteristic length (ro–ri). conclusion.becomes bigger In fact, and the has reason a tendency is that to the move change upward. of AR It leads is worth to a changenoting thatof the the characteristic color scales on theThe right isotherms have a distribution downward istrend changed with fromthe increase vertical of to RR horizontal,, which is which inverse means to the that above the heat length (ro–ri). The isotherms distribution is changed RRfrom vertical to horizontal, which conclusion.transfer In is dominatedfact, the reason by natural is that convection.the change of When AR leads= to 1.2, a multicellularchange of the flow characteristic structures are meansformed. that the Figure heat 20 transfer displays is thedominated effect of by radius natural ratio convection. on the overall When heat RR transfer = 1.2, ratemulticel- along the length (ro–ri). The isotherms distribution is changed from vertical to horizontal, which lularinner flow wall.structures It can are be foundformed. that Figure heat transfer20 displa inys the the annulus effect of is radius enhanced ratio with on the the overall increase of means that the heat transfer is dominated by natural convection. When RR = 1.2, multicel- heat radiustransfer ratio rate from alongRR the= 1.3inner to RR wall.= 6,It butcan a be fall found occurred that inheat the transfer range of inRR the= annulus 1.2 to RR is= 1.3. lular flow structures are formed. Figure 20 displays the effect of radius ratio on the overall enhancedA conclusion with the can increase be drawn of radius that a bifurcationratio from RR point = 1.3 exists to RR when = 6, the but radius a fall ratiosoccurred is in in the the range heat transfer rate along the inner wall. It can be found that heat transfer in the annulus is rangeof ofRR RR= = 1.2 1.2 to toRR RR= = 1.3 1.3. for A the conclusion considered can parameters. be drawn that a bifurcation point exists when theenhanced radius withratios the is inincrease the range of radius of RR ratio = 1.2 from to RR RR = 1.3= 1.3 for to the RR considered = 6, but a fall parameters. occurred in the range of RR = 1.2 to RR = 1.3. A conclusion can be drawn that a bifurcation point exists when the radius ratios is in the range of RR = 1.2 to RR = 1.3 for the considered parameters.

Nanomaterials 2021, 11, x FOR PEER REVIEW 21 of 24

Nanomaterials 2021, 11, 990 21 of 24

Nanomaterials 2021, 11, x FOR PEER REVIEW 22 of 24

Figure 19. Isotherms (left) and streamlines (right) for Cu–water nanoﬂuid for different radius ratios (RR) at Ra = 5 × 104, Figure− 19.2 Isotherms (left) and streamlines (right) for Cu–water nanofluid for different radius ratios (RR) at Ra = 5 × 104, Da = 10 , φ = 0.05, dsp = 50 nm, and ε = 0.5. Da = 10−2, ϕ = 0.05, dsp = 50 nm, and ε = 0.5.

10 9 8 7 6

avg 5 2.4 Nu 2.2 4 2.0 1.8

3 avg 1.6 Nu 1.4 2 1.2 1.1 1.2 1.3 1.4 1.5 1 12345678910 Figure 20. Average Nusselt number (Nuavg) of the inner wall for different radius ratios (RR) at Figure 20. Average4 Nusselt−2 numberRR (Nuavg) of the inner wall for different radius ratios (RR) at Ra = Ra = 5 × 10 , Da = 10 , φ = 0.05, dsp = 50 nm, and ε = 0.5. 4 −2 5 × 10 , Da = 10 , ϕ = 0.05, dsp = 50 nm, and ε = 0.5. 5.Figure Conclusions 20. Average Nusselt number (Nuavg) of the inner wall for different radius ratios (RR) at Ra = 4 −2 5. Conclusions5 × 10Natural, Da = 10convection , ϕ = 0.05, heatdsp = 50 transfer nm, and in εa = porous0.5. annulus filled with a Cu-Nanofluid has beenNatural investigated convection numerically. heat transfer The effects in a porous of Brownian annulus motion, filled solidwith volumea Cu-Nanofluid fraction, has nanoparticle diameter, Rayleigh number, Darcy number, porosity on the flow pattern, tem- been investigated numerically. The effects of Brownian motion, solid volume fraction, nanoparticle diameter, Rayleigh number, Darcy number, porosity on the flow pattern, tem-

perature distribution, and heat transfer characteristics are discussed in detail. The following conclusions could be drawn: (1) Brownian motion should be considered in the natural convection heat transfer of a nanofluid. The effect of Brownian motion becomes more remarkable with the increase of Ray- leigh number and nanoparticle volume fraction, while it is less pronounced with the increase of Darcy number and nanoparticle diameter. (2) The increase of nanoparticle volume fraction results in an improvement of the overall heat transfer rate, and the effect is more remarkable when the Rayleigh and Darcy numbers are at high value. (3) Increasing the nanoparticle diameter has a negative effect on the overall heat transfer rate and the effect has a limit when the nanoparticle diameter reaches a high value. (4) The porosity affects the flow pattern, temperature distribution, and heat transfer rate. The flow motion is limited when the porosity is too low. The heat transfer rate is strengthened with the increase of porosity, especially with high Rayleigh and Darcy numbers. (5) The radius ratio has a significant influence on the isotherms, streamlines, and heat transfer rate. The rate is greatly enhanced with the increase of radius ratio. Additionally, when the radius ratio is too low, multicellular flow structures are formed, and a bifurcation point exists.

Author Contributions: Conceptualization, Y.H. and M.L.; Methodology, L.Z.; Software, L.Z.; Vali- dation, Y.H., L.Z. and M.L.; Formal Analysis, L.Z.; Investigation, Y.H.; Resources, Y.H.; Data Cura- tion, L.Z.; Writing—Original Draft Preparation, L.Z.; Writing—Review & Editing, L.Z.; Visualiza- tion, L.Z.; Supervision, Y.H.; Project Administration, M.L.; Funding acquisition Administration, M.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Nature Science Foundation of China, grant number [51706213]. Data Availability Statement: Data available in a publicly accessible repository. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature RR radius ratio cp thermal capacity, J/(kg × K)

Nanomaterials 2021, 11, 990 22 of 24

perature distribution, and heat transfer characteristics are discussed in detail. The following conclusions could be drawn: (1) Brownian motion should be considered in the natural convection heat transfer of a nanofluid. The effect of Brownian motion becomes more remarkable with the increase of Rayleigh number and nanoparticle volume fraction, while it is less pronounced with the increase of Darcy number and nanoparticle diameter. (2) The increase of nanoparticle volume fraction results in an improvement of the overall heat transfer rate, and the effect is more remarkable when the Rayleigh and Darcy numbers are at high value. (3) Increasing the nanoparticle diameter has a negative effect on the overall heat transfer rate and the effect has a limit when the nanoparticle diameter reaches a high value. (4) The porosity affects the flow pattern, temperature distribution, and heat transfer rate. The flow motion is limited when the porosity is too low. The heat transfer rate is strengthened with the increase of porosity, especially with high Rayleigh and Darcy numbers. (5) The radius ratio has a significant influence on the isotherms, streamlines, and heat transfer rate. The rate is greatly enhanced with the increase of radius ratio. Additionally, when the radius ratio is too low, multicellular flow structures are formed, and a bifurcation point exists.

Author Contributions: Conceptualization, Y.H. and M.L.; Methodology, L.Z.; Software, L.Z.; Valida- tion, Y.H., L.Z. and M.L.; Formal Analysis, L.Z.; Investigation, Y.H.; Resources, Y.H.; Data Curation, L.Z.; Writing—Original Draft Preparation, L.Z.; Writing—Review & Editing, L.Z.; Visualization, L.Z.; Supervision, Y.H.; Project Administration, M.L.; Funding acquisition Administration, M.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Nature Science Foundation of China, grant number [51706213]. Data Availability Statement: Data available in a publicly accessible repository. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

RR radius ratio cp thermal capacity, J/(kg × K) d nanoparticle diameter, m Da Darcy number g acceleration due to gravity, m/s2 k thermal conductivity, W/(m × K) K medium permeability, m2 Nu Nusselt number p pressure, Pa P dimensionless pressure Pr Prandtl number r radius, m Ra Rayleigh number T temperature, K (u, v) velocity, m/s (U, V) dimensionless velocity (x, y) cartesian coodinates (X, Y) dimensionless cartesian coodinates Nanomaterials 2021, 11, 990 23 of 24

Greek symbols α thermal diffusivity, m2/s β thermal expansion coefficient, K−1 γ angle, deg ε porosity θ dimensionless temperature ρ density, Kg/m3 µ dynamic viscosity, kg/(m × s) ν kinematic viscosity, m2/s φ nanoparticle volume fraction Subscripts avg average bf base fluid i, o inner, outer loc local mnf porous medium filled with nanofluid nf nanofluid s porous medium sp nanoparticle

References 1. Esfe, M.H.; Bahiraei, M.; Hajbarati, H.; Valadkhani, M. A comprehensive review on convective heat transfer of nanofluids in porous medium: Energy-related and thermalhydraulic characteristics. Appl. Them. Eng. 2020, 178, 115487. [CrossRef] 2. Das, D.; Roy, M.; Basak, T. Studies on natural convection within enclosures of various (non-square) shapes—A review. Int. J. Heat Mass Transf. 2017, 106, 356–406. [CrossRef] 3. Dawood, H.K.; Mohammed, H.A.; Sidik, N.A.C.; Munisamy, K.M.; Wahid, M.A. Forced, Natural and Mixed-convection Heat Transfer and Fluid Flow in Annulus: A Review. Int. Commun. Int. Commun. Heat Mass Transf. 2015, 62, 45–57. [CrossRef] 4. Caltagirone, J.P. Thermoconvective instabilities in a porous medium bounded by two concentric horizontal cylinders. J. Fluid Mech. 1976, 76, 337–362. [CrossRef] 5. Rao, Y.F.; Fukuda, K.; Hasegawa, S. Steady and transient analyses of natural convection in a horizontal porous annulus with the Galerkin method. In Proceedings of the 4th AIAA/ASME Thermophysics Heat Transfer Conference, Boston, MA, USA, 2–4 June 1986. 6. Rao, Y.F.; Fukuda, K.; Hasegawa, S. A numerical study of three-dimensional natural convection in a horizontal porous annulus with Galerkin method. Int. J. Heat Mass Transf. 1988, 31, 695–707. [CrossRef] 7. Himasekhar, K. Two-dimensional bifurcation phenomena in thermal convection in horizontal concentric annuli containing saturated porous media. J. Fluid Mech. 1988, 187, 267–300. [CrossRef] 8. Leong, J.C.; Lai, F.C. Natural convection in a concentric annulus with a porous sleeve. Int. J. Heat Mass Transf. 2006, 49, 3016–3027. [CrossRef] 9. Braga, E.J.; Lemos, M. Simulation of turbulent natural convection in a porous cylindrical annulus using a macroscopic two equation Model. Int. J. Heat Mass Transf. 2006, 49, 4340–4351. [CrossRef] 10. Khanafer, K.; Al-Amiri, A.; Pop, I. Numerical analysis of natural convection heat transfer in a horizontal annulus partially filled with a fluid-saturated porous substrate. Int. J. Heat Mass Transf. 2008, 51, 1613–1627. [CrossRef] 11. Alloui, Z.; Vasseur, P. Natural convection in a horizontal annular porous cavity saturated by a binary mixture. Comput. Therm. Sci. 2011, 3, 407–417. [CrossRef] 12. Belabid, J.; Cheddadi, A. Multicellular flows induced by natural convection in a porous horizontal cylindrical annulus. Phys. Chem. News 2013, 70, 67–71. 13. Belabid, J.; Allali, K. Influence of gravitational modulation on natural convection in a horizontal porous annulus. J. Heat Transf. 2017, 139, 022502.1–022502.6. [CrossRef] 14. Belabid, J.; Allali, K. Effect of temperature modulation on natural convection in a horizontal porous annulus. Int. J. Therm. Sci. 2020, 151, 106273. [CrossRef] 15. Rostami, S.; Aghakhani, S.; Pordanjani, A.H.; Afrand, M.; Shadloo, M.S. A review on the control parameters of natural convection in different shaped cavities with and without nanofluid. Processes 2020, 8, 1011. [CrossRef] 16. Choi, S.U.S. Enhancing thermal conductivity of fluids with nanoparticles. In Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, CA, USA, 12–17 November 1995; Volume 66, pp. 99–105. 17. Zhou, W.N.; Yan, Y.Y.; Xu, J.L. A lattice Boltzmann simulation of enhanced heat transfer of nanofluids. Int. Commun. Heat Mass Transf. 2014, 55, 113–120. [CrossRef] 18. Abouali, O.; Ahmadi, G. Computer simulations of natural convection of single phase nanofluids in simple enclosures: A critical review. Appl. Therm. Eng. 2012, 36, 1–13. [CrossRef] Nanomaterials 2021, 11, 990 24 of 24

19. Jou, R.Y.; Tzeng, S.C. Numerical research of nature convective heat transfer enhancement filled with nanofluids in a rectangular enclosure. Int. Commun. Heat Mass Transf. 2006, 33, 727–736. [CrossRef] 20. Ghasemi, B.; Aminossadati, S.M. Natural convection heat transfer in an inclined enclosure filled with a water–CuO nanofluid. Numer. Heat Transf. Part A 2009, 55, 807–823. [CrossRef] 21. Abu-Nada, E.; Oztop, H.F. Effects of inclination angle on natural convection in enclosures filled with Cu-Water nanofluid. Int. J. Heat Fluid Flow 2009, 30, 669–678. [CrossRef] 22. Soleimani, S.; Sheikholeslami, M.; Ganji, D.D.; Gorji-Bandpay, M. Natural convection heat transfer in a nanofluid filled semi- annulus enclosure. Int. Commun. Heat Mass Transf. 2012, 39, 565–574. [CrossRef] 23. Seyyedi, S.M.; Dayyan, M.; Soleimani, S.; Ghasemi, E. Natural convection heat transfer under constant heat flux wall in a nanofluid filled annulus enclosure. Ain Shams Eng. J. 2015, 6, 267–280. [CrossRef] 24. Boualit, A.; Zeraibi, N.; Chergui, T.; Lebbi, M.; Boutina, L.; Laouar, S. Natural convection investigation in square cavity filled with nanofluid using dispersion model. Int. J. Hydrogen Energy 2017, 42, 8611–8623. [CrossRef] 25. Liao, C. Heat transfer transitions of natural convection flows in a differentially heated square enclosure filled with nanofluids. Int. J. Heat Mass Transf. 2017, 115, 625–634. [CrossRef] 26. Wang, L.; Yang, X.G.; Huang, C.S.; Chai, Z.H.; Shi, B.C. Hybridlattice Boltzmann-TVD simulation of natural convection of nanofluids in a partially heated square cavity using Buongiorno’s model. Appl. Therm. Eng. 2019, 146, 318–327. [CrossRef] 27. Mi, A.; Ts, B.; Ymcc, D.; Hma, E.; Gc, F.; Rk, G. Comprehensive study concerned graphene nano-sheets dispersed in ethylene glycol: Experimental study and theoretical prediction of thermal conductivity. Powder Technol. 2021, 386, 51–59. 28. Sheremet, M.A.; Pop, I. Free convection in a porous horizontal cylindrical annulus with a nanofluid using Buongiorno’s model. Comput. Fluids 2015, 118, 182–190. [CrossRef] 29. Sheremet, M.A.; Pop, I. Natural convection in a horizontal cylindrical annulus filled with a porous medium saturated by a nanofluid using tiwari and das’ nanofluid model. Eur. Phys. J. Plus 2015, 130, 107. [CrossRef] 30. Belabid, J.; Belhouideg, S.; Allali, K.; Mahian, O.; Abu-Nada, E. Numerical simulation for impact of copper/water nanofluid on thermo-convective instabilities in a horizontal porous annulus. J. Therm. Anal. Calorim. 2019, 138, 1515–1525. [CrossRef] 31. Abu-Nada, E. Effects of variable viscosity and thermal conductivity of Al2O3-water nanofluid on heat transfer enhancement in natural convection. Int. J. Heat Fluid Flow 2009, 30, 679–690. [CrossRef] 32. Cianfrini, M.; Corcione, M.; Quintino, A. Natural convection heat transfer of nanofluids in annular spaces between horizontal concentric cylinders. Appl. Therm. Eng. 2011, 31, 4055–4063. [CrossRef] 33. Goudari, S.; Shekaramiz, M.; Omidvar, A.; Golab, E.; Karimipour, A. Nanopaticles migration due to thermophoresis and Brownian motion and its impact on Ag-MgO/Water hybrid nanofluid natural convection. Powder Technol. 2020, 375, 493–503. [CrossRef] 34. Bourantas, G.C.; Skouras, E.D.; Loukopoulos, V.C.; Burganos, V.N. Heat transfer and natural convection of nanofluids in porous media. Eur. J. Mech. B Fluid 2014, 43, 45–56. [CrossRef] 35. Hu, Y.P.; Li, Y.R.; Lu, L.; Mao, Y.J.; Li, M.H. Natural convection of water-based nanofluids near the density maximum in an annulus. Int. J. Thermal. Sci. 2020, 152, 106309. [CrossRef] 36. Wang, B.; Shih, T.M.; Chen, X.; Chang, R.G.; Wu, C.X. Cascade-like and cyclic heat transfer characteristics affected by enclosure aspect rations for low Prandtl numbers. Int. J. Heat Mass Transf. 2018, 124, 131–140. [CrossRef] 37. Singh, A.K.; Basak, T.; Nag, A.; Roy, S. Heatlines and thermal management analysis for natural convection within inclined porous square cavities. Int. J. Heat Mass Transf. 2015, 87, 583–597. [CrossRef]