Numerical Investigation on Forced Hybrid Nanofluid Flow and Heat Transfer Inside a Three-Dimensional Annulus Equipped with Hot and Cold Rods: Using Symmetry Simulation

Abstract

A 3D computational fluid dynamics method is used in the current study to investigate the hybrid nanofluid (HNF) flow and heat transfer in an annulus with hot and cold rods. The chief goal of the current study is to examine the influences of dissimilar Reynolds numbers, emissivity coefficients, and dissimilar volume fractions of nanoparticles on hydraulic and thermal characteristics of the studied annulus. In this way, the geometry is modeled using a symmetry scheme. The heat transfer fluid is a water, ethylene–glycol, or water/ethylene–glycol mixture-based Cu-Al₂O₃ HNF, which is a Newtonian NF. According to the findings for the model at Re = 3000 and ${\varphi }_1$ = 0.05, all studied cases with different base fluids have similar behavior. ϕ₁ and ϕ₂ are the volume concentration of Al₂O₃ and Cu nanoparticles, respectively. For all studied cases, the total average Nusselt number (Nu_ave) reduces firstly by an increment of the volume concentrations of Cu nanoparticles until ${\varphi }_2$ = 0.01 or 0.02 and then, the total Nu_ave rises by an increment of the volume concentrations of Cu nanoparticles. Additionally, for the case with water as the base fluid, the total Nu_ave at ${\varphi }_2$ = 0.05 is higher than the values at ${\varphi }_2$ = 0.00. On the other hand, for the other cases, the total Nu_ave at ${\varphi }_2$ = 0.05 is lower than the values at ${\varphi }_2$ = 0.00. For all studied cases, the case with water as the base fluid has the maximum Nu_ave. Plus, for the model at Re = 4000 and ${\varphi }_1$ = 0.05, all studied cases with different base fluids have similar behavior. For all studied cases, the total Nu_ave reduces firstly by an increment of the volume concentrations of Cu nanoparticles until ${\varphi }_2$ = 0.01 and then, the total Nu_ave rises by an increment of the volume concentrations of Cu nanoparticles. The Nu_ave augments are found by an increment of Reynolds numbers. Higher emissivity values should lead to higher radiation heat transfer, but the portion of radiative heat transfer in the studied annulus is low and therefore, has no observable increment in HNF flow and heat transfer.

1. Introduction

Heat exchangers are types of equipment that have many industrial applications and this factor makes their performance improvement an attractive subject for researchers. Extensive research in this field has led to the development of new methods to improve the performance of heat exchangers, some of the most important of which are the use of corrugated surfaces [1,2,3], turbulators [4,5,6], fins [7,8,9], magnetic fields [10,11,12], and the replacement of common coolants with nanofluids (NFs) [13,14,15]. The suspension of nanoparticles in conventional coolants improves their cooling performance, and this fact has been confirmed in many laboratory-based and numerical studies [16,17,18].

All of the above methods, in turn, improve the performance of heat exchangers, but researchers are always looking for ideas to improve the effectiveness of these methods. A group of researchers posed the question: if adding one nanoparticle to a coolant improves its cooling performance, does adding two types of nanoparticles to the coolants not lead to a further improvement in cooling performance? Numerous experimental [19,20,21] and numerical [22,23,24] studies were conducted to answer this question, and it was finally found that the cooling performance of hybrid nanofluids (HNFs) was generally better than that of conventional NFs. Li et al. [25] investigated numerically different features of convection heat transfer inside a cavity equipped with a twin-web turbine disk and pin fins. Their obtained results indicate that there is a clockwise circular fluid flow inside the studied cavity which influences the front and back web for the improvement of the local convection heat transfer coefficients. The back web, Nu, is strangely higher than the Nu of the front web, and the back web temperature is perceptibly lower than the front web temperature. Chorin et al. [26] studied experimentally free convective heat transfer in a differentially heated cavity equipped with localized turbulators. The turbulator was located in the warm boundary coating of the cavity fluid flow. Experiments were carried out in heat transfers, temperature profiles, and velocity fields terms. The effect of the vertical location and the length for a conducting turbulator and an insulator was investigated. For the insulator turbulator, a fluid flow part diverged inside the colder turbulator region, leading to a downstream heat transfer upsurge. Giwa et al. [27] investigated experimentally and numerically different uniform magnetic induction effects on heat transfer presentation of a rectangular cavity equipped with aqueous HNF. The aqueous HNF’s thermal properties had been studied for different nanoparticle volume fractions and various temperature ranges. Their obtained results show that by employing the vertically magnetic field on the cavity-side walls, the maximum Nu improvement was attained in compression with the model which did not have a magnetic field. Furthermore, it was realized that an increment in the magnetic field can improve the heat transfer characteristics significantly. Mansouri et al. [28] considered numerically conjugate conduction–convection heat transfer in a cavity filled with air and equipped with a rhombus conducting block exposed to subdivision, such as opposing and cooperating roles. The temperature and flow results were obtained by employing the Lattice Boltzmann technique with the finite volume method and the multi-relaxation time collision scheme. The influences of the initial solid block thermal conductivity on heat transfer and fluid flow inside the cavity were studied. Their obtained results present that the initial block subdivision leads to a reduction in heat transfer inside the cavity. Thiers et al. [29] studied numerical heat transfer improvement inside a rectangular differentially heated cavity with different thermal perturbations on cold and hot walls. The different effects of wave characteristics such as phase shift, frequency, and amplitude, and also the disturbance area’s vertical location, were studied in the range of different Rayleigh numbers. The key aim of their study was to find the optimum location of differentially heated sources for heat transfer improvement. Ataei-Dadavi et al. [30] considered experimental fluid flow and heat transfer characteristics in a differential side-heated coarse porous media cavity. Their findings illustrated that the presence of a porous medium in the cavity leads to heat transfer reduction compared to the cavity without porous media. They attained a novel correlation method to predict the Nu for coarse porous media-based cavities. Farsani et al. [31] investigated heat transfer improvement in cavities using baffles melting, which were filled with gallium. They modeled the process of the phase using the fixed grid-based enthalpy–porosity technique coupled with the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm. They reported the streamlines and isotherm lines, and also, Nu values, on hot walls to investigate and analyze the problem. Vishnu et al. [32] examined numerically different conditions of heat transfer inside an oblique cavity located in supersonic flow. They simulated a 2D compressible unsteady turbulent flow field using the finite volume method and the Harten-Lax-van Leer-Contact (HLLC) scheme. They also validated their numerical procedure with available empirical and mathematical data. Sadaghiani et al. [33] examined parametrically and experimentally different influences of bubble coalescence on critical heat flux and pool boiling heat transfer in a cavity. They examined different surface wettability effects on the structured surfaces’ performance by employing a Teflon film with a thickness of 50 nm. Additionally, they examined bubble dynamics by employment of a high-speed camera. Pan et al. [34] experimentally considered the heat transfer features of a heat exchanger equipped with microchannel and fan-shaped cavities. Their obtained data illustrated that the microchannel heat exchanger’s performance, which is equipped with fan-shaped cavities, is sharply better than the heat exchangers without fan-shaped cavities and also, their pressure drop is lower than the heat exchangers without fan-shaped cavities. Balotaki et al. [35] modeled numerically, and by employing the lattice Boltzmann method, the heat transfer of free natural convections in a triangular cavity filling with different operating fluids, including water, TiO₂-H₂O, Al-H₂O, and Cu-H₂O, and equipped with double distribution functions. They reported various interesting results such as average predicted Nu, heat lines, an isothermal line, streamlines, and entropy generation values. Liu and Huang [36] studied numerically different effects using different linear thermal forcing models on fluid flow and convective heat transfer in a rectangular finned enclosure. The results showed fluctuations in the flow above the fin. Seo et al. [37] numerically analyzed the performance of a naturally cooled rectangular enclosure with a sinusoidal cylinder. The outcomes revealed the significant impact of cylinder diameter and Rayleigh number on the performance features of the enclosure. Razzaghpanah and Sarunac [38] simulated the free convection from a bundle of heated cylinders submerged in molten salt. They proposed a correlation set for Nusselt number in terms of Rayleigh number. Krakov and Nikiforov [39] employed numerical data to reveal the impact of interior cylinder shape on thermomagnetic convection heat transfer through a horizontal annulus. It was demonstrated that the shape of the interior cylinder can increase the heat transfer by 40–50%. Pawar et al. [40] numerically investigated convection heat transfer and forced laminar steady state flow inside a cylindrical finned cavity at various incidence angles. Due to analyzing aerodynamic characteristics, factors such as local time-averaged flux and vorticity, length of recirculation, Strouhal number and moment, and lift and drag coefficients were reported by them. Alam et al. [41] considered the flow around a cylinder with variable cross-sections and analyzed the dependence of velocity and temperature field on corner radius and attack angle. The dramatic role of boundary layers in improving heat transfer was reported. The main reason for this behavior is that the primary wake bubbles are mostly linked to the forces. Vyas et al. [42] conducted an experimental numerical assessment on fluid flow in the wake region of a cylinder submerged in a conduit, considering the impacts of blockage ratio. It was realized that the highest downstream distance traveled by the vortices was at a blockage ratio of 0.38. Hadžiabdić et al. [43] simulated the flow and heat transfer around a rotating cylinder. They reported the high local rates of heat transfer in the cylinder. Furthermore, they found that the overall mean predicted Nu values of rotational cylinder did not have a significant variation inside the cylinder but its time-average presented some changes in comparison with the basic cylinder.

The literature review clarifies that even though the influence of using NF for electrical heat exchangers was evaluated, to the best knowledge of the authors, there is not any research which examines the symmetry simulation of forced HNF flow and heat transfer in a three-dimensional annulus equipped with hot and cold rods. The chief goal of the current study is to examine the influences of dissimilar Reynolds numbers, emissivity coefficients, and dissimilar volume fractions of nanoparticles on hydraulic and thermal characteristics of the studied annulus. Moreover, in this study, the geometry is modeled using symmetry scheme.

2. Methodology

2.1. Problem Statement and Governing Equations

Figure 1 demonstrates a basic and simulated three-dimensional cylindrical shape cavity (annulus) equipped with hot and cold rods in the present study. In the present paper, this geometry is analyzed using a symmetry boundary condition. Figure 1 shows, on the right side, a schematic diagram of the studied configuration. The height of the cylinder is changed from 300 mm. Additionally, the height of the square is changed from 46.03 to 70.71 mm, the length of the square is 1000 mm, and the gravity influences are determined in z direction. This is a hypothetical geometry and was designed by the authors of the article. However, it has many applications in electrical components and electronics industries. The system is made of an insulator in the center and outer sides of the cylinder, with a boundary condition of zero heat flux. However, this material is presumed with two dissimilar emissivity values of 0.2 and 0.5 for analyzing its influences on radiation heat transfer. Moreover, two dissimilar Reynolds numbers, Re = 3000 and 4000, are in transient regime. The initial NF temperature is T_initial = 350 K and the cold side temperature is T_c = 300 K. The hot side temperature can be T_h = 430 K. The heat transfer fluid is a water-, ethylene–glycol-, or water/ethylene–glycol mixture-based Cu-Al₂O₃ HNF, which creates a Newtonian NF. To attain the most proficient Newtonian NF in the current research, Cu and Al₂O₃ nanoparticles were added to the base fluid in different volume concentrations of 0.01 to 0.05 with diameters of 24 nm.

Table 1. Base fluid and nanoparticle numerical values for thermophysical properties (T = 298 K) [44].

Thermophysical Properties	k (W/m·K)	cp (J/kg·K)	ρ (kg/m3)
Cu	400	385	8933
Al2O3	40	765	3970
Ethylene-glycol	0.252	2415	1114.4
H2O	0.613	4179	997.1
H2O/ethylene-glycol mixture	0.3799	3300	1067.5

presents the thermophysical features of the base fluid and nanoparticles.

The equations presented in

Table 2. Useful equations to model the studied nanofluid (NF) in the present work [44].

Density	${\rho }_{hnf}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\rho }_{bf}+{\varphi }_1{\rho }_{np,1}\right]+{\varphi }_2{\rho }_{np,2}$
Specific heat	${\left(\rho c_p\right)}_{hnf}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\left(\rho c_p\right)}_{bf}+{\varphi }_1{\left(\rho c_p\right)}_{np,1}\right]+{\varphi }_2{\left(\rho c_p\right)}_{np,2}$
Thermal conductivity	$k_{hnf}=k_{nf}\left[k_{np,2}+2k_{nf}-2{\varphi }_2\frac{k_{nf}-k_{np,2}}{k_{np,2}}+2k_{nf}+{\varphi }_2\left(k_{nf}-k_{np,2}\right)\right]$
	$k_{nf}=k_{bf}\left[k_{np,1}+2k_{bf}-\frac{2{\varphi }_1\left(k_{bf}-k_{np,1}\right)}{k_{np,1}}+2k_{bf}+{\varphi }_1\left(k_{bf}-k_{np,1}\right)\right]$
Dynamic viscosity	${\mu }_{hnf}=\frac{{\mu }_{bf}}{{\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}}$

have been used to calculate the thermophysical properties of NF. The density $({\rho }_{hnf}$) and specific heat $(c_{P,hnf})$ of the HNF at each section temperature $\left(T_m\right)$ were calculated using the equations presented in Table 2 [44], where ${\varphi }_1$ is the volume concentration of Al₂O₃ and ${\varphi }_2$ is the volume concentration of Cu nanoparticles. Moreover, the effective dynamic viscosity and thermal conductivity of HNF were obtained by the equations in Table 2.

A 3D numerical analysis has been used in the current research to examine the transient HNF flow and heat transfer in an annulus with hot and cold rods. Given that the Reynolds number is more than 2300, the flow regime is considered transient. The Reynolds-averaged Navier–Stokes (RANS) equations with the shear-stress (SST) $k–\omega$ turbulence model were used for simulating the turbulence regime. The used equations can be defined as reported in

Table 3. Governing equations and parameters of interest [45].

Continuity equation	$\frac{\partial }{\partial x_i}\left(\rho u_i\right)=0$
Momentum equation	$\frac{\partial }{\partial x_i}\left(\rho u_i{u}_j\right)=-\frac{\partial p}{\partial x_i}+\frac{\partial {\tau }_{ij}}{\partial x_j}+\rho g_i$
	${\tau }_{ij}=\frac{{\mu }^{*}\partial u_i}{\partial x_j}$
	${\mu }^{*}=\mu +{\mu }_t$
Energy equation	$\frac{\partial }{\partial x_i}\left(\rho c_p{u}_{i}T\right)=\frac{\partial }{\partial x_i}\left(\frac{{\lambda }^{*}\partial T}{\partial x_i}\right)+Qi$
	${\lambda }^{*}=\lambda +{\lambda }_t$
	${\lambda }_t=\frac{c_p{\mu }_t}{{\sigma }_t}$
Turbulence equations	$\frac{\partial }{\partial x_i}\left(\rho u_{i}k\right)=\frac{\partial }{\partial x_i}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_i}\right]+G_k-Y_k$
	$\frac{\partial }{\partial x_i}\left(\rho u_i\omega \right)=\frac{\partial }{\partial x_i}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\omega }}\right)\frac{\partial k}{\partial x_i}\right]+G_{\omega }-Y_{\omega }+D_{\omega }$
	${\mu }_t=\frac{\rho k}{\omega }\frac{1}{max\left\{\frac{1}{{\alpha }^{*}},\frac{\Omega F_2}{a_1\omega }\right\}}$
	$\Omega =\sqrt{2{\Omega }_{ij}{\Omega }_{ij}}$
	${\Omega }_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\right)$
	${\sigma }_k=\frac{1}{\frac{F_1}{{\sigma }_{k,1}}+\frac{1-F_1}{{\sigma }_{k,2}}}$
	${\sigma }_{\omega }=\frac{1}{\frac{F_1}{{\sigma }_{\omega ,1}}+\frac{1-F_1}{{\sigma }_{\omega ,2}}}$
	$F_1=tanh\left({\Phi }_1^4\right)$
	$F_2=tanh\left({\Phi }_2^2\right)$
	${\Phi }_1=min\left\{max\left\{\frac{\sqrt{k}}{0.09\omega y},\frac{500\mu }{\rho y^2\omega }\right\},\frac{4\rho k}{{\sigma }_{\omega ,2}{D}_{\omega }^{+}{y}^2}\right\}$
	${\Phi }_2=max\left\{\frac{2\sqrt{k}}{0.09\omega y},\frac{500\mu }{\rho y^2\omega }\right\}$
	$D_{\omega }^{+}=max\left\{\frac{2\rho }{\omega {\rho }_{\omega ,2}}\frac{\partial k}{\partial x_i}\frac{\partial \omega }{\partial x_i},{10}^{-20}\right\}$
	${\alpha }^{*}={\alpha }_{\infty }^{*}\left(\frac{{\alpha }_0^{*}+\frac{{\text{Re}}_t}{R_k}}{1+\frac{{\text{Re}}_t}{R_k}}\right),{\alpha }_0^{*}=\frac{{\beta }_i}{3},{\text{ Re}}_t=\frac{\rho k}{\mu \omega },{\beta }_i=F_1{\beta }_{i,1}+\left(1-F_1\right){\beta }_{i,2}$
	$G_k={\tau }_{t,ij}\frac{\partial u_i}{\partial x_j},{\tau }_{t,ij}={\mu }_t\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)-\frac{2}{3}\rho k{\delta }_{ij}$
	$Y_k=\rho {\beta }^{*}k\omega ,G_{\omega }=\frac{\rho \alpha }{{\mu }_t}{G}_k,Y_{\infty }=\rho {\beta }_i{\omega }^2$
	$\alpha =\frac{{\alpha }_{\infty }}{{\alpha }^{*}}=\left(\frac{{\alpha }_0^{*}+\frac{{\mathrm{Re}}_t}{R_k}}{1+\frac{{\mathrm{Re}}_t}{R_k}}\right),{\alpha }_{\infty }=F_1{\alpha }_{\infty ,1}+\left(1-F_1\right){\alpha }_{\infty ,2}$
	${\alpha }_{\infty ,1}=\frac{{\beta }_{i,1}}{{\beta }_{\infty }^{*}}-\frac{{\kappa }^2}{{\sigma }_{\omega ,1}\sqrt{{\beta }_{\infty }^{*}}},{\alpha }_{\infty ,2}=\frac{{\beta }_{i,2}}{{\beta }_{\infty }^{*}}-\frac{{\kappa }^2}{{\sigma }_{\omega ,2}\sqrt{{\beta }_{\infty }^{*}}}$
Parameters of interest	${\alpha }_{nf}=\frac{k_{nf}}{\left(\rho c_p\right){}_{nf}},\mathrm{Re}=\frac{U_{ref}{D}_h}{{\nu }_f},\mathrm{Nu}=\frac{k_{nf}}{k_f}{\displaystyle \int }\frac{\partial T}{\partial x}$

. As mentioned earlier, in the current modeling, the SST $k–\omega$ turbulence model [45] has been presumed as the turbulence model [46,47,48,49,50,51,52,53]. This is used to have a better forecast of the flow conditions, such as large normal strain, strong acceleration, reverse pressure gradients, and separating flow, which can occur in the HNF flow appearances adjacent to the rods and walls. Based on the SST $k–\omega$ model, the turbulence kinetic energy ($k$) and specific dissipation rate ($\omega$) can be attained by the equations reported in Table 3, where ${\Omega }_{ij}$ is the mean rate-of-rotation tensor and F₁ and F₂ are the blending functions. Additionally, $D_{\omega }^{+}$ is the positive portion of the cross-diffusion term. The constant values reported in Table 3 are defined in

Table 4. Constant values of the shear-stress (SST) $k–\omega$ turbulence model [45].

$${\mathit{\sigma }}_{\mathit{t}}$$	$${\mathit{\alpha }}_{\mathit{k},1}$$	$${\mathit{\alpha }}_{\mathit{k},2}$$	$${\mathit{\alpha }}_{\mathit{\omega },1}$$	$${\mathit{\alpha }}_{\mathit{\omega },2}$$	$${\mathit{a}}_1$$	$${\mathit{\beta }}_{\mathit{i},1}$$	$${\mathit{\beta }}_{\mathit{i},2}$$	$${\mathit{\beta }}_{\mathit{\infty }}^{*}$$	$$\mathit{\kappa }$$	$${\mathit{R}}_{\mathit{k}}$$
0.85	1.176	1.000	2.000	1.168	0.31	0.0750	0.0828	0.0900	0.41	6

. All equations were resolved until the maximum residual of grid control volume became smaller than 10⁻⁷.

The Monte Carlo method has been used for modeling the radiation in the annulus gap. In this model, the radiation has been determined to affect the domain by heating its surface, without radiant energy transfer directly to the domain (Surface-to-Surface transfer mode (S2S)). The spectral dependency of the radiative heat transfer equation was estimated using the Gray Model (GM), which determines whether all radiation amounts are closely uniform through the spectrum. The inner rod and outer wall are determined as insulators and no heat transfer occurs in these sections.

It was also predictable that the radiation mechanism could not play a decisive role in heat transfer. However, the goal was to model the problem with the least assumptions and the most accurate details. Therefore, despite the low share of radiation in heat transfer, its effect was considered.

In order to simulate the nanofluid flow inside the cylinder, the single-phase technique was employed. Just the thermophysical characteristics were calculated at different temperatures and nanoparticle volume fractions and have been entered into software.

2.2. Validation

As displayed in Figure 2, a grid independency study was implemented for the studied cavity containing Al₂O₃-Cu/water HNF to examine the influences of mesh sizes on the findings. For the validation, the emissivity of the cylinder is constant and equal to 0.2. Additionally, the volume fraction of the Al₂O₃ nanoparticles is 0.05. A non-uniform structured mesh was used for the channel. Moreover, the near-wall mesh was fine enough to be able to resolve the viscous sublayer ($y+\le 1$). As observed, several meshes have been tested versus different Reynolds numbers and volume concentrations of Cu nanoparticles. Finally, it was determined that the mesh configurations, including 113,285 nodes, had acceptable precision with a maximum error of 3%.

Figure 3 shows code validation in the present study. Figure 3a illustrates dimensionless temperature dissimilarity versus dimensionless distance to validate the obtained results in the present work and the empirical data of Kuehn and Goldstein [54]. Due the lack of empirical results for natural heat transfer in a cylinder, the numerical process has been validated with the presented empirical data of Kuehn and Goldstein [54] for ΔT = 0.371 K, Pr = 5.45, Ra_L = 2.09 × 10⁵, L/2ri = 0.8, and T_avg = 303.18 K, providing T_i > T_o. The judgments for different temperature profiles at various angles are indicated in Figure 3. It is seen clearly that there is a remarkable coincidence between the obtained results from reference [54] and the numerical data in the present investigation.

In addition, Figure 3b illustrates Nusselt number variation versus Reynolds number (2500 < Re < 3900) to validate the obtained results in the present work and the numerical data of Liu and Yu [55] in the transient flow regime. It is seen clearly that there is a good agreement between the obtained results from reference [55] and the numerical data in the present investigation.

Due to the specific geometric conditions of the problem, the boundary conditions, and the explanations in the manual of ANSYS-Fluent software, using the symmetry boundary condition in the present work is a very effective and appropriate method for modeling. In order to validate the boundary condition, Figure 3c is plotted. In this diagram, the results of modeling a full cylinder and a half cylinder modeled with the symmetry condition are presented. The good agreement of the answers is well observed, which indicates the correctness of the boundary conditions used.

3. Results and Discussion

Figure 4 illustrates streamlines and isotherm lines at different cross-sections of the studied cylinder for the case filled with Al₂O₃-Cu/water HNF at ${\varphi }_1$ = 0.05, ${\varphi }_2$ = 0.01, $\epsilon$ = 0.2, and Re = 3000. As seen in this figure, HNF flow and heat transfer through the z-direction of the cylinder are developed. Additionally, the calculations show that different vortexes are through the annulus. The main heat transfer mechanism in the annulus is convection and the next one is radiation. However, the portion of radiation heat transfer is sharply lower than the convective mechanism. In addition, the presence of rods leads to more heat transfer because of vortexes being made. Rods are like obstacles that destroy the laminar sublayers and make local turbulence which enhance the convective heat transfer coefficient in the annulus and intensify the Nu. Figure 5 illustrates streamlines and isotherm lines at different cross-sections of the studied cylinder for the case filled with Al₂O₃-Cu/water HNF at ${\varphi }_1$ = 0.05, ${\varphi }_2$ = 0.01, $\epsilon$ = 0.2, and Re = 4000. As seen in this figure, HNF flow and heat transfer through the z-direction of the cylinder are developed. Additionally, the calculations show that different vortexes are through the annulus. The main heat transfer mechanism in the annulus is convection and the next one is radiation. However, the portion of radiation heat transfer is sharply lower than the convective mechanism. Additionally, the presence of rods leads to more heat transfer because of vortexes being made. Rods are like obstacles that destroy the laminar sublayers and make local turbulence which enhance the convective heat transfer coefficient in the annulus and augment the Nu. Additionally, it is realized that higher Reynolds numbers lead to more heat transfer coefficients because of more vortexes and turbulence through the cylindrical annulus. By comparing the streamlines and isotherm lines in Figure 4 and Figure 5, some similarities between them are clearly seen.

Figure 6 illustrates the streamlines and isotherm lines at different cross-sections of the studied cylinder for the case filled with Al₂O₃-Cu/water HNF at ${\varphi }_1$ = 0.05, ${\varphi }_2$ = 0.05, $\epsilon$ = 0.2, and Re = 4000. As seen in this figure, HNF flow and heat transfer through the z-direction of cylinder are developed. Additionally, the calculations show that different vortexes are through the annulus. The main heat transfer mechanism in the annulus is convection and the next one is radiation. However, the portion of radiation heat transfer is sharply lower than the convective mechanism. In addition, the presence of rods leads to more heat transfer because of vortexes being made. Rods are like obstacles that destroy the laminar sublayers and make local turbulence which enhance the convective heat transfer coefficient in the annulus and augment the Nu. Furthermore, it is realized that higher volume concentrations of nanoparticles lead to more heat transfer coefficients because of the higher thermal conductivity of HNF flow and also more vortexes and turbulence through the cylindrical annulus because of more dynamic viscosity values. By comparing the streamlines and isotherm lines in Figure 5 and Figure 6, some similarities between them are clearly seen.

Figure 7 illustrates the streamlines and isotherm lines at different cross-sections of the studied cylinder for the case filled with Al₂O₃-Cu/water HNF at ${\varphi }_1$ = 0.05, ${\varphi }_2$ = 0.01, $\epsilon$ = 0.7, and Re = 3000. As seen in this figure, HNF flow and heat transfer through the z-direction of the cylinder are developed. Additionally, the calculations show that different vortexes are through the annulus. The main heat transfer mechanism in the annulus is convection and the next one is radiation. However, the portion of radiation heat transfer is sharply lower than the convective mechanism. In addition, the presence of rods leads to more heat transfer because of vortexes being made. Rods are like obstacles that destroy the laminar sublayers and make local turbulence which enhance the convective heat transfer coefficient in the annulus and augment the Nu. Figure 8 illustrates streamlines and isotherm lines at different cross-sections of the studied cylinder for the case filled with Al₂O₃-Cu/water HNF at ${\varphi }_1$ = 0.05, ${\varphi }_2$ = 0.01, $\epsilon$ = 0.7, and Re = 4000. As seen in this figure, HNF flow and heat transfer through the z-direction of cylinder are developed. Additionally, the calculations show that different vortexes are through the annulus. The main heat transfer mechanism in the annulus is convection and the next one is radiation. However, the portion of radiation heat transfer is sharply lower than the convective mechanism. Additionally, the presence of rods leads to more heat transfer because of vortexes being made. Rods are like obstacles that destroy the laminar sublayers and make local turbulence which enhance the convective heat transfer coefficient in the annulus and augment the Nu. Additionally, it is realized that higher Reynolds numbers lead to more heat transfer coefficients because of more vortexes and turbulence through the cylindrical annulus. By comparing the streamlines and isotherm lines in Figure 7 and Figure 8, some similarities between them are clearly seen. As noted previously, the portion of radiation heat transfer inside the studied cylinder is significantly lower than the convection heat transfer. Therefore, variations in emissivity values do not have an important influence on HNF flow and heat transfer distribution inside the cylindrical annulus, which is shown clearly by comparing Figure 4, Figure 5, Figure 7, and Figure 8 with each other. Higher emissivity values should lead to higher radiation heat transfer, but the portion of radiative heat transfer in the studied annulus is low and therefore, does not have an observable increment in HNF flow and heat transfer.

Figure 9 illustrates streamlines and isotherm lines at different cross-sections of the studied cylinder for the case filled with Al₂O₃-Cu/water HNF at ${\varphi }_1$ = 0.05, ${\varphi }_2$ = 0.05, $\epsilon$ = 0.7 and Re = 4000. As seen in this figure, HNF flow and heat transfer through the z-direction of the cylinder are developed. Additionally, the calculations show different vortexes are through the annulus. The main heat transfer mechanism in the annulus is convection and the next one is radiation. However, the portion of radiation heat transfer is sharply lower than the convective mechanism. Additionally, the presence of rods leads to more heat transfer because of vortexes being made. Rods are like obstacles that destroy the laminar sublayers and make local turbulence which enhance the convective heat transfer coefficient in the annulus and augment the Nu. In addition, it is realized that higher volume concentrations of nanoparticles lead to more heat transfer coefficients because of more thermal conductivity of HNF flow and also, more vortexes and turbulence through the cylindrical annulus because of more dynamic viscosity values. By comparing the streamlines and isotherm lines in Figure 6 and Figure 9, some similarities between them are clearly seen. As noted previously, the portion of radiation heat transfer inside the studied cylinder is significantly lower than the convection heat transfer. Therefore, variations in emissivity values do not have an important influence on HNF flow and heat transfer distribution inside the cylindrical annulus, which is shown clearly by comparing Figure 6 and Figure 9 with each other. Higher emissivity values should lead to higher radiation heat transfer, but the portion of radiative heat transfer in the studied annulus is low and therefore, does not have an observable increment in HNF flow and heat transfer.

Figure 10 demonstrates the effects of using different base fluids on variation of predicted Nu_ave versus different volume fractions of Cu nanoparticles for the studied cylinder for the case filled with Al₂O₃-Cu HNF at ${\varphi }_1$ = 0.05 or 0.10, $\epsilon$ = 0.2, and Re = 3000 or 4000. As seen in Figure 10a, for the model at Re = 3000 and ${\varphi }_1$ = 0.05, all studied cases with different base fluids have similar behavior. For all studied cases, the total Nu_ave reduces firstly by an increment of the volume concentrations of Cu nanoparticles until ${\varphi }_2$ = 0.01 or 0.02 and then, the total Nu_ave rises by an increment of the volume concentrations of Cu nanoparticles. Additionally, it is seen that for the case with water as the base fluid, the total Nu_ave at ${\varphi }_2$ = 0.05 is more than the values at ${\varphi }_2$ = 0.00, while for the other cases, the total Nu_ave at ${\varphi }_2$ = 0.05 is less than the values at ${\varphi }_2$ = 0.00. Furthermore, for all studied cases, the case with water as the base fluid has the maximum Nu_ave. As seen in Figure 10b, for the model at Re = 4000 and ${\varphi }_1$ = 0.05, all studied cases with different base fluids have similar behavior. For all studied cases, the total Nu_ave reduces firstly by an increment of the volume concentrations of Cu nanoparticles until ${\varphi }_2$ = 0.01 and then, the total Nu_ave rises by an increment of the volume concentrations of Cu nanoparticles. Additionally, it is seen that for the case with water as the base fluid, the total Nu_ave at ${\varphi }_2$ = 0.05 is more than the values at ${\varphi }_2$ = 0.00, while for the other cases, the total Nu_ave at ${\varphi }_2$ = 0.05 is less than the values at ${\varphi }_2$ = 0.00. Additionally, it is seen that by increment of Reynolds numbers, the Nu_ave augments, as found by the comparison of Figure 10a,b.

As seen in Figure 10c, for the model at Re = 3000 and ${\varphi }_1$ = 0.10, all studied cases with different base fluids have similar behavior. For all studied cases, the total Nu_ave increases firstly by an increment of the volume concentrations of Cu nanoparticles until ${\varphi }_2$ = 0.01, 0.02, or 0.03 and then, the total Nu_ave diminishes by the intensification of volume concentrations of Cu nanoparticles. Additionally, it is seen that for the case with water as the base fluid, the total Nu_ave at ${\varphi }_2$ = 0.05 is more than the values at ${\varphi }_2$ = 0.00, while for the other cases, the total Nu_ave at ${\varphi }_2$ = 0.05 is less than the values at ${\varphi }_2$ = 0.00. Furthermore, for all studied cases, the highest Nu_ave belongs to the case with water as the base fluid. As seen in Figure 10d, for the model at Re = 4000 and ${\varphi }_1$ = 0.10, all studied cases with different base fluids have similar behavior. For all studied cases, the total Nu_ave reduces firstly by an increment of the volume concentrations of Cu nanoparticles until ${\varphi }_2$ = 0.01, 0.02, and 0.03, and then, the total Nu_ave rises by an increment of the volume concentrations of Cu nanoparticles. Additionally, it is seen that for the case with water as the base fluid, the total Nu_ave at ${\varphi }_2$ = 0.05 is more than the values at ${\varphi }_2$ = 0.00, while for the other cases, the total Nu_ave at ${\varphi }_2$ = 0.05 is less than the values at ${\varphi }_2$ = 0.00. Furthermore, for all studied cases, the case with water as the base fluid has the maximum Nu_ave. Additionally, it is seen that by increment of Reynolds numbers, the Nu_ave augments, as found by the comparison of Figure 10c,d.

Figure 11 shows different portions of the predicted Nu versus different volume fractions of Cu nanoparticles of the studied cylinder for the case filled with ethylene glycol-based HNF at ${\varphi }_1$ = 0.05, $\epsilon$ = 0.2, and Re = 3000. It is seen that just about 5 to 7% of the total average Nusselt number is related to radiation heat transfer and a huge portion is related to convection heat transfer. This is why variations of emissivity coefficient do not have a significant effect on the final results.

4. Conclusions

The present paper investigates symmetry simulation of transient forced HNF flow and heat transfer in a three-dimensional annulus equipped with hot and cold rods. The chief goal of the current study is to model the geometry using a symmetry scheme and also examine the influences of dissimilar Reynolds numbers, emissivity coefficients, and dissimilar volume fractions of nanoparticles on the thermal and hydraulic characteristics of the studied annulus. The height of cylinder is changed from 300 mm. Additionally, the height of square is changed from 46.03 to 70.71 mm, the length of square is 1000 mm, and the gravity influences are determined in z direction. The system is made of an insulator in the center and outer sides of the cylinder, with a boundary condition of zero heat flux. However, this material is assumed with two different emissivity values, ε = 0.2 and ε = 0.5, in order to analyze the effects of emissivity values on radiation heat transfer, in addition to two different Reynolds numbers, Re = 3000 and 4000. For all studied models, the initial NF temperature is T_initial = 350 K and the cold side temperature is T_c = 300 K. The hot side temperature can be T_h = 430 K. The heat transfer fluid is a water-, ethylene–glycol- or water/ethylene–glycol mixture-based Cu-Al₂O₃ HNF, which makes a Newtonian NF. The RANS equations with the shear-stress (SST) $k–\omega$ turbulence model have been used for simulating the turbulence regime. The most important obtained results are as follows:

• For the model at Re = 3000 and ${\varphi }_1$ = 0.05, all studied cases with different base fluids have similar behavior. For all studied cases, the total Nu_ave reduces firstly by an increment of the volume concentrations of Cu nanoparticles until ${\varphi }_2$ = 0.01 or 0.02 and then, the total Nu_ave rises by an increment of the volume concentrations of Cu nanoparticles.
• For the case with water as the base fluid, the total Nu_ave at ${\varphi }_2$ = 0.05 is more than the values at ${\varphi }_2$ = 0.00, while for the other cases, the total Nu_ave at ${\varphi }_2$ = 0.05 is less than the values at ${\varphi }_2$ = 0.00.
• For all studied cases, the case with water as the base fluid has the maximum Nu_ave.
• For the model at Re = 4000 and ${\varphi }_1$ = 0.05, all studied cases with different base fluids have similar behavior. For all studied cases, the total Nu_ave reduces firstly by an increment of the volume concentrations of Cu nanoparticles until ${\varphi }_2$ = 0.01 and then, the total Nu_ave rises by an increment of the volume concentrations of Cu nanoparticles.
• By increment of Reynolds numbers, the Nu_ave augments.
• Higher emissivity values should lead to higher radiation heat transfer, but the portion of radiative heat transfer in the studied annulus is low and therefore, does not have an observable increment in HNF flow and heat transfer.