Free convection heat transfer and entropy generation analysis of water-Fe 3 O 4 /CNT hybrid nanofluid in a concentric annulus

Purpose- This work aims to numerically investigate the heat transfer and entropy generation characteristics of water-based hybrid nanofluid in natural convection flow inside a concentric horizontal annulus. Design/Methodology/approach-The hybrid nanofluid is prepared by suspending tetramethylammonium hydroxide (TMAH) coated Fe 3 O 4 (magnetite) nanoparticles and gum arabic (GA) coated carbon nanotubes (CNTs) in water. The effects of nanoparticles volume concentration and Rayleigh number on the streamlines, isotherms, average Nusselt number as well as the thermal, frictional and total entropy generation rates are investigated comprehensively. Findings- Results show the advantageous effect of hybrid nanofluid on the average Nusselt number. Furthermore, the study of entropy generation shows the increment of both frictional and thermal entropy generation rates by increasing Fe 3 O 4 and CNTs concentrations at various Rayleigh numbers. Increasing Rayleigh number from 10 3 to 10 5 , at Fe 3 O 4 concentration of 0.9% and CNT concentration of 1.35%, increases the average Nusselt number, thermal entropy generation rate, and frictional entropy generation rate by 224.95%, 224.65%, and 155.25%, respectively. Moreover, increasing the Fe3O4 concentration from 0.5 to 0.9%, at Rayleigh number of 10 5 and CNT concentration of 1.35%, intensifies the average Nusselt number, thermal entropy generation rate, and frictional entropy generation rate by 18.36%, 22.78%, and 72.7%, respectively. Originality/Value- To the best knowledge of the authors, there are not any archival publications considering the detailed behaviour of the natural convective heat transfer and entropy generation of hybrid nanofluid in a concentric annulus.


Introduction
The problem of natural convection heat transfer in concentric and eccentric cylindrical annulus has been attracting lots of attention due to its wide applications including heat exchangers, thermal storage systems, solar collectors, water distillation and underground electric transmission cables (Garg and Szeri, 1992;Khanafer and Chamkha, 2003;Ghernoug et al., 2016;Mahmoud Aly and Asai, 2016;Afrand et al., 2017, Alipour et al, 2017. However, modifying the heat transfer characteristics in the natural convection flows in the annulus is always sought out due to a large number of applications (Afrand, 2017;Abhilash and Lab, 2018;Zhao et al., 2018. As one of the first studies, Crawford and Lemlich (Crawford and Lemlich, 1962) studied the laminar natural convection flow between concentric circular annulus for several diameter ratios and Grashof numbers at a constant Prandtl number. Kuehn andGoldstein (1976, 1978) performed an investigation on natural convection within a horizontal annulus. Different fluids of water and air were examined for various Rayleigh numbers considering constant thermophysical properties. They reported good agreement between the numerical and experimental results. Dutta et al. (2018) investigated the impacts of tilt angle and fluid yield stress on the natural convection from an isothermal square bar in a square annulus. They considered the effects of Rayleigh number, Prandtl number, Bingham number, aspect ratio, and angle of inclination. The results showed that the Nusselt number increases with increasing the Rayleigh number and decreases with increasing Bingham number. Imtiaz and Mahfouz (2018) studied the natural convection flow in an eccentric annulus containing heat-generating fluid. They found that the average dimensionless temperature of the fluid intensifies with the increase in the heat generation parameter. Besides, it was reported that the rate of heat transfer reduces for a fixed Rayleigh number as the inner cylinder moves upward from negative eccentricity to positive eccentricity.
Heat transfer enhancement, especially in natural convection systems, is important for energy saving purposes in industries (Eckert et al., 1987, Rezaei et al., 2017. The main limitation of conventional heat transfer fluids is their low thermal conductivity which has a high effect in natural convection systems (Parvin et al., 2012, Akbari et al., 2016, Sajadifar et al., 2017. Nanofluids have been used widely in the last two decades as one of the advantageous methods to increase the heat transfer in different energy systems (Putra et al., 2003, Akbari et al., 2015, Arabpour et al., 2018a, Shamsi et al., 2017. By using nanoparticles in a base fluid, the heat transfer characteristics of the employed fluid can be modified (Vadasz et al., 2005, Arabpour et al., 2018b, Heydari et al,, 2017, Zadkhast et al., 2017. There have been lots of studies on the heat transfer enhancement in natural convection problems in the literature using nanofluids. Cadena-de la Peña et al. (2017) examined different mineral oil-based nanofluids as a cooling fluid in an annulus between two cylinders and showed heat transfer enhancement by the newly generated nanofluids. The inner cylinder acted as a heat source while the outer one remained at a constant temperature. AIN and TiO2 nanoparticles were added to the oil-based fluid with various concentrations with and without oleic acid treatment. They presented correlations for the Nusselt number in terms of Rayleigh number. Selimefendigil and Oztop (2017) numerically assessed the natural convection heat transfer of water-CuO nanofluid in a horizontally partitioned annulus in the presence of an inclined magnetic field. They considered a conductive partition with different thickness and thermal conductivity. The results demonstrated that the heat transfer enhances by increasing the thickness of the conductive partition, Rayleigh number and inclination angle while reduces by decreasing Hartmann number. Siavashi and Rostami (2017) numerically examined the natural convection heat transfer of non-Newtonian water-Al2O3 nanofluid within a cylindrical annulus with a concentric circular heat source covered with a conductive porous layer. The results revealed that the non-Newtonian nanofluid has a higher Nusselt number than the other studied cases. Li et al. (2018) investigated the influence of radiative heat transfer on magnetohydrodynamic free convection of a water-Fe3O4 nanofluid in a sinusoidal annulus. To the best knowledge of the authors, there are not any archival publications considering the detailed behaviour of the natural convective heat transfer and entropy generation of hybrid nanofluid in a concentric annulus. In this paper, the problem of two-dimensional natural convection of water-Fe3O4/CNT hybrid nanofluid between two horizontal concentric cylinders is investigated. The inner cylinder surface is kept at a constant temperature which is higher than the outer cylinder temperature. The effects of nanoparticles volume concentration and Rayleigh number on the streamlines, isotherms, average Nusselt number as well as the local and global thermal entropy generation rate, frictional entropy generation rate and total entropy generation rate are analysed.

Physical properties of nanofluid
The water-based hybrid nanofluid containing TMAH coated Fe3O4 nanoparticles and GA coated CNTs is synthesized by mixing the required amount of water-Fe3O4 nanofluid and water-CNT nanofluid, followed by sonication of the mixture for 5 min. The water-Fe3O4 nanofluid was prepared by utilizing the technique proposed by Berger et al. (1999) and the water-based CNT nanofluid was synthesized via the method described by Garg et al. (2009).
The details of the preparation and characterization of the hybrid nanofluid can be found in the author's previous work (Shahsavar et al., 2016). The interaction between the TMAH and GA molecules results in the physical attachment of the Fe3O4 and CNT nanoparticles.
After careful preparation and characterization, a series of experiments were carried out to obtain the thermophysical properties of the hybrid nanofluids containing different concentrations of the Fe3O4 and CNT nanoparticles. The volume concentration of the Fe3O4 and CNT nanoparticles in the prepared nanofluid samples as well as the density ( ), specific heat ( , ), viscosity ( ), thermal conductivity ( ) and thermal expansion coefficient ( ) of these hybrid nanofluids are presented in Table 1.

Geometry and boundary conditions
The nanofluid is considered incompressible and the flow is considered to be laminar and steady-state. The outer cylinder with the radius of is cooled at the temperature of and the inner cylinder with the radius of is kept at the hot temperature of ℎ . The schematic of the problem is demonstrated in Fig. 1.

Governing equations
In order to numerically investigate the natural convection heat transfer behaviour of the studied nanofluids in a concentric annulus, the conservation of mass, momentum and energy should be solved. The assumptions considered in the study are as follows: (a) The thermophysical properties of the hybrid nanofluid are constant except for the density variation.
(b) The Boussinesq approximation is employed to consider the effect of natural convection.
(c) The hybrid nanofluid behaves like a homogeneous single phase fluid.
The governing equations for conservation of mass, momentum, and energy, based on the assumptions given above can be expressed as below: (i) Conservation of mass, (iii) Conservation of momentum in the y-direction, (iv) Conservation of energy, where is the x component velocity, is the y component velocity, is the pressure, is the gravitational acceleration, and is the temperature.

Data reduction
The Rayleigh number for pipe flow is defined by the following equation: The average convective heat transfer coefficient at the inner annular wall is defined as: where ̇ is the heat transfer rate.

8
The convective heat transfer coefficient is also defined in the form of Nusselt number as: where is the thermal conductivity of water at 25 ℃.

Entropy generation
The entropy generation in the flow field comprises of two main parts; (i) frictional factors, and (ii) thermal irreversibility. The local thermal and frictional entropy generation rates can be obtained as, Moreover, the total entropy generation rate (̇, ′′′ ) is obtained by the following equation: The global entropy generation rates are calculated by the integration of the local entropy generation rates over the whole domain as follows:

Numerical method and validation
In this contribution, ANSYS-FLUENT computational fluid dynamics (CFD) software incorporated with a finite volume method is utilized to solve the governing equations along with the mentioned boundary conditions. The second order upwind method is used for solving the momentum and energy equations. Pressure and velocity are coupled using Semi-Implicit convergence criteria is set to 10 -6 .
Five different grids were considered to check grid independency of the numerical results.
Owing to severe velocity and temperature gradients adjacent to the walls, smaller elements were applied in the near wall region. Eventually, a grid with 400 nodes in the circumferential direction and 45 nodes in the radial direction was selected as the best one (see Fig. 2).

Fig. 2
Validation of the numerical analysis is accomplished by comparing the obtained numerical data for the temperature profile with the experimental results of Kuhen and Goldstein (1976) for the flow of water in an annulus using = 4.57 × 10 4 and = 0.7. The comparison for three temperature profiles at three different angles is illustrated in Fig. 3. It can be seen that there is good agreement between the results.

Results and discussion
In the current investigation, the analysis is carried out to evaluate the influences of nanoparticle ( θ=0° (Kuhen and Goldstein, 1976) θ=90° (Kuhen and Goldstein, 1976) θ=270° (Kuhen and Goldstein, 1976)     (8), the thermal entropy generation rate is a function of thermal conductivity, the average fluid temperature and temperature gradient. As previously mentioned, increasing the Rayleigh number leads to an increase in the heat transfer and, consequently, an increase in the average temperature of the nanofluid, which results in a reduction in the rate of thermal entropy generation. On the other hand, the increase in the Rayleigh number leads to the reduction of the velocity boundary layer thickness and, thus, the reduction of the thermal boundary layer thickness (Bergman et al., 2011). This causes an increase in the temperature gradient and, consequently, an increase in the thermal entropy generation rate. Fig. 8 demonstrates that the higher temperature gradient effect prevails over the higher average temperature effect and, therefore, the thermal entropy generation rate augments by increasing the Rayleigh number. In addition, Fig. 8 shows that increasing the Fe3O4 and CNT concentrations in a constant Rayleigh number leads to an increase in the rate of entropy generation. As an example, at Rayleigh than that for 0.9%FF, respectively. According to Eq. (9), the frictional entropy generation rate is a function of viscosity, the average fluid temperature, and the velocity gradient. At a constant concentration of nanoparticles, increasing the Rayleigh number leads to a decrease in the velocity boundary layer thickness and, consequently, the increase of the velocity gradient, which results in an increase in the frictional entropy generation rate. Furthermore, as mentioned, increasing the Rayleigh number leads to an increase in the average temperature and consequently, a reduction in the frictional entropy generation rate. The results presented in Fig.   9 show that the higher velocity gradient effect prevails over the higher average temperature effect and, therefore, the frictional entropy generation rate augments by increasing the Rayleigh number. In addition, at a constant Rayleigh number, increasing the concentration of nanoparticles leads to an increase in the nanofluid viscosity and the average fluid temperature, which respectively leads to an increase and decrease in the frictional entropy generation rate.
However, the effect of increasing the viscosity is higher and therefore the frictional entropy generation rate increases with increasing the concentration of nanoparticles. The contours of the thermal entropy generation rate (left) and frictional entropy generation rate Frictional entropy generation rate Thermal entropy generation rate  concentrations, the thermal entropy generation rate is much higher than the frictional entropy generation rate. Hence, the trend of changes in the total entropy generation rate is similar to the thermal entropy generation rate. According to the results presented in Fig. 13, it can be concluded that using water-Fe3O4/CNT hybrid nanofluid in natural convection flow inside a concentric horizontal annulus is not advantageous from the second-law point of view. Hence, the use of water-Fe3O4/CNT nanofluids in natural convection flow in a concentric horizontal annulus depends on whether the designer's goal is to use the nanofluid to increase heat transfer or reduce entropy generation or both. number on the average Nusselt number as well as the thermal, frictional and total entropy generation rates were investigated. Besides, the effect of these parameters on the streamlines, isotherms and contours of the thermal and frictional entropy generation rates were examined.
The results showed that the average Nusselt number increases by increasing the Rayleigh number, Fe3O4 concentration and CNT concentration. Furthermore, it was found that the rate of thermal, frictional and total entropy generation augments with an increase in the Fe3O4 and CNT concentrations as well as Rayleigh number. However, the rate of thermal entropy generation was much higher than that for the frictional entropy generation. The results of this paper could provide a guideline on the application of newly generated nanofluids in natural convection flows inside the annulus according to both first and second laws of thermodynamics.       CNT concentration.