Natural Convection in an Annular Enclosure: Influence of Magnetic Field-dependent Thermal Conductivity on Heat Transfer

In this paper, the natural convection heat transfer characteristics are investigated inside an annular enclosure containing Magnetic Nanofluid (MNF) (i.e., Fe3O4 nanoparticles are dispersed in Kerosene). A uniform magnetic field (H) is applied along the axial direction of the enclosure. Magnetic field-dependant thermal conductivity (k) of the MNF is considered as a nonlinear function which is interpolated from the experimental results. Finite element method is utilized to solve the governing equations for various magnetic field strengths, volume fractions of MNF, and Rayleigh numbers. Average Nusselt numbers along the hot wall are calculated and compared for different scenarios. The results show that the applied magnetic field has a significant effect on the heat transfer rate, more specifically on the Nusselt number, in the enclosure for higher volume fractions of nanoparticles. Thermal conductivity enhancement as a result of using magnetic field can be used for various applications such as thermal energy storage in which the heat transfer needs to be accurately controlled.


Introduction
Heat transfer enhancement has been a challenging topic for researchers for many years to enhance the efficiency of many applications such as cooling and electronic equipment and solar systems.Numerous studies have been carried out on passive and active heat transfer systems to increase their efficiencies [1].One of the effective ways to improve the heat transfer rate in fluids is adding conductive nanoparticle to the fluid [2].
Adding nanoparticles to the base fluid can change the thermophysical properties of nanofluids.To have more control on the properties of nanofluids, magnetic nanofluids (MNF), which consist of magnetic nanoparticles in a base fluid, are introduced.The application of magnetic fields caused nanoparticles to aggregate in the carrier fluid, creating highly conductive paths for the heat flow inside the convective heat transfer environment.Therefore, the thermal conductivity of the nanofluid enhances.Several studies reported that having nanoparticles with higher thermal conductivity can enhance the heat transfer rate [3,4].However, in another study when Fe nanoparticles and Cu nanoparticles were examined and compared, it was indicated that adding conductive nanoparticle is not always the most effective way to improve the conductivity [5].
The main advantage of having magnetic nanoparticles is the capability to control thermophysical properties using external magnetic fields [6].The use of MNF creates opportunities in different applications such as electrical application, thermal engineering systems, heat exchanger designs, thermal energy storage systems, and thermal batteries or medical applications such as Magnetic Resonance Imaging (MRI) enhancement and bioseparation for targeted cancer treatment [7][8][9][10].Magnetic nanoparticles are suspended in the carrier/base fluid and subjected to the gravitational and magnetic fields; thus, the particle sedimentation is a factor that has to be taken into account [11].The percentage of nanoparticles inside the carrier fluid is very important.Moreover, the interaction forces between nanoparticles, which cause Brownian motion, directly depend on size of the nanoparticle.
Tynjala and Ritvanen [12] studied the thermomagnetic convection of MNF in annulus enclosure.The effect of gravity, which causes the buoyancy force, and magnetic field were numerically investigated.The results showed that the average Nusselt number increased when Rayleigh number increased [12].Experimental and theoretical studies were conducted on thermal conductivity and viscosity by Murshed et.al., [13].In [13], it was found that the thermal conductivity strongly depends on the temperature.They also proposed the model to accurately estimate the thermal conductivity with only 2% error [13].In an experimental study by Pastoriza-Gallego et.al. [14], the volumetric behaviours of iron oxide nanofluids and enhancement of thermal conductivity were investigated.It was reported that the enhanced thermal conductivity is a function of volume fraction and does not depend on temperature (up to 1% volume fraction of the nanoparticle) [14].Ashorynejad et.al., numerically investigated the effects of magnetic field on a cylindrical annulus filled with Agwater nanofluid when a radial magnetic field was used parallel to temperature gradient [15].It was reported that the average Nusselt number increased when volume fraction/Rayleigh number increased; however, it was a decreasing function of Hartmann number [15].Twophase simulation of nanofluid in an annulus subjected to the axial magnetic field was studied by Sheikholeslami and Abelman [16].A reduced model (ODE) of the geometry in their study was solved numerically and the results showed a direct relationship between Nusselt number and Hartmann number, but reverse relationship with Reynolds number and Brownian parameter [16].In much of literature, the applied magnetic field was assumed constant and uniform.However, in the study conducted by Sheikholeslami et.al., forced convection heat transfer in a semi-annulus geometry was studied when a variable magnetic field was used [17].In [17], Fe3O4-water was used as MNF and the thermal conductivity was estimated using Maxwell-Garnett (MG) model.Results showed that for high Rayleigh numbers, the effects of Kelvin body forces are notable [17].
Based on the literature on MNF, thermal conductivity is an important parameter to improve the heat transfer rate.To the best knowledge of the authors, no numerical study is conducted in which thermal conductivity was a function of both magnetic field and volume fraction  nf (, ).Thus, as a continuation of our previous work [18], the motivation of this study was to investigate the development of heat transfer inside the annular enclosure filled with MNF in which the thermal conductivity of MNF is a nonlinear function of both magnetic field and volume fraction.
In this study, we proposed the annular enclosure filled with kerosene-Fe3O4 when the thermal conductivity was a nonlinear function of both magnetic field and volume fraction.The proposed enclosure is subjected to the uniform magnetic field.Natural convection of MNF inside the enclosure for various Rayleigh numbers was studied.
The rest of the paper is organized as follows: Section 2 defines the problem, the geometry of enclosure, and assumptions.Section 3 provides governing equations and boundary conditions.Section 4 describes the numerical procedure in details to solve the governing equations followed by Section 5 that reports the results and discusses the findings in this study.Finally, Section 6 concludes the paper.

Geometry and Properties
Figure 1 shows the enclosure filled with MNF (kerosene and Fe3O4).The material of the enclosure is acrylic glass and the inner pipe is copper.The bottom and top of the annulus are adiabatic.The properties of pure kerosene and Fe3O4 are provided in Table .1The thermophysical properties of MNF, except thermal conductivity ( nf ), are calculated using classical models as follows [20][21][22]: 2.5  (1) Note that in Eq. 4,  nf is a function of magnetic field and volume fraction of nanoparticles and obtained experimentally by [23].

Mathematical Modelling and Boundary Conditions
In this study, it is assumed that the flow is laminar and steady, and the MNF is incompressible.In addition, the effect of Brownian motion is neglected.The cylindrical coordinates system of (, , ) that represents the velocity components of (, , ) are illustrated in Figure 1.To simulate the heat transfer through the magnetic nanofluid and by assuming Boussinesq approximation, the conservation of mass, momentums, and energy equations are as follows: ρ nf (.∇) = −∇ p + µ nf ∇ 2  + ρ nf  +  (6) The Electric potential and Lorentz force which depend on the electrical conductivity of the MNF and velocity of nanoparticles, are calculated as follows: where  is the magnetic field intensity (A/m) and B is the magnetic flux density ().It is assumed that the magnetic field is uniform and azimuthally symmetric only along .In this case, we only have  in  direction as the magnetic field is applied perpendicular to the temperature gradient.In our simulation, we assumed no slip-velocity condition on the walls.The hot surface is the copper pipe where the hot fluid passes through and the cold surface is the acrylic glass.Having steady single phase flow inside the enclosure and thermal insulations on the top and the bottom of the enclosure, we have the following boundary condition: where L is the height of the enclosure (3.5 in) and the thickness of the MNF in a radial direction ( out −  in ) is 22.24 mm .Using polynomial curve fitting method, we obtained the thermal conductivity as a function of both volume fraction () and magnetic field () from the literature [24].In our numerical study, we chose five different volume fractions (1.115%, 1.7%, 2.23%, 3.465%, and 4.7%).Also, the reference velocity and thermal thickness are expressed as follows [25]: where   is the thermal thickness and  nf is a thermal diffusivity of MNF.Moreover, the stream function  is defined as  = = 0 for initial condition.Note that the formulation in Eq. 11 satisfies the continuity where . ⃗ =0.To compare and validate our simulation results with previous studies in which Ra versus Nu is provided (no magnetic field and no nanoparticles), we defined the Rayleigh number as Ra =  3   and the local Nusselt number on the hot surface is defined as follows: where  is effective length, ℎ is the heat transfer coefficient, and  n is normal heat flux from the wall.Finally, by substituting Eq. 13 into Eq.12, we calculated the average Nusselt number along the hot wall as follows: where  =  ℎ −   ,  f is thermal conductivity of fluid obtained from Table .1,and  nf (, ) is a polynomial function of both magnetic field and volume fraction of nanoparticles.

Numerical Procedure and Grid Testing
In this study, we used finite element method (FEM) to solve the governing equations.We used direct solver method for three fully coupled modules (heat transfer in solid, fluid and laminar flow) and solved numerically with the commercial COMSOL Multiphysics software (4.3b).The convergence was set to be 10 −3 for termination criteria, and linear approximation was assumed within the elements.In order to validate the numerical scheme and accuracy of the simulation, the Nusselt number is calculated when the Prandtl number (Pr) is 0.7.Our simulation results showed the good agreement when they were validated against the previous literature in Figure .2. To ensure the grid independency in our simulation results, we used different element sizes (various grid in z and r directions) and computed the average Nusselt number.Figure 3 illustrates the average Nusselt number along the hot wall versus total number of elements.Based on our results in Figure 3, the total number of 22275 (75 × 297) provided the sufficient accuracy and beyond this point, the results did not change significantly.Thus, we used this number of elements to compute our results in this study.

Results and Discussion
Effects of the uniform transverse magnetic field on heat transfer distribution inside the annular enclosure filled with kerosene-Fe3O4 is investigated.Figure 4-Figure 6 show the effect of magnetic field on the average Nusselt number when Rayleigh is changing from 10 4 to 10 7 for various volume fractions (1.115%, 1.7%, 2.23%, 3.465% , and 4.7%), and magnetic fields (0-80×103 [A/m]).As it can be seen from Figure 4 -Figure 6, the Nu is an increasing function of volume fraction.Also, increasing the magnetic field with higher volume fractions result in increasing the Nusselt number.However, a slight decrease is observed for high volume fractions at/about 70×10 3 [A/m] which can be explained by the atomic motion of the particles and induction heating which was occurred inside the enclosure exposed to the external magnetic field.The main contribution of this study is depicted in Figure 7 and Figure 8 for two specific volume fractions (φ=1.115% and φ=4.7%. Figure 7 shows the relative Nusselt number which is calculated as follow: where Nu0 is the average Nusselt number without any magnetic field ( = 0).Both figures show that the average Nusselt number increases with increasing Rayleigh number.The variation of Nu in the presence of the magnetic field is notable for magnetic nanofluid with a high volume fraction of nanoparticle.Due to the fact that the magnetic field affects the thermal conductivity; this effect became noticeable for the high volume fraction of MNF.Note that the MNF can be saturated due to the high applied magnetic field.This phenomenon can be seen easily in Figure 7 where the highest relative Nu obtained when the applied magnetic field is 60×10 3 [A/m].In Figure 10, the variation of the dimensionless velocity inside the enclosure is depicted along the radial distance at the centre of the enclosure.It can be observed that the variation of velocity for higher Rayleigh numbers is more pronounced.The applied magnetic field affected the velocity and consequently convective heat transfer inside the enclosure.Since this effect is very small, it cannot be visualized.Thus, the average Nusselt numbers at the hot wall for different Ra and H are tabulated in Table 2.According to Table 2, the average Nusselt number is an increasing function of the magnetic field.Note that the Nusselt number slightly dropped at/beyond 90×10 3 [A/m]; this decrease in Nu can be explained by the deformation of chain-like structures of nanoparticles inside the enclosure in the presence of the high-intensity magnetic fields.Another reason for the decrease in Nu beyond 90×10 3 [A/m] is the application of strong magnetic fields which can internally heat up the MNF (Joule heating) and has an impact on the average Nusselt number.Figure 11 shows the comparison between streamlines for different Rayleigh numbers and volume fraction of nanoparticles when magnetic field varies.Although the streamline patterns stayed the same for each Rayleigh number, the stream function values decreased in the presence of magnetic field (Table .3).Note that the changes of stream function values in low Ra are small; this decrease is quite notable in high Rayleigh numbers.Table 3 shows the variation of stream function () for various Ra and magnetic fields.From Table 3 and Figure 11, it can be concluded that strong magnetic field affected the Buoyancy driven flow and significantly decreased the velocity of the flow.
Figure 12 illustrates the isotherms for Ra=103, Ra=104, Ra=105, Ra=106, and Ra=107.The results show the conduction and convection regimes for low and high values of Ra, respectively.From Figure 12, it can be concluded that for small values of Rayleigh the conduction like flow was dominant, while for higher values of Rayleigh the convection like regime was observed.Figure12.Isotherms for different Ra and  Kerosene with Fe3O4 =3.465%(H=70×10 3 ).

Conclusion
In this paper, we investigated the effect of the transverse magnetic field on the annular enclosure filled with magnetic nanofluid (kerosene-Fe3O4).Nonlinear relationships between thermal conductivity and magnetic field were obtained and used as polynomial functions for MNF property.Variation of the average Nusselt number along the hot wall was studied for different magnetic field intensities.Heat transfer rate enhanced significantly when the thermal conductivity was a function of magnetic field and volume fraction instead of a constant value.The results show the increase of magnetic fields strength and Rayleigh number both lead to increase the Nusselt number, significantly.

Figure 1 .
Figure 1.Axisymmetric view along the z direction and mesh distribution of the enclosure filled with kerosene-Fe3O4.
, the stream function which satisfies the governing equations for the annular geometry is as follows:

Figure 9
Figure9illustrates the temperature distribution for different Rayleigh numbers when φ = 4.7% and H=80×10 3 [A/m].In Figure9(a), when Ra is 10 2 , the uniform temperature distribution is observed along the radius of the enclosure while in (b), where Ra=10 7 , the effect of buoyancy force on natural convection inside the enclosure is dominant.