Galerkin finite element analysis of thermal aspects of Fe3O4-MWCNT/water hybrid nanofluid filled in wavy enclosure with uniform magnetic field effect

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Abstract

In this contribution, a numerical investigation was made for heat transfer and steady magneto-hydrodynamic natural convection in a fined cold wavy-walled porous enclosure with a hot elliptic inner cylinder occupied by hybrid Fe3O4-MWCNT /water nanofluid. The enclosure is also placed in a uniform magnetic field. The governing equations are verified by using the Galerkin Finite Element Method (GFEM). Our numerical conclusions are expressed in terms of distributions of isotherms, Nusselt number, and streamlines, which are important, control parameters for the heat convection, and flow in the enclosure. The findings elucidate that the Nu avg rises with the growth of Ra, porosity ratio, and Da ratio increase, though it reduces with the growth of Hartmann number.

Introduction

The mastery of heat transfer is always sought out by researchers and engineers on the grounds of its extensive employment in industrial systems [[1], [2], [3], [4], [5], [6]]. Knowing that the heat transfer improvement affected by several factors and the best convenient approach is to customize the thermal properties of the working fluid in a system for instance using nanofluid first established by Choi [7] which is fluid obtained by suspending a small dose of nanoparticles usually metal ones with a thermal conductivity with high-level in reference liquids which have weak thermal conductivity (ethylene glycol, H2O or oil).

Correspondingly, natural convection has been significantly studied in nanofluids within a cavity. The free convection was studied by Ghasemi [8] in a quadrilateral enclosure loaded with a H2O-Al2O3 nanoliquid likewise Jahanshahi et al. [9] simulated the free convection using H2O-SiO2 nanoliquid the properties of this nanofluid were measured using an experimental setup. The results demonstrated that with the growth of the Ra, the Nuavg increases. However, it decreases whenever the Ha increases. Additionally, Ghasemi found that on account of the value of Ra and Ha numbers, the growth in the nanoparticles concentration may stimulate improvement or depreciation of the heat transfer rate. Other researchers added different shaped obstacles in the square cavity like Selimefendigil et al. [10] added square, diamond, and circular obstacles to study the impact of these obstacle forms on the flow behavior and the thermal performance. They observed deterioration in the heat transport rate when the obstacles are added at higher Ra. However, in the presence of these obstacles, it was observed that there was less deterioration in the mean Nusselt number when we raised the Hartmann number.

Sheikholeslami [11], Hussain [12], and Bararnia [13] performed studies where the added obstacles (rectangular; elliptic; triangular; circular; rhombic) in the square cavity are defined as heat sources as for the results they depicted that the Nuavg has a direct relation with nanoparticle concentration, Ra number and for the consequence of the shape of the obstacles it is determined that the best heat transfer is associated with using the triangular cylinder shape, other parameters have been investigated like the position of the inner elliptic heat source in Bararnia et al. [13] study where they found noteworthy effects on the stream function, isotherm lines and eddies formation in Sheikholeslami et al. [11] noted the decrease in entropy generation when the height to width ratio of the rectangular is increased. Mihoubi [14] numerically examined the laminar free convection in the region between two coaxial cylindrical enclosures filled with a nanoliquid. Sheikholeslami [15] investigate the influence of Rayleigh number on the thermal conductivity of a nanofluid and its viscosity in a cavity with a tilted elliptic inner cylinder and the impact of the nanoparticles concentration on the flow attributes and thermal performance. They have discovered that the Nusselt growth is associated with augmenting the nanoparticles concentration and Rayleigh number. Recently a numerical simulation has been adopted by Selimefendigil et al. [16] using the FVM to investigate the entropy and free convection within a slanted cavity loaded with nanofluid. With the same method, Mebarek-Oudina [17] investigated the effect of heat transport of the Titania nanofluid in a cylindrical annulus. Ashorynejad et al. [18] examined the thermal behavior in an open undulations enclosure filled with (Al2O3-Cu) nanoliquid. The natural convection of a nanoliquid was examined by Dogonchi [19] in a wavy enclosure by implementing the finite element method with controlled volume and taking into account the effect of Brownian motion. In these latter studies, the nanofluid is subjected to the magnetic field so the researcher applied magneto-hydrodynamic in their studies. The findings revealed that the rate of heat transport is directly correlated with the Ra and also it is disproportionate to the intensity of the Lorentz force that we observed a significant decline in the middle value of the Nusselt number whenever the Hartman number increased. For further references, the reader can refer to [[20], [21], [22], [23], [24], [25], [26], [27], [28], [29]].

We have noted also that cavities filled with porous media have captivated the interest of many scientists because of the several functional uses. Raza et al. [30] examined the natural convection inside a permeable enclosure loaded with nanoliquid. They used Darcy–Boussinesq approximation and the 2nd order finite difference approach to solve the principal equations numerically Sheikholeslami et al. [31] analyzed the thermal performance of a CuO-H2O nanoliquid in a permeable enclosure, they took into consideration Darcy law, the Brownian motion influence, and the consequence of Lorentz forces on the nanoliquid behavior. Sun et al. [32] inspected the natural convection conduct of nanofluids numerically within a porous medium saturated cavity with a right-angle triangular shape. Furthermore, the investigations were performed with three types of nanofluids. As a consequence, the results produced by these studies pointed out that the optimum value of Nuavg is attained by reducing the Ha increasing φ, Ra, and the porosity of the permeable media. For the mixed convection, Rashed et al. [33] and Karbasifar et al. [34] examined Al2O3 nanoliquid inside a lid-driven cavity incorporating a centered elliptical heater. Dogonchi et al. [35] used an inclined elliptical heater and added a magnetic field to investigated its influence. The obtained results showed a rise of thermal conductivity and convection coefficients due to the claustration phenomena when they raised the solid volume also the thermal performance rise due to augmentation in the temperature gap (ΔT) between the hot and cold wall led to improvement in the heat transfer rate and Nusselt number furthermore Bijan's results revealed that growing Richardson number decreases the nanofluid velocity causing a reduction in the Nusselt number Dogonchi's study outcomes showed that at steady Ra the ascendant of Ha brings about degradation in the average Nusselt number. For further details, the readers are recommended to study [[36], [37], [38], [39], [40], [41], [42]].

As seen, our present investigation is the first effort to examine the natural convection among an enclosure with cold corrugated walls incorporating a hot and centered elliptical cylinder and partitioned into two layers. In the first layer, the nanofluid is freely flowing and in the second one, it saturates a porous medium using the GFEM method for different dimensionless parameters, such as Ra (103 ≥ Ra ≥ 106), Ha (0 ≥ Ha ≥ 100), Da (10−5 ≥ Da ≥ 0.1) and several undulations in the sidewalls of the cavity (1 ≥ N ≥ 9), and nanoparticle concentration (0 ≥ φ ≥0.05).

Section snippets

Problem description

In the present work, we have studied the MHD free convection inside a quadrilateral enclosure that has two wavy cold sidewalls and a heated ellipse-shaped hole in the center the enclosure was divided into two equaled layers the first is porous media saturated with the hybrid nanofluid Fe3O4- MWCNT /water the rest of the cavity is filled with the same hybrid nanoliquid. The enclosure is illustrated in Fig. 1. The hybrid nanofluid flow is presumed to be laminar, incompressible and an external

Results and discussion

In the current investigation, a numerical approach has been undertaken to review the impact of different controlling parameters the first is the geometrical parameter the undulation number (N = 1, 3, 5, and 9) the others are fluid and heat parameters, for instance, Ra (106 ≥ Ra ≥ 103), Da (10−2 ≥ Da ≥ 10−5) and Ha (100 ≥ Ha ≥ 0).

Fig. 3, illustrates the impacts of the number of undulations on the isotherms and streamlines at Ra = 105 and Ha = 0. It could be observed that when N = 1 there are two

Conclusion

The impact of applying an external magnetic field on heat convection flow of hybrid Fe3O4- MWCNT/water nanoliquid within the cavity with cold corrugated walls incorporating a hot and centered elliptical cylinder and partitioned into two layers is investigated. Along with the impact of various associated parameters such as the Ra, Da, and strong concentration on the isotherms, streamlines. The main findings from the simulation are as follows:

  • 1.

    The intensity of the nanofluid flow circulation

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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