MHD natural convection in an inclined square porous cavity with a heat conducting solid block

https://doi.org/10.1016/j.jmmm.2016.11.112Get rights and content

Highlights

  • MHD natural convection in an tilted porous cavity with a centered solid block is analyzed.

  • The finite volume method with SIMPLE algorithm is used to solve the dimensionless governing equations.

  • Magnetic field dampens the buoyancy induced fluid motion and the heat transfer rates in the cavity.

  • The average Nusselt number is a decreasing function of Hartmann number and non-monotonic function of inclination angles.

Abstract

This paper deals with natural convection in an inclined porous cavity with a heat conducting solid body placed at its center under the influence of the applied magnetic field of different orientations. The left and right vertical walls of the cavity are maintained at different temperatures Th and Tc, respectively, while the horizontal walls are adiabatic. The governing coupled partial differential equations were solved using a finite volume method on a uniformly staggered grid system. The effects of the inclination angles of the magnetic field and cavity and the Hartmann number on the flow and thermal fields are investigated in detail. Numerical results are presented in terms of isotherms, streamlines and average Nusselt numbers. In general, the results indicate that the inclusion of the magnetic field reduces the convective heat transfer rate in the cavity. It is also found that an increase in the angle of the applied magnetic field produces a non-linear variation in the average Nusselt numbers.

Introduction

Convective flow and heat transfer inside cavities filled with an electrically-conducting fluid has many applications in technology and nature. For example, using the effect of magnetic field it is possible to control hydrodynamic behavior and heat transfer during the processes of crystal growth, smelting, solidification, float glass production, or magnetic fields are utilized for food cleaning in separators, in lubrication and drying technologies [1], [2].

MHD natural convection in different cavities has been investigated widely analytically, numerically and experimentally [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Thus, an effect of inclined uniform magnetic field on natural convection in a rectangular cavity has been analyzed numerically by Yu et al. [3] for low Prandtl number fluid (Pr=0.025). It has been found that the inclination angle of magnetic field and the cavity aspect ratio play an important role for fluid flow and heat transfer, namely, for large Hartmann number the flow structure is highly dependent on the magnetic field inclination angle. At the same time, variation of aspect ratio leads to different hydrodynamic effects. Benos et al. [4] have studied numerically and analytically natural convection within an internally heated horizontal shallow cavity under an influence of uniform vertical magnetic field. The authors have shown that the fluid is decelerated by the magnetic field leading to the dominance of heat conduction and reduction of the heat transfer rate. Selimefendigil and Oztop [5] have investigated numerically natural convection in a flexible sided triangular cavity filled with an electrically-conducting and heat generation fluid under the effect of inclined uniform magnetic field. They have shown that the average Nusselt number is an increasing function of external Rayleigh number and a decreasing function of internal Rayleigh and Hartmann numbers. Now there are many papers devoted to numerical analysis of MHD natural convection in cavities filled with nanofluids [7], [8], [9], [10], [11], [12], [13], [14], [15].

It is well known that a presence of solid blocks or solid walls inside the cavity can lead to modification of fluid flow and heat transfer [16], [17]. In the case of additional effect of uniform magnetic field on natural convection in a square cavity with an adiabatic body [18] it has been revealed that the effect of magnetic field on flow structure is more essential for high Prandtl number values and a presence of internal adiabatic body can lead to an intensification or damping of heat transfer that depends on the heat conductivity ratio. Nasrin [19] has investigated MHD mixed convection in a wavy lid-driven cavity with a heat-conducting square cylinder. The obtained results have shown that thermal conductivity ratio and inner square cylinder location affect essentially the velocity and temperature fields.

An influence of porous medium inside the cavity on fluid flow and heat transfer is widely spread in different engineering and natural applications including geothermal reservoirs, underground water flow, granary, heating and drying processes, sterilization, etc [20], [21], [22]. MHD natural convection in a porous medium has been analyzed extensively. Mansour et al. [23] have numerically studied transient MHD free convection in an inclined porous cavity under an effect of internal heating. The authors have found that it is possible to increase the temperature inside the cavity by increasing both the Lorentz force magnitude and an inclination angle of the magnetic field. Hussain [24] has numerically investigated double-diffusive MHD natural convection and entropy generation in an inclined porous cavity with wavy walls. It has been shown that the cavity inclination angle has an essential influence on fluid structures and heat transfer inside the cavity. At the same time to enhance the flow circulation inside the cavity it is necessary to exclude the vertical position of the enclosure and horizontal magnetic field. Grosan et al. [25] and Revnic et al. [26] have analytically and numerically examined the effect of magnetic field and internal heat generation on transient free convection in a rectangular porous cavity. It has been demonstrated that the applied horizontal magnetic field is more effective in suppressing the convective flow in comparison with vertical orientation of uniform magnetic field and for all considered values of governing parameters the average Nusselt number reaches the steady case faster when magnetic field is parallel to the vertical walls of the cavity. Jiang et al. [27] have numerically studied free convection in a porous cavity under the effect of magnetic quadrupole field. For mathematical description of the porous medium the authors have utilized the Forchheimer–Brinkman–extended Darcy model. The authors have shown that magnetic field intensity, Darcy and Rayleigh numbers have an essential influence on the fluid flow and heat transfer within the porous enclosure. Nayak et al. [28] have numerically investigated using the Brinkman–extended Darcy model transient MHD natural convection in a porous cavity with two local heat sources of constant and periodic temperatures. It has been found the heat transfer enhancement for high values of Prandtl number when Da <1.0.

However these abovementioned papers did not take into account the effect of heat-conducting solid blocks on magnetohydrodynamic natural convection in an inclined porous enclosure. The main purpose of the present paper is a numerical simulation of natural convection in an inclined non-Darcy porous square cavity with a heat-conducting solid body under the effect of inclined uniform magnetic field. It is worth noting that a presence of heat-conducting solid blocks or solid walls of finite thickness and conductivity can essentially modify fluid flow and heat transfer [29], [30]. In the present study we have analyzed an interaction of conjugate natural convection in a porous cavity with uniform magnetic field and found an essential effect of magnetic field intensity and orientation on temperature and velocity fields. Based on author's knowledge and author's survey the considered work is the first attempt in this field.

Section snippets

Mathematical formulation

The physical model of natural convection in an inclined differentially heated square porous cavity under the effect of an inclined magnetic field and the coordinate system are schematically shown in Fig. 1. The domain of interest includes the fluid-saturated porous medium and a heat conducting solid block. The horizontal walls (y=0,y=L) are assumed to be adiabatic while vertical walls (x=0,x=L) are kept at constant temperatures Th and Tc, respectively where Tc<Th. It is assumed in the analysis

Numerical procedure

The governing Eqs. (8), (9), (10), (11), (12) with corresponding initial and boundary conditions (13) were discretized by the finite volume method on a uniform staggered grid system using the SIMPLE algorithm of Patankar [32]. The third order QUICK scheme of Hayase et al. [33] and the second order central difference scheme were, respectively, used for the convective and diffusive terms. In order to keep consistent accuracy over the entire computational domain, a third order accurate boundary

Results and discussion

A present numerical study has been carried out to investigate the MHD convective flow in an inclined porous cavity containing a heat conducting solid body placed at the center of the cavity. All results are computed for a fixed values of the Prandtl number (Pr =1.0), the Rayleigh number (Ra =105), the Darcy number (Da =10–2), the dimensionless solid body size (D =1/3), the thermal diffusivity ratio α*=1 and the thermal conductivity ratio λ*=1. The effect of the pertinent parameters such as:

Conclusions

Numerical study of MHD natural convection in an inclined porous cavity with a heat conducting solid body has been performed under the influence of different orientations of the applied magnetic field. Governing equations formulated in dimensionless primitive variables with appropriate boundary conditions have been solved by finite volume method. In the present study for the first time we have analyzed an interaction between conjugate natural convection in a non-Darcy porous medium and external

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