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Latin American applied research

Print version ISSN 0327-0793On-line version ISSN 1851-8796

Lat. Am. appl. res. vol.43 no.1 Bahía Blanca Jan. 2013

 

Numerical study of MHD natural convection in an inclined rectangular cavity with internal heat generation filled with a porous medium under the influence of joule heating

S.E. Ahmed

Department of Mathematics, Faculty of Sciences, South Valley University, Qena, Egypt. E-mail: sameh_sci_math@yahoo.com

Abstract— Using an implicit finite difference method, the effect of Joule heating on the magnetohydrodynamic natural convection in an inclined rectangular enclosure with internal heat generation has been performed. The left and right walls of the enclosure are maintained at a constant temperature while the bottom and top walls are kept insulated. The values of the governing parameters are the Hartmann number, Ha=0, 5, 10 and 50, Joule heating parameter J=0, 0.005, 0.01 and 0.02, the aspect ratio a=0.01, 0.2, 0.5 and 1, the inclination angle φ=0, π/6 and π/3 and Rayleigh number Ra=102 and 103. Comparisons with previously published work are performed and excellent agreement is obtained. A parametric study of the physical parameters is conducted and a representative set of numerical results for the streamlines and isotherms as well as the local Nusselt number is illustrated graphically to show interesting features of the solutions. It is found that, the local Nusselt number increases on the bottom wall as the enclosure aspect ratio increases.

Keywords— Internal and Joule Heating; MHD Natural Convection; Heat Generation; Numerical Simulation; Inclined Cavity; Porous Medium.

I. INTRODUCTION

The study of natural convection in fluid saturated porous media has several applications in engineering and nature. Such applications include exothermic reactions in packed bed reactors, heat transfer associated with the deep storage of nuclear waste, flow past heat exchanger tubes and to study the effect of the metabolic heat generation in grains causing hot spots that induce fungal growth (Nield and Bejan, 2006; Ingham and Pop, 2005; Vafai, 2005; Bejan et al., 2004; Pop and Ingham 2001; Prasad, 1987; Jimenez et al., 1999; Nithiarasu et al., 2000). There are many important effects should not be neglected due to investigating heat transfer inside the enclosures filled with porous media. Such effects include magnetic force effect, Joule heating effect and viscous dissipation effect. Recently, the equally important problem of hydromagnetic convective flow of a conducting fluid through a porous medium has been investigated. In fact, when an electrically conducting fluid is subjected to a magnetic field, the fluid motion induces an electric current and, in general, the fluid velocity is reduced due to the induced Lorentz force. There are many open literature related to natural convection with magnetic force. Garandet et al. (1992) investigated buoyancy-driven convection in a rectangular enclosure with a transverse magnetic field. Hydromagnetic free convection flow through a porous medium between two parallel plates was investigated by Raptis et al. (1982a). Raptis and Vlahos (1982) further extended their investigation to study the free convection flow of a conducting fluid through a porous medium bounded by two horizontal plates. Singh and Dikshit (1987) studied the free convection of the Couette motion of an electrically conducting fluid through a porous medium. Exact solutions for the velocity field, skin-friction, and the volume flux of the fluid were obtained in terms of the governing parameters of the problem. Haajizadeh et al. (1984) investigated natural convection in a vertical porous enclosure with internal heat generation. Grosan et al. (2009) extended the Haajizadeh et al. (1984) problem by considering the effect of an inclined magnetic field.

In this study, the effect of Joule heating on MHD natural convection in an inclined rectangular cavity filled with a porous medium is considered. These types of the problems are well known natural phenomenon and have attracted interest of many researchers due to its many practical situations. Among these insulation materials, the electronic packages and microelectronic devices. The problem of steady conjugate heat transfer through an electrically-conducting fluid for a vertical flat plate in the presence of transverse uniform magnetic field taking into account the effects of viscous dissipation, Joule heating, and heat generation was formulated by Azim et al. (2010). Zhao and Yang (2011) presented an analysis of Joule heating induced heat transfer for electroosmotic flow (EOF) of power-law fluids in a microcapillary. A comprehensive investigation on hydrodynamic and thermal transport properties of mixed electroosmotically and pressure driven flow in microtubes was presented by Yavari et al. (2012). Rahman et al. (2010) studied the conjugate effect of Joule heating and magnetic force, acting normal to the left vertical wall of an obstructed lid-driven cavity saturated with an electrically conducting fluid.

II. MATHEMATICAL FORMULATION

Consider steady laminar natural convection in an inclined rectangular cavity with an electrically conducting fluid saturated porous medium. It is assumed that, the height of the cavity is denoted by h, the width is denoted by l and the inclination angle by φ. The left and right walls are maintained at a constant temperature T0, while the top and bottom walls are insulated. The enclosure is permeated by a uniform magnetic field. A uniform source of heat generation with constant volumetric rate is placed in the flow region. The Joule heating effect is considered while the viscous and radiation effects are neglected. The effect of buoyancy through the well-known Boussinesq approximation and the Darcy approximation are considered in the current investigation. It is, also, assumed that the properties of the fluid and the porous medium are isotropic and homogeneous everywhere and the local thermal equilibrium between the fluid and medium is applied everywhere inside the enclosure.

The geometric and the Cartesian coordinate system are schematically shown in Fig. 1. Under the above assumptions, the continuity, momentum and energy equations are given by:

(1)
(2)
(3)
(4)


Fig.1. Physical model and coordinate system.

Subject to the boundary conditions:

(5a)
(5b)

where u and v are the velocity components, T is the fluid temperature, k is permeability of the porous medium, μ is the dynamic viscosity, g is the gravitational accelaration, v is the kinematic viscosity, β is the coefficient of thermal expansion, ρ is the density, σ is the electrical conductivity, α is the effective thermal diffusivity and cp is the specific heat at constant pressure.

Eliminating the pressure term in Eqs. (2) and (3), we obtain the following equation:

(6)

Introducing the following dimensionless variables:

(7)

Also, the stream function can be defined in the usual way as:

(8)

Using Eqs. (7) and (8) in Eqs. (4) and (6), the following dimensionless equations are obtained

(9)
(10)

Further, the dimensionless form of the boundary conditions is written as:

(11a)
(11b)

In the above equations, a=l/h is the aspect ratio of the enclosure, is the Rayleigh number, is the Hartmann number and is the Joule heating parameter.

The rate of heat transfer in terms of the local Nusselt number along the vertical walls is defined as:

(12)

III. NUMERICAL PROCEDURE

Finite difference method was adopted to solve numerically the governing Eqs. (9) and (10) together with the boundary conditions in Eq. (11). The diffusion terms within Eq. (10) was replaced by the second order central differencing schemes, while the second order upwind differencing scheme was chosen to approximate the convective term in order to make the numerical procedure stable. The solution for the corresponding linear algebraic equations was obtained through the tridiagonal method algorithm (TDMA). The numerical method was implemented in a FORTRAN software. The unknowns θ and ψ were calculated iteratively until the following criteria of convergence was fulfilled:

(13)

where χ is the general dependent variable. Table 1 shows an accuracy tests using the finite difference method using five sets of grid: 31×31, 41×41, 61×61, 81×81 and 101×101 at Ra=103, Ha=2, J=0.01, a=1 and φ=π/6. These tests are clearly presented in Table 1. It is found that, there is a good agreement was found between grids 41×41 and 61×61, so the numerical computations were carried out for these grids. Further, in order to verify the accuracy of the code, the obtained results are compared with those obtained by Haajizadeh et al. (1984) and Grosan et al. (2009) for Ha=0, J=0 and a=0.5. These results are shown in Table 2. It can be concluded that the results in a very good agreement.

Table 1. Results of grid independence examination at Ra=103, Ha=2, J=0.01, a=1 and π=π/6.

Table 2. Comparison of ψmax and θmax for Ha=0, J=0 and a=0.5

IV. RESULTS AND DISCUSSION

In this section, numerical results for MHD natural convection in an inclined rectangular cavity filled with a porous media with internal heat generation under the influence of Joule heating are presented and discussed. A primary attention is given to the effects of four parameters for this investigation. These are Joule heating parameters J, Hartmann number Ha, the aspect ratio a and the inclination angle φ.

Figure 2 illustrates the contours of streamlines and isotherms for different values of Joule heating parameter (J=0, 0.005, 0.02) with Ra=103, a=1, φ=π/6 and Ha=2. The non-symmetrical clockwise and anticlockwise cells are formed within the enclosure. The non-symmetrical behaviors of the streamlines can be attributed to the cavity inclination (φ=π/6). In the absence of Joule heating effect (J=0), a weak convective motion with ψmin=-1.642084 and ψmax=0.8785512 is seen inside the cavity. As J increases, the fluid temperature increases and the maximum temperature records the values θmax=0.1412717 and θmax=0.203763 at J=0.005 and J=0.01, respectively, which in turn increases the fluid motion as well. In fact, the Joule heating parameter having magnetic field strength increases the temperature and eventually the fluid motion is accelerated inside the cavity. Also, this temperature profiles causes an increase in thermal boundary layer thickness. As results, the temperature gradients at the left wall decrease, which in turn, decreases the local Nusselt number (Fig. 3).


Fig. 2. Streamlines (left) and isotherms (right) for J=0, 0.005, 0.02 at Ra=103, a=1, φ=π/6 and Ha=2. Increasing from top towards bottom.


Fig. 3.Profiles of local Nusselt number at the left wall at Ra= 103, a=1, φ=π/6 and Ha=2.

In Figs. 4 and 5 the effect of presence of magnetic force (Ha= 0,5,10,50) inside the flow region on the streamlines, isotherms and Nusselt number profiles at Ra=102, a=1, φ=π/6 and J=0.01 are presented. It is seen from this figure that the intensity of the convection in the core is considerable affected by the magnetic force. The increased magnetic field acting transversely to the cavity retards the fluid motion. Also, the interaction between the magnetic field and the fluid motion decreases the temperature of the fluid within the enclosure. These can be easily observed from the maximum and the minimum values of stream function and the maximum temperature which records the data ψmin = -0.7270522, ψmax= 0.543026, and θmax=0.1404365 at Ha=0 and ψmin= -0.000277, ψmax=0.000277 and θmax= 0.1250044 at Ha= 50. In addition, the non-symmetric behaviors of stream function tends to be symmetric at the large value of Hartmann number. Further, when the magnetic force is considered (Ha=5,10, 50), the core vortex is elongated vertically and the isotherms are seen almost parallel. On the other hand, Figure 5 shows that, an increase in Ha leads to increase the heat transfer rate.


Fig. 4. Streamlines (left) and isotherms (right) for Ha=0, 5, 10, 50 at Ra=102, a=1, φ=π/6 and J=0.01. Increasing from top towards bottom.


Fig. 5. Profiles of local Nusselt number at the left wall at Ra=102, a=1, φ=π/6 and J=0.01.

Figure 6 shows plots of the streamlines and isotherms for different values of the cavity inclination angle (φ=0, π/6, π/3) at Ha=2, Ra=103, a=1 and J=0.01. The fluid flows as two clockwise and anticlockwise circular cells inside the enclosure. These cell are symmetrical with respect to the central plane for non-inclined cavity (φ=0) with ψmin =-1.6182, ψmax= 1.618. Titling the enclosure by φ=π/6, removes this symmetries and increases activity of the fluid motion in the right half of the cavity. However, increase the cavity inclination angle to the valueφ=π/3 tends to decrease the convective motion and the values of stream function change from ψmin = -1.902207, ψmax= 1.023106 at φ=π/6 to ψmin = -1.558822, ψmax=0.4190141 at φ=π/3. In addition, the isotherms pattern is changed remarkably due to different values of φ. As it can be observed, the isotherms lines follow the problem geometry and take an inclined paths within the cavity. Further, the maximum temperature decreases as φ increase. This temperature reduction leads to increase the temperature gradients, which in turn, increases the rate of heat transfer at the enclosure left wall (Fig.7).


Fig. 6. Streamlines (left) and isotherms (right) for φ=0, π/6, π/3 at Ha=2, Ra=103, a=1 and J=0.01. Increasing from top towards bottom.


Fig. 7. Profiles of local Nusselt number at the left wall at Ha=2, Ra=103, a=1 and J=0.01.

Figure 8 presents the variations of the local Nusselt number Nul with Y for Ha=2, Ra=103, φ=π/6, J=0.01 and for several values of the enclosure aspect ratio (a=0.01, 0.2, 0.5, 1). It is found that, at the bottom half of the left wall, the local Nusselt number increases as enclosure aspect ratio a increases. However, at the top half, the local Nusselt takes the opposite behaviors.


Fig. 8. Profiles of local Nusselt number at the left wall at Ha=2, Ra=103, φ=π/6 and J=0.01.

V. CONCLUSIONS

The present numerical study exhibits many interesting features concerning the effect of Joule heating on the MHD free convection flow and heat transfer characteristics in an inclined rectangular cavity filled with a porous medium. Detailed numerical results for the temperature distribution and heat transfer have been presented in graphical and tabular form. The main conclusions of the present analysis are as follows:

  • As the Joule heating parameter increases, both of the convective motion and the fluid temperature increase, whereas the local Nusselt number decreases.
  • The local Nusselt number at the left wall increases with increase the inclination angle and decreases with increase the aspect ratio.
  • As the magnetic filed parameter increases, both of the fluid motion and the maximum temperature decreases, whereas it increases the local Nusselt number.

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Received: March 23, 2012.
Accepted: May 18, 2012.
Recommended by Subject Editor Cecil Coutinho.

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