Hydromagnetic Natural Convection Flow in a Fluid-Saturated Porous Medium with Nonuniform Heat Generation

Abstract

Convective flows can develop naturally within the porous materials if they are subject to external heating with/without internal heat source. In this work, the heat is assumed to be generated internally within the porous enclosure at a rate proportional to a power of the temperature difference. This relation is an approximation of the state of some exothermic chemical reaction. An inclined magnetic field is imposed to the porous enclosure. Darcy, model is used to formulate the porous layer and finite difference method is applied to solve the governing equations. The obtained results indicate that strong internal heating can generate significant maximum fluid temperatures above the heated wall temperature, and location of the maximum fluid temperature moves towards the center of the top wall by strengthening the magnetic field. Local heating exponent does not have a major effect on the flow and temperature distributions as well as the heat transfer performance within the porous medium. The large magnetic fields, regardless of direction, are effective to suppress the convective flows and reduce the rate of heat transfer.

1. Introduction

A porous medium means a material consisting of a solid matrix with an interconnected void [1]. The interconnectedness of the void (the pores) allows the flow of one or more fluids through the material. Examples of natural porous media are beach sand, sandstone, limestone, rye bread, wood, rocks, and soil. Fluid which flows in porous media has occupied the central stage in many fundamental heat transfer analysis and has received considerable attention over the last few decades. This interest is due to its wide range of applications, for example, high-performance insulation for buildings, chemical catalytic reactors, packed sphere beds, grain storage, and geophysical problems as frost heave. Porous media are also of interest in relation to the underground spread of pollutants, solar power collectors, and to geothermal energy systems. The literature concerning convective flow in porous media is abundant and representative studies in this may be found in the recent books by Ingham and Pop [2], Pop and Ingham [3], Ingham et al. [4], Ingham and Pop [5], Vafai [6], de Lemos [7], and Vadasz [8].

Convective flows can develop naturally within the porous materials if they are subject to some form of external heating. In some situations this material provides its own source of heat such as spontaneous combustion in coal stockpiles, heat removal from nuclear fuel debris in nuclear reactors, underground disposal of radioactive waste materials, and exothermic chemical reactions in packed-bed reactors. Moreover, this phenomenon can be encountered during the storage of agricultural products where heat is generated as a result of metabolism of the products [9]. Uncontrolled the spontaneous combustion cause a destruction and financial losses [10] and [11]. Thus, understanding and controlling the convective flows are clearly important for prevention of fires in the coal stockpiles.

The study of the interaction of the geomagnetic field with the fluid, such as oil, gas, and water, in the geothermal regions arises in geophysics. We can determine the flow and temperature distribution by solving the Navier-Stokes and energy equations. Early studies on hydromagnetic natural convection flow through a porous medium are due to Raptis et al. [12] and Raptis and Vlahos [13]. The effect of the transverse magnetic field within an inclined porous tall enclosure were investigated numerically by Vasseur et al. [14] and Bian et al. [15]. They found that the temperature and the velocity fields are significantly modified with the application of the magnetic. Khanafer and Chamkha [16] studied a transverse magnetic field and fluid heat generation effects. Grosan et al. [17] and Saleh et al. [18] extended this to include an inclined magnetic field. They reported an optimum reducing of the heat transfer performance that was obtained for a large magnetic field in the horizontal direction. Recently, analytical solutions of the magnetohydrodynamic problem were obtained by Raftari and Yildirim [19] and Raftari et al. [20]. The present work investigates the effect of an inclined magnetic field on natural convection in an enclosure filled with a porous medium saturated with an electrically conducting fluid having nonuniform internal heat generation. The special purpose is to find the optimum parameters in controlling the convective flows within the enclosure.

2. Mathematical Formulation

We consider the steady, two-dimensional natural convection flow in a square region filled with an electrically conducting fluid-saturated porous medium (see Figure 1). The coordinate system employed is also depicted in this figure. The top and bottom surfaces of the convective region are assumed to be thermally insulated, the face $x = ℓ$ is held at the constant reference temperature T_c and the face $x = 0$ has a constant prescribed temperature T_h above T_c. Heat is assumed to be generated internally within the porous medium at a rate proportional to (T – T_c)^p (p  1), T is the local temperature. This relation, as explained by Mealey and Merkin [21], is an approximation of the state of some exothermic chemical reaction. Further, the angle of inclination of the magnetic field $B$ from the horizontal plane and measured positively in the counterclockwise direction is denoted by ϕ.

Figure 1:

Schematic representation of the model.

The viscous, radiation and Joule heating effects are neglected. The resulting convective flow is governed by the combined mechanism of buoyancy force, internal heat generation, and magnetic field. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected compared to the applied magnetic field. Under these assumptions, the conservation equations for mass, momentum, energy, and electric transfer are as follows:

\nabla \cdot V = 0,

(1)

V = \frac{K}{μ} (- \nabla P + ρ g + J \times B),

(2)

(V \cdot \nabla) T = α_{m} \nabla^{2} T + \frac{q_{0}^{′′′}}{ρ_{0} c_{p}} {(T - T_{c})}^{p},

(3)

\nabla \cdot J = 0,

(4)

J = σ (- \nabla φ + V \times B),

(5)

ρ = ρ_{0} [1 - β (T - T_{c})],

(6)

where $V$ is the fluid velocity, T is the fluid temperature, P is the pressure, $B$ is the external magnetic field, $J$ is the electric current, φ is the electric potential, $g$ is the gravitational acceleration, K is the permeability of the porous medium, α_m is the effective thermal diffusivity, ρ is the density, μ is the dynamic viscosity, β is the coefficient of thermal expansion, T_c is the reference temperature, $q_{0}^{′′′}$ is the local heat generation, p is the local-heating exponent, c_p is the specific heat at constant pressure, σ is the electrical conductivity, $ρ_{0}$ is the reference density, and $- \nabla φ$ is the associated electric field. As discussed by Garandet et al. [4, 5, 22] reduce to $\nabla^{2} φ = 0$ . The unique solution is $\nabla φ = 0$ since there is always an electrically insulating boundary around the enclosure. Thus, it follows that the electric field vanishes everywhere [23].

The above equations correspond to having the porous medium-modeled according to Darcy's law. Because of the complexity of pore geometries in a porous medium, Darcy's law has to be used to obtain any meaningful insights into the physics of flow in porous media. Darcy's law has been verified by the results of many experiments. Theoretical backing for it has been obtained in various ways, with the aid of either deterministic or statistical models [1]. Vafai [6] have described that Darcy's approximation is reasonable for the geothermal flows, except perhaps near boreholes [6]. Darcy's law is valid only when the pore Reynolds number is of the order of 1 and for many practical applications, Darcy's law is not valid, and boundary and inertial effects need to be accounted for [6]. For example, modeling fluid flows in the fuel cells. This is because the fuel cell consists of a number of distinct layers of different porosities. Darcy's law assumes no effect of boundaries and the fluid velocity in Darcy's equation is determined by the permeability of the matrix. Therefore at the interface between the regions of different porosity in the fuel cell, particularly between the free fluid flow region, such as a gas flow channel and a permeable medium, then a discontinuity in the fluid velocity and/or the shear stress could emerge [5].

Eliminating the pressure term in [2] in the usual way, the governing equations [1–3] can be written as

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,

(7)

\frac{\partial u}{\partial y} - \frac{\partial v}{\partial x} = \frac{- g K β}{ν} \frac{\partial T}{\partial x} + \frac{σ K B_{0}^{2}}{μ} \times (- \frac{\partial u}{\partial y} si n^{2} ϕ + 2 \frac{\partial v}{\partial y} \sin ϕ \cos ϕ + \frac{\partial v}{\partial x} \cos^{2} ϕ),

(8)

u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = α_{m} (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}}) + \frac{q_{0}^{′′′}}{ρ c_{p}} {(T - T_{c})}^{p},

(9)

where $B_{0}$ is the magnitude of $B$ and ν is the kinematic viscosity of the fluid. The boundary conditions are

\begin{matrix} \begin{matrix} on x = l, u = 0, T = T_{c} (0 < y < l), \\ on x = 0, u = 0, T = T_{h} (0 < y < l), \\ on y = 0, y = l, v = 0, \frac{\partial T}{\partial y} = 0 (0 < x < l) . \end{matrix} \end{matrix}

(10)

Further, we introduce the following nondimensional variables:

\begin{matrix} X = \frac{x}{l}, Y = \frac{y}{l}, U = \frac{l}{α_{m}} u, V = \frac{l}{α_{m}} v, θ = \frac{T - T_{c}}{Δ T}, \end{matrix}

(11)

where $Δ T = T_{h} - T_{c} > 0$ . Introducing the stream function ψ defined as $U = \partial ψ / \partial Y$ and $V = - \partial ψ / \partial X$ , and using expressions (11) in (7) and (9), we obtain the following partial differential equations in nondimensional form:

\begin{matrix} \frac{\partial^{2} ψ}{\partial X^{2}} + \frac{\partial^{2} ψ}{\partial Y^{2}} = - Ra \frac{\partial θ}{\partial X} - H a^{2} \times (\frac{\partial^{2} ψ}{\partial Y^{2}} si n^{2} ϕ + 2 \frac{\partial^{2} ψ}{\partial X \partial Y} \sin ϕ \cos ϕ + \frac{\partial^{2} ψ}{\partial X^{2}} \cos^{2} ϕ), \frac{\partial^{2} θ}{\partial X^{2}} + \frac{\partial^{2} θ}{\partial Y^{2}} + G θ^{p} = \frac{\partial ψ}{\partial Y} \frac{\partial θ}{\partial X} - \frac{\partial ψ}{\partial X} \frac{\partial θ}{\partial Y}, \end{matrix}

(12)

subject to the boundary condition

\begin{matrix} \begin{matrix} X = 1, ψ = 0, θ = 0 (0 < Y < 1), \\ X = 0, ψ = 0, θ = 1 (0 < Y < 1), \\ Y = 0, Y = 1, ψ = 0, \frac{\partial θ}{\partial Y} = 0 (0 < X < 1), \end{matrix} \end{matrix}

(13)

where $Ra = g K β ℓ Δ T / α_{m} ν$ is the Rayleigh number, G = q₀′″ ℓ² (ΔT/α_mν is the internal heat generation parameter, and $Ha = σ K B_{0}^{2} / μ$ is the Hartmann number for the porous medium. It should be mentioned that $ϕ = 0$ corresponds to a horizontal magnetic field and $ϕ = π / 2$ corresponds to a vertical magnetic field, respectively.

Once we know the temperature we can obtain the rate of heat transfer from each of the vertical walls, which are given in terms of the mean Nusselt number as Nu_h = ∞₀¹(∂θ/∂X)_X=0 at the hot wall and Nu_c = ∞₀¹(∂θ/∂X)_X=1 at the cold wall.

3. Numerical Method and Validation

An iterative finite difference procedure will be applied to solve (12), (13). Central difference method is applied for discretization of equations. Forward and backward difference schemes are used for the Neumann boundary conditions. The solution of algebraic equations was performed using Gauss-Seidel iteration. The unknown θ and ψ were calculated until the following criteria of convergence were fulfilled:

\begin{matrix} \frac{\sum_{i, j} ‍ | ζ_{i, j}^{n + 1} - ζ_{i, j}^{n} |}{\sum_{i, j} ‍ | ζ_{i, j}^{n + 1} |} \leq ϵ, \end{matrix}

(14)

where ζ is either ψ or θ, n represents the iteration number, and ϵ is the convergence criterion.

In this study, the convergence criterion is set at $ϵ = 1 0^{- 5}$ . The effect of grid resolution was examined in order to select the appropriate grid density as Tabulated in Table 1 for $Ha = 1$ , $ϕ = 0$ , $G = 3.0$ , $p = 1$ , and $Ra = 300$ . The results indicate that an $101 \times 101$ mesh can be used in the final computations. As a validation, our results for the mean Nusselt number compare well with that obtained by Revnic et al. [24] for case $Ha = 1$ , $G = 1.0$ , $p = 0$ , and $Ra = 1000$ as shown in Table 2.

Table 1:

Grid sensitivity check at $Ha = 1$ , $ϕ = 0$ , $G = 3.0$ , $p = 1$ , and $Ra = 300$ .

Grid size	${Nu}_{h}$	${Nu}_{c}$	$\| ψ_{\min} \|$	T _max

$21 \times 21$	3.475	5.067	7.147	1.002
$41 \times 41$	3.753	5.358	7.361	1.004
$61 \times 61$	3.831	5.446	7.403	1.005
$81 \times 81$	3.868	5.489	7.417	1.005
$101 \times 101$	3.89	5.516	7.423	1.005
$121 \times 121$	3.905	5.534	7.425	1.006
$141 \times 141$	3.916	5.547	7.427	1.006
$161 \times 161$	3.925	5.557	7.427	1.006
$181 \times 181$	3.931	5.564	7.428	1.006
$201 \times 201$	3.936	5.571	7.428	1.006

Table 2:

Comparison of the ${Nu}_{h}$ between the present and the literature result for case $Ha = 1$ , $G = 1.0$ , $p = 0$ , and $Ra = 1000$ .

ϕ	Present	Revnic et al. [24]

0	9.1823	9.1489
$π / 6$	9.8334	9.7976
$π / 4$	10.4818	10.4437
$π / 2$	11.9631	11.9196

4. Results and Discussion

In this section, we present numerical results for the streamlines and isotherms for various values of the magnetic field parameter, Ha, the inclination angle, ϕ, the internal heat generation parameter, G, and the exponent p in the local-heating term as well as the Rayleigh number, Ra. Mealey and Merkin [21] have investigated that the steady solutions exist only for a finite range of Ra and G. In this study, we performed calculation for range G from 0 until 5 and Ra from 0 until 300. The low Ra is also parallel with the fact of coefficient permeability K values from $1 0^{- 7} m^{2}$ until $1 0^{- 15} m^{2}$ inside the Ra formula. Therefore, for physical reality, Ra should be small.

Figure 2 shows the streamlines and isotherms for the no internal heating case ( $G = 0$ ) and horizontal magnetic field ( $ϕ = 0$ ) with $Ra = 300$ and a range of values of Ha. In the absence of a magnetic field, the fluid motion as shown in Figure 2(a) is described as follows. Since the temperature of the left wall is higher than that of the fluid inside the enclosure, the wall transmits heat to the fluid and raises the temperature of fluid particles adjoining the left wall. As the temperature rises, the fluid moves from the left wall (hot) to the right wall (cold), falling along the cold wall, then rising again at the hot wall. This movement creates a clockwise vortex cell inside the enclosure and the isotherms start either from the hot wall or from the bottom wall and end at the top wall or cold wall. When the magnetic field is relatively strengthened (Figure 2(b)), the central streamlines cell are becoming upright and the isotherms stratification in the core diminishes. Further for large Ha (Figure 2(c)), the core vortex is elongated vertically, that is, the flow circulation is progressively inhibited by the retarding effect of the electromagnetic body force $J \times B$ (Lorentz force). While the isotherms are almost parallel to the vertical wall, this implies that conduction is dominant.

Figure 2:

Contour plots of the stream function and temperature for $Ra = 300$ , ϕ = 0, G = 0, and (a) $Ha = 0$ , (b) $Ha = 5$ , (c) $Ha = 20$ .

Figure 3 shows the streamlines and isotherms for the strong heating case ( $G = 5$ ) and horizontal magnetic field ( $ϕ = 0$ ) with $Ra = 300$ , $p = 1$ and a range of values of Ha. The situation for the absence of a magnetic field ( $Ha = 0$ ), is similar to that of the previous case, but with a small increase in intensity of convective motion (as the value of each ψ_min and ψ_max increases). It notes that symbol ψ_max denoted a secondary counterclockwise flow, but relative to small to be displayed in the figure. The internal heat generation enhances the flow near the cold wall and forces the streamlines in a denser distribution. On the other hand, the negative buoyancy caused by G produces a vertical downward flow motion in the vicinity of the top corner hot wall. The maximum temperature increase above that on the heated wall, that is, $T_{\max} > 1$ . When the magnetic field is relatively strengthened (Figure 3(b)), the central streamline cells are also becoming upright and the maximum temperature drifts towards the center of the top wall. This is due to the fact that the hot fluid reaching the top left corner of the enclosure is unable to reject energy since the velocities are small. Finally, for large Ha (Figure 3(c)), the core vortex is also elongated vertically and the isotherms are almost parallel to the vertical wall, this implies that again conduction is dominant.

Figure 3:

Contour plots of the stream function and temperature for $Ra = 300$ , p = 1, ϕ = 0, G = 5, and (a) $Ha = 0$ , (b) $Ha = 5$ , (c) $Ha = 20$ .

The streamlines and isotherms for a range of values of p with $Ra = 300$ , $Ha = 5$ , $ϕ = 0$ , $G = 3$ are shown in Figure 4. The values of T_max decreases slightly as p is increased. The trends for moderate Ha and G shown in Figure 4 suggest that p does not have a major effect on the flow and heat transfer within the porous medium.

Figure 4:

Contour plots of the stream function and temperature for $Ra = 300$ , $Ha = 5$ , ϕ = 0, G = 3 with (a) p = 1, (b) p = 2, (c) p = 8.

Figure 5 shows the influence of the inclination angle ϕ of the applied magnetic field on flow and temperature distributions for $Ra = 300$ , $G = 3$ , $p = 1$ , $Ha = 5$ . The influence of ϕ on flow and temperature distributions is apparent from these figures. The flow fields become distorted with stronger convective flows developing on the vertical walls. The maximum temperature is above $T = 1$ and is located on the top wall, though the value of T_max is reduced as ϕ is increased, from $T_{\max} \approx 1.042$ for $ϕ = 0$ to $T_{\max} \approx 1.037$ for $ϕ = π / 2$ .

Figure 5:

Contour plots of the stream function and temperature for $Ra = 300$ , G = 3, p = 1, $Ha = 5$ , and (a) ϕ = 0, (b) $ϕ = π / 4$ , (c) $ϕ = π / 2$ .

The effect of Ra on the stream function and temperature are shown in Figure 6 for $G = 3$ , $p = 1$ , $Ha = 5$ , $ϕ = 0$ . As Ra is increased, the streamline pattern becomes mildly distorted. This in turn distorts the isotherms, giving higher temperatures in a region towards the top of the heated wall.

Figure 6:

Contour plots of the stream function and temperature for G = 3, p = 1, $Ha = 5$ , ϕ = 0, and (a) $Ra = 40$ , (b) $Ra = 100$ , (c) $Ra = 300$ .

Figure 7 presents relationship between the mean Nusselt number along the hot wall Nu_h and along the cold wall Nu_c against the Rayleigh number Ra for the case $p = 1$ , $ϕ = 0$ , and several values of G and Ha. The variations of Nu_h and Nu_c with Ha for several values of G are shown in Figure 8 for the case $Ra = 100$ , $ϕ = 0$ , $p = 1$ . Naturally, the heat transfer is maximum in the absence of a magnetic field, because the convection is maximum for this situation. In general, Nu_h and Nu_c initially decrease steeply with Ha. As the value of Ha is made larger, the strength of the convective motion is progressively suppressed and $Nu \to 1$ for $G = 0$ . For a fixed moderate value of Ha, increasing G has the effect of decreasing Nu_h and increasing Nu_c.

Figure 7:

Plots of the mean Nusselt number along the hot wall (a), ${Nu}_{h}$ , and along the cold wall (b), ${Nu}_{c}$ , against Rafor the values of G and Ha labelled on the figure with p = 1, ϕ = 0.

Figure 8:

Plots of the mean Nusselt number along the hot wall (a), ${Nu}_{h}$ , and along the cold wall (b), ${Nu}_{c}$ , against Hafor the values of Glabeled on the figure with $Ra = 100$ , ϕ = 0, p = 1.

Figure 9 shows the variations of Nu_h and Nu_c with Ha when p is varied for the case $Ra = 100$ , $G = 3$ , $ϕ = 0$ . Generally, for all values of Ha, increasing p has the effect of increasing Nu_h and decreasing Nu_c, contrasting to the effect of increasing G above. The inclination angle of the magnetic field ϕ has only a moderate effect on Nu_h and Nu_c for a fixed small value of Ha as shown in Figure 10.

Figure 9:

Plots of the mean Nusselt number along the hot wall (a), ${Nu}_{h}$ , and along the cold wall (b), ${Nu}_{c}$ , against Ha for the values of plabelled on the figure with $Ra = 100$ , G = 3, ϕ = 0.

Figure 10:

Plots of the mean Nusselt number along the hot wall (a), ${Nu}_{h}$ , and alongthe cold wall (b), ${Nu}_{c}$ , against ϕfor the values of Ha labelled on the figure with $Ra = 300$ , p = 1.

5. Conclusions

The present numerical study exhibits many interesting features concerning the effect of the inclined magnetic fields on natural convection in square enclosure filled with a porous medium having nonuniform internal heat source. Detailed numerical results for flow field, temperature distribution, and heat transfer performance have been presented in graphical form. The results of the numerical analysis lead to the following conclusions.

Strong internal heating can generate significant maximum fluid temperatures above the heated wall temperature and location of the maximum fluid temperature moves towards the center of the top wall by strengthening the magnetic field. Local heating exponent does not have a major effect on the flow and temperature distributions as well as the heat transfer performance within the porous medium. In general, effect of magnetic field is to retard the convective flows circulation within the porous enclosure. Combination of the large of applied magnetic field in any direction having a strong internal heat source with a small local heating exponent was found to be most effective in suppressing the heat transfer performance along the hot wall. However, combination of the large of applied magnetic field in any direction having a low internal heat source with a high local heating exponent was found to be most effective in suppressing the heat transfer performance along the cold wall.

The findings may be useful to find an effective way in controlling the convective flows due to external and internal heating. The results of the problem are also useful in the study of movement of oil or gas and water through the reservoir of an oil or gas field in the migration of underground water.

Footnotes

Nomenclature

Greek Symbols

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