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The free convection flow of radiating gas between two vertical thermally conducting walls through porous medium in the presence of a uniform gravitational field has been studied. Closed form solutions for the velocity and temperature have been obtained in the optically thin limit case when the wall temperatures are varying linearly with the vertical distance. It is observed that the fluid velocity increases and the temperature difference between the walls and the fluid decreases with an increase in the radiation parameter. It is also observed that both the fluid velocity and temperature in the flow field increase with an increase in the porosity parameter. It is found that the fluid velocity decreases while the temperature increases with an increase in the thermal conductance of the walls. Further, it is found that radiation causes to decrease the rate of heat transfer to the fluid, thereby reducing the effect of natural convection.

In recent years, free convection flow of viscous fluids through porous medium has attracted the attention of a number of researchers in view of its wide application to geophysics, astrophysics, meteorology, aerodynamics, boundary layer control and so on. In addition, convective flow through a porous medium has the application in the field of chemical engineering for filtration and purification processes. In petroleum technology, to study the movement of natural gas oil and water through oil channels/reservoirs and in the field of agriculture engineering to study the underground resources, the channel flows through porous medium have numerous engineering and geophysical applications. However, these studies are confined to normal temperatures of the surrounding medium. If the temperature of the surrounding fluid is rather high, radiation effects play an important role and this situation does exist in space technology. In such cases, one has to take into account the effects of radiation and free convection. Channels are frequently used in various applications in designing ventilating and heating of buildings, cooling electronic components, drying several types of agriculture products grain and food, and packed bed thermal storage. Convective flows in channels driven by temperature differences at bounding walls have been studied and reported extensively in literature. Radiative convective flows are frequently encountered in many scientific and environmental processes such as astrophysical flows, water evaporation from open reservoirs, heating and cooling of chambers and solar power technology. Several researchers have investigated convective flow in porous medium such as Nield and Bejan [

In this paper, we have studied the fully developed free convection flow of radiating gas between two vertical thermally conducting walls embedded in porous medium. The governing equations are solved analytically. The effects of the permeability of the porous medium and the influences of radiation parameter and thermal wall conductances on velocity filed and temperature distribution are investigated and analyzed with the help of their graphical representations. It is observed that the fluid velocity increases whereas the temperature distribution decreases with an increase in the radiation parameter F. It is also observed that both the fluid velocity and temperature in the flow field increase with an increase in the porosity parameter. It is found that the fluid velocity decreases while the temperature increases with an increase in the thermal conductance. Further, it is found that radiation causes to decrease the rate of heat transport to the fluid thereby reducing the effect of natural convection. The rate of flow increases with an increase in either radiation parameter F or Rayleigh number

Consider a fully developed flow of a viscous incompressible fluid flow in a vertical channel embedded in porous medium. The distance between the channel walls is 2L. Employ a Cartesian coordinates system with zaxis vertically upwards along the direction of flow and y-axis perpendicular to it. The origin of the axes is such that the channel walls are at positions and (see

For the fully developed laminar flow in porous medium, the velocity and the temperature field have only a vertical component and all of the physical variables except temperature and pressure are functions of y. The temperature inside the fluid can be written as

where N is the vertical temperature gradient.

On the use of (1), the momentum and energy equations are simplified to the following form

where, v is the kinematic coefficient of fluid viscosity, g the acceleration due to gravity, k permeability of the porous medium and the thermal conductivity.

In the optically thin limit, the fluid does not absorb its own emitted radiation. This means that there is no selfabsorption but the fluid does absorb radiation emitted by the boundaries. Cogley et al. [

where, is the absorption coefficient, is the Planck function and the subscript w refers to values at the wall. Further simplifications can be made concerning the spectral properties of radiating gases ([

On the use of (5), Equation (4) becomes

where

Subscript “0” indicates that all quantities have been evaluated at the entrance temperature which is the temperature of the wall at.

Integrating Equation (3) we get

On use of (8), Equation (2) becomes

where

Introducing the non-dimensional variables

and on using (11), Equations (9) and (6) become

where is the porosity parameter,

is the Rayleigh number and is the radiation parameter.

The dimensionless velocity and the temperature boundary conditions are

where is the thermal conductance ratio.

Eliminating from (12) and (13), we obtain

The solution of satisfying the boundary conditions (14) is easily obtained. Achieving, one can determine from (12) using the boundary conditions (14).

The solutions for and subject to the boundary conditions (14) are

(16)

(17)

where

It is observed from the Equations (16) and (17) that the velocity and temperature depend on the parameters, F, and.

Case-I: Constant wall temperature ().

The temperature distribution and velocity for constant wall temperature are given by

where and are given by (19).

Case-II: Thermally insulated walls ().

The temperature distribution and velocity for thermally insulated walls are given by

where and are given by (19).

To study the effects of radiation and porosity of the porous medium on the velocity field and temperature distribution, we have presented the non-dimensional velocity and the temperature against for various values of radiation parameter F, Rayleigh number, porosity parameter and the thermal conductance parameter in Figures 2-9. It is observed from

increase in thermal conductance parameter. Figures 6 and 7 reveal that the temperature decreases with an increase in either radiation parameter F or Rayleigh number. Radiation tends to increase the rate of heat transport to the fluid. Thus the effect of radiation reduces the influence of natural convection by causing a reduction in the temperature difference between the fluid and the channel walls. The increase in radiation parameter F means the release of heat energy from the flow region and so the fluid temperature significantly decreases. It is found from Figures 8 and 9 that the temperature increases with an increase in either porosity parameter or thermal conductance parameter.

The non-dimensional shear stress at the right wall of the channel is given by

Numerical values of non-dimensional shear stress at the right wall of the channel are plotted against F for different values of Ra, and in Figures 10-12. It is observed from

The rate of heat transfer across the channel’s wall is given as

Numerical values of the rate of heat transfer are shown graphically against F for different values of, and in Figures 13-15. It is observed from

The non-dimensional flow rate is given by

The non-dimensional flow rate, W has been plotted against F for different values of, and in Figures 16 - 18. It is observed from

The fully developed free convection flow of a radiating gas between two vertical thermally conducting walls embedded in porous medium has been studied. The effects of the permeability of the porous medium and the influences of radiation parameter and thermal wall conductances on velocity and temperature fields are investigated and analyzed with the help of their graphical representa-

tions. It is observed that the fluid velocity increases and the temperature distribution decreases with an increase in the radiation parameter F. It is also observed that both the fluid velocity and temperature in the flow field increase with an increase in the porosity parameter. It is found that the fluid velocity decreases while the temperature increases with an increase in the thermal conductance of the walls. Further, it is found that radiation causes to decrease the rate of heat transfer to the fluid thereby reducing the effect of natural convection. The rate of flow increases with an increase in either F or Rayleigh number.