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.
Radiation Porous Medium Heat Transfer and Thermal Conductance1. Introduction
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 [1], Sparrow and Cess [2], Burmeister [3], Bejan [4], Kaviany [5] and Vafai [6]. Raptis [7,8] has studied the effect of radiation on free convection flow through a porous medium. The natural convection cooling of vertical rectangular channels in air considering radiation and wall conduction has been studied by Hall et al. [9]. Al-Nimr and Haddad [10] have described the fully developed free convection in open-ended vertical channels partially filled with porous material. Thermal dispersion-radiation effects on non-Darcy natural convection in a fluid saturated porous medium have been investigated by Mohammadein and El-Amin [11]. The effect of wall conductances on free convection between asymmetrically heated vertical plates has been studied by Kim et al. [12]. Greif et al. [13] have made an analysis on the laminar convection of a radiating gas in a vertical channel. Effect of wall conductances on convective magnetohydrodynamic channel flow has been investigated by Yu and Yang [14]. Gupta and Gupta [15] have studied the radiation effect on hydromagnetic convection in a vertical channel. Datta and Jana [16] have studied the effect of wall conductances on hydromagnetic convection of a radiation gas in a vertical channel. Makinde and Mhone [17] investigated the effect of thermal radiation on MHD oscillatory flow in a channel filled with saturated porous medium and non-uniform wall temperatures. Narahari [18] has investiagted the effects of thermal radiation and free convection currents on the unsteady Couette flow between two vertical parallel plates with constant heat flux at one boundary.
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
2. Formulation of the Problem and Its Solutions
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 Figure 1).
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. [19] showed that in the optically thin limit for a non-grey gas near equilibrium, the following relation holds
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 ([20]) but are not necessary for our investigation.
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).
3. Results and Discussion
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 Figure 2 that the velocity increases with an increase in radiation parameter F. Increasing the radiation parameter F produces significant increase in the thermal condition of the fluid. This increase in the fluid temperature induces more flow causing the velocity of the fluid there to increase. Figure 3 shows that the velocity at any point in the flow region decreases with an increase in Rayleigh number. The Rayleigh number is viewed as the ratio of buoyancy forces and the product of thermal and momentum diffusivities. Increasing value of Rayleigh number opposes the natural convection and hence the fluid velocity decreases. It is observed from Figure 4 that the velocity increases with an increase in porosity parameter. Porosity is a measure of the void (or empty) spaces in a porous medium and is a fraction of the volume of voids over the total volume. Porosity influences the convection flow and so the fluid velocity increases. It is seen from Figure 5 that the velocity decreases with an
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 Figure 10 that for fixed values of F the magnitude of the shear stress at the right wall decreases with an increase in Rayleigh number Ra. Figure 11 reveals that the magnitude of increases with an increase in porosity parameter. On the other hand, it is seen from Figure 12 that for fixed values of F, the magnitude of decreases with an decrease in.
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 Figure 13 that for fixed values of F, the rate of heat transfer decreases with an increase in Rayleigh number. Figure 14 reveals that the increases with an increase in porosity parameter. On the other hand, it is seen from Figure 15 that for fixed values of F, decreases with an decrease in.
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 Figure 16 that for fixed values of F, the flow rate W decreases with an increase in Rayleigh number. Figure 17 reveals that flow rate W increases with an increase in porosity parameter. On the other hand, it is seen from Figure 18 that for fixed values of F, W decreases with an decrease in.
4. Conclusions
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.
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doi:10.1023/A:1006654309980S. H. Kim, N. K. Anand and W. Aung, “Effect of Wall Conduction on Free Convection between Asymmetrically Heated Vertical Plates, Uniform Wall Heat Flux,” International Communications in Heat and Mass Transfer, Vol. 33, No. 5, 1990, pp. 1013-1023.
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doi:10.1016/0017-9310(74)90053-2O. D. Makinde and P. Y. Mhone, “Heat Transfer to MHD Oscillatory Flow in a Channel Filled with Porous Me- dium,” Romanian Journal of Physics, Vol. 50, No. 9-10, 2005, pp. 931-938.M. Narahari, “Effects of Thermal Radiation and Free Con- vection Currents on the Unsteady Couette Flow between Two Vertical Parallel Plates with Constant Heat Flux at One Boundary,” WSEAS Transactions on Heat and Mass Transfer, Vol. 1, No. 5, 2010, pp. 21-30.A. C. Cogley, W. C. Vincenti and S. E. Gilles, “Differen- tial Approximation for Radiative Transfer in a Non-Grey Gas near Equilibrium,” AIAA Journal, Vol. 6, No. 3, 1968, pp. 551-553. doi:10.2514/3.4538C. L. Tien, “Thermal Radiation Properties of Gases,” Advan- ces in Heat Transfer, Vol. 5, 1968, pp. 253-324.
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
and 
(see 
, v is the kinematic coefficient of fluid viscosity, g the acceleration due to gravity, k permeability of the porous medium and 
the thermal conductivity.
, 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 ([
which is the temperature of the wall at
.
is the porosity parameter, 
is the radiation parameter.
is the thermal conductance ratio.
from (12) and (13), we obtain
satisfying the boundary conditions (14) is easily obtained. Achieving
, one can determine 
from (12) using the boundary conditions (14).
and 
subject to the boundary conditions (14) are
(16)
(17)
, F, 
and
.
).
and velocity 
for constant wall temperature are given by
and 
are given by (19).
).
and velocity 
for thermally insulated walls are given by
and 
are given by (19).
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 
increases with an increase in radiation parameter F. Increasing the radiation parameter F produces significant increase in the thermal condition of the fluid. This increase in the fluid temperature induces more flow causing the velocity of the fluid there to increase. 
. The Rayleigh number is viewed as the ratio of buoyancy forces and the product of thermal and momentum diffusivities. Increasing value of Rayleigh number opposes the natural convection and hence the fluid velocity decreases. It is observed from 
increases with an increase in porosity parameter
. Porosity is a measure of the void (or empty) spaces in a porous medium and is a fraction of the volume of voids over the total volume. Porosity influences the convection flow and so the fluid velocity increases. It is seen from 
. 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
.
of the channel is given by
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 
at the right wall decreases with an increase in Rayleigh number Ra. 
increases with an increase in porosity parameter
. On the other hand, it is seen from 
decreases with an decrease in
.
are shown graphically against F for different values of
, 
and 
in Figures 13-15. It is observed from 
decreases with an increase in Rayleigh number
. 
increases with an increase in porosity parameter
. On the other hand, it is seen from 
decreases with an decrease in
.
, 
and 
in Figures 16 - 18. It is observed from 
. 
. On the other hand, it is seen from 
.
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
.