Thermal Radiation Effect on a Porous Media under Optically Thick Approximation
1Department of Mathematics, University of Ilorin, Kwara State, Nigeria
2Department of Physical and Computer Sciences, College of Natural and Applied Science McPherson University, Abeokuta, Ogun State Nigeria
3Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria
4Sokoto Energy Research Centre, Usmanu Danfodiyo University, Sokoto, Nigeria
The purpose of this paper is to investigate thermal radiation effect on a porous media under optically thick approximation using Newton scheme method from Taylor series implemented numerically on MATLAB. The velocity profiles and temperature profile are studied for different physical parameters like, Porous term P, Radiation F and thermal Grashof number Ga. The results shows that Porous parameter increases with increasing velocity, while the trend reverses with thermal radiation and thermal Grashof number under optically thick approximation. The flow rate increases asymmetrically due to conduction.
At a glance: Figures
Keywords: porous media, recurrence relation, radiation, flow rate
American Journal of Numerical Analysis, 2013 1 (1),
Received September 16, 2013; Revised November 05, 2013; Accepted November 25, 2013Copyright © 2013 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Ibrahim, M.O, et al. "Thermal Radiation Effect on a Porous Media under Optically Thick Approximation." American Journal of Numerical Analysis 1.1 (2013): 15-21.
- Ibrahim, M. , Asogwa, K. , Uwanta, I. , & Shehu, B. D. (2013). Thermal Radiation Effect on a Porous Media under Optically Thick Approximation. American Journal of Numerical Analysis, 1(1), 15-21.
- Ibrahim, M.O, K.K. Asogwa, I.J Uwanta, and B.G Dan Shehu. "Thermal Radiation Effect on a Porous Media under Optically Thick Approximation." American Journal of Numerical Analysis 1, no. 1 (2013): 15-21.
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Fluid flow through a porous channel has been studied theoretically and experimentally by numerous authors due to its wide applications in various fields such as diffusion technology, transpiration cooling, hemodialysis processes, flow control in nuclear reactors, etc. In a pioneering work, Berman  studied an exact solution of the Navier-Stokes equations that describes the steady two-dimensional flow of an incompressible viscous fluid along a channel with parallel rigid porous walls, the flow being driven by uniform suction or injection at the walls. henceforth, many authors such as Terrill , Makinde , Robinson  and Brady  have extended and reconsidered the problem under various physical conditions. Minkowycz and Cheng , Cheng and Minkowycz , and Badr and Pop  were based on Darcy’s law which states that the volume-averaged velocity is proportion to the pressure gradient. Kaviany  used the line integral method to study the heat transfer from a semi-infinite flat plate embedded in a fluid-saturated porous medium. Jang and Shiang  studied the mixed convection along a vertical adiabatic surface embedded in a porous medium. Few studies of convective boundary-layer flows in porous media using the Darcy-Brinkman equation model are considered for the momentum equation, for example, Hsu and Cheng , Rees and Vafai , Nazar et al. [13, 14], Ishak et al. , and Harris et al .
Sreekanth et al.  examined the effect of permeability variation on free convective flow past a vertical porous wall in a porous medium when the permeability varies with respect to time. Subsequently, Singh et al.  studied hydro magnetic free convective and mass transfer flow of a viscous stratified fluid considering the variation in permeability with direction. The problem of unsteady oscillatory laminar free convection flow of an electrically conducting fluid through a porous medium along a porous hot vertical plate with time dependent suction in the presence of heat source or sink was analyzed by Kumar et al.  while Acharya et al.  discussed the magnetic field effects on the free convention and mass transfer flow through a porous medium with constant suction and constant heat flux. An experimental and analytic investigation was carried out by Fand et al.  to study the free convection heat transfer from a horizontal cylinder embedded in porous medium consisting of randomly packed glass spheres and the medium is saturated by water or silicon oil. Murthy and Singh  have modified the viscous dissipation effect on the flow of an incompressible fluid in a saturated porous medium. They applied the Forchheimer-Darcy model for momentum equation. Neild  suggested a modified formula to model viscous dissipation in non-Darcy porous medium. He concluded that modelling of viscous dissipation is related with local drag modelling. Recently Taiwo and Ogunlaran  studied Numerical solution of fourth order linear ordinary differential equations by cubic spline collocation tau method.
Makinde  studied Hermite-Pad´e approximation approach to nonlinear wall driven steady flow in a tube, Recently Ibrahim et al  presented and analysed Radiative Effect on MHD Fluid Flow in a Vertical Channel under Optically Thick Approximation.
This research work is an improved work of Ibrahim  to include porous term. Hence it is proposed to study thermal radiation effect on porous media under optically thick approximation.
2. Formulation of the Problem
The geometry of the problem shows thermal radiation effect on a porous media under optically thick approximation formulated, analysed and solved numerically. The - axis is taken along the plate in the vertically upward direction and also the -axis is taken normal to the plate. That the flow is fully developed, the velocity and temperature fields are symmetrical about the central line of the channel in a porous media. The temperature of the walls is the same and is maintained at a constant temperature. The viscosity, the thermal conductivity and specific heat are independent of temperature and the essential influence of the variation in density is included in the body force term. Steady flow equations are momemtum and energy equation.
where u and are the velocity of the fluid, T is the fluid temperature, β is the thermal expansion of the fluid is the fluid density, is the specific heat capacity, v is the viscosity of the fluid, k is the thermal conductivity and the temperature of the wall , permeability of the porous medium parameter and the kinematic viscosity.
In this research work the mathematical formulation has Porous term which is not included in the work of Ibrahim (1997) where the thermal diffusivity and is the radiative, using the Rosseland differential approximation.
The boundary conditions are;
On introducing the following non-dimensional quantities
Substituting the non-dimensional quantities of equation (5) into (1) to (2), leads to
Equation (6) and (7) leads to
3. Solution to the Problem
To solve equations (8), subjected to the boundary conditions of (9), the solutions are obtained for temperature and velocity flow. using the Newton scheme method from Taylor series to solve the fourth order non linear problem.
Thus from (8) we have
Substituting equations (11) to (15) into equation (10) we obtain
Expanding equations (16) where
Collecting the terms involving
On the L.H.S and the terms involving neglecting
On the R.H.S gives
Where is our initial solution and is the assumed solution.
Considering N = 7
Putting (20) into (19)
Simplifying by collecting terms in we obtain.
We now collate equations (22) at point , where
In the problem under consideration, N=7, a =-1 and b=+1.
The boundary conditions are
Thus, we obtain five equations from (22).
Let be our initial solution
We now substitute for into (22) and then solve the equation simultaneously to obtain the unknowns using Gaussian elimination method with partial pivoting using MATLAB software to obtain the unknown constants
Determination of the Temperature profile, to obtain the solution for the temperature
in the interval since the various values of and are known from the MatLab Program
Determination of the Velocity profile
The velocity distribution of the flow can be obtained from equation (6)
We have our assumed solution to be
Substituting the expression into (6) gives
Evaluating equation (6) in the interval will enable us to determine the effect of different radiation parameter.
Determination of the Non-dimensional flow rate
The non-dimensional flow rate through the channel per unit width is given by
Using the equation (6) and evaluating the integral will enable us to determine the flow rate.
Determination of Heat Transfer coefficient
Thus heat transfer coefficient due to thermal conduction is given by
4. Results and Discussion
The problem of thermal radiation effect on a porous media under optically thick approximation formulated, analysed and solved numerically. In order to point out the effects of physical parameters namely; thermal Grashof number Ga, radiation parameter F, Porous term P. on the flow patterns, the computation of the flow fields are carried out.The values of velocity and temperature are obtained for the physical parameters as mention. The velocity profiles has been studied and presented in Figure 2 to Figure 4 The effect of velocity for different values of Porous parameter (P = 2, 5, 10, 20) is presented in Figure 2. The trend shows that the velocity increases with increasing Magnetic parameter. The effect of velocity for different values of radiation (F = 2, 5, 10) is also presented in Figure 3. It is then observed that the velocity decreases with increasing values of radiation. The effect of velocity for different values of thermal Grashof number (Ga= 1, 2, 3) is also presented in Figure 4. It is then observed that the velocity decreases with increasing values of thermal Grashof number
In Figure 5 It is observed that temperature rise with increasing values of while Figure 6 shows that trend of heat transfer coefficient due to thermal conduction h increases with the rate of change of at less than 5 cooling sets in.
5. Summary and Conclusion
Thermal radiation effect on a porous media under optically thick approximation has been studied. .In order to point out the effects of physical parameters namely; thermal Grashof number Ga, radiation parameter F, Porous parameter P. are presented graphically. It is observed that the velocity increases with increasing Porous parameter, while radiation and thermal Grashof number decreases with increasing values of radiation and thermal Grashof number respectively.
It also observed that temperature rise with increasing values of and trend of heat transfer coefficient due to thermal conduction h increases with the rate of change of at less than 5 cooling sets in.
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