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The effects of hydrodynamic anisotropy on the mixed-convection in a vertical porous channel heated on its plates with a thermal radiation are investigated analytically for fully developed flow regime. The porous medium is anisotropic in permeability whose principal axes are oriented in a direction that is oblique to the gravity. The generalized Brinkman-extended Darcy model which allows the no-slip boundary-condition on solid wall is used in the formulation of the problem. The flow reversal, the thermal radiation influence for natural, and forced convection are considered in the limiting cases for low and high porosity media. It was found that the anisotropic permeability ratio, the orientation angle of the principal axes of permeability and the radiation parameter affected significantly the flow regime and the heat transfer.

Convection heat transfer in porous media is a fundamental importance in such technologies as geothermal exploitation, oil recovery, radioactive waste management, insulation of building and cold storage, drying processes, transpiration cooling, powder metallurgy, agricultural engineering, solidification and binary alloys, etc. It is also important to geophysics and environmental sciences. Much of this activity, both numerical and experimental, has been summarized by Nield and Bejan (1999) [

Thermal radiation always exists and can strongly interact with convection in many situations of engineering interest. The convection heat transfer in a porous channel (or in an enclosed space) in the presence of thermal radiation continues to receive considerable attention because of its importance in many practical applications such as furnaces, combustion chambers, cooling towers, rocket engines and solar collectors. During the last decade, many experimental and numerical investigations on the phenomenon of the interaction of natural or mixed convection with thermal radiation in vertical porous channels or enclosures have been presented. Mahmud and Fraser (2003) [

Most of the existing theories and experimental investigations on the topic, are concerned isotropic porous media. However, in several applications the porous materials are anisotropic. Such porous media are in fact encountered in numerous systems in industry and nature. As examples, we can cite fibrous materials, biological materials, geological formations, and oil extraction. The inclusion of more physical realism in the matrix properties of the medium is important for the accurate modeling of the anisotropic media. Anisotropy is generally a consequence of a preferential orientation and asymmetric geometry of the grain or fibbers which constitute the porous medium. Despite its vast range of applications, convection in such anisotropic porous media has received relatively little attention.

Thermal convection in a porous medium with anisotropic permeability was first considered by Castinel and Combarnous (1974) [

The contemporary trend in the field of heat transfer and thermal design is to apply a second law of thermodynamics analysis, and its design-related concept of entropy generation minimization (see Bejan (1996) [

The present work deals with an analytical study of coupled fluid flow and heat transfer by mixed convection and radiation in a vertical channel opened at both ends and filled with a fluid-saturated porous medium. The bounding walls of the channel are isothermal and gray. The effects of hydrodynamic anisotropy of the porous medium will be investigated, since the physical problem is of significant importance to many engineering-related applications.

The problem, under different considerations, concerns an optically thin and electrically conducting fluid flowing through a vertical channel opened at both ends and filled with a porous medium as shown by the physical model considered in

Under the above approximations, the equations governing the conservation of mass, momentum (generalized Brinkman-extended Darcy’s law) and energy can be written as follows (see [

where

The energy Equation (3) takes into account the radiative heat flux term

where

Considering that the absorption is negligible for a thin gas,

where

Solving the system of equations, Equations (7a) and (7b), the solutions for

Substituting results obtained for these intensities into Equation (6) and noting that

where

Taking into account that the emissivities of the left and the right walls are identical, (i.e.,

In the above equation,

Truncating the above series after the second term and using the definitions of

Introducing the Boussinesq approximation

and assuming that when the flow is fully developed in the channel, the axial (x’-direction) velocity depends only of the transverse coordinate y’ (i.e.,

where

The hydrodynamic and thermal boundary conditions for the vertical channel are

Taking

where

In the above equation,

the reference temperature difference,

The boundary conditions, Equations (19) and (20) become

where

From the dimensionless Equations (21) and (24) and the boundary conditions Equations (26) and (27), it is seen that the present problem is governed by eight dimensionless parameters, namely

Using the boundary conditions for the temperature, Equation (24) can be integrated to give the following fully developed temperature profile

By substituting Equation (28) into Equation (22), and using the boundary conditions, Equations (26) and (27), the velocity profile is obtained as follows

where

In the above expression of the distribution of the velocity, the parameter

It is noticed that when

This result is in agreement with that which has been found by Degan and Vasseur [

The wall friction is defined by the following expression

where the plus and minus signs correspond to the left and the right walls. Hence, on the left wall the friction is expressed as follows

while, on the right wall, for the friction one can have

such that the average friction defined by

where

Concerning theory of fully developed confined convection including flow reversal in vertical channels, when buoyancy effects are increased (i.e., when the heat flux is increased), the fluid will accelerate near the walls. Then, mass conservation requires that the fluid decelerates in the center of the channel. Consequently, if buoyancy effects are strong enough, a minimum will form in the velocity distribution at the channel centerline. For even stronger buoyancy effects, a flow reversal will form at the channel centerline. In the case of aiding mixed convection through the porous channel, when a reverse flow occurs, the relatively lower velocity negative flow passes along side the cold wall hence carries a lower level of thermal energy. Since a net dimensionless mass flow is fixed, an equal quantity of fluid is added to the fluid flowing in the positive (upward) direction in this fluid flows adjacent to the hot wall thereby carrying a larger amount of energy.

Following Aung and Worku [

Applying Equation (29), the above condition translates into

where

Accordingly, one can deduce the flow reversal function

Two cases are of interest, one with

・ Case with

Consequently, as

Then, the velocity profile, Equation (29) and its limit, as

and

The average friction and its limit, as

and

Also, the flow reversal function, Equation (41) and its limit can be written as

and

Consequently, as

Similar results has been obtained by Degan and Vasseur [

・ Case with

and

Also, in this limiting situation, the pressure gradient, the average friction and the flow reversal function are expressed as follows

and

The limiting case of forced convection solution is obtained by setting

Here, the pressure variation is determined by the following expression

The results presented above will be specified for two cases of interest (

・

The pressure gradient is written

・

and for the pressure gradient

This important limiting case will be studied by setting

and, as

The solution for the velocity distribution may be written in terms of variables utilized herein, giving

In the above velocity profile, the corresponding expressions to B and C are those indicated in Equation (30).

Taking

The volume flow rate b per unit channel width is defined as

that must be calculated by the expression

When

The total heat absorbed by the fluid in traversing the channel is

Writting

where B and C are the corresponding expressions indicated in Equation (30).

An average Nusselt number may be defined as

where

The flow reversal condition for the limiting case of natural convection is

such that the flow reversal criterion becomes

Taking

We notice here two cases of interest, the first one with

・

and

・

It is noticed that when

Similar result has been found by Degan and Vasseur [

The effects of varying

In

From

in the flow field when

The effects of the anisotropic parameters of the porous matrix and the thermal radiation parameter on the gradient of the pressure are presented in

_{d} = 6 respectively. From

In

varying anisotropic parameters of the porous matrix are observed significant, since

The effects of the radiation parameter R_{d} and the permeability ratio K^{*} on the Nusselt number is illustrated in ^{−}^{2}, _{d}, the convective heat transfer increases as R_{d} is made smaller. For a fixed value of R_{d}, Nu tends asymptotically toward a constant value as K^{*} is made small enough

heat transfer for this limiting case, it is observed that

when

The influence of the anisotropic orientation _{d} and K^{*} when

Curves plotted in

permeability ratio

A study has been made of mixed convection through a parallel-plate vertical porous channel submitted to a thermal radiative flux on its wall. The porous medium is assumed to be hydrodynamically anisotropic with its principal axes oriented in a direction that is oblique to the gravity.

Analytical expressions valid for fully developed flow and based on the generalized Brinkman-extended Darcy are obtained. The main conclusions of the present analysis are:

・ Both thermal radiation and anisotropic parameters have a strong influence on the fluid motion and the heat transfer through the parallel-plate vertical porous channel.

・ In the pure Darcy medium

・ In the fluid medium (

・ The effects of increasing values of the anisotropic parameters and the thermal radiation parameter tend to decrease the temperature and the gradient of the pressure. Moreover, the decrease of values of thermal radiation reduces the fluid velocity and makes the reversal flow occurs.

・ For a given value of the thermal radiation parameter, a maximum (minimum) heat transfer rate through the parallel-plate vertical porous channel is obtained when the porous matrix is oriented in such a way that the principal axis with higher permeability is parallel (perpendicular) to the gravity.

GérardDegan,ChristianAkowanou,LatifFagbemi,JoëlZinsalo, (2016) Hydrodynamic Anisotropy Effects on Radiation-Mixed Convection Interaction in a Vertical Porous Channel. Applied Mathematics,07,22-39. doi: 10.4236/am.2016.71003