International Journal of Scientific & Engineering Research, Volume 2, Issue 12, December-2011 1
ISSN 2229-5518
Effects of anisotropy on convection in porous media subject to nonuniform thermal average gradient
F. P. Codo, V. Adanhounme , A. Adomou
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direction
3. Darcy number ,
Superscript
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Natural convection heat transfer in a fluid saturated porous horizontal cavity subjected to nonuniform thermal gradients with the hydrodynamic anisotropy of the porous matrix and the rigid/rigid and stress-free/stress-free horizontal boundaries remains one of the most important problems for modern theoretical physics and applied mathematics. This problem is motivated by engineering applications as: convection in the Earth's crust, flows in soils, aquifers, storage of agriculture products and so on.
Since complicated convective motions appear in the layers near the surface, many scientific papers aim at investigating the conditions for stability or instability. Most of the work on onset of convection in a porous medium is based on linear theory.The critical Rayleigh number derived by such a theory gives a necessary condition for stability(or equivalently, a sufficient condition for instability). Degan and Vasseur [6] have studied the influence of anisotropy on convection in porous media with nonuniform thermal gradient by assuming the flow parallel in the x-direction. The stability of convection in a horizontal porous layer subjected to an inclined gradient of finite amplitude was investigated by Weber [12] and Nield [9] . The results showed by the critical Rayleigh number are always higher than 4π2. The stability of horizontal porous and viscous layer, when the thermal gradient is not uniform, has been considered by Nield [9].
Walker and Homsy [11] have used the Brinkman model to determine the critical Rayleigh number against Darcy number for the case of conducting no-slip boundaries. Vasseur and Robillard [10] have used the Brinkman model to investigate the effects of nonlinear temperature distribution on stability and natural convection in a horizontal porous layer heated from below. All previous studies have usually been concerned with homogeneous porous structures. But the inclusion of more physical realism in the matrix properties of the medium is important for the accurate modeling of the anisotropic media. Anisotropy, which is generally a consequence of a preferential orientation or asymmetric geometry of the grain of fibers, is in fact encountered in numerous systems in industry and nature. The critical Rayleigh number for the onset of convection was first considered by Castinel and Combarnous [4] who conducted an experimental and theoretical investigation for a layer with impermeable boundaries.Mckibbin [8] conducted an extensive study on the effects of anisotropy on the convective stability of a porous layer.
Also, it was demonstrated that the critical Rayleigh number was always reduced when compared with a perpendicular or
parallel orientation of boundaries.From many papers it was
found that anisotropic medium is most stable while either the isotropic situation or the horizontally isotropic situation is the most unstable one depending on the horizontal Rayleigh number and anisotropy parameters.
In this paper we aim at determining the critical Rayleigh numbers for the onset of convection on the basis of the generalized Brinkman-extended Darcy model which allows the no-slip boundary condition on a solid wall,to be satisfied.
The rest of this paper is organized as follows: in the next section we present the details of the model we will analyze. In a section 3 we look for the solutions to the governing equations, depending on the anisotropic parameters in permeability of the porous matrix and we compute the critical Rayleigh number. It is demonstrated that anisotropic parameters have considerable influence on the onset of convection. The last section contains the discussion and the conclusion .
The physical model considered here consists of a two dimensional horizontal rectangular enclosure of elongated shape filled with a porous medium composed of sparse distribution of particles completely surrounded by Boussinesq fluid and bounded by two rigid vertical side walls and two long horizontal boundaries at that may be both rigid or both stress-free. The anisotropy in flow permeability of the porous medium is characterized by the permeability ratio (where and are the permeabilities along the two principal axes of the porous matrix respectively) and the orientation angle , defined as the angle between the horizontal direction and the principal axis with the permeability . The layer is heated from the bottom by a constant heat flux and the other surfaces are insulated. We assume that the viscous,incompressible and saturating fluid and the porous medium are everywhere in local thermodynamic equilibrium.
Under the above approximations,the equations describing the laminar and two-dimensional convective flow in an anisotropic porous medium can be written in the form as in
6 and 1 :
(1)
(2) (3)
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(4)
where is the superficial flow velocity, the temperature, the gravitational acceleration, the time, and the heat capacities of the saturated porous medium
and the fluid respectively , the dynamic viscosity,
apparent dynamic viscosity for Brinkman's model, the thermal conductivity, the density,
assuming that , governing equations become at quasi-steady-state
(8)
(9)
where is a heat sink term.
We will consider the following boundary conditions:
both horizontal boundaries rigid
the symmetrical second-order permeability tensor and the Boussinesq approximation.
We take and
at (10)
at (11)
both horizontal boundaries stress-free
as respective dimensional scales for length, velocity, temperature, stream function and time.
Taking into account the components of the velocity
and applying the operator rot to the
at (12)
at (13)
equation (2) we can write the governing equations in nondimensional form as
The appropriate solutions to the above equations can be
sought in the form as in 6 and 3 :
(5)
(6)
(14)
Where
(7)
In the above equations, is the Darcy number, the Darcy-Rayleigh number
based on permeability the permeability ratio
and the relative viscosity.
In the present study as a first approximation in Brinkman's extension for which . The quasi-state of the resulting transient natural convection heat transfer in the present study will be reached if the heating process is maintained long enough. So, all quantities governing the phenomenon become nearly independent of time ,except the
temperature which continues to increase with. Consequently,
(15)
Where is the unknown but constant temperature gradient in the - direction, is the thickness of the boundary layer at
[12] and
(16) Substituting the functions (14) and (15) in the equations (8)
and (9) , we obtain for
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(25)
(17)
Where and denotes the derivative. Without loss of generality we will consider
in the following discussion.
The solutions to the above equations are in the form
(18) (19)
(20)
Where
(26)
(27)
(28)
(21)
Where and are functions depending upon the hydrodynamic and thermal boundary conditions imposed on the porous layer in the direction and is the temperature profile for pure conduction regime.
The solutions to equations (17) satisfying the boundary conditions (10),(11) can be defined by
, (22)
Then the stream function and temperature fields are known from the equations (14),(15) , (22) , (23), (24), (25) and (26).
Let us define the stream functions gradient in the direction and the function depending only on as
follows
(29)
(30) (31)
Therefore
(32)
(33) (34)
Where
, (23)
; (24)
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When or and ,
1 cosh s
1 y
cosh sy
1 y
the equation (36) admits the real root and there is no convection.
2
s cosh 2s
Obviously,the marginal state determining the critical
Rayleigh number for the onset of convection is reached
when
(42)
Following Bejan [2], an equivalent energy flux condition in the x direction can be imposed such that
The Nusselt number Nu is given by
(43)
(35)
Substituting the equations (22), (23), (24), (25) and (26) into
(35) we can write the following equation
(36)
Where
(37) (38) (39)
. (40)
Consider the discriminant
(41)
of the equation (36), where
When (it is sufficient that the equation
(36) admits the real roots :
,
Where is the wall-to-wall dimensionless temperature difference and
the corresponding value of pure conduction regime.
Taking into account the equations (32),(33) and (34) we can express as follows
(44)
It gives rise to convection cells in opposite directions.
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; (46)
; (45)
(47)
Therefore the temperature gradient may be evaluated from
(42) for given values of and
We can express the critical Rayleigh number in the form
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Now we can point out two limiting cases:
When approaches (corresponding to the anisotropic porous medium),
When approaches (i.e.viscous fluid case where anisotropic effects are irrelevant),
. For
This result is obtained by [6].
The solutions to the equations (17) satisfying the boundary conditions (12) and (13) are given by
; (49)
, (50)
(48)
, (51)
(52)
(53)
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Where will be defined with respect to Nusselt number.
Taking into account the equations (49), we can express the equations (37) and (40) as follows
; (54)
(56)
(55)
Therefore the temperature gradient may be evaluated from
(42) for given values of and .
We can express the critical Rayleigh number in the form
In this paper we have investigated the anisotropic effect of the porous medium.
When the dimensionless Darcy parameter is infinitely large (corresponding to the anisotropic porous medium), the critical Darcy-Rayleigh is a function of the permeability ratio and the orientation angle . When the dimensionless Darcy parameter approaches 0 (corresponding to the viscous fluid
where anisotropic effect is irrelevant) the critical Darcy-
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Rayleigh is constant. Moreover for the appropriated value of the thickness the critical Darcy-Rayleigh is that obtained in [6]
[1] J.Bear, "Dynamics of fluids in porous media".Dover
Publication, Elsevier,New York,1972.
[2] A.Bejan, "Convection heat transfer". A Wiley- Interscience Publication, John Wiley and Sons. New York,1984.
[3] M.E.Eglit et al, Problems in mechanical theory of continua.Tom 2, Moscow Lycee, Moscow 394pp.,1996.
[4] G.Castinel, M.Combarnous, Critères d'apparition de la convection naturelle dans une couche poreuse anisotrope", J.C.R. Hebd.Seanc.Acad.Sci.Paris B 278, 701-704,1974.
[5] G. Degan, P. Vasseur, "Boundary layer regime in a vertical porous layer with anisotropic permeability and boundary effects",Int.J.Heat Fluid Flow 18,334-343,1997.
[6] G. Degan, P. Vasseur , "Influence of anisotropy on convection in porous media with nonuniform thermal gradient".Int.J.Heat Mass Transfer 46,781-789,2003.
[7] L.Lu, C.R.Doering, F.H.Busse, "Bounds on convection driven by internal heating". J.Math.Phys.Vol.45,No.7,2968-
2985,2004.
[8] R. Mckibbin, "Thermal convection in a porous layer: effects of anisotropy and surface boundary conditions".Trans.Porous Media 1, 271-292,1984.
[9] D.A. Nield, "Convection in a porous medium with inclined temperature gradients",Int.J.Heat Mass Transfer
34,87-92,1991.
[10] P. Vasseur, L. Robillard ,"The Brinkman model in a porous layer: effects of nonuniform thermal gradient", Int.J.HeatMass Transfer 36, 4199-4206,1993.
[11] K.L. Walker, Homsy G.M., "A note of convective instability in Boussinesq fluids and porous media", J.Heatttransfer 99,338-339.1977.
[12] J.E. Weber , "Convection in a porous medium with horizontal and vertical temperature gradients", Int.J.Heat Mass Transfer17, 241-248,1974.
Francois de Paule Codo received the Mining
Engineer, M.Sc. and Ph.D.degrees from the Heavy
Industries Technical University of Miskolc, Hungary.
He is currently Assistant Professor of Applied Fluid Mechanics and Hydraulics in Department of Civil Engineering at the University of Abomey- Calavi,Benin.
His principal research interests are applied fluid mechanics and hydraulics at the Applied Mechanics and Energy Laboratory.
e-mail:fdepaule2003@yahoo.fr
Villevo Adanhounme received the M.Sc. and Ph.D. degrees in Mathematics from the Russian People University of Moscow, Federation of Russia.
He is currently Assistant Professor of Variational Calculus and Advanced Probability at the International Chair of Mathematical Physics and Applications-University of Abomey-Calavi, Benin.
His principal research interests are applied mechanics, partial differential equations and optimal control in the International Chair of Mathematical Physics and Applications.
e-mail:adanhounm@yahoo.fr
Alain Adomou received the M.Sc. and Ph.D.degrees in Theoretical Physics from the Russian People University of Moscow,Federation of Russia.
He is currently Assistant Professor of mechanical theory of continua at the Technological Institute of Lokossa-University of Abomey-Calavi.\\
His principal research interests are applied mechanics and theory of gravitation.
e-mail:denisadomou@yahoo.fr
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