J. Comp. & Math. Sci. Vol.2 (3), 483-492 (2011)
Effect of Radiation on Non-Darcy Convective Heat Transfer through a Porous Medium in a Vertical Channel B.DEVIKA RANI, P.VIJAYA BHASKAR REDDY and D.R.V. PRASADA RAO Department of Mathematics, Sri Krishnadevaraya University, S.V. Puram, Anatapur-515 003, Andhra Prasedh, India. ABSTRACT We analyse the effect of radiation on non-darcy convective heat transfer through a porous medium in a vertical channel in the presence of heat sources. The non-linear, coupled equations governing the flow and heat transfer have been solved by employing a perturbation technique with ﾎｴ the porous parameter as perturbation parameter. The velocity and temperature dissipation are analysed for various parameters. Keywords: Non-darcy generating sources.
convection,
1. INTRODUCTION Any substance with a temperature above zero transfers heat in the form of radiation. Thermal radiation always exits and can strongly interact with convection in many situations of engineering interest. The influence of radiation on natural or mixed convection is generally stronger than that on forced convection because of the inherent coupling between temperature and flow field. Convection in a channel (or 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. The interaction of thermal radiation with conduction and
porous
medium,
heat
convection in thermally developing, absorbing-emitting, non gray gas flow in a circular tube is investigated by Tabanfar and Modest15. Combined radiation窶田onvection is studied by Lauriat9 for a vertical cavity, and Chang et al.4 for a complex enclosure. Akiyama and Chang2 numerically analysed the influence of gray surface radiation on the convection of nonparticipating fluid in a rectangular enclosed space. Hunt7 analysed the fluid flow problem in a magnetic field with or without buoyancy effects for a conducting fluid , Shercliff13 analysed the fluid flow characteristics in a pipe under a transverse magnetic field. Alboussiere et al.1 did an asymptotic analysis to study buoyancy driven convection in a uniform magnetic field. Recently, Ghaly6 and Chamkha et al.3
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B. Devika Rani, et al., J. Comp. & Math. Sci. Vol.2 (3), 483-492 (2011)
analyze the effect of radiation heat transfer on flow and thermal fields in the presence of a magnetic field for horizontal and inclined plates. Non-Darcy effects on natural convection in porous media have received a great deal of attention in recent years because of the experiments conducted with several combinations of solids and fluids covering wide ranges of governing parameters which indicate that that the experimental data for systems other than glass water at low Rayleigh members do not agree with theoretical predictions based on the Darcy flow model. This divergence in the heat transfer results has been reviewed in detail in Cheng5 and Prasad et al.,11 among others. Detailed accounts of recent efforts on non-Darcy convection have been recently reported in Tien and Hong16, Prasad et al.,12 and Kalidas and Prasad8.The BrinkmannExtended Darcy model was considered in Tong and Subramanian17 and Laurait and Prasad10 to examine the boundary effects on free convection in a vertical cavity. Recently Sivasenkar Reddy14 has discussed the effects of radiation on a non-Darcy convective flow in a circular duct under different conditions. In this paper we investigate the effect of radiation on the Non-Darcy convection heat transfer flow of a viscous electrically conducting fluid through a porous medium confined in a vertical channel in the presence of heat sources. The non-linear, coupled equations governing the flow and heat transfer have been solved by employing a perturbation technique with δ the porous parameter as perturbation parameter. The velocity and temperature dissipation are analysed for different values of G,D-1,α,M and N1.
2. FORMULATION OF THE PROBLEM We consider a fully developed laminar convective heat transfer flow of a viscous, electrically conducting fluid through a porous medium confined in a vertical channel bounded by flat walls. We choose a Cartesian co-ordinate system O(x,y,z) with x- axis in the vertical direction and y-axis normal to the walls. The walls are taken at y= ± 1.The walls are maintained at constant temperature and concentration. The temperature gradient in the flow field is sufficient to cause natural convection in the flow field .A constant axial pressure gradient is also imposed so that this resultant flow is a mixed convection flow. The porous medium is assumed to be isotropic and homogeneous with constant porosity and effective thermal diffusivity. The thermo physical properties of porous matrix are also assumed to be constant and Boussinesq approximation is invoked by confining the density variation to the buoyancy term. In the absence of any extraneous force flow is unidirectional along the x-axis which is assumed to be infinite.
∇.q = 0
( Equation of continuity)
ρ ∂q ρ µ + 2 (q .∇)q = −∇p + ρg − ( )q k δ ∂t δ ρF − q .q . + µ ∇ 2 q + µ e JxH k ( Equation of linear momentum)
ρC p (
(2.1)
(2.2)
∂ (qr ) ∂T + (q .∇ )T ) = λ ∇ 2T + Q (To − T ) − ∂t ∂y ( Equation of energy)
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
(2.3)
B. Devika Rani, et al., J. Comp. & Math. Sci. Vol.2 (3), 483-492 (2011)
Since the flow is unidirectional, continuity of equation (2.1) reduces to
the
∂u = 0 Where u is the axial velocity implies ∂x
u = u(y) The momentum, energy and diffusion equations in the scalar form reduces to ρδF 2 ∂p µ ∂ 2 u µ − + ( ) 2 − ( )u − u ∂x δ ∂y k k (2.5) σµ e2 H o2 −( )u − ρg = 0 ρ0 ρ 0C p u
∂q ∂T ∂ 2T = λ 2 + Q(To − T ) − r ∂x ∂y ∂y
(2.6)
The boundary conditions are u=0
, T = T1
on y = − L
u=0
, T = T2
on y = + L
The axial temperature gradients
(2.7) ∂T ∂x
is
assumed to be a constant, say, A . Invoking Rosseland approximation for radiation flux we get 4σ • ∂ (T ′ 4 ) qr = − (2.8) β R ∂y and linearising T ′4 about Te by using Taylor’s expansion and neglecting higher order terms we get (2.9) T ′ 4 ≅ 4Te3T − 3Te4 We define the following non-dimensional variables as u , ( x ′, y ′) = ( x, y ) / L , (ν / L ) T − T2 pδ p′ = θ= T1 − T2 ( ρν 2 / L2 )
u′ =
d 2u dy 2
485
= π + δ ( D −1 + M 2 )u − δ G(θ + NC ) (2.11)
d 2θ 4 d 2θ α θ − + = ( PN T )u 3 N1 dy 2 dy 2
(2.12)
where Α = FD −1/ 2 (Inertia or Fochhemeir parameter),
βg (T1 − T2 ) L3 (Grashof Number) ν2 L2 D −1 = (Darcy parameter). k µCp (Prandtl Number) P= λ QL2 α= (Heat source parameter), λ 4σ •Te3 N1 = (Radiation parameter) βR G=
The corresponding boundary conditions are u = 0 , θ = 1 on y = −1 (2.13) u = 0 , θ = m on y = +1 3. SOLUTION OF THE PROBLEM The governing equations of flow and heat transfer are coupled non-linear differential equations. Assuming the porosity δ to be small we write u = u 0 + δ u1 + δ 2 u 2 + .......... ..
(2.10)
Introducing these non-dimensional variables the governing equations in the dimensionless form reduce to (on dropping the dashes)
θ = θ 0 + δ θ 1 + δ 2θ 2 + ..........
(3.1)
Substituting the above expansions in the equations (2.11) & (2.12) and equating like powers of δ ,we obtain equations to the zeroth order as
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
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B. Devika Rani, et al., J. Comp. & Math. Sci. Vol.2 (3), 483-492 (2011)
d 2 u0 =π dy 2
(3.2)
d 2u2 − ( M 2 + D −1 )u 2 = −G ( θ1 ) − Αu02 dy 2
(3.6)
d 2θ 0 − α1θ 0 = ( P1 N T )u 0 dy 2
(3.3)
d 2θ 2 − α1θ 2 = ( P1 N T )u 2 dy 2
(3.7)
The equations to the first order are
The corresponding conditions are
d 2u1 − ( M 2 + D −1 )u1 = −G ( θ 0 ) 2 dy
(3.4)
d 2θ1 − α1θ1 = ( P1 N T )u dy 2
(3.5)
u0 (1) = u0 (−1) = 0 , θ0 (+1) = 0, θ0 (−1) = 1 u1 (1) = u1 (−1) = 0 , θ1 (+1) = 0,
The equations to the second order are
θ1 (−1) = 0, C1 (+1) = 0
θ 0 = a1 ( u1 = a14 (
π
2
( y 2 − 1)
Ch(β1 y ) Ch(β1 y) Ch( β1 y) Sh(β1 y) − y 2 ) + a1a2 ( − 1) + 0.5( − ) Ch(β1 ) Ch( β1 ) Ch(β1 ) Sh( β1 )
Ch ( β 2 y ) Ch ( β 2 y ) Ch ( β1 ) − Ch ( β1 y )) + a16 ( − y2 ) − Ch( β 2 ) Ch ( β 2 )
− a18 (
Ch ( β 2 y ) Sh ( β 2 y ) Sh ( β 2 y ) ) − 1) + + a15 ( Sh( β1 y ) − Sh( β1 ) + a17 ( y − Ch ( β 2 ) Sh( β 2 ) Sh ( β 2 )
θ1 = (a24 Sh( β1 ) − a23 − a21 − a19Ch( β1 ))
Ch ( β1 y ) Sh( β1 y ) + (a22 − a25Ch ( β1 )) + Ch ( β1 ) Sh( β1 )
+ a19Ch ( β 2 y ) + a20 Sh( β 2 y ) + a21 y 2 − a22 y + a23 − a24 ySh( β1 y ) + a25 yCh( β1 y )
u2 = a60 (
Ch( β 2 y ) Ch( β 2 y ) Ch( β1 ) − Ch( β1 y )) + a62 ( Sh( β1 ) − ySh( β1 y )) + Ch( β 2 ) Ch( β 2 )
+ a64 (
Ch( β 2 y ) Ch( β 2 y ) Ch( β 2 y ) Sh( β 2 ) − ySh( β 2 y )) + a66 ( − y 4 ) − a68 ( − y2) + Ch( β 2 ) Ch( β 2 ) Ch( β 2 )
+ a70 (
Ch( β 2 y ) Sh( β 2 y ) Sh( β 2 y ) − 1) + a61 ( Sh( β1 ) − ySh( β1 y )) − a63 ( Ch( β 2 ) − Ch( β 2 ) Sh( β 2 ) Sh( β 2 )
− yCh( β 2 y )) − a65 ( − a69 (
(3.9)
u2 (1) = u2 (−1) = 0 , θ 2 (+1) = 0, θ 2 (−1) = 0 (3.10)
Solving the equations (3.2)-(3.7) subject to the boundary conditions (3.8-3.10) we get u0 ( y ) =
(3.8)
Sh( β 2 y ) Sh( β 2 y ) Ch( β1 ) − yCh( β1 y )) − a67 ( − y3 ) − Sh( β 2 ) Sh( β 2 )
Sh( β 2 y ) − y) Sh( β 2 )
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
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B. Devika Rani, et al., J. Comp. & Math. Sci. Vol.2 (3), 483-492 (2011)
θ 2 = a71 (Ch( β 2 y ) − Ch( β 2 )) + a74 ( ySh( β 2 y ) − Sh( β 2 ) + a75 ( ySh( β1 y ) − Sh( β1 )
Ch( β 2 y ) ) − a79 ( y 2 − 1)Ch( β1 y ) + Ch( β 2 )
+ a79 ( y 4 − 1) + a81 ( y 2 − 1) + a83 (1 − −
Ch( β 2 y ) )+ Ch( β 2 )
Ch( β1 y ) ) + a72 ( Sh( β 2 y ) − Ch( β1 )
Sh( β1 y ) Sh( β1 y ) Sh( β 2 )) + a73 ( yCh( β1 y ) − Ch( β 2 )) + Sh( β1 ) Sh( β1 )
+ a76 ( yCh( β1 y ) − + a80 ( y 3 −
Sh( β1 y ) Ch( β1 ) + a78 ( y 2 − 1) Sh( β1 y ) + Sh( β1 )
Sh( β1 y ) Sh( β1 y ) ) + a82 ( y − ) Sh( β1 ) Sh( β1 )
where a1, a2 - - -
a82 are constants.
4 . NUSSELT NUMBER The rate of heat transfer (Nusselt Number) is given by Nu y =± i = (
dθ ) y = ±1 dy
and corresponding expressions are Nu y = + 1 = b 27 + δ b 33 + δ 2 b 39 Nu
y = −1
= b 28 + δ b 34 + δ 2 b 40
where b1, b2 - - - - ,b40 are constants. Table 1 Nusselt Number(Nu) at y=1 P=0.71,N1=0.4
g
I
II
III
IV
V
VI
VII
103
-4.69955
-4.74269
-4.76758
-4.73867
-4.7547
-2.94674
-2.29748
2x103
-4.53249
-4.66412
-4.73637
-4.65174
-4.70027
-2.88639
-2.26177
3c103
-4.38299
-4.59495
-4.70993
-4.5751
-4.65272
-2.83307
-2.23032
-103
-5.08633
-4.92807
-4.84426
-4.94341
-4.88419
-3.08853
-2.37987
-2x103
-5.30605
-5.03486
-4.88973
-5.06121
-4.95927
-3.16998
-2.42716
-3x103
-5.54333
-5.15107
-4.93996
-5.18931
-5.04122
-3.25845
-2.47841
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
488
B. Devika Rani, et al., J. Comp. & Math. Sci. Vol.2 (3), 483-492 (2011) Table 2 Nusselt Number(Nu) at y=-1 P=0.71,N1=0.4
G 103 2x103 3x103 -103 -2x103 -3x103 M D-1
Îą
I 4.7 4.557 4.455 5.11 5.377 5.685 2 103 2
II 4.729 4.643 4.573 4.948 5.082 5.232 4 103 2
III 4.735 4.673 4.618 4.879 4.961 5.051 6 103 2
IV 4.76 4.635 4.562 4.963 5.109 5.272 2 2x103 2
V 4.734 4.663 4.602 4.909 5.012 5.126 2 3x103 2
VI 2.942 2.883 2.837 3.099 3.198 3.309 2 103 4
VII 2.292 2.253 2.221 2.388 2.445 2.509 2 103 6
Table 3 Nusselt Number(Nu) at y=1 P=0.71,M=2 G 103 2x103 3103 -103 -2x103 -3x103
I -2.84072 -1.77207 -0.33779 -3.88114 -3.85292 -3.45906
II -4.69955 -4.53249 -4.38299 -5.08633 -5.30605 -5.54333
III -6.09001 -5.98558 -5.90418 -6.36793 -6.54142 -6.73794
IV -7.18753 -7.08302 -6.98729 -7.42287 -7.55371 -7.69332
Table 4 Nusselt Number(Nu) at y=-1 P=0.71,M=2 G 103 2x103 3x103 -103 -2x103 -3x103 N1
I 2.89 1.887 0.534 3.847 3.801 3.405 0.1
II 4.7 4.557 4.455 5.11 5.377 5.685 0.5
III 6.065 5.963 5.911 6.419 6.671 6.972 1.5
IV 7.14 7.014 6.924 7.497 7.729 7.996 5
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
B. Devika Rani, et al., J. Comp. & Math. Sci. Vol.2 (3), 483-492 (2011)
Fig. (2) Variation of u with D-1
Fig. (1) Variation of u with G
I G 102
N1=0.5,M=2,D-1=102 G=1 II III IV V VI 2x102 3x102 -102 -2x102 -3x102
Fig. (3) Variation of u with α
N1=0.5,M=2,D-1=102 G=102 I II III α 2 4 6
489
D-1
N1=0.5,M=2,G=102, α=2 I II III IV V 102 102 102 102 3x102
Fig. (4) Variation of u with M
N1=0.5,D-1=102, G=102 I II III M 2 4 6
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
VI 5x102
490
B. Devika Rani, et al., J. Comp. & Math. Sci. Vol.2 (3), 483-492 (2011)
Fig(6) Variation of θ with G
Fig. (5) Variation of u with N1
M=2,D-1=102 G=102, α=2 I II III IV V N1 0.1 0.3 0.5 1.5 5
VI 10
VII 100
Fig. (7) Variation of θ with D-1 2
D-1
N1=0.5,M=2,G=10 , α=2 I II III IV V VI 102 102 102 102 3x102 5x102
I G 102
N1=0.5,M=2,D-1=102 G=102, α=2 II III IV V VI 2x102 3x102 -102 -2x102 -3x102
Fig. (8) Variation of θ with α
N1=0.5,M=2,D-1=102 G=102 I II III α 2 4 6
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
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B. Devika Rani, et al., J. Comp. & Math. Sci. Vol.2 (3), 483-492 (2011)
Fig. (9) Variation of θ with M
N1=0.5,D-1=102, G=102, α=2 I II III M 2 4 6
5. CONCLUSIONS The effect of radiation on non-darcy convection heat transfer flow of a viscous electrically conducting fluid through a porous medium in a vertical channel is analysed.The important conclusions are (i) The region of transition of velocity from negative to positive enlarges with increase in |G|(≶0). (ii) Lesser the permeability of the porous medium/ higher the Lorentz force larger |u| in the entire flow region. (iii) An increase in the radiation parameter α, results in enhancement in |u| except in a narrow region abutting y=1. (iv) The temperature enhances with |G|, D-1, M ≥ 6 and reduces with G>0 and M ≤ 4.Also it reduces with N1. (v) The rate of heat transfer reduces wih G>0 and enhances G<0. |Nu| at y= ±
Fig. (10) Variation of θ with N1
N1
M=2,D-1=102 G=102, α=2 I II III IV V VI 0.1 0.3 0.5 1.5 5 10
VII 100
1enhances with M & D-1 for G>0 and reduces with M & D-1 for G<0. (vi) |Nu| exhibits an increasing tendency with radiation prarmeter N1 at both the walls. 6. REFERENCES 1. Alboussiere T. Garandet J.P, Moreau R. Buoyancy-driven convection with a uniform magnetic field. Part1, asymptotic analysis. J Fluid Mech; 253:545-63 (1983). 2. Akiyama M, Chong Q.P. Numerical analysis of natural convection with surface radiation in a square enclosure. Numeric Heat Transfer Part A; 32:41933 (1997). 3. Chamkha A.J., Issa C., Khanfer K. Natural convection from an inclined plate embedded in a variable porosity
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