- Email: [email protected]
A Numerical Study of Convec1tive Heat Transfer through a Porous Medium in a Vertical Channel with Radiation Effect
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 127 (2015) 1285 – 1291
International Conference Computational Heat and Mass Transfer - 2015
A Numerical Study of Convec1tive Heat Transfer through a Porous Medium in a Vertical Channel with Radiation Effect S.T.Dinesh Kumara, Dr.P.Raveendra Nathb a
Department of Mathematics, Govt.Science college, Chitradurga, Karnataka - 577 501, India Email:[email protected] b
Sri Krishnadevaraya University College of Engineering and Technology, S.K. University, Anantapur - 515 00, India Email: [email protected]
Abstract We analyze the effect of Radiation on the Convective Heat Transfer flow of a viscous electrically conducting fluid in a vertical channel bounded by the flat plates at Heat sources taking the slope
δ
x = ± L. Which are maintained at Non uniform Temperature in the presence of
of the boundary temperature the Non-linear coupled equations governing the flow and Heat
Transfer are solved by employing a regular perturbation technique .We notice an enhancement in the vicinity of the boundary
η = ±1
and depreciates marginally in the central region of the flow. In this analysis we investigate the effect of Radiation on
Convective Heat Transfer flow of a Viscous Incompressible electrically conducting fluid in a Non-Uniformly vertical channel. We take the Prandtl Number P=0.71 and į = 0.01.The Velocity components u ,w and The Non-dimensional Temperature distribution ș and the rate of heat transfer has been calculated numerically for different values of the governing parameters. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
© 2015 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review underresponsibility responsibility of the organizing committee of ICCHMT Peer-review under of the organizing committee of ICCHMT – 2015 – 2015. Key words: Heat Transfer, Radiation effect, Vertical channel.
1. Introduction. The magneto hydrodynamic heat transfer has gained significance in recent times owing to its applications in recent advancement of space technology. The process of free convection as a mode of heat transfer has wide applications in the fields of chemical Engineering, Aeronautics and Nuclear power generation.
* Corresponding author. Tel.: +91 9980868622 E-mail address: [email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT – 2015
doi:10.1016/j.proeng.2015.11.485
1286
S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
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 [1]. 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. During the past several decades, a number of experiments and numerical computations have been presented for describing the phenomenon of natural (or mixed) convection in channels or enclosures. These studies aimed at clarifying the effect of mixed convection on flow and temperature regimes arising from variations in the shape of the channel (or enclosure),in fluid properties, in the transition to turbulence, etc Recently, Ghaly [2], Chamkha et al.[3], Yussyo El-Dib and Ghaly[4], analyze the effect of radiation heat transfer on flow and thermal fields in the presence of a magnetic field for horizontal and inclined plates. Shohel Mahmud [5] studied the effects of radiation heat transfer on magneto hydrodynamic mixed convection through a vertical channel packed with fluid saturated porous substances. The unsteady flow of a rotating viscous fluid in a channel maintained by non-tensional oscillations of one or both the boundaries has been studied by several authors to analyze the growth and development of boundary layers associated with geothermal flows for possible applications in geophysical fluid dynamics [6-10]. From the literature survey we can see that the majority of published papers use only the convective heat transfer through a porous medium with Radiation effect mode in their research whereas the effect of Radiation exists in vertical channel. The study of the Radiation effect offers a good comprehension on magneto hydrodynamic convection through vertical channel. The present work deals with the problem of heat transfer through a porous medium in a vertical channel with Radiation effect under the influence of an inclined magnetic field of intensity. 2. Formulation of the problem We consider the steady flow of an incompressible, viscous ,electrically conducting fluid confined in a vertical channel bounded by two flat walls under the influence of an inclined magnetic field of intensity Ho lying in the plane (y-z).The magnetic field is inclined at an angle α to the axial direction k and hence its components are
(0, H 0 Sin(α ), H 0 Cos (α )) .The walls are maintained at non-uniform temperature. In view of the non-uniform boundary temperature the velocity field has components(u,0,w)The magnetic field in the presence of fluid flow induces the current( ( J x ,0, J z ) .We choose a rectangular Cartesian co-ordinate system O(x,y,z) with z-axis in the vertical direction and the walls at x = ± L . When the strength of the magnetic field is very large we include the Hall current so that the generalized Ohm’s law is modified to
J + ω eτ e JxH = σ ( E + μ e q xH )
(1)
where q is the velocity vector, H is the magnetic field intensity vector, E is the electric field, J is the current density vector,
ω e is the cyclotron frequency, τ e
is the electron collision time,σ is the fluid conductivity and
μ e is the
magnetic permeability. Neglecting the electron pressure gradient, ion-slip and thermo-electric effects and assuming the electric field E=0. Then We choose a Cartesian co-ordinate system O(x,y,z) such that the plates are at z=0 and z=L and the z-axis coincide with the axis of rotation of the plates .The steady hydro magnetic boundary layer equations of motion with respect to a rotating frame moving with angular velocity Ω under Boussinesq approximations are
J x − m H 0 J z Sin( α ) = −σμ e H 0 wSin(α )
(2)
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S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
J z + m H 0 J x Sin( α ) = σμ e H 0 u Sin(α )
(3)
where m= ω eτ e is the Hall parameter. On solving equations (2)&(3) we obtain
§ σμ e H 0 Sin(α ) · ¸(mH 0 Sin(α ) − w) J x = ¨¨ 2 2 2 ¸ © 1 + m H 0 Sin (α ) ¹ § σμ e H 0 Sin(α ) · ¸ (u + mH 0 w Sin(α ) ) J z = ¨¨ 2 2 2 ¸ © 1 + m H 0 Sin (α ) ¹
(4)
(5)
where u,w are the velocity components along x and z directions respectively, On introducing the following non-dimensional variables
( x ′, z ′) =
T − Te ψ ( x, z ) ,ψ ′ = ,θ = qL ΔTe L
The equations of momentum and energy in the presence of heat generating sources in the non-dimensional form are
§ § ∂ψ § § G ·§ ∂θ · · ∇ 4ψ − M 12 ∇ 2ψ + ¨¨ ¨ ¸¨ ¸ ¸¸ = R¨¨ ¨ © © R ¹© ∂x ¹ ¹ © © ∂z
2 ·§¨ ∂ (∇ ψ ) ·¸ § ∂ψ ¸¨ ¸−¨ ¹© ∂x ¹ © ∂x
2 ·§¨ ∂ (∇ ψ ) ·¸ ·¸ ¸¨ ¸ ¹© ∂z ¹ ¸¹
§ § ∂ψ · § ∂θ · § ∂ψ ·§ ∂θ · · 2 PR ¨¨ ¨ ¸¨ ¸ − ¨ ¸¨ ¸ ¸¸ = ∇ θ − αθ ∂ x ∂ z ∂ z ∂ x ¹© ¹ © ¹© ¹ ¹ ©©
(7)
where
§ βg ΔTe L3 · ¸ G = ¨¨ 2 ¸ ν © ¹
(Grashof Number)
§ qL · R = ¨ ¸ (Reynolds Number) ©ν ¹ § QL2 α =¨ ¨K © f
§ σμ e2 H o2 L2 · ¸ (Hartman Number) M = ¨¨ 2 ¸ ν © ¹ § μ Cp · ¸ (Prandtl Number) P=¨ ¨ K ¸ f ¹ © 2
· ¸ (Heat Source Parameter) ¸ ¹
The corresponding boundary conditions are
ψ ( f ) − ψ (− f ) = 1 ∂ψ ∂ψ = 0, = 0, θ = γ (δz ) ∂z ∂x ∂ψ ∂ψ = 0, = 0, θ = γ (δz ) ∂z ∂x
at x = − L
at x = + L
3. Method of solution Introduce the transformation such that z = δ z , Then
∂ ∂ ≈ O(δ ) → ≈ O(1) ∂z ∂z
(6)
∂ ∂ =δ ∂z ∂z
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S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
For small values of δ <<1,the flow develops slowly with axial gradient of order δ and hence we take
∂ ≈ O(1). ∂z
Using the above transformation the equations governing equations reduces to
§ ∂ψ ∂ ( F 2ψ ) ∂ψ ∂ ( F 2ψ ) · § § G · § ∂θ · · ¸¸ F ψ − M F ψ + ¨¨ ¨ ¸ ¨ ¸ ¸¸ = δR¨¨ − ∂ ∂ R x z x x z ∂ ∂ ∂ © ¹ © ¹ © ¹ © ¹ 4
2 1
2
§ § ∂ψ ·§ ∂θ · § ∂ψ · § ∂θ · · 2 ¸¨ ¸ − ¨ ¸ ¨ ¸ ¸¸ = F θ − αθ x z z x ∂ ∂ ∂ ∂ ¹© ¹ © ¹ © ¹¹ ©© ∂ ∂ 2 Where F = +δ 2 2 2 ∂x ∂z
δPR¨¨ ¨
(8)
(9)
Assuming the slope δ of the wavy boundary to be small we take
ψ ( x, z ) = ψ 0 ( x, y ) + δψ 1 ( x, z ) + δ 2ψ 2 ( x, z ) + ................ θ ( x, z ) = θ o ( x, z ) + δθ 1 ( x, z ) + δ 2θ 2 ( x, z ) + ...................
(10)
Substituting (10) in equations (8)&(9) and equating the like powers of δ the equations and the respective boundary conditions to the zeroth order are
∂ 2θ 0 ∂y 2
− α1 θ 0 = 0
2 § ∂ 4ψ 0 2 ∂ ψ0 ¨ − M ( ) 1 ¨ ∂y 4 ∂y 2 ©
(11)
· § G § ∂θ ¸ = −¨¨ §¨ ·¸¨ 0 ¸ © © R ¹© ∂z ¹
·· ¸ ¸¸ ¹¹
(12)
with
ψ 0 (+1) − ψ 0 (−1) = 1 ∂ψ 0 ∂ψ 0 = 0, = 0, ∂y ∂z ∂ψ 0 ∂ψ 0 = 0, = 0, ∂z ∂y
θ 0 = γ (z )
at x = −1
θ 0 = γ (z )
at x = +1
(13)
and to the first order are
§ § ∂ψ ·§ ∂θ · § ∂ψ ·§ ∂θ · · § ∂ 2θ1 · ¨¨ 2 − αθ1 ¸¸ = PR ¨¨ ¨ 0 ¸¨ 0 ¸ − ¨ 0 ¸¨¨ 0 ¸¸ ¸¸ © ∂y ¹ © © ∂x ¹© ∂z ¹ © ∂z ¹© ∂y ¹ ¹ § § ∂ψ ·§ ∂ 3ψ · § ∂ψ ·§ ∂ 3ψ 0 · · § ∂ 4ψ 1 ∂ 2ψ 1 · § § G ·§ ∂θ · · ¸¸ ¨¨ 4 − ( M 12 ) ¸¸ = −¨ ¨ ¸¨ 1 ¸ ¸ + R ¨¨ ¨¨ 0 ¸¸¨¨ 30 ¸¸ − ¨ 0 ¸¨¨ 2 ¸¸ ∂y ¹ © © R ¹© ∂z ¹ ¹ © ∂y © © ∂y ¹© ∂z ¹ © ∂z ¹© ∂x∂y ¹ ¹
(14) (15)
with
ψ 1 (+1) − ψ 1 (−1) = 0 ∂ψ 1 ∂ψ 1 = 0, = 0, ∂y ∂z ∂ψ 1 ∂ψ 1 = 0, = 0, ∂y ∂z
θ1 = 0
at x = −1
θ1 = 0
at x = +1
(16)
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S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
Where N 2 =
3N 1 , α 1= αN 2 , P1 = PN 2 3N 1 + 4
4. Rate of Heat Transfer The rate of Heat transfer (Nusselt Number) on the walls has been calculated using the formula 1 § ∂θ · 1 ¨ ¸ Nu = where θ m = 0.5 ³ θ dy (θ m − θ w ) ¨© ∂y ¸¹ y = ±1 −1
and the corresponding expressions are
( Nu ) y = −1 =
1 1 (− a92 + δa94 ) ( N u ) y =+1 = (a92 + δ a93 ) (θ m − 1) θm
5. Analysis of the Numerical results In this analysis we investigate the effect of Radiation on Convective Heat Transfer flow of a Viscous Incompressible electrically conducting fluid in a Non-Uniformly vertical channel. We take the Prandtl Number P=0.71 and į = 0.01.The Velocity components u, w and The Non-dimensional Temperature distribution ș are shown in Figs.1-3 with Radiation Parameter N1. The Secondary velocity(u) which is due to the Non-uniform Temperature on the boundary is depicted in Fig.1.It shows that an increase in N1 results in an enhancement in u . The effect of radiation parameter N1 on the axial flow w is shown in Fig.2.It is found that an increase in N15, w depreciates in the left region and enhances in the right region and for higher N110. w depreciates in the flow region except in the narrow region adjacent the lower boundary y=-1 and this region reduces with increase in N1. The variation of ș with Radiation parameter N1 .It is found that ș reduces with increase in N1 the inclusion of the radiative Heat Transfer leads to the remarkable depreciation in the Non-dimensional Temperature ș (Fig.3). The average Nusselt Number (Nu) which represents the measures the rate of Heat Transfer at η = ± 1 is shown in Tables.1 and 2 for different variations of G,Į1, and N1 . We find that the rate of Heat Transfer experiences an enhancement with increase in the amplitude Į 1 and the Radiation parameter N1 (Tables.1 and 2) 0.4
0
0.04
-1 0.03
-0.5
0
0.5
1
-0.05
0.35
0.02 I
0.01 u
II
0 -1
-0.5
-0.01
-0.1
0
0.5
1
v
-0.15
II
III
III
IV
IV
-0.2
I
I
0.3
-0.04
Fig. 1 : u with N1 I II III IV N1 2.5 5 10 100
III
0.25
IV
0.2
-0.02 -0.03
II
θ
-0.25
0.15 -0.3
Fig. 2 : w with N1 I II III IV N1 2.5 5 10 100
Table - 1
-1
-0.5
0
Fig. 3 : ș with N1 I II III IV N1 2.5 5 10 100
0.5
1
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S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
η =+1 P=0.71, R=35, α=2, M=10, m=0.5, x= π /4
Table.1 Average Nusselt Number (Nu) at
G
I
II
III
IV
V
103
-0.2144
0.8542
0.9548
3.0022
13.9736
2X103
-0.2363
0.8018
0.9403
2.9918
13.9603
-103
-0.1694
0.9589
0.9840
3.0230
14.0002
-2X103
-0.1461
1.0112
0.9987
3.0334
14.0135
α1
0.30
0.70
0.50
0.50
0.50
N1
1.5
1.5
2.5
5
10
Table - 2 Average Nusselt Number (Nu) at
η =-1
P=0.71, R=35, α=2, M=10, m=0.5, x= π /4 G
I
II
III
IV
V
103
0.2965
-0.7777
-0.4210
1.0729
11.5777
2X103
0.3176
-0.7252
-0.4076
1.0796
11.5776
-103
0.2531
-0.8824
-0.4479
1.0594
11.5781
-2X103
0.2308
-0.9347
-0.4615
1.0526
11.5783
α1
0.30
0.70
0.50
0.50
0.50
N1
1.5
1.5
2.5
5
10
6. Conclusions The variation of u with Radiation Parameter N1 shows that an increase in N1 results in an enhancement in The effect of radiation parameter N1 on the axial flow w shows that an increase in N15, region and enhances in the right region and for higher N110.
u .
w depreciates in the left
w depreciates in the flow region except in the
narrow region adjacent the lower boundary η = −1 and this region reduces with increase in N1. The variation of ș with Radiation parameter N1 .It is found that ș reduces with increase in N1 the inclusion of the radiative Heat Transfer leads to the remarkable depreciation in the Non-dimensional Temperature ș .
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References [1] S.Vedat Arpaci,A.Selamet,Shu-HsinKao, Introduction to heat transfer, New York, PrenticeHall(2000). [2] AY.Ghaly, Radiation effects on a certain MHD free-convection flow. Chaos Solitons and Fractals ,13 (2002)13:1843-1850. [3] A.J.Chamkha, C.Issa, K.Khanfer, Natural convection from an inclined plate embed in a variable porosity porous medium due to solar radiation, Int..Thermal Sci.41( 2002)73-81. [4] Yussyo El-Dib,AY.Ghaly, Nonlinear interfacial stability for magnetic fluids in porous media, Chaos Solitons and Fractals,18,Issue1,(2003) 5568. [5] Shohel Mahmud, Roydon Andrew Fraser, Mixed convection-radiation interaction in a
vertical channel Entropy generation, Energy 28
(2003) 1557-1577. [6] Krishna,D.V,Prasada rao,D.R.V,Ramachandra Murty,A.S:Hydromagnetic convection flow through a porous medium in a rotating channel., J.Engg.Phy. and Thermo.Phy,V.75(2),pp.281-291(2002) [7] Murthy,K,N,V.S:Oscillatory MHD flow past a flat plate.,Ind,J,Pure and Appl,Mathematics V.9,p.501(1979). [8]
Mohan,M
and
Srivastava,K.K:Combined
convection
flows
through
a
porous
channel
rotating
with
angular
velocity,Proc.Ind.Acad.Sci,V.87,p.14(1978). [9] Rao,D.R.V,Krishna,D.V and Debnath,L: Combined effect of free and forced convection on MHD flow in a rotating porous channel, Int. J.Maths and Math.Sci,V.5,pp.165-182(1982). [10]Sarojamma,G and Krishna,D.V: Transient hydro magnetic convective flow in a rotating channel with porous boundaries, Acta Mechanica, V.39, p. 277(1981).
ScienceDirect Procedia Engineering 127 (2015) 1285 – 1291
International Conference Computational Heat and Mass Transfer - 2015
A Numerical Study of Convec1tive Heat Transfer through a Porous Medium in a Vertical Channel with Radiation Effect S.T.Dinesh Kumara, Dr.P.Raveendra Nathb a
Department of Mathematics, Govt.Science college, Chitradurga, Karnataka - 577 501, India Email:[email protected] b
Sri Krishnadevaraya University College of Engineering and Technology, S.K. University, Anantapur - 515 00, India Email: [email protected]
Abstract We analyze the effect of Radiation on the Convective Heat Transfer flow of a viscous electrically conducting fluid in a vertical channel bounded by the flat plates at Heat sources taking the slope
δ
x = ± L. Which are maintained at Non uniform Temperature in the presence of
of the boundary temperature the Non-linear coupled equations governing the flow and Heat
Transfer are solved by employing a regular perturbation technique .We notice an enhancement in the vicinity of the boundary
η = ±1
and depreciates marginally in the central region of the flow. In this analysis we investigate the effect of Radiation on
Convective Heat Transfer flow of a Viscous Incompressible electrically conducting fluid in a Non-Uniformly vertical channel. We take the Prandtl Number P=0.71 and į = 0.01.The Velocity components u ,w and The Non-dimensional Temperature distribution ș and the rate of heat transfer has been calculated numerically for different values of the governing parameters. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
© 2015 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review underresponsibility responsibility of the organizing committee of ICCHMT Peer-review under of the organizing committee of ICCHMT – 2015 – 2015. Key words: Heat Transfer, Radiation effect, Vertical channel.
1. Introduction. The magneto hydrodynamic heat transfer has gained significance in recent times owing to its applications in recent advancement of space technology. The process of free convection as a mode of heat transfer has wide applications in the fields of chemical Engineering, Aeronautics and Nuclear power generation.
* Corresponding author. Tel.: +91 9980868622 E-mail address: [email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT – 2015
doi:10.1016/j.proeng.2015.11.485
1286
S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
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 [1]. 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. During the past several decades, a number of experiments and numerical computations have been presented for describing the phenomenon of natural (or mixed) convection in channels or enclosures. These studies aimed at clarifying the effect of mixed convection on flow and temperature regimes arising from variations in the shape of the channel (or enclosure),in fluid properties, in the transition to turbulence, etc Recently, Ghaly [2], Chamkha et al.[3], Yussyo El-Dib and Ghaly[4], analyze the effect of radiation heat transfer on flow and thermal fields in the presence of a magnetic field for horizontal and inclined plates. Shohel Mahmud [5] studied the effects of radiation heat transfer on magneto hydrodynamic mixed convection through a vertical channel packed with fluid saturated porous substances. The unsteady flow of a rotating viscous fluid in a channel maintained by non-tensional oscillations of one or both the boundaries has been studied by several authors to analyze the growth and development of boundary layers associated with geothermal flows for possible applications in geophysical fluid dynamics [6-10]. From the literature survey we can see that the majority of published papers use only the convective heat transfer through a porous medium with Radiation effect mode in their research whereas the effect of Radiation exists in vertical channel. The study of the Radiation effect offers a good comprehension on magneto hydrodynamic convection through vertical channel. The present work deals with the problem of heat transfer through a porous medium in a vertical channel with Radiation effect under the influence of an inclined magnetic field of intensity. 2. Formulation of the problem We consider the steady flow of an incompressible, viscous ,electrically conducting fluid confined in a vertical channel bounded by two flat walls under the influence of an inclined magnetic field of intensity Ho lying in the plane (y-z).The magnetic field is inclined at an angle α to the axial direction k and hence its components are
(0, H 0 Sin(α ), H 0 Cos (α )) .The walls are maintained at non-uniform temperature. In view of the non-uniform boundary temperature the velocity field has components(u,0,w)The magnetic field in the presence of fluid flow induces the current( ( J x ,0, J z ) .We choose a rectangular Cartesian co-ordinate system O(x,y,z) with z-axis in the vertical direction and the walls at x = ± L . When the strength of the magnetic field is very large we include the Hall current so that the generalized Ohm’s law is modified to
J + ω eτ e JxH = σ ( E + μ e q xH )
(1)
where q is the velocity vector, H is the magnetic field intensity vector, E is the electric field, J is the current density vector,
ω e is the cyclotron frequency, τ e
is the electron collision time,σ is the fluid conductivity and
μ e is the
magnetic permeability. Neglecting the electron pressure gradient, ion-slip and thermo-electric effects and assuming the electric field E=0. Then We choose a Cartesian co-ordinate system O(x,y,z) such that the plates are at z=0 and z=L and the z-axis coincide with the axis of rotation of the plates .The steady hydro magnetic boundary layer equations of motion with respect to a rotating frame moving with angular velocity Ω under Boussinesq approximations are
J x − m H 0 J z Sin( α ) = −σμ e H 0 wSin(α )
(2)
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S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
J z + m H 0 J x Sin( α ) = σμ e H 0 u Sin(α )
(3)
where m= ω eτ e is the Hall parameter. On solving equations (2)&(3) we obtain
§ σμ e H 0 Sin(α ) · ¸(mH 0 Sin(α ) − w) J x = ¨¨ 2 2 2 ¸ © 1 + m H 0 Sin (α ) ¹ § σμ e H 0 Sin(α ) · ¸ (u + mH 0 w Sin(α ) ) J z = ¨¨ 2 2 2 ¸ © 1 + m H 0 Sin (α ) ¹
(4)
(5)
where u,w are the velocity components along x and z directions respectively, On introducing the following non-dimensional variables
( x ′, z ′) =
T − Te ψ ( x, z ) ,ψ ′ = ,θ = qL ΔTe L
The equations of momentum and energy in the presence of heat generating sources in the non-dimensional form are
§ § ∂ψ § § G ·§ ∂θ · · ∇ 4ψ − M 12 ∇ 2ψ + ¨¨ ¨ ¸¨ ¸ ¸¸ = R¨¨ ¨ © © R ¹© ∂x ¹ ¹ © © ∂z
2 ·§¨ ∂ (∇ ψ ) ·¸ § ∂ψ ¸¨ ¸−¨ ¹© ∂x ¹ © ∂x
2 ·§¨ ∂ (∇ ψ ) ·¸ ·¸ ¸¨ ¸ ¹© ∂z ¹ ¸¹
§ § ∂ψ · § ∂θ · § ∂ψ ·§ ∂θ · · 2 PR ¨¨ ¨ ¸¨ ¸ − ¨ ¸¨ ¸ ¸¸ = ∇ θ − αθ ∂ x ∂ z ∂ z ∂ x ¹© ¹ © ¹© ¹ ¹ ©©
(7)
where
§ βg ΔTe L3 · ¸ G = ¨¨ 2 ¸ ν © ¹
(Grashof Number)
§ qL · R = ¨ ¸ (Reynolds Number) ©ν ¹ § QL2 α =¨ ¨K © f
§ σμ e2 H o2 L2 · ¸ (Hartman Number) M = ¨¨ 2 ¸ ν © ¹ § μ Cp · ¸ (Prandtl Number) P=¨ ¨ K ¸ f ¹ © 2
· ¸ (Heat Source Parameter) ¸ ¹
The corresponding boundary conditions are
ψ ( f ) − ψ (− f ) = 1 ∂ψ ∂ψ = 0, = 0, θ = γ (δz ) ∂z ∂x ∂ψ ∂ψ = 0, = 0, θ = γ (δz ) ∂z ∂x
at x = − L
at x = + L
3. Method of solution Introduce the transformation such that z = δ z , Then
∂ ∂ ≈ O(δ ) → ≈ O(1) ∂z ∂z
(6)
∂ ∂ =δ ∂z ∂z
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S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
For small values of δ <<1,the flow develops slowly with axial gradient of order δ and hence we take
∂ ≈ O(1). ∂z
Using the above transformation the equations governing equations reduces to
§ ∂ψ ∂ ( F 2ψ ) ∂ψ ∂ ( F 2ψ ) · § § G · § ∂θ · · ¸¸ F ψ − M F ψ + ¨¨ ¨ ¸ ¨ ¸ ¸¸ = δR¨¨ − ∂ ∂ R x z x x z ∂ ∂ ∂ © ¹ © ¹ © ¹ © ¹ 4
2 1
2
§ § ∂ψ ·§ ∂θ · § ∂ψ · § ∂θ · · 2 ¸¨ ¸ − ¨ ¸ ¨ ¸ ¸¸ = F θ − αθ x z z x ∂ ∂ ∂ ∂ ¹© ¹ © ¹ © ¹¹ ©© ∂ ∂ 2 Where F = +δ 2 2 2 ∂x ∂z
δPR¨¨ ¨
(8)
(9)
Assuming the slope δ of the wavy boundary to be small we take
ψ ( x, z ) = ψ 0 ( x, y ) + δψ 1 ( x, z ) + δ 2ψ 2 ( x, z ) + ................ θ ( x, z ) = θ o ( x, z ) + δθ 1 ( x, z ) + δ 2θ 2 ( x, z ) + ...................
(10)
Substituting (10) in equations (8)&(9) and equating the like powers of δ the equations and the respective boundary conditions to the zeroth order are
∂ 2θ 0 ∂y 2
− α1 θ 0 = 0
2 § ∂ 4ψ 0 2 ∂ ψ0 ¨ − M ( ) 1 ¨ ∂y 4 ∂y 2 ©
(11)
· § G § ∂θ ¸ = −¨¨ §¨ ·¸¨ 0 ¸ © © R ¹© ∂z ¹
·· ¸ ¸¸ ¹¹
(12)
with
ψ 0 (+1) − ψ 0 (−1) = 1 ∂ψ 0 ∂ψ 0 = 0, = 0, ∂y ∂z ∂ψ 0 ∂ψ 0 = 0, = 0, ∂z ∂y
θ 0 = γ (z )
at x = −1
θ 0 = γ (z )
at x = +1
(13)
and to the first order are
§ § ∂ψ ·§ ∂θ · § ∂ψ ·§ ∂θ · · § ∂ 2θ1 · ¨¨ 2 − αθ1 ¸¸ = PR ¨¨ ¨ 0 ¸¨ 0 ¸ − ¨ 0 ¸¨¨ 0 ¸¸ ¸¸ © ∂y ¹ © © ∂x ¹© ∂z ¹ © ∂z ¹© ∂y ¹ ¹ § § ∂ψ ·§ ∂ 3ψ · § ∂ψ ·§ ∂ 3ψ 0 · · § ∂ 4ψ 1 ∂ 2ψ 1 · § § G ·§ ∂θ · · ¸¸ ¨¨ 4 − ( M 12 ) ¸¸ = −¨ ¨ ¸¨ 1 ¸ ¸ + R ¨¨ ¨¨ 0 ¸¸¨¨ 30 ¸¸ − ¨ 0 ¸¨¨ 2 ¸¸ ∂y ¹ © © R ¹© ∂z ¹ ¹ © ∂y © © ∂y ¹© ∂z ¹ © ∂z ¹© ∂x∂y ¹ ¹
(14) (15)
with
ψ 1 (+1) − ψ 1 (−1) = 0 ∂ψ 1 ∂ψ 1 = 0, = 0, ∂y ∂z ∂ψ 1 ∂ψ 1 = 0, = 0, ∂y ∂z
θ1 = 0
at x = −1
θ1 = 0
at x = +1
(16)
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S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
Where N 2 =
3N 1 , α 1= αN 2 , P1 = PN 2 3N 1 + 4
4. Rate of Heat Transfer The rate of Heat transfer (Nusselt Number) on the walls has been calculated using the formula 1 § ∂θ · 1 ¨ ¸ Nu = where θ m = 0.5 ³ θ dy (θ m − θ w ) ¨© ∂y ¸¹ y = ±1 −1
and the corresponding expressions are
( Nu ) y = −1 =
1 1 (− a92 + δa94 ) ( N u ) y =+1 = (a92 + δ a93 ) (θ m − 1) θm
5. Analysis of the Numerical results In this analysis we investigate the effect of Radiation on Convective Heat Transfer flow of a Viscous Incompressible electrically conducting fluid in a Non-Uniformly vertical channel. We take the Prandtl Number P=0.71 and į = 0.01.The Velocity components u, w and The Non-dimensional Temperature distribution ș are shown in Figs.1-3 with Radiation Parameter N1. The Secondary velocity(u) which is due to the Non-uniform Temperature on the boundary is depicted in Fig.1.It shows that an increase in N1 results in an enhancement in u . The effect of radiation parameter N1 on the axial flow w is shown in Fig.2.It is found that an increase in N15, w depreciates in the left region and enhances in the right region and for higher N110. w depreciates in the flow region except in the narrow region adjacent the lower boundary y=-1 and this region reduces with increase in N1. The variation of ș with Radiation parameter N1 .It is found that ș reduces with increase in N1 the inclusion of the radiative Heat Transfer leads to the remarkable depreciation in the Non-dimensional Temperature ș (Fig.3). The average Nusselt Number (Nu) which represents the measures the rate of Heat Transfer at η = ± 1 is shown in Tables.1 and 2 for different variations of G,Į1, and N1 . We find that the rate of Heat Transfer experiences an enhancement with increase in the amplitude Į 1 and the Radiation parameter N1 (Tables.1 and 2) 0.4
0
0.04
-1 0.03
-0.5
0
0.5
1
-0.05
0.35
0.02 I
0.01 u
II
0 -1
-0.5
-0.01
-0.1
0
0.5
1
v
-0.15
II
III
III
IV
IV
-0.2
I
I
0.3
-0.04
Fig. 1 : u with N1 I II III IV N1 2.5 5 10 100
III
0.25
IV
0.2
-0.02 -0.03
II
θ
-0.25
0.15 -0.3
Fig. 2 : w with N1 I II III IV N1 2.5 5 10 100
Table - 1
-1
-0.5
0
Fig. 3 : ș with N1 I II III IV N1 2.5 5 10 100
0.5
1
1290
S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
η =+1 P=0.71, R=35, α=2, M=10, m=0.5, x= π /4
Table.1 Average Nusselt Number (Nu) at
G
I
II
III
IV
V
103
-0.2144
0.8542
0.9548
3.0022
13.9736
2X103
-0.2363
0.8018
0.9403
2.9918
13.9603
-103
-0.1694
0.9589
0.9840
3.0230
14.0002
-2X103
-0.1461
1.0112
0.9987
3.0334
14.0135
α1
0.30
0.70
0.50
0.50
0.50
N1
1.5
1.5
2.5
5
10
Table - 2 Average Nusselt Number (Nu) at
η =-1
P=0.71, R=35, α=2, M=10, m=0.5, x= π /4 G
I
II
III
IV
V
103
0.2965
-0.7777
-0.4210
1.0729
11.5777
2X103
0.3176
-0.7252
-0.4076
1.0796
11.5776
-103
0.2531
-0.8824
-0.4479
1.0594
11.5781
-2X103
0.2308
-0.9347
-0.4615
1.0526
11.5783
α1
0.30
0.70
0.50
0.50
0.50
N1
1.5
1.5
2.5
5
10
6. Conclusions The variation of u with Radiation Parameter N1 shows that an increase in N1 results in an enhancement in The effect of radiation parameter N1 on the axial flow w shows that an increase in N15, region and enhances in the right region and for higher N110.
u .
w depreciates in the left
w depreciates in the flow region except in the
narrow region adjacent the lower boundary η = −1 and this region reduces with increase in N1. The variation of ș with Radiation parameter N1 .It is found that ș reduces with increase in N1 the inclusion of the radiative Heat Transfer leads to the remarkable depreciation in the Non-dimensional Temperature ș .
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S.T.Dinesh Kumar and P.Raveendra Nath / Procedia Engineering 127 (2015) 1285 – 1291
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and
Srivastava,K.K:Combined
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flows
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a
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