Cmjv04i04p0282 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

Soret and Dufour Effect on Convective Heat and Mass Transfer Through a Porous Media in a Rectangular Cavity G. VENU GOPALA KRISHNA1, D. CHITTI BABU2 and D.R.V. PRASADA RAO3 1

Department of Mathematics, VRS & YRN College of Engg. & Tech., Chirala, INDIA. 2 Department of Mathematics, Government College(A), Rajahmundry, A.P., INDIA. 3 Department of Mathematics, S.K.University, Anantapur, A.P., INDIA. (Received on: July 12, 2013) ABSTRACT In this paper an attempt has been made to discuss the combined influence of Soret and Dufour effect on the convective heat and mass transfer flow of a viscous fluid through a porous medium in a rectangular cavity using Darcy model. Making use of the incompressibility the governing non-linear coupled equations for the momentum, energy and diffusion are derived in terms of the non-dimensional stream function, temperature and concentration. These coupled matrices are solved using iterative procedure and expressions for the stream function, temperature and concentration are obtained as linear combinations of the shape functions. The behaviour of temperature, concentration, Nusselt number and Sherwood number are discussed computationally for different values of the governing Parameters. Keywords: Convective heat transfer, mass transfer, porous medium, radiation. Mathematics Subject Classification Nos. 76R10,76S05.

1. INTRODUCTION The investigation of heat transfer in enclosures containing porous media begin

with the experimental work of Verschoor and Greebler16. Verschoor and Greebler16 were followed by several other investigators interested in porous media heat transfer in

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

rectangular enclosures. In particular Bankvall2-4 has published a great deal of practical work concerning heat transfer by natural convection in rectangular enclosures completely filled with porous media. Burns, P.J.Chow5 have described a porous medium heat transfer flow in a rectangular geometry. Cheng et al.6 have studied the flow and heat transfer rate in a rectangular box with solid walls using a Brinkman model the box is differentially heated in the horizontal direction. Chan et al.7 have considered enclosures with aspect ratio greater than or equal to one. Their numerical computations indicate that when Darcy number based on the width of the enclosures is less than 10-9, Darcyâ&#x20AC;&#x2122;s law and the Brinkman equation virtually the same results for the heat transfer rate. Joseph et al.10 have considered laminar forced convection in rectangular channels with unequal heat addition on adjacent sides. Teomann Ayhan, Hayati et al.15 have considered heat transfer and flow structure in a rectangular channel with wing-type Vortex Generator. Ham-Chien Chiu et al.9 have discussed mixed convection heat transfer in horizontal rectangular ducts with radiation effects. Chittibabu et al.8 has discussed convective flow in a porous rectangular duct with differentially heated side wall using Brinkman model. Badruddin et al.1 have investigated the radiation and viscous dissipation on convective heat transfer in porous cavity. Recently Padmavathi11 has analyzed the connective heat transfer through a porous medium in a rectangular cavity with heat sources and dissipation under varied conditions. Rangareddy12 has discussed the natural convective Heat and Mass transfer in Porous Rectangular Cavity

with a differentially heated side walls using Brinkman model. By using Galerkin finite element analysis, the governing equations are solved. Sivaiah et al.14 have investigated double-diffusive convective Heat transfer flow of a viscous fluid through a porous medium with rectangular duct with thermodiffusion by using finite element technique. Reddaih et al.13 have analyzed the effect of viscous dissipation on convective heat and mass transfer flow of a viscous fluid in a duct of rectangular cross section by employing Galerkin finite element analysis. In this paper an attempt has been made to discuss the combined influence of Soret and Dufour effect on the convective heat and mass transfer flow of a viscous fluid through a porous medium in a rectangular cavity using Darcy model. The Galerkin finite element analysis with linear triangular elements is used to obtain the Global stiffness matrices for the values of stream function, temperature and concentration. These coupled matrices are solved using iterative procedure and expressions for the stream function, temperature and concentration are obtained as linear combinations of the shape functions. The behaviour of temperature, concentration, Nusselt number and Sherwood number are discussed computationally for different values of the governing Parameters. 2. FORMULATION We consider the mixed convective heat and mass transfer flow of a viscous incompressible fluid in a saturated porous medium confined in the rectangular duct (Fig. 1) whose base length is a and height b.

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

The heat flux on the base and top walls is maintained constant. The Cartesian coordinate system O(x,y) is chosen with origin on the central axis of the duct and its base parallel to x-axis. We assume that i) The convective fluid and the porous medium are everywhere in local thermodynamic equilibrium. ii) There is no phase change of the fluid in the medium. iii) The properties of the fluid and of the porous medium are homogeneous and isotropic. iv) The porous medium is assumed to be closely packed so that Darcy’s momentum law is adequate in the porous medium. v) The Buoyancy approximation is applicable. Under these assumption the governing equations are given by

∂u ′ ∂v ′ + =0 ∂x ′ ∂y ′ k  ∂p ′  u′ = −   µ  ∂x′   k  ∂p ′ v′ = −  + ρ′g  µ  ∂y ′  

ρ σ c p  u ′

∂(q r ) µ +  (u 2 + v 2 ) − ∂x K 2 2 D kT  ∂ T ∂ T   + 1 1  2 + C s C p  ∂x ′ ∂y ′ 2 



D1 kT Tm

 ∂ 2C ∂ 2C  ∂C  ∂C +  = D1  2 + + v′ ∂x ′ ∂y ′  ∂y ′ 2  (2.5)  ∂x ′

 ∂ 2T ∂ 2T   2 +  − k ′C ∂y ′ 2   ∂x ′

ρ ′ = ρ 0 {1 − β (T ′ − T0 ) − β • (C ′ − C 0 )}

T0 =

(2.6)

Th + Tc C + Cc , C0 = h 2 2

The boundary conditions are u′ = v′ = 0 on the boundary of the duct T′ = Tc ,C = Cc on the side wall to the left T′ = Th ,C = Ch on the side wall to the right ∂T ′ =0, ∂y

∂C =0 ∂y

(2.7)

on the top ( y = 0) and bottom

u=v=0

(2.1)

Walls(y = 0) which are insulated. Invoking Rosseland approximation radiation

(2.2)

qr =

(2.3)

Expanding T4 in Taylor’s series about Te and neglecting higher order terms

for

4σ ∗ ∂T ′ 4 3β R ∂y

T ′ 4 ≅ 4Te3T − 3Te4

 ∂ 2T ′ ∂ 2T ′  ∂T ′ ∂T ′    = K 1  + v′ + 2 ∂x ′ ∂y ′  ∂y ′ 2   ∂x ′

 + Q(T0 − T ) +



ρ σ c p  u ′

284

(2.4)

We now introduce the following nondimensional variables x′ = ax; ; y′ = by ; c = b/a u′ = (ν/a) u ; v′ = (ν/a)v ; p′ = (ν2ρ/a2)p T′ = T0 + θ (Th – Tc) C′ = C0 + φ (Th – Tc) (2.8) The governing equations in the nondimensional form are

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

 K  ∂p u = − 2   a  ∂x v=−

(2.9)

k ∂p kag kagβ (Th − Tc )θ − + a 2 ∂y v 2 v2

kagβ • (Ch − Cc )φ + v2  ∂θ ∂θ P  u +v ∂y  ∂x

  4N  =  1 + 3  

2 2  ∂ θ ∂ θ  2 + 2 ∂y  ∂ x

∂ φ ∂ φ + E C u 2 + v 2 + DuP  2 + 2 ∂y  ∂x

(

)

   (2.11)

  

 ∂φ ∂φ   ∂ 2φ ∂ 2φ   Sc u + v  =  2 + 2  ∂ x ∂y   ∂x ∂y    ∂ 2θ ∂ 2θ  + Sc So 2 + 2  − kφ ∂y   ∂x

(2.10)

(2.12)

∂ψ ; ∂y

v=−

∂ψ ∂x

(2.13)

Eliminating p from the equation (2.9) and (2.10) and making use of (2.11) the equations in terms of ψ and θ are

∇ 2ψ = − Ra (

∂θ ∂φ +N ) ∂x ∂x

 ∂ψ ∂θ ∂ψ ∂θ   − P ∂x ∂y   ∂y ∂x  4  ∂ 2θ ∂ 2θ   2 + 2  + = 1 + ∂y   3N 1  ∂x   ∂ψ   ∂ψ  2   + + EC       ∂y   ∂x     2 2 ∂ φ ∂ φ  + Du P 2 + 2  ∂y   ∂x 2

(2.16)

3. FINITE ELEMENT ANALYSIS AND SOLUTION OF THE PROBLEM The region is divided into a finite number of three node triangular elements, in each of which the element equation is derived using Galerkin weighted residual method. In each element fi the approximate solution for an unknown f in the variational formulation is expressed as a linear combination of shape function. i N k k = 1,2,3, Which are linear polynomials in x and y. This approximate solution of the unknown f coincides with actual values at each node of the element. The variational formulation results in a 3 x 3 matrix equation (stiffness matrix) for the unknown local nodal values of the given element. These stiffness matrices are assembled in terms of global nodal values using inter element continuity and boundary conditions resulting in global matrix equation. In each case there are r distinct global nodes in the finite element domain and fp (p = 1,2,……r) is the global nodal values of any unknown f defined over the domain then

( )

In view of the equation of continuity we introduce the stream function ψ as

 ∂ψ ∂φ ∂ψ ∂φ   Sc − ∂x ∂y   ∂y ∂x  ∂ 2φ ∂ 2φ   ∂ 2θ ∂ 2θ  =  2 + 2  + ScSo 2 + 2  − kφ ∂y  ∂y   ∂x  ∂x

(2.14)

(2.15)

f =∑ i =1

∑f p =1

Φ ip ,

Where the first summation denotes summation over s elements and the second one represents summation over the independent global nodes and

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

Φ ip = N Ni , if

p is one of the local nodes

say k of the element ei = 0, otherwise. fp’ s are determined from the global matrix equation. Based on these lines we now make a finite element analysis of the given problem governed by (2.14)- (2.16) subjected to the conditions. Let ψi , θi and φi be the approximate values of ψ ,θ and φ in an element θi.

ψ i = N 1i ψ 1i + N 2i ψ 2i + N 3i ψ 3i

(3.1a)

θ i = N 1i θ 1i + N 2i θ 2i + N 3i θ 3i

(3.1b)

φ = N 1i φ1i + N 2i φ 2i + N 3i φ 3i =

(3.1c)

Substituting the approximate value ψi , θi and φi for ψ ,θ and φ respectively in (2.13), the error  4  ∂ 2θ i ∂ 2θ i  2 + E1i = 1 + ∂y 2  3N 1  ∂x  ∂ψ i ∂θ i ∂ψ i ∂θ i  + − P − ∂x ∂y   ∂y ∂x

(3.2)

2 2   ∂ψ    +      ∂x   ∂ 2φ i ∂ 2φ i + DuP( 2 + ) ∂x ∂y 2

i 1

N ki dΩ = 0 ,

∫E

i 2

N ki dΩ = 0

 4  ∂ zθ i ∂ zθ i   2 +  N ki (1 + ∂y 2   3 N 1  ∂x ∫  ∂ψ i ∂θ i ∂ψ i ∂θ i  ei =  − P − ∂x ∂y   ∂y ∂x   ∂ψ +  E C    ∂y

2 2   ∂ψ    +      ∂x  

(3.4)

∂ 2φ i ∂ 2φ i + ))dΩ = 0 ∂x 2 ∂y 2  ∂ zφ i ∂ zφ i   N ki ( 2 + ∂y 2   ∂x

+ DuP (

∫

 ∂ψ i ∂φ i ∂ψ i ∂φ i   − Sc − ∂x ∂y   ∂y ∂x

 ∂ zθ i ∂ z θ i + ScSo 2 + ∂y 2  ∂x

(3.5)

  − kφ i ) dΩ = 0 

Using Green’s theorem we reduce the surface integral (3.4) & (3.5) without affecting ψ terms and obtain  4  ∂N ki ∂θ i ∂N ki ∂θ i 

∂ 2φ i ∂ 2φ i + ∂x 2 ∂y 2

 ∂ψ i ∂φ i ∂ψ i ∂φ i   − Sc − ∂ y ∂ x ∂ x ∂ y   2 i 2 i ∂θ ∂θ + ScSo( 2 + 2 ) − kφ i ∂x ∂y

∫E

ei =

 ∂ψ + EC   ∂y

E2i =

Under Galerkin method this error is made orthogonal over the domain of ei to the respective shape functions (weight functions) where

(3.3)

 + 1 + ∂y ∂y  3N 1  ∂x ∂x  i i  ∂ψ ∂θ ∂ψ i ∂θ i  i    − p N −  k i  ∂y ∂x ∂ x ∂ y N    k ∫ ei   2 2   ∂ψ   ∂ψ   + E C   +     ∂y   ∂x  

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

     dΩ      

287 =

∫

Γi

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

 ∂θ i ∂θ i  N  nx + n y dΓi ∂y  ∂x  i k

(3.6)

1 ei

 ∂N ki ∂φ i ∂N ki ∂φ i +  ∂y ∂y  ∂x ∂x   ∂ψ i ∂φ i ∂ψ i ∂φ i − Sc i N k  − ∂x ∂y i   ∂y ∂x ∫eiei N k  i i  ∂N k ∂θ i ∂N k ∂θ i + ) + ScSo( ∂x ∂x ∂y ∂y   − k φ i N i dΩ k  ∫  ei

∫

Γi

      +   dΩ      

 ∂θ i ScSo ∂φ i   + )n x  N ∂x ∂x dΓ N ki  i i  i ∂θ ∂θ  + ( + ScSo n y  ∂y ∂y  

1 ei

(3.7)

∂N Li ∂y

(l, m, k = 1,2,3) (3.8)

 dΩ  

∂N ki )dΩ ∂y

 ∂θ i  ∂θ i  =∫N nx + n y dΓi = Qki  ∂x  ∂y Γi   i k

 ∂N i ∂NLi ∂Nmi ∂NLi  dΩ − Sc∑ ψ mi ∫  m − ∂x ∂x ∂y  1 ei  ∂y i

∂N ki ∂N Li ∂x ∂x ∂N Li ∂N ki ) dΩ i − dΩ i + ScSo∑ θ i ∫ + ∂y ∂y ei (

k ∑ ∫ φ i N ki N Li = 0 1 ei

∫

Γi

∂N Li ∂N ki ∂y ∂y

 ∂N i ∂N Li ∂N mi − P ∑ ψ mi ∫  m − ∂x ∂x 1 ei  ∂y 2   ∂ψ   ∂ψ  2   +  + E C ∫      ∂ y ∂ x     ei   ∂N i ∂N Li ∂N Li + DuP ∑ ∫ φ i ( k + ∂x ∂x ∂y 1 ei

∂Nki ∂NLi ∂NLi ∂Nki + ) ∂x ∂x ∂y ∂y

 ∂θ  ∂θ i  + ScSo )n x  ∂x  ∂x  N ki  dΓi = QiC i ∂θ ∂θ i   + ( + ScSo n y  ∂y ∂y   i

Substituting L.H.S. of (3.1a)- (3.1c) for ψi , θi and φi in (3.6)&(3.7) we get i i  4 N  ∂N k ∂N L 1 +   3  ∂x ∂x 

∑∫

∑∫ φ i (

Repeating the above process with each of s elements, we obtain sets of such matrix equations. Introducing the global i coordinates and global values for θ p and making use of inter element continuity and boundary conditions relevant to the problem the above stiffness matrices are assembled to obtain a global matrix equation. This global matrix is r x r square matrix if there are r distinct global nodes in the domain of flow considered. In the problem under consideration, for computational purpose, we choose uniform mesh of 10 triangular elements. The domain has vertices whose global coordinates are (0,0), (1,0) and (1,c) in the non-dimensional form. Let e1, e2…..e10 be the ten elements and let θ1, θ2, …..θ10 be the global values of θ and ψ1, ψ2,……ψ10 be the global values of ψ at the ten global nodes of the domain.

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

The global matrix equation for θ is

A3 X 3 = B3

(4.1)

+ Ψ6 (

The global matrix equation for φ is

A4 X 4 = B4

(4.3)

Ψ =( Ψ3N53+ Ψ4N54+ Ψ5N55) H(1- τ3) + (Ψ3N43+ Ψ5N45+

c In the horizontal strip 0 ≤y ≤ 3 1

= Ψ1 (1-4x)+ Ψ24(x-

y 4y )+ Ψ7 ( (1- τ1 ) c c

1 ) 3

Ψ = ( Ψ2N32+ Ψ3N33+ Ψ6N36) H(1- τ2)+ (Ψ2N22+ Ψ7N27+ Ψ6N26)H(1- τ3) (

2 ≤ x≤1) 3 2y 4y =( Ψ3 (3-4x) + Ψ42(2x-1)+ Ψ6 ( c c 4y )+ Ψ5 (4x-3)+ 4x+3))H(1- τ3) + Ψ3 (1c 4y Ψ6 ( ))H(1- τ4) c 4y Ψ =( Ψ7 2(1-2x) + Ψ6 (4x-3)+ Ψ8 ( c Ψ6N46)H(1- τ4)

(

1))H(1- τ3) + Ψ6 (2(1-

2y 4y 4y )+ Ψ9 ( -1)+ Ψ8 (1+ c c c

4x))H(1- τ4) + Ψ6 (4(1-x)+ Ψ5 (4x-

1 1 ≤ x≤ ) 3 3

4y 2y -1)+ Ψ92( c c

1))H(1- τ5)

Ψ= (Ψ1N 1+ Ψ2N 2+ Ψ7N 7) H(1- τ1)

(0≤ x≤

4y 4y ))H(1- τ2 +( Ψ2 (1)+ Ψ7 (1+ c c

4y -4x)+ Ψ6 (4x-1))H(1- τ3) c

The global matrix equations are coupled and are solved under the following iterative procedures. At the beginning of the first iteration the values of (ψi) are taken to be zero and the global equations (4.1)&(4.2) are solved for the nodal values of θ and φ.These nodal values (θi) and (φi) obtained are then used to solve the global equation (4.3) to obtain(ψi).In the second iteration these (ψi)values are obtained are used in (4.1)&(4.2) to calculate (θi) and (φi) and vice versa. The three equations are thus solved under iteration process until two consecutive iterations differ by a pre assigned percentage.The domain consists three horizontal levels and the solution for Ψ & θ at each level may be expressed in terms of the nodal values as follows,

4y -1) c

(4.2)

The global matrix equation for ψ is

A5 X 5 = B5

=( Ψ22(1-2x) + Ψ3 (4x-

288

Along the strip

2c ≤ y≤1 3

Ψ =( Ψ8N98+ Ψ9N99+ Ψ10N910) H(1- τ6) (

2 ≤ x≤1) 3

Ψ8

(4(1-x)+

Ψ94(x-

y 4y )+ Ψ102( c c

3))H(1- τ6) where τ1= 4x ,τ2 = 2x ,

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

y y 4x , τ4= 4(x- ), τ5= 2(x- ) , c c 3 y 4 τ6 = (x- ) and H represents c 3 τ3 =

the Heaviside function.

In the horizontal strip 0≤ y≤

H(1- τ1)

(0≤ x≤

c 3

y 4y )+ θ7 ( )) c c

1 ) 3

θ = (θ 2(2(1-2x)+ θ3 (4x-

4y 4y -1)+ θ 6( )) c c

4y H(1- τ2) + θ 2(1)+ c 4y θ 7(1+ -4x)+ θ 6(4x-1))H(1- τ3) c 1 2 ( ≤ x≤ ) 3 3 2y 4y θ = θ 3(3-4x) +2 θ 4(2x-1)+ θ 6( c c 4x+3) H(1- τ3) +( θ 3(1-

θ 6(

4y )+ θ 5(4x-3)+ c

4y )) H(1- τ4) c

Along the strip

2y 4y 4y )+ θ 9( -1)+ θ 8(1+ c c c

4x)) H(1- τ4) +( θ 6(4(1-x)+

4y -1)+ c 4y θ9 2( -1)) H(1- τ5) c 2c Along the strip ≤ y≤1 3 y 4y θ = (θ84 (1-x) + θ 94(x- )+ θ 10( -3) c c 2 H(1- τ6) ( ≤ x≤ 1) 3 θ 5(4x-

The expressions for θ are

θ = [θ1(1-4x)+ θ2 4(x-

+( θ 6(2(1-

(

2 ≤ x≤1) 3

2c c ≤ y≤ 3 3

θ = (θ 7(2(1-2x)+ θ 6(4x-3)+

4y 1 2 θ 8( -1)) H(1- τ3) ( ≤ x≤ ) c 3 3

The expressions for φ are φ = [φ1(1-4x)+ φ2 4(xH(1- τ1) (0≤ x≤

y 4y )+ φ7 ( )) c c

1 ) 3

4y 4y -1)+ φ 6( )) c c 4y 4y )+ φ7(1+ -4x)+ φ H(1- τ2 + φ 2(1c c 1 2 ( ≤ x≤ ) 6(4x-1))H(1- τ3) 3 3 φ = (φ 2(2(1-2x)+ φ3 (4x-

The dimensionless Nusselt Numbers (Nu) and Sherwood Numbers (Sh) on the non-insulated boundary walls of the rectangular duct are calculated using the formula Nu = (

∂θ ∂φ ) x=1 and Sh = ( ) x=1 ∂x ∂x

Nusselt Number on the side wall x=1in different regions are Nu1=2-4θ3 ( 0 ≤ y ≤ h / 3)

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

Nu2=2-4θ6

( h / 3 ≤ y ≤ 2h / 3)

Nu3=2-4θ8

( 2 h / 3 ≤ y ≤ h)

Sherwood Number on the side wall x=1in different regions are Sh1=2-4φ3

( 0 ≤ y ≤ h / 3)

Sh2=2-4φ6

( h / 3 ≤ y ≤ 2h / 3)

Sh3=2-4φ8

( 2 h / 3 ≤ y ≤ h)

4. DISCUSSION OF THE NUMERICAL RESULTS In this analysis we discuss the effect of Soret and Dufour effects on the

0 0.666 -0.1

0.732

0.798

0.864

0.93

convective heat and mass transfer flow of a viscous fluid through a porous medium in a rectangular cavity. The equations governing the flow, heat and mass transfer are solved by employing Galerkin finite element analysis with linear triangular elements. The shape functions are bilinear in x and y. The non-dimensional temperature (θ) is shown in figures at different horizontal and vertical levels for different values of Schmidt number Sc, Soret parameter S0, Dufour parameter Du, buoyancy ratio N. We follow the convention that the nondimensional temperature is positive or negative according as the actual temperature is greater/lesser than the mean temperature T 0.

0 0.33 0.39 0.46 0.53 0.59 0.66 0.72 0.79 0.86 0.92 0.99 3 9 5 1 7 3 9 5 1 7 3 -0.05

0.996

-0.2 -0.1

-0.3 -0.4

I II

-0.5

III

II III -0.2

-0.6

-0.15

-0.7

-0.25

-0.8 -0.3

-0.9 -1

-0.35

Fig. 1 : Variation of θ with S0 at S0

I -1

2h 3 II -0.5

Fig. 2: Variation of θ with S0 at III 1

h 3

IV 0.5

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level

291

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013) 0.5

0.4

-0.1

0.066 0.132 0.198 0.264 0.33 0.396 0.462 0.528 0.594 0.66

-0.2

0.3

-0.3

0.2

-0.4

I 0.1

III

0 0

0.067

0.134

0.201

0.268

0.335

-0.1

I II

-0.5

III IV

-0.6 -0.7

-0.2

-0.8

-0.3

-0.9

-0.4

-1

Fig. 3 : Variation of θ with S0 at I 0.24

S0 0 0.666 -0.1

x= II 0.6

1 3

Fig. 4 : Variation of θ with S0 at III 1.3

IV 2.01

2 3

0.2

0.732

0.798

0.864

0.93

0.996 0 0.333 0.399 0.465 0.531 0.597 0.663 0.729 0.795 0.861 0.927 0.993

-0.2 -0.3

-0.2

-0.4

I II

-0.5

III

-0.6

-0.4

III IV

IV -0.6

-0.7 -0.8 -0.8

-0.9 -1

-1

Fig. 5 : Variation of θ with N at N 1

I II 2

2h 3

III -0.5

Fig. 6 : Variation of θ with N at

IV -0.8

h 3

0 0

0.067

0.134

0.201

0.268

0.066 0.132 0.198 0.264 0.33 0.396 0.462 0.528 0.594 0.66

-0.1

-0.05

-0.2 -0.1

-0.3 -0.15

-0.4

I II

-0.2

III

-0.25

I II

-0.5

III IV

-0.6 -0.7

-0.3

-0.8 -0.35

-0.9 -1

-0.4

Fig. 7 : Variation of θ with N at x = 1

I 1

II 2

III -0.5

Fig. 8 : Variation of θ with N at x = IV -0.8

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2 3

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013) 0.7

0 0.333 0.399 0.465 0.531 0.597 0.663 0.729 0.795 0.861 0.927 0.993

0.6

-0.05 0.5 I 0.4

III 0.3

-0.1

III IV -0.15

0.2

-0.2

0.1

0 0.666

0.732

0.798

0.864

0.930

-0.25

0.996

Fig. 9 : Variation of θ with Du at y = I 0.05

Du 0 0.000

II 0.08

2h 3

III 0.12

Fig. 10 : Variation of θ with Du at y = IV 0.15

h 3

V 0.25 0.4

0.067

0.134

0.201

0.268

0.335

-0.05

0.2

-0.1

0 0.000 0.066 0.132 0.198 0.264 0.330 0.396 0.462 0.528 0.594 0.660 I

-0.15

-0.2

I II

III

III IV

-0.4

-0.25

-0.6

-0.3

-0.8

-1

-0.35

Fig. 11 : Variation of θ with Du at I II 0.05

Du 0 0.666

0.732

0.798

0.864

III 0.08 0.930

1 3

IV 0.12

Fig. 12 : Variation of θ with Du at V 0.15

2 3

0.25 0 0.333 0.399 0.465 0.531 0.597 0.663 0.729 0.795 0.861 0.927 0.993

0.996

-0.1 -0.5 -0.2 -1

-0.3

III -0.4

I II

III -1.5

-0.5 -2 -0.6

-0.7

-2.5

Fig. 13 : Variation of C with Sc at I

y= II

2h 3

Fig. 14 : Variation of C with Sc at III

h 3

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293

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

1 3

Fig. 15 : Variation of C with Sc at x = I 0.24

II 0.6

Fig. 16 : Variation of C with Sc at x = III 1.3

0.8

IV 2.01

2 3

0.5

0.6

0 0.333 0.399 0.465 0.531 0.597 0.663 0.729 0.795 0.861 0.927 0.993

0.4

-0.5 I

0.2

0.732

0.798

0.864

0.930

III -1.5

0.996

III 0 0.666

-1

-0.2

-2

-0.4

-2.5

-0.6

-3

Fig. 17 : Variation of C with S0 at I -1

2h 3

II -0.5

Fig. 18 : Variation of C with S0 at III 1

IV 0.5

0.5

h 3

2.5 2

0 0.000

0.066

0.132

0.198

0.264

1.5

0.330

1 -0.5

0.5

-1

III IV -1.5

I II

0 0.000 0.066 0.132 0.198 0.264 0.330 0.396 0.462 0.528 0.594 0.660 -0.5

III IV

-1 -1.5

-2

-2 -2.5

-2.5

Fig. 19 : Variation of C with S0 at S0

I 0.24

II 0.6

III 1.3

1 3

Fig. 20 : Variation of C with S0 at

IV 2.01

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2 3

294

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013) 0 0.666

2.5 0.732

0.798

0.864

0.930

0.996

-0.2

1.5

-0.4

-0.6 -0.8

III

-1

0.5

IV -1.2

IV V

-1

-1.4

-1.5

-1.6

-2

-1.8

-2.5

Fig. 21 : Variation of C with Du at I 0.05

II 0.08

2h 3

III 0.12

Fig. 22 : Variation of C with Du at IV 0.15

h 3

V 0.25 -1.58 0.000 0.066 0.132 0.198 0.264 0.330 0.396 0.462 0.528 0.594 0.660

0.35 0.3

-1.6

0.25

-1.62

0.2

I -1.64

III

0 0.333 0.399 0.465 0.531 0.597 0.663 0.729 0.795 0.861 0.927 0.993 -0.5

0.15

III IV

-1.66

IV 0.1

III

V -1.68

0.05 0 0.666

-1.7

0.733

0.800

0.867

0.934 -1.72

-0.05

Fig. 23 : Variation of C with Du at Du

I 0.05

II 0.08

1 3

III 0.12

Fig. 24: Variation of C with Du at IV 0.15

2 3

V 0.25

From figs.1-4, we observe that the actual temperature reduces with increase in

levels and enhances at y =

2h and 3 1 h x = levels, while it enhances at y = 3 3

the buoyancy forces act in the same direction and for the forces acting in

levels with |S0|(><0). When the molecular buoyancy force N dominates over the thermal buoyancy force the actual

2 2h levels and reduces at y = level. 3 3 1 At x = level it reduces with N irrespective 3

S0>0 and enhances with S0<0 at y =

temperature reduces at y =

h 2 and x = 3 3

2h level when 3

opposite directions it enhances at y =

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h and 3

295

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

of the directions of the buoyancy forces (figs.5-8). The variation of θ with Dufour effect is shown in figs (9-12) at different level. It is found that the actual temperature

h 2 and x = levels 3 3 2h 1 and x = levels it y= 3 3

depreciates at y = while at

experiences an enhancement with increase in the Dufour parameter Du. The concentration distribution(C) is shown in figures for different parametric values. The variation of C with Schmidt Number Sc shows that lesser the molecular diffusivity larger the actual concentration at all levels (figs.13-16). With respect to Soret parameter S0, we find that the actual concentration reduces S0>0 and enhances with S0<0 at all horizontal levels (figs.17&18). At the vertical level x = depreciates with |S0|, while at x =

1 it 3

2 level, 3

the actual concentration reduces in the horizontal strip (0, 0.333) and enhances in the region (0.396, 0.594) and it experiences an enhancement with S0<0 in the entire flow region(figs.19&20). With respect to Dufour parameter Du, we find that the actual concentration at all horizontal levels enhances with increase in Du ≤ 0.05 and depreciates with higher Du ≥ 0.07 (figs.21&22). At the vertical level x = 1 / 3 , it depreciates with Du (fig. 23) and at higher vertical level x = 2 / 3 , it enhances with Du (fig.24). The rate of heat transfer at x=1 is evaluated at different levels. It is found that the rate of heat transfer enhances with

increase in Rayleigh number Ra at all quadrants (table. 1). Lesser the molecular diffusivity larger Nu at the lower and middle quadrants and at the upper quadrant lesser Nu and for further lowering of the diffusivity larger Nu (table 2). With respect to Soret parameter we find that Nu reduces with S0>0 at the lower and middle quadrants and enhances at the upper quadrant while an increase in |S0| enhances Nu at the lower quadrant and reduces at the middle and upper quadrants. When the molecular buoyancy force dominates over the thermal buoyancy force the rate of heat transfer enhances when the buoyancy forces act in the same directions and for the forces acting in opposite directions, it reduces at all quadrants (table.3). Higher the radiative heat flux larger Nu at all quadrants. With respect to Dufour parameter we find that the rate of heat transfer enhances with increase in Du ≤ 0.03 while at the upper quadrant it reduces with Du ≤ 0.03 and enhances with Du ≥ 0.05. The variation of Nu with Ec shows that higher the dissipative heat larger the rate of heat transfer all quadrants (table. 4). The rate of mass transfer (Sherwood number) is evaluated at different quadrants for different parametric values and is shown in tables 5-8. It is found that he rate of mass transfer enhances with increase in Ra>0 and reduces with |Ra| at all levels (table. 5). Sh reduces at the lower and middle quadrants and enhances at the upper quadrant. Lesser the molecular diffusivity lesser the rate of mass transfer at all quadrants (table. 6). The rate of mass transfer enhances with S0>0 and depreciates with S0<0 at all the quadrants. When the molecular buoyancy force dominates over the thermal buoyancy force the rate of mass transfer experiences an

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296

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

enhancement at all the three quadrants irrespective of the directions of the buoyancy forces. An increase in the radiation parameter N1 results in an enhancement in Nu at all quadrants (table. 7). With respect to Du we find that the rate of mass transfer depreciates at the lower and middle quadrants and enhances it at the

Table – 1

upper quadrant with increase in Du. Also higher the dissipative heat larger Nu at the lower and middle quadrants while at the upper quadrant it enhances in the region (0, 0.003) and depreciates in the remaining region. The rate of mass transfer depreciates at all the quadrants in both degenerating and generating chemical reaction cases (table. 8).

Nusselt nulber at x = 1 level

III

Nu1

2.264

2.254

2.2414

2.2833

2.288

2.3211

Nu2

2.1938

2.1837

2.1695

2.1792

2.2538

2.3603

Nu3

2.1008

2.1104

2.1168

2.0751

2.0596

2.0242

-3x102

2x10

3x10

-10

-2x10

Table – 2 Nusselt nulber at x = 1 level G

III

Nu1

2.264

5.501

4.844

5.667

Nu2

2.1938

4.1716

1.4720

4.1960

Nu3

2.1008

2.8324

1.9012

2.7352

0.01

0.03

0.05

0.07

Table – 3

Nusselt nulber at x = 1 level

III

VII

VIII

Nu1

5.4464

5.7732

3.5808

5.5808

5.7340

5.2608

5.0960

5.4832

5.7081

5.7312

Nu2

3.6288

4.1716

3.8360

4.1872

4.3404

3.8708

3.7136

4.1444

4.2056

4.2068

Nu3

2.7108

2.7710

3.1008

2.7936

2.9460

3.9668

2.3312

2.8260

2.7032

2.8410

0.24

0.6

1.3

2.01

1.3

0.5

1.0

-0.5

-1.0

0.5

-0.5

-0.8

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

Table – 4

Nusselt nulber at x = 1 level

III

VII

Nu1

5.0672

5.0720

5.0780

5.0824

5.5796

5.5804

5.6868

Nu2

3.0572

3.1801

3.1308

3.1928

4.1840

4.2676

4.2768

Nu3

2.7328

2.7115

2.8168

2.8948

2.7788

2.7958

2.8408

0.01

0.05

0.07

0.12

0.01

0.001

0.003

0.007

0.009

Table – 5

Sherwood number (Sh) at x = 1 level

III

Sh1

8.3808

8.6616

8.8480

8.9228

8.7340

8.6508

Sh2

2.2001

2.3168

2.4416

2.0112

1.8814

1.6809

Sh3

-3.8804

-4.0180

-4.0648

-3.9456

-3.9316

-3.9108

102

2x102

3x102

-102

-2x102

-3x102

Table – 6 Sherwood number (Sh) at x = 1 level G

III

VII

Sh1

8.3808

7.7840

1.6145

3.2448

8.3829

8.4309

8.4678

Sh2

2.2001

2.2191

1.7721

1.8495

2.2004

2.2165

2.2284

Sh3

-3.8804

-3.0536

-2.0652

0.5344

-3.9804

-3.9971

-4.2017

-0.5

-0.8

0.5

1.5

Table – 7

Sherwood number at x = 1 level

III

VII

Nu1

9.8446

8.7508

8.3808

8.3354

10.7624

3.9856

7.8754

Nu2

2.8088

2.3645

2.2001

2.1772

2.3646

1.9806

1.8962

Nu3

-4.2264

-4.0478

-3.8804

-3.9762

-6.0311

-0.0254

1.9185

0.24

0.6

1.3

2.01

1.3

0.5

1.0

-0.5

-1.0

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.4 (4), 282-299 (2013)

Table – 8

298

Sherwood number at x = 1 level

III

VII

Nu1

8.42

8.3961

8.3680

8.3972

9.24

8.4396

7.6264

Nu2

2.45

2.1684

2.1669

2.1651

2.32

2.2160

2.1360

Nu3

1.485

-4.5596

-4.0628

-4.0684

-2.424

-3.9988

-4.9948

0.01

0.05

0.07

0.12

0.01

0.001

0.003

0.007

0.009

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11.

12.

ASME Journal of Heat Transfer, V. 92, pp, 21-27,(1970). Chittibabu. D., Prasada Rao. D.R.V., Krishna D.V.: Convection flow through a porous medium in ducts. Act science Indica Vol. 30 2M, pp, 635-642, (2006). Ham – Chien Chiu, Jer-Huan Jang, WeiMonyan,: Mixed convection heat transfer in Horizontal rectangular ducts with radiation effects. Int. Journal of Heat Mass Transfer,50, pp. 2874-2882, (2007). Joseph, J. Savino and Robert Siegel,: Laminar forced convection in Rectangular channels with unequal Heat addition on adjacent sides. Int. J. Heat Mass Transfer Vol. 71, pp, 733-741 (1964). Padmavathi, A: Finite element analysis of the Convective heat transfer flow of a viscous in compressible fluid in a Rectangular duct with radiation,viscous dissipation with constant heat source, Jour. Phys and Appl. Phys.,V.2, (2009). Rangareddy, M: Heat and Mass transfer by Natural convection through a porous medium in ducts, Ph.D thesis, S.K. University, Anantapur (1997).

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13. Reddaiah, P : Heat and Mass Transfer flow of a viscous fluid in a duct of rectangular cross section by using finite element analysis, European J. of prime and applied mathematics (Accepted) (2010). 14. Sivaiah, S: Thermo-Diffusion effects on convective heat and mass transfer through a porous medium in Ducts, Ph.D thesis, S.K University, Anantapur, India (2004).

15. Teoman Ayhan, Hayati Olgum : Betul Ayhan : Heat transfer and flow structure in a Rectangualr channel withwing -1. type vortex Generator. Tr. J. of Engineering and Environmental Science, pp, 185-195,(1998). 16. Verschoor, J.D, and Greebler, P:Heat Transfer by gas conduction and radiation in fibrous insulation Trans. Am. Soc. Mech. Engrsmp. p. 961-968, (1952).

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