Cmjv01i04p0469

J. Comp. & Math. Sci. Vol. 1(4), 469-476 (2010).

Combined effect of Dissipation and Soret effect on convective Heat and Mass transfer in a vertical channel with Non-uniform temperature SUDHA MATHEWa, P.RAVEENDRA NATHb* and P. SRINIVASA RAOc a

Research scholar, Department of Mathematics, S.K. University, Anantapur, A.P., India b Sri Krishnadevaraya University College of Engineering and Technology, S.K. University, Anantapur - 515 055, A.P., India. c Research scholar, Department of Physics, S.K. University, Anantapur, A.P., India ABSTRACT We analyse the combined of dissipation and Soret effect on the convective Heat and Mass Transfer, flow of a viscous, incompressible fluid through a porous medium confined in a vertical channel with flat walls at y  1 . The velocity and the temperature have been evaluated for variations in the different governing parameters. The effect of the various parameters on the flow and heat transfer has been exhibited through profiles of velocity and temperature. The shear stress and the rate of heat transfer are evaluated for different variations of parameters.

Key words: Non-uniform Temperature, Heat Transfer, Dissipation, Porous medium, Concentration.

1. INTRODUCTION Coupled heat and mass transfer by natural convection in a fluid-saturated porous medium has attracted considerable attention in recent years due to many important engineering and geophysical applications such as cooling of nuclear

fuel in shipping flasks and water filled storage bays, insulation of high temperature gas-cooled reactor vessels, drums containing heat generating chemicals in the earth, thermal energy storage tanks, regenerative heat exchangers containing catalytic reaction.

A review of combined heat and

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Sudha Mathew et al., J.Comp.&Math.Sci. Vol.1(4), 469-476 (2010)

mass transfer by free convection in porous medium is given by Trevisan and Bejan8. Angirasa et al.1 have presented the analysis for combined heat and mass transfer by natural convection for aiding and opposing buoyancies in fluidsaturated porous enclosures.

This problem of combined buoyancy driven thermal and mass diffusion has been studied in parallel plate geometries by a few authors in the recent times, Lai4, Poulikakos6, Mehta and Nanda Kumar5 and Angirasa et al.1.

The Volumetric heat generation has been assumed to be constant2,3. The Heat and Mass transfer through porous medium has been carried out by several authors7,8 under different conditions.

2. Formulation of the Problem We analyse the steady motion of viscous, electrically conducting fluid through a porous medium in a vertical channel bounded by flat walls which are maintained at a non-uniform wall temperature in the presence of a constant heat source. A uniform magnetic of strength H0 is applied transverse to the walls. The Boussinesq approximation is used so that the density variation will be considered only in the buoyancy force. The viscous and Darcy dissipa-tions are taken into account in the energy equation. Also the kinematic viscosity , the thermal conducting k are treated as constants. We choose a rectangular Cartesian system 0( x ,y ) with x-axis in the vertical direction and y-axis normal

to the walls. The walls of the channels are at y = ± L. The equations governing the steady hydromagnetic flow, heat and mass transfer are In the equilibrium state

0

p e  e g x

Where

(2.1)

p  p e  p D , pD being the

hydrodynamic pressure.

The flow is maintained by a constant volume flux for which a characteristic velocity is defined as

Q

1 L  u d y.

2L L

(2.2)

The boundary conditions for the velocity and temperature fields are u = 0, v = 0

on y =  L

(2.3)

 x  T  Te     on y =  L (2.4)  L  C = C1 C = C2

on on

y = - L (2.5) y = +L (2.6)

 is chosen to be twice differentiable function, is a small parameter characterizing the slope of the temperature variation on the boundary.

In view of the continuity equation we define the stream function  as

u   y , v   x,

Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)

(2.7)

Sudha Mathew et al., J.Comp.&Math.Sci. Vol.1(4), 469-476 (2010) the equation governing the flow in terms of  are

  (2 )  (2 )    =   x  y  y  x  

   2    x 2 

   

   T   C     g     * g     y   y   

  2  2  y

   e2 H 02    2    2    y e  

     2  2      2  2      k  x y  (2.8)

T  Te ' C  C1 ,C  Te C2  C1 pD  ' p  ,   Te  eU 2

471



(2. 11)

(under the equilibrium state

QL2 ) Te  Te ( L)  Te ( L)   the governing equations in the nondimensional form (after dropping the dashes) are

( ,  2 ) G        R   4    y  NCy  2      Q   eC p    ( x, y ) R   y  x  x  y   2 1 2 2   2 2 D    M   2    2    (2.12)   2 2 y 2     2    2    e H 0    y   x    k  and the energy diffusion equations in    the non-dimensional form are

2

              x   y   2

(2.9)

 PR2 Ec        2      1  PR    y x x y   G 

2   2 C  2C    C  C  2 2       D1  2  2      2   2    y    2    2    M 2  D1    y x x y   x     

2

  y 

2

     k11  2  2  y   x

(2.10)

Introducing the non-dimensional variables in (2.8)- (2.10) as

 x, y   u, v  ( x, y)   , (u' , v' )    , L U    

       y 

 x  

2



   





x 

(2.1 3)

  C  C    2 C  2 C     2  2  RSc  y   y x x y   x

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Sudha Mathew et al., J.Comp.&Math.Sci. Vol.1(4), 469-476 (2010)

s s  c 0  N R

C= 0 on y = -1 C = 1 on y = 1

   2  2    2  2  y    x

(2. 14)

UL L2 (Reynolds Number) D 1   k

(Darcy parameter)

gL3 Ec  Cp M2 

(Eckert Number)

 e2 H 02 L2  0

 * C N  T gTe L3 G 2  sc 

(Hartmann Number)

(Buoyancy Number)

(Grashof Number)

(Prandtl

D k11  * s0  

at

y 0

(2.20)

The value of  on the boundary assumes the constant volumetric flow consistent with the hypothesis (2.1).

Also the wall temperature varies in the axial direction in accordance with the prescribed arbitrary function (x).

3. Shear Stress and Nusselt Number

 cp k1 

 C  0, 0 y y

(2.18) (2.19)

Number)

( Schmidt Number )

The shear stress on the channel walls is given by

 u v         y x 

y  L

which in the non- dimensional form reduces to

 U  ( )  ( yy  2 x x ) a  [ 00, yy  Ec 01, yy   (10, yy  Ec11, yy  O( 2 )]y1

( Soret Parameter) and the corresponding expressions are

The corresponding boundary conditions are (2.15)  ( 1)   (1)  1

   0, 0 x y

at y  1

 ( x, y )  f (x )

on y   1 (2.17)

(2.16)

( ) y   1  d 3  Ecd 4  d 5  O( 2 ) ( ) y   1  d 6  Ecd 7  d 8  O( 2 ) The local rate of heat transfer coefficient (Nusselt number Nu) on the walls has been calculated using the formula

Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)

Sudha Mathew et al., J.Comp.&Math.Sci. Vol.1(4), 469-476 (2010)

Nu 

1  ( ) y  1  m   w y

473

2.5

2

and the corresponding expressions are

1.5

( N u ) y  1 

(d 10   ( d11  d12 ) ( d 8   ( x)  d 9 )

u

i ii

1

iii iv

0.5

( N u ) y  1 

( d10   (d12  d11 ) (d 8   ( x)  d 9 )

0 -1

0

0.5

1

-0.5

The local rate of mass transfer coefficient on the walls has been calculated using the formula

 1  c    Sh   c  c w  y  y  1  m

-0.5

y

Fig. 1 u with SC SO=0.5, N=1, M=2 I II III IV 1.3 2.01 0.24 0.6

where SC

1

3

c m  (0.5)  c dy

1

1

-1

And the corresponding expressions are

-0.5

-1

0

0.5

1

v

i

( Sh) y  1

(0.5  d15   (d 17  d16 )  (d13  1  d14 )

ii

-3

iii in

-5

-7

( Sh ) y  1 

(0.5  d15   (d17  d16 ) (d13  d14 )

-9 y

Where d3—————d17 are constants 4. Discussion of the numerical results The effect of different parameters

SC

Fig. 2 V with SC SO=0.5, N=1, M=2 I II III IV 1.3 2.01 0.24 0.6

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Sudha Mathew et al., J.Comp.&Math.Sci. Vol.1(4), 469-476 (2010) 0.8

10 0.7

8

0.6

6

0.5 i 0.4

ii

I

C

II

4

III

iii 0.3

IV

iv

2

0.2

0 0.1

-1

-0.5

0 -2

0 -1

-0.5

0

0.5

0.5

1

y

1

y

SC

Fig. 3  with SC SO=0.5, N=1, M=2 I II III IV 1.3 2.01 0.24 0.6

governing the flow, heat and mass transfer on the velocity, temperature distributions are analysed and exhibited in Figures (1-5). The axial velocity is in the vertically down wards direction and hence u<0 is the actual velocity and u>0 is the reversal flow. We observe that lesser the molecular diffusivity, larger |u| in the region (Fig. 1).

Fig. 2 exhibits that lesser the molecular diffusivity, higher |v| in the region -0.8  y  0.2 and depreciates in the remaining region.

Fig. 3 shows that lesser the molecular diffusivity, smaller the tempe-

SC

Fig. 4 C with SC SO=0.5, N=1, M=2 I II III IV 1.3 2.01 0.24 0.6

rature in the flow region. Also, lesser the molecular diffusivity, smaller the concentration in the flow region. (Fig. 4).

Table 1. shows that lesser the molecular diffusivity, larger |τ|. The variation of τ with soret parameter S0 shows that |τ| enhances with increase in |S0| (>0) and depreciates with |S0| (<0).

It is found that on y=1,the rate of heat transfer depreciates with |S0| (>0) and enhances with |S0| (<0) in both heating and cooling cases, while on y= -1, it enhances with all values S0 for all G (Table 2).

Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)

Sudha Mathew et al., J.Comp.&Math.Sci. Vol.1(4), 469-476 (2010) Table 1. Shear Stress ( τ ) at y =1P=0.71, N=, Sc=1.3 I II III IV V VI

G/τ 103 3

3x10 -103

3

-3x10

475

VII

-4.4014

-4.9526 -3.5783

-3.8578 -5.4102

-2.3823 -1.3736

35.4785

-38.131 -31.518

-32.862 -40.354

-25.968 -20.905

-3.0357

-3.8178 -3.3612

-3.6509 -3.6366

-3.8346 -3.2341

-31.382

-31.727 -30.867

-31.042 -32.014

-30.118 -29.487

Table 2 Shear Stress ( τ ) at y = -1P=0.71, N=1, M=2, Sc=1.3 =0.5 G/τ I II III IV V VI VII 103

6.6117

7.1631

5.7887

6.0683

7.6212

4.5931

3.5838

3x103

42.109

44.763

38.148

39.494 46.967

32.393

27.536

-10

0.8255

0.6075

1.1512

1.0405 0.4262

1.6242

2.0235

-3x103

24.751

25.097

24.235

24.410

23.487

22.855

3

G/Nu 3

10

25.383

Table 3 Nusselt Number (Nu) at y =1P=0.71,N=1,Sc=1.3, I II III IV V VI

VII

-0.1128

-0.1038 -0.1292

-0.1232 -0.0975

-0.1639 -0.2121

-0.0195

-0.0184 -0.0222

-0.0209 -0.0175

-0.0255 -0.0321

-0.9273

-1.3577 -0.6292

-0.7064 -0.6099

-0.2488 -0.3379

-3x103 -0.0222

-0.0218 -0.0226

-0.0225 -0.2165

-0.0234 -0.0239

3x103 3

-10

Table 4. Nusselt Number (Nu) at y = -1P=0.71,N=1,M=2, =0.5,Sc=1.3 G/Nu

I

II

III

IV

V

VI

VII

0.0985

0.0908

0.1129

0.1076

0.0852

0.1434

0.1854

3x10

0.0131

0.0122

0.0144

0.0139

0.0116

0.0171

0.0205

3

-10

0.4608

0.4232

0.7198

0.6081

0.4279

0.4906

0.4865

-3x103

0.0333

0.0329

0.0341

0.0338 0.0325

0.0351

0.0351

S0

I 0.5

II 0.5

III 0.5

VI -0.5

VII -1.0

3

10

3

IV 0.5

V 1.0

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Sudha Mathew et al., J.Comp.&Math.Sci. Vol.1(4), 469-476 (2010) 5. REFERENCES

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40, 2755-2773 (1997).

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4. F.C. Lai, Coupled heat and mass transfer by mixed convection from a vertical plate in a saturated porous medium, Int. Comm. Heat Mass Transfer, 18, 93-106 (1991).

5. K.N. Mehata and K. Nanda Kumar,

Natural convection with combined heat and mass transfer buoyancy effects in non-homogeneous porous medium, Int. J. Heat Mass transfer 30, 2651-2656 (1987).

6. D. Poulikakos, On buoyancy induced heat and mass transfer from a concentrated source in an infinite porous medium, Int. J. Heat Mass Transfer, 28, 621-629 (1985).

7. P.Raveendra Nath, P.M.V. Prasad, D.R.V. Prasada Rao, Computational hydromangetic mixed convective heat and mass transfer through a porous medium in a non-uniformly heated vertical channel with heat sources and dissipation, Int. J. Computers and Mathematics with Applications, 59, 803-81(2010).

8. O.V. Trevisan and A. Bejan, Coupled heat and mass transfer by natural convection in a porous medium, Adv. Heat transfer, 20, 315-352 (1990).

Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)