Heat Transfer in A Coupled-Fluid Flow Over A Highly Porous Medium Layer in the Presence of Heat Source

Heat Transfer in A Coupled-Fluid Flow Over A Highly Porous Medium Layer in the Presence of Heat Source

Dr. Shilpi Saxena

Department of Mathematics, Poornima University, IS-2027-31, Ramchandrapura, Sitapura Extension, Jaipur – 303905. India.

Abstract

A viscous fluid flow over a highly porous layer of thickness a is considered. Porous layer is fluid saturated and has a permeable bottom where a transverse sinusoidal suction velocity is applied and the permeable bottom of the porous layer is kept at constant temperature Ta. Since the porous layer is infinite in the x – direction, all physical quantities will be independent of x, however, the flow remains three – dimensional due to the variation of the suction velocity which is applied at the permeable bottom. The governing equations are solved using a perturbation series expansion method. The effects of various flow parameters such as Prandtl number (Pr), suction parameter (), Permeability of the porous medium (K), Heat source parameter (S) and viscosity ratio (1), are investigated on temperature distribution and rate of heat transfer at the porous medium interface, and discussed graphically.

Key words: Heat transfer, coupled flow, porous medium, permeability, heat source.

1. Introduction

   The Viscous fluid flow through and across porous media is a subject of common interest and has emerged as a separate intensive research area because heat and mass transfer in porous medium is very much prevalent in nature and can also be encountered in many technological processes. It has its applications in a variety of engineering processes such as heat exchanger devices, petroleum reservoirs, chemical catalytic reactors and processes, geothermal and geophysical engineering, moisture migration in a fibrous insulation and nuclear waste disposal and others. Such problems of flow and heat transfer through a wall- bounded porous medium in various types of ducts and channels, have been modeled using some variation of extended Darcys equation, which describes a balance among pressure gradient, viscous transfer of momentum, linear or/Quadratic drag forces, by several researchers. E.g. Durlofsky and Brady

   (1987), Kladias and Prasad (1991), Vafai and Kim (1989), Nakayama et al. (1988), Nield at al. (1996), Al- Hadhrami et al. (2002), Kim and Russell (1985), Nield at al. (2004), Hooman et al. (2007) and Chauhan and Kumar (2009). Bejan and Khair (1985) investigated the free convection boundary layer flow in a porous medium owing to combined heat and mass transfer. Lai and Kulacki (1990) used the series expansion method to investigate coupled heat and mass transfer in natural convection from a sphere in a porous medium. The suction and blowing effects on free convection coupled heat and mass transfer over a vertical plate in a saturated porous medium were studied by Raptis et al. (1981) and Lai and Kulacki (1991), respectively.

   Free convective flow in presence of heat source has been a subject of interest of many researchers because of its possible application to geophysical sciences, astrophysical sciences, and in cosmical studies. Such flows arise either due to unsteady motion of the boundary or the boundary temperature. Therefore, many researchers have paid their attention towards the fluctuating flow of viscous incompressible fluid past an infinite plate. Singh et al. (2003) have analyzed the heat and mass transfer in MHD flow of viscous fluids past a vertical plate under oscillatory suction velocity. Sharma and Singh (2008) have reported the unsteady MHD-free convective flow and heat transfer along a vertical porous plate with variable suction and internal heat generation. The effect of temperature-dependent heat sources has been studied by Moalem (1976) taking into account the steady state heat transfer within porous medium. Aziz (2009) theoretically examined a similarity solution for a laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Recently, the combined effects of an exponentially decaying internal heat generation and a convective boundary condition on the thermal boundary layer over a flat plate are investigated by Olanrewaju et al. (2012). Similar analysis had been carried out by with heat source Bakr, A. A(2011) neglecting chemical reaction effect. There has been considerable interest in studying the effect of chemical reaction and heat source effect on the boundary layer flow problem with heat and mass transfer of an

   electrically conducting fluid in different geometry Gangadhar et al. (2011-2012).

   v v w v p 2v 2v

   y z y 2 2

   (3)

   y z

   w w p 2w 2w

   v y w z z 2 2

   (4)

   y z

   t t 2t 2t

   Cp v y w z k 2 2

   y z

   v 2

   2

   w 2

   y y

   (5)

   u 2 w v 2

   u 2

   y y z z

   Q t-t

   And for porous region-II(-ayo) are :

   V W 0

   (6)

2. Formulation of the problem

   y z

   y

   2U

   2U

   A viscous fluid flow over a highly porous layer of thickness a is considered. Porous layer is fluid

   0

   2 z

   2 K U

   (7)

   saturated and has a permeable bottom where a transverse sinusoidal suction velocity is applied and the permeable bottom of the porous layer is kept at constant temperature Ta. The surface of the porous

   1 P 2V 2V

   0 y y2 z2 K V

   (8)

   layer is taken horizontal in x-z plane. The x-axis is taken in the direction of the flow, and the y-axis is taken normal to the porous surface directed into the

   fluid flowing with free stream velocity U. Let

   1 P 2W

   z

   0

   y2

   2W

   • W

   z

   2 K

   (9)

   (u,v,w,t) and (U,V,W,T) are the velocity and temperature components in free fluid and porous

   Cp T W T k 2T 2T

   V

   y z 2 2

   regions in the directions x, y, z respectively. Since the porous layer is infinite in the x – direction, all physical quantities will be independent of x, however, the flow

   y

   V 2

   2

   z

   W 2

   y y

   (10)

   remains three – dimensional due to the variation of the

   suction velocity distribution which is applied at the

   U 2 W V 2 U 2

   permeable bottom. This applied suction velocity is

   consisting of a basic steady distribution with a superimposed weak transversally varying distribution. The governing equations for the free fluid region-I (0y), are:

   y y z

   Q T-t

   The boundary conditions are:

   z

   v w 0

   (1)

   y z

   at y 0, u U , w W ,

   yx I

   yx II ,

   u u 2u 2u

   (2)

   t T

   v y w z 2 2

   v V , p P,t T , k k

   y z

   y y

   * U z r y

   aty a,U 0, v V0 1 cos ,

   y

   2 2 3

   A e 1

   p ( y, z) A e r r r 3 cos z

   w 0,T Ta ,

   1 2 1 1 1

   2

   y y

   y ,u U

   ,v V*, w 0, p p ,t t

   U ( y, z) B e 1 a e 1 cos z

   0

   1 7 40

   (11)

   y

   y

   y y

   V ( y, z) B e

   • B e

     • B e 1 B e 1 cos z

   Here, p and P are the pressures in the free and porous 1

   regions respectively, , the density; , the viscosity;

   , the effective viscosity in the porous region; , the

   3 4 5 6

   B e y B e y

   Kinematic viscosity; K, the permeability of the porous

   W ( y, z) 1 3 4

   sin z

   1

   medium; V0*0, is the mean suction velocity and 1

   1, is the modulation parameter. Cp, k, k, are the

   y

   B e

   y

   B e 1

   specific heat at constant pressure, thermal conductivity, effective thermal conductivity in porous region and Q is the heat source/sink respectively.

   Solution for the flow problem is taken from Chauhan and Sahai [2004]. Thus we have

   P1 ( y, z)

   5 1 6 1

   4 3

   1 B e y B e y cos z

   K

   1. 1 A e y , v

      , w

      0, p p

      Making use of the following non-dimensional

      0 1 0 0 0

      1 y 1 y

      K K

      quantities:

      t t

      2

      C p U

      U B e 1 B e 1 , V , W 0 ,

      t

      , Pr

      , Ec

      0 1 2 0 0

      T t

      k C T T

      P y P

      a P a

      0 K

      V *

      T t Qt

      0 ,T

      , S

      And

      U T t

      C U 2

      r y

      A A e( ) y A A e( r ) y

      a P

      u ( y, z) A e 1

      1 2 1 3

      1

      cos z

      1 4

      y

      r y

      2r

      1

      where , suction parameter, Pr, Prandtl number, Ec, Eckert number and S, the heat source parameter.

   2. ( y, z) A e A e 1 cos z

   1 2 3

   The non-dimensional energy equations for the free fluid regionI, is

   t t

   1 2t 2t

   r y

   v y w 2 2

   r A e 1

   z Pr y

   z

   w ( y, z) A e y 1 3 sin z

   2 2

   (12)

   1 2

   2 v

   w

   y z

   Ec

   St

   u 2 w v 2 u 2

   y y

   z

   z

   and for the porous region II is :

   T T

   2 2T

   2T

   2y 2y

   y

   V W

   z Pr 2 2

   a y a y K

   K 2B B Ec

   y z

   T (y, z) B e 53 B e 54 a e 1 a e

   1 1 2

   V 2

   2

   W 2

   (13)

   9 10 55 56

   KS

   1

   y

   z

   Ec

   ST

   1 y

   1. y

      U 2

      W

      V 2

      U 2

      a y a y

      1 K 1

      K

      y

      y z

      z

      [B e 57

      • B e 58

      • a e

      1 a e 1

      12 13 59 60

      The boundary conditions are

      1 1

      y y

      1 K 1

      K a y

      at y ,t 0

      a e 1 a e

      1 a e 53

      61 62 63

      at y 0, t T, t

      T

      ,

   2. y

   2 y

   y 2 y

   a y

   K

   K

   a e

   54 a e

   1 a e 1

   at y a,T 1,

   (14)

   64 65

   66

   2 y

   k, a y

   a y

   K

   where

   a e

   53 a e

   54 a e 1

   2 k 67 68 69

3. Solution of the Problem

   2 y

   K

   a y

   1 a y

   The energy equations (12) to equations (13) can be solved by perturbation series method, for very small

   values of the parameter (1). We write t(y,z)=to(y)+ t1(y)cosz,

   a e 1 a e 1 53 a e 54 70 71 72

   2 2

   y y

   1 K 1

   K a y

   T(y,z)=T (y)+ T (y)cosz, (15)

   a e 1 a e

   1. a e

      1 53

      o 1 73 74 75

      Using equations (15) in the equations (12) and (13) and

   2. y

   2 y

   the corresponding boundary conditions, comparing the

   a y

   1 K 1

   K

   coefficients of equal power of on both sides, and then solving the resulting set of ordinary differential equations under the corresponding boundary conditions, we obtain

   a e 1 54 a e 1 a e

   76 77 78

   1 ]cos z

   a y

   t( y, z) B e 46

   8

   Ec Pr 2 A2e2 y

   1

   42 22 Pr S Pr

   Where A1, A2, A3, A4, B1, B2, B3, B4, B5, B6, B7,

   B8, B9, B10, B11, B12, B13 are constants of integrations. These constants have been obtained by the boundary conditions and matching conditions and are

   a y

   B e 49

   • a e

     • r y

     1 a e

     2 y

   • a e

     2 r y

     1 cos z

     reported in the appendix.

     The dimensionless rate of heat transfer at the

     11 50 51 52

     permeable surface is given by

     t

     2Ec Pr 3 A2

     B a

     1

     y y 0

     8 46

     4 2 22 Pr S Pr

     B a r a

     11 49 1 50

     cos z

     2 a 2 r a

     51 1 52

4. Results and discussions

   The variations in the temperature profiles for different values of the permeability K is shown in Figure 2. On comparing the various curves in the figure it is observed that the effect of permeability is to decrease the temperature at all points in the flow field. In fact, the thermal conduction, in flow field of both the regions, is lowered as we increase the value of permeability parameter K and consequently the temperature falls.

   Fig. 2. Temperature Distribution vs y for

   K=0.1 K=1 K=2 K=3 t
   .2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	T 1	1.

   =0.2, 1=1.25, 2=1.6, =0.05, Ec=0.01, Pr=0.71, S=-1.5 and z=0

   1.4

   1.2

   1

   0.8

   value of causes a decrease in temperature in both regions.

   Fig. 4. Temperature Distribution vs y for

   1 2

   =0.2, =1.25, =1.6, Ec=0.01, Pr=0.71, S=-1.5, K=1 and z=0

   =0.01 =0.05 =0.1 =0.3 t
   18	0.38	0.58	0.78	0.98	1.18	1.38 T	1.58

   1.8

   1.6

   1.4

   1.2

   1

   0.8

   y

   0.6

   0.4

   0.2

   0

   0.

   -0.2

   y

   0.6

   0.4

   0.2

   0

   0

   -0.2

   Figure 5, depicts the temperature distribution for different values of Prandtl number Pr. On comparing various curves in the figure, it is observed tht the effect of Prandtl number is to decrease temperature at

   1 all points in the region.

   Fig. 5. Temperature Distribution vs y for

   Pr=.71 Pr=1.5 Pr=3 Pr=5 t
   0	0.2	0.4	0.6	0.8	T	1	1.

   Figure 3, depicts the temperature distribution for different values of viscosity ratio 1. It is observed that temperature increases by increasing the viscosity ratio

   1.

   1.4

   1.2

   1

   =0.2, 1=1.25, 2=1.6, Ec=0.01, =0.05, S=-1.5, K=1 and z=0

   0.7

   0.6

   0.5

   0.4

   0.3

   y

   0.2

   0.1

   Fig. 3. Temperature Distribution vs y for

   =0.2, 2=1.6, =0.05, Ec=0.01, Pr=0.71, S=-1.5, K=1 and z=0

   0.8

   0.6

   y

   0.4

   0.2

   0

   2

   -0.2

   0

   0.

   -0.1

   Figure 6, however shows that the source parameter S

   1=1.25 1=3 1=6 t
   5	0.6	0.7	0.8	0.9 T

   1 increases the temperature in the porous and free fluid region at all points.

   -0.2

   Figure 4, shows the effect of suction parameter on the temperature distribution. When is very small, the temperature profile is nearly linear. Increase in the

   1

   0.8

   0.6

   0.4

   0.2

   0

   -0.2

   Fig. 6. Temperature Distribution vs y for

   Fig. 8. Rate of heat transfer vs for

   =0.2, 2=1.6, Ec=0.01, S=-1.5, Pr=0.71, K=1 and y=0

   14

   (ot/oy)y=0

   S=-0.6 S=-0.8 S=-1.2 S=-1 S=-1.5 t
   4	0.5	0.6	0.7	0.8	T	0.9	1

   =0.2, 1=1.25, 2=1.6, Ec=0.01, =0.05, Pr=0.71, K=1 and z=0

   12

   10

   8

   6

   1=1.25 1=2 1=4

   1=6

   4

   2

   0

   y

   0.

   0

   0.5

   1

   1.5

   2

   2.5

   -2

   K=1

   (ot/oy)y=0

   In Figure 7, Rate of heat transfer at the porous interface is plotted against the suction parameter for various values of K. It is observed that rate of heat transfer increases with the increase in the suction parameter . It also increases with the increase in K.

   Fig. 7. Rate of heat transfer vs for

   =0.2, 1=1.25, 2=1.6, Ec=0.01, S=-1.5, Pr=0.71 and y=0

   35

   30

   25

   20

   k=0.0001

   10

   k=3

   5

   0

   0 0.1

   0.2 0.3

   0.4

   0.5 0.6 0.7

   -5

   15

   Figure 8, shows variation in rate of heat transfer at porous interface for different values of 1. The effect of viscosity parameter 1 is to decrease rate of heat transfer.

   Rate of heat transfer for different values of Pr is shown in Figure 9. It is observed that increase in Prandtl number results in the increase in rate of heat transfer.

   300

   Fig. 9. Rate of heat transfer vs for

   =0.2, 1=1.25, 2=1.6, Ec=0.01, S=-1.5, Pr=0.71, K=1 and y=0

   2.5

   2

   1.5

   1

   0.5

   0

   150

   100

   50

   0

   Pr=5 Pr=7

   200

   Pr=0.7

   Pr=1.5

   Pr=3

   250

   (ot/oy)y=0

5. Conclusion

Heat transfer characteristics in the three dimensional steady flow of a viscous incompressible fluid over a highly porous layer is investigated in the presence of heat source, when a transverse sinusoidal suction velocity is applied at the permeable bottom of porous medium. Figures 2 to 6 shows variation of temperature distribution for various values of permeability K, suction parameter, viscosity ratio 1, Prandtl number Pr and Heat source parameter S. It is found that temperature decreases in both porous region and free fluid region with the increase in K or or Pr, whereas it increases by increasing 1 or S. In figures 7 to 9, rate of heat transfer at the porous interface is plotted against

the suction parameter for various values of K, Pr and

1. It is observed that the magnitude of the rate of heat transfer increases with the increase in K or Pr whereas it decreases with increase in 1.

References

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Appendix

A1 B1 B2 1

B11

a90 a89 B12

a49

a a

a e0.2a58 a

B3 r1 B4 r1 B5 r1 1 B6 r1 1

B 90 86 88 49

57 58

A2

r

12 a

1 e0.2a

e0.2a

a

89

49

1

B a

B e0.2a57 e0.2a58

2 B3 B5 1 B6 1

13 88 12

A3

r

1

A4 B7 a40 a38

0.4

1

1

r

2 4 2

2

B B e 1K

1 2

2 1

B2

1 K

0.4 1

1

K

1 e 1K

K

a11

0.2

B B ea11 B ea12 B ea12 ea11

a 0.2

3 4 5 6

12 1

a22B5 a33B6

a 2 2r 2r 2 2 r 2

B4

a21 .

13 1 1 1 1

a 2 2 r 1

B5

a34 a31B6

a

14 1 1

30

a 2 2

2 1

6

B a37

a36

15 2r1 1 r1 1 1 r1 2

K

1

0.4

a 2 2r r2 r 2 2 1

B a e 1

16 1 1 1 1 1 1

7 40

K1

B8 a81 B9 B10

a 2 r r

a83 a85a44

B

17 1 K 1

9 a a a

82 84 44

a 2 r

19 1 1

18 K 1

B a

B e0.2a

e0.2a

10

79 9

53 54

a 2 r

a 2 r

A1 A2

r1 A1 A3

20 1 1

a39

2r1

a21

2 ea11

a39 r1a38

a40 r r e0.4

1

a ea12

1 1 1 1

22 1

a23 1 e

a12

a41

Ec Pr B2

1

4 2 Pr

12 K K

a a ea11 a ea11

1 2

1K

24 14 13

a a ea11 a ea12

Ec Pr B2

25 15 13

a42

2

4 2 Pr

a a

12 K K

a26 a16e

11 a13e 12

1 2

1 K

a a ea11 a ea11 2EcB1B2

27 18 17

a43

1K

a a ea11 a ea12

1

28 19 17

a

Ec PrA2

a a ea a ea

44 22 Pr

11 12

29 20 17

a30 a21a25 a22 a24

Pr 2 Pr2 4S Pr

a45

2

a31 a21a26 a23a24

Pr 2 Pr2 4S Pr

a32 a21a28 a22 a27

a46

2

a33 a21a29 a23a27

a34 a13a21 a24

a47

1

23 Ec Pr A2

42 22 Pr S Pr

a a a a

Pr 2 Pr2 4S Pr 2

35 17 21 27

a

48 2

a a a a a

36 30 33 31 32

a a a a a

Pr 2 Pr2 4S Pr 2

37 30 35 32 34

a

49 2

a A1 A2 A1 A3

38

2r1

a

2 A1 A4 Ec Pr r1

a 2B2 B7 1Ec Pr

50 r 2 Pr r S Pr 2

61 2

1 1

K

1

Pr

1 2 S Pr

a51

2 A 2 A Ec Pr

1 2

2 2 Pr 2 S Pr 2

2 1 1

1K

1

2

1 K 2

2B a

Ec Pr

a62

2

2 40 1

K

1 Pr

1 2 S Pr

a52

A 2 A Ec Pr r

r 2 r 2 Pr 2 r S Pr 2

2 1 1

1 3 1

1K

1

2

1 K 2

1

Pr

1 1

2 Pr2 4S Pr

a63

B3 B9 a53 Pr

a 2

2 Pr

2 S Pr

53

22

2 a53

a53

2

2

Pr

2 Pr2 4S Pr

a B3 B10 a54

a 2 64

54 2

2 Pr

2 S Pr

2

2 a54 a54

2 2

a55

Ec Pr B2

1

4 2 Pr

S Pr

a65

2B3a55 Pr

2

12 K

K

2 Pr

1 2 S Pr

1K

2 1K

2

2 1

K

K

1 2 1 2

2

a

Ec Pr B2

a

2B3a55 Pr

56

4 2 Pr

S Pr

66 2

12 K

K

2 Pr

1 2 S Pr

1K

2 1 K

2

2 1

K K

1 2 1 2

Pr

2 Pr2 4 2 S Pr

a

2 2

B B a Pr

57 2

a 4 9 53

67

a

2

Pr

a

2

S Pr

2 53 53

Pr

2 Pr2 4 2 S Pr

2 2

a

2 2

B B a Pr

58 2

a 4 10 54

68

a

2

Pr

p> a

2

S Pr

2 54 54

a

2B1B2 1Ec Pr

2 2

59 2

2B a Pr

K

1 Pr

1 2 S Pr

a 4 55

2 1 1

K

1

K 69

2

1 2 1 2

K

2 Pr

2 2 S Pr

2 1

K

K

a 2B1a40 1Ec Pr

1 2 1 2

60 2

K

1 Pr

1 2 S Pr

2 1 1

1K

1

2

1K 2

2B4 a56 Pr

23 Ec Pr A2

2 a 2 a

a70 2

a 1

2 55 2 56

2

Pr

2 2

S Pr

80 42 22 Pr S Pr K K

2 1K

1 2

K

1

1 1

2

K

2 Ec Pr A2

a 1 a a

• 2B1B2 Ec

  a71

  B5 B9 a53 Pr

  Pr

  S Pr

  81 42 22 Pr S Pr

  55 56

  1KS

  a

  2 a 2

  2 1 53 1 53

  2 2

  a a a a e0.2a54

  B5 B10 a54 Pr

  83 80 2 54 79

  a72

  a

  2 Pr a

  2 S Pr

  a84

  e0.2a54 a53 1

  2 1 54 1 54

  2 2

  a a

  a e0.2a54

  a 2B5 a55 Pr

  85 81 79

  73 2

  a a a

  … a

  a

• a a

K

2

Pr

2 2 S Pr

86 59 60 78 50 51 52

2 1 1

1K

1

2

1 K 2

2

a87 a50 r1 a51 2 a52 2 r1 2 a59 a60 … a80 1

K

a74

2B5 a56 Pr

1

1

1

1

1

2

2

0.2 1

0.21

0.2 1

0.2 1

0.2 1

2 Pr 2

S Pr

1K 1K

1K 1K

1K

K

2

a a e

88 59

a60e a61e

a62e a78e

2 1 1 K

1 K

1 2 1 2

a a

e0.2a58 a57 a

a75

B6 B9 a53 Pr

Pr

S Pr

89 2 57 58

2 1 a53 1 a53

a a a e

58

2 2

2 2

a90 87 2 58 88

0.2a

a76

B6 B10 a54 Pr

Pr

S Pr

a

2

a 2

2 1 54 1 54

2 2

a77

2

2B6 a56 Pr

K

2 Pr

2 2 S Pr

2 1 1

1K

1

2

1K 2

a78

2

2B6 a56 Pr

K

2 Pr

2 2 S Pr

2 1 1

1K

1

2

1 K 2

0.4

0.4

2B B Ec

a 1 a e 1K a e

1K 1 2

79 55 56

1KS