DOI : 10.17577/IJERTV1IS6386
- Open Access
- Total Downloads : 783
- Authors : G.Soudjada, Dr.Subbulakshmi
- Paper ID : IJERTV1IS6386
- Volume & Issue : Volume 01, Issue 06 (August 2012)
- Published (First Online): 30-08-2012
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
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This work is licensed under a Creative Commons Attribution 4.0 International License
Free Convection Flow of Non-Newtonian Fluids in an Anisotropic Porous Medium
G.Soudjada1 Dr.Subbulakshmi2
1Assistant Professor, Department of Mathematics, AAGA&S College, Karaikal.
2 Assistant Professor, Department of Mathematics, D.G.G.A College(W),
Mayiladuthurai
Abstract
Convection in porous media plays a vital role in recent advancements. The applications of porous media are found in different areas like geophysics, petroleum processes, and air conditioning porosity. In this study, the anisotropic effects of porous medium are investigated for suitable range of parameters. The governing partial nonlinear differential equations were transformed into a set of coupled ordinary differential equations, which was solved using the fourth-order Runge-Kutta method. Nusselt number increases almost linearly with increasing porosity.
Key Words: Non-Newtonian flow, free convection, anisotropic porous medium.
1. Introduction
There has been an increase in interest in the effect of anisotropic porous media, because of their extensive practical application in many areas. A porous medium is a material containing pores. The skeletal portion of the material is often called the metric or frame. The pores are typically filled with a fluid. Many studies related to non-Newtonian fluids saturated in an anisotropic porous medium have been carried out. Non-Newtonian fluids are characterized by a non-linear relationship between the shear stress and shear velocity of the flow. These fluids are often encountered in nature and industrial technologies (volcanic, lava, mudflavs, oil, plastics, oil-based points and polymer solutions) Chen & Chen (1988) investigated the free convection flow along a vertical plate embedded in a porous medium. Vafai et.al (1983), carried out an experimental investigation into valuable porosity, finding that the Nusselt number depends on the Reynolds number and the free convection flow of Non-Newtonian fluids in an anisotropic porous medium is investigated
numerically. Sekar. R et. al. (1996) have investigated ferro convection in an anisotropic porous medium. Sengupta.T.K (2004) have investigated foundation of computational fluid dynamics. G. Degan et. al(2007). have studied transient natural convection of non- Newtonian fluids about a vertical surface embedded in an anisotropic porous medium. Han-Taw Chen et. al. (1988) have investigated the free convection flow of non-Newtonian fluids along a vertical plate embedded in a porous medium. Rajendra et. al. (2001) investigated the flow of non-Newtonian fluids in fixed and fluidized beds. Prakash Chandra and V.
-
Satyamurty et. al. (2011) have studied the non- Darcian and Anisotropic Effects on Free Convection in a Porous Enclosure. Gorla et. al. (2001) have studied the free convection in non-Newtonian fluids along a horizontal plate in a porous medium. E. Kim (1997) have investigated the natural convection along a wavy vertical plate to non-Newtonian fluids A similarity solution is sought for the governing equations. Then the effect of variable porosity on the temperature distribution and Nusselt number in both cases is stated.
2. Mathematical formulation
The following figure represents a Non- Newtonian power law fluid flow along a constant temperature vertical plate embedded in an anisotropic porous medium.
The Governing equations are: The continuity equation is
(1)
The power law fluid is
The temperature equation is
The density equation is
(2)
(3)
(4)
The boundary conditions for these equations are
(5)
(6)
where K, n and k2(n) are the power law constant, power law index and the permeability of the porous medium in the vertical direction respectively.
k2(n) = k1(n)
where is the anisotropic parameter and k1 is the
permeability along the horizontal direction. R
(7)
(15)
(16)
(8)
The non-dimensionless form of governing equations are
(9)
(17)
(10)
(18)
The dimensionless terms used are
(11)
(12)
(13)
(19)
The above partial differential equations are transformed into ordinary differential equations using the following dimensionless variables defined by,
(20)
(21)
Ra
(14)
(22)
Consequently, the velocity components become,
(23)
(24)
(29)
Simplifying, using Runge-Kutta method, we get the local Nusselt number as
The boundary conditions are
(25) (30)
Substituting eq. 30 into 29 gives
(26)
(27)
(28)
(31)
The local heat flux at the wall is
Table -1
Variation of with = for a value of
=
0.0000
1.0000
1.0000
0.5871
1
4.0000
0.6631
7.0000
0.00609
0.0000
10.0000
10
1.0000
5.8719
4.0000
0.6631
7.0000
0.0609
0.0000
30.0000
1.0000
17.6157
30
4.0000
1.9893
7.0000
0.1827
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2 =1
0
-2
B C D
=30
=10
-1 0 1 2 3 4 5 6 7 8
Fig : 1, Variation of with = for a value of
Table – 2
Variation of with = for a value of
=
0.0000
1.0000
1.0000
0.7663
1
4.0000
0.2575
7.0000
0.0789
0.0000
3.1623
10
1.0000
2.4232
4.0000
0.8143
7.0000
0.2468
0.0000
5.4772
1.0000
4.1971
30
4.0000
1.4104
7.0000
0.4274
B C D
6
5 =30
4
3
2
=1
=10
1
0
-1 0 1 2 3 4 5 6 7 8
Fig.- 2 Variation of with = for a value of
Table – 3
Variation of with = for a value of
=
0.0000
1.0000
1.0000
0.8374
1
4.0000
0.4047
7.0000
0.1826
0.0000
2.1544
10
1.0000
1.8041
4.0000
0.8720
7.0000
0.3934
0.0000
3.1072
1.0000
2.6019
30
4.0000
1.2576
7.0000
0.5674
B
3.4
3.2
3.0
2.8
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
D
C
=10
=1
=30
-1 0 1 2 3 4 5 6 7 8
Fig – 3, Variation of with = for a value of
4. Result and discussion
The results of equation (25) are shown in the above three tables. The above three figures shows the dimensionless temperature versus the dimensionless similarity variable for different n.
5. Conclusion
In this study the effects of porosity on free convection flow of Non-Newtonian fluids in an anisotropic porous medium was investigated. When n=1, the results is in agreement with a Newtonian fluid.
When n>1,
-
As porosity increases Nusselt number increases.
-
As temperature variation becomes steeper, heat transfer rate increases and Nusselt number increases with increasing porosity.
6. References
-
Chen.H and Chen C, Free convection flew of non Newtonian Embedded in a porous medium Journal of Heat Transfer. Vol 110 PP. 257 260, 1988
-
Sekar.R. Vaidanathan.G Ramanathan.A. Ferro Convection in an anisotropic porous medium. Int. J. Engng.Sci 1996, Vol 34(4) PP. 399-405
-
Sengupta.T.K. 2004 Foundation of computational fluid dynamics (Hyderabad University press)
-
Vafai K. Et.al An Experimental investigation of heat transfer in Variable porosity medium, ADME.J. Heat transfer, Vol 107, PP. 642-647, 1985
-
G.Degan, C.Akowanou, N.C.Awanou, Transient natural convection of Non- Newtonian fluids about a vertical surface embedded in an anisotropic porous medium, I.J of Heat and Mass Transfer, 50 (2007),4629-4639.
-
Prakash Chandra and V.VSatyamurthy, Non-Darcian and Anisotropic Effects on Free Convection in a porous Enclosure, Transp porous Med, 90 (2011),301-320.
-
Rajendra.P, Chhabra, Jacques Comiti,Ivan Machad, Flow of non-Newtonian fluids in fixed and fludised beds,Chemical Engineering Science, 56 (2001),1-27.
-
Han-Taw Chen, Chao-Kuang, Chen,Free convection flow of non-Newtonian fluids along a vertical plate embedded in a porous medium, Journal of Heat Transfer, 110-1 (1988),257-260.
-
Gorla, Rama Subba Reddy and Kumari, Mahesh, Free convection in non- Newtonian fluids along a horizontal plate in a porous medium, In: Heat and Mass Transfer, 39 (2) (2001), pp.101-106.
-
E.Kim, Natural convection along a wavy vertical plate to non-Newtonian fluids ,In: Heat and Mass Transfer, 40(13) (1997), pp.3069-3078.
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