effects of heat and mass transfer on mhd free...

IJREAS VOLUME 6, ISSUE 4 (April, 2016) (ISSN 2249-3905) International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR – 6.573)

International Journal of Research in Engineering & Applied Sciences Ema il:- [email protected], http://www.euroasiapub.org

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EFFECTS OF HEAT AND MASS TRANSFER ON MHD FREE CONVECTION FLOW OF A VISCOUS CONDUCTING FLUID IN A POROUS VERTICAL CHANNEL WITH HALL

CURRENT

M.Jena1,

Deptt. of Physics, College of Basic Science and Humanities, OUAT, Bhubaneswar-3, Odisha, India

S.Biswal2

Former Professor of Physics, SERC, Bhubaneswar, Odisha, India

Abstract

MHD power generation is a new system of electric power generation which is said to be of high efficiency and low pollution. In advanced countries MHD generators are widely used but in developing countries like INDIA, it is still under construction. The work is in progress at TRICHI in TAMIL NADU, under the joint efforts of BARC (Bhabha Atomic Research Center), Associated Cement Corporation (ACC) and Russian technologists.80 % of total electricity produced in the world is hydal, while remaining 20% is produced from nuclear, thermal, solar, geothermal energy and from magnetohydrodynamic (mhd) generator.MHD is concerned with the flow of a conducting fluid in the presence of magnetic and electric field. The fluid may be gas at elevated temperatures or liquid metals like sodium or potassium.Heat and mass transfer play a vital role in fluid flow with or without application of an external transverse magnetic field.Thermal energy transmission and mass transport of the species are more prominent in the rotational flow of a liquid or gas.In case of porous media, the permeability of the pores modifies the flow behaviour to a great extent.This paper deals with the study of heat and mass transfer effects on magnetohydrodynamic free convection flow of a viscous conducting fluid in a porous vertical channel with Hall current. Expressions for primary velocity, secondary velocity, temperature and concentration have been obtained by solving the constitutive equations. It is observed that the primary velocity increases with the increase of Hall parameter and decreases with the rise of Hartmann number. The temperature falls with Reynolds number R and the concentration falls with Schmidt number Scas well as Reynolds number R.

Keywords :Heat & mass transfer, MHD, Channel, Viscous fluid, Hall current.



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Introduction

The study of the problem of heat and mass transfer of an electrically conducting fluid through a channel has many engineering applications such as cooling of nuclear reactors, rocket technology, etc. Sachetiet. al. [1] studied hydro-magnetic free convection flow of a viscous incompressible fluid. Soundalgekar [2], Ramankumari and Reddy [3], Dash and Biswal [4] studied free convective of flow with mass transfer in the presence or absence of magnetic field. Sattar and Alam [5], Datta and Majumdar [6] studied the effect of Hall current on a steady hydro-magnetic free convective flow. Mohapatra and Tripathy [7] analysed the Hall effect on oscillatory hydro-magnetic free convection flow past an infinite vertical porous flat plate. However, in the above investigations they have not considered mass transfer and Hall effect simultaneously. Also in the above studies the effect of viscous dissipation which plays an important role has been neglected. Gebhert [8] studied the effect of viscous dissipation in natural convection in the absence of magnetic field. The objective of the present study is to consider the effect of dissipation and Hall current on the MHD free convective flow and mass transfer in a porous vertical channel.

Mathematical Formulation

Consider the free convection flow of an electrically conducting fluid in a vertical channel where the plates are separated by a distance ‘h’ from each other. We take x-axis parallel to the plates and y-axis normal to it. There is a uniform suction V0 on the wall y=0 and a uniform injection V0 on the wall y=h. A uniform magnetic field H0 acts normal to the channel neglecting the induced magnetic field, we can write, the magnetic field as:



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(0, ,0)yH H

(2.1)

This assumption is justifiable only when the magnetic Reynolds number is very small (of Cowling 9). For a fully developed steady free convection flow and mass transfer, the governing equations are

22*0

0 2 2

( )

(1 )

B u mwdu d uV g T g c

dy dy m

(2.2)

220

0 2 2

( )

(1 )

B mu wdw d wV

dy dy m

(2.3)

2

0 2p

d T k d TV

dy C dy (2.4)

2

0 2

dc d cV D

dy dy

(2.5)

The boundary conditions are



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1 1

2 2

0 , 0 , , at 0

0 , 0 , , at

u w T T C C y

u w T T C C y h

(2.6)

On introducing the following dimensionless variables

2 2

0 0 1 2 1 2

, , , ,T T C Cy u w

u w Ch V V T T C C

(2.7)

The equation (2.2-2.5), can be written in dimensionless form as

2

2 2

1( ) 0

1c

d u du RMu mw RG RG C

R dd m

(2.8)

2

2 2

1( ) 0

1

d w dw RMmu w

R dd m

(2.9)

2

2

10

. r

d d

R P dd

(2.10)

2

2

10

. c

d C dC

R S dd

(2.11)

Where the dimensionless parameters are 0V hR

,

1 230

( )g T TG

V

,

*1 2

30

( )c

g C CG

V

, P

r

CP

k

,

cSD

, M is the Hartmann number

The modified boundary conditions are:

0 , 1 , 1 at 0

0 , 1 , 0 at 1

u w c

u w c

(2.12)

Solutions of the equations

The solution of the governing equations (2.8-2.11) has been made using successive approximation. The solutions for and C satisfying the approximate boundary conditions from (2.12) are given by

(1 )1

1

r

r

R P

R P

e

e

(3.1)

And (1 )

1

1

c

c

R S

R S

eC

e

(3.2)



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Introducing u iw , equations (2.8) and (2.9) can be written together as

2 22

2 2

(1 )( )

(1 )c

d d MR imR R G G C

dd m

(3.3)

Substituting and C from equations (3.1) and (3.2) in equation (3.3) and solving for, with separating the real and imaginary parts, we have the primary and secondary velocity as:

5 5 7

7 1 2

1 6 2 6 3 8

4 8 1 1 1

cos( ) sin ( ) cos( )

sin ( )

a a a

a a a

u b e a b e a b e a

b e a P e q e S

(3.4)

5 5 7

7 1 2

1 6 2 6 3 8

4 8 1 2 2

sin ( ) cos( ) sin ( )

cos( )

a a a

a a a

w b e a b e a b e a

b e a P e q e S

(3.5)

Where the expressions for constants are omitted here in order to save space.

Fig. 1 : Primary velocity profile for different values of m,

Sc, R and Mwhen Pr = 1.0, Ec=0.01, Gc = 2.0, G = 5.0

Fig. 2 :Secondary velocity profile for different values of

m, Sc, R and Mwhen Pr = 1.0, Ec=0.01, Gc = 2.0, G = 5.0



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Results and Discussion

Figure 1 shows that the primary velocity u increases with increase in Hall parameter (m) but decreases with increase in Hartmann number (M), concentration parameter (Sc) and Reynolds number (R), where as it is found that for higher values of R (R>50), the primary velocity (u) approaches to zero.

Figure 2 shows the similar effects on the secondary velocity as in the primary velocity profiles. However, the secondary velocity has larger values than the primary one for the same value of above parameters.

Figure 3 shows the temperature profiles for various values of Reynolds number (R). It is observed that the temperature falls with increase in Reynolds number.

Figure 4 shows the concentration profiles for different values of concentration parameters (Sc) and Reynolds number (R). It is observed that concentration of the fluid decreases with increase in the value of Sc or R.

Conclusion

From the above study it is observed that primary velocity increases with the increase of Hall parameter & decreases with the rise of Hartmann number. The temperature falls with increase of Reynolds number & concentration falls with Sc as well as R.

Fig. 3 :Temperature profile for different values of R

whenG =5.0, Gc=2.0, Sc=0.6, Pr=1.0, m=1.0, M=8.0

Fig. 4 :Concentration profile for different values of Sc and R

when G = 5.0, Gc=2.0, m=1.0, M=8.0, Ec=0.01, Pr=1.0



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Applications

Agricultural engineering (to study underground water resources, seepage of water in river beds)

• MHD Power generation

• Geophysics(production of electricity through moving steam turbine)

• Space research(MHD accelerator)

• Thermonuclear fusion(nuclear energy)

• Aeronautical engineering(MHD accelerator)

• Biotechnology(magnetotherapy)

• Chemical engineering(purification and filtration)

• Petroleum engineering(oil recovery techniques)

• Pollution studies(descaling of water supply metallic pipes, unburned gasoline exhausted into environment)

• Industrialization applications

References

[1] Sacheti N.C.et. al., 1994,Int. Journal Comm. Heat & Mass Transfer. , 21(1), pp 131-142.

[2] Pop I., andSoundalgekar V.M., 1976,Astrophysics and Space Science,62, pp 389-396.

[3] Ramankumari C.V., andBhasker Reddy N.J,1994,Energy, Heat & Mass Transfer. 16, pp 279-287.

[4] Dash G.C., and Biswal S.,1988,Modeling, Simulation and Control B, AMSE Press,21,pp25-26.

[5] Sattar A., andAlam Md., 1995,Int.Journal of Pure and Applied Math., 26(2), pp 157-167.

[6] Mazumdar B.S, Gupta A.S, andDatta N., 1976,Int. J. Engg. Science,14, pp285-292.

[7] Mohapatra P., and Tripathy A., 1988,Journal of Mathematical Science,7, pp 49-63.

[8] Gebhert B.J., 1962,Fluid Mech., 14, pp 225-229.

[9] Cowling T.G.,1957,Magneto-hydrodynamics, New York: Interscience Publication, Inc., pp.13 & pp.100.

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effects of heat and mass transfer on mhd free...

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