Unsteady heat and mass transfer in mhd flow past a...
Transcript of Unsteady heat and mass transfer in mhd flow past a...
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7.1 INTRODUCTION
Heat and mass transfer phenomenon occurs in a variety of problems. In nature this
occurs in buoyancy induced motions in the atmosphere. This finds variety of applications
in science and technology. Singh and Rana (1992) investigated the three–dimensional
flows and heat transfer through a porous medium. They observed that heat transfer
decreases with increase in Prandtl number and permeability parameter, while it increases
with increase in Eckert number. Sacheti et al (1994) investigated the unsteady
hydromagnetic flow past a vertical plate subject to constant heat flux. They obtained an
exact solution of this problem using Laplace transform. Velocity and skin friction of the
flow have been presented for water. They noted that the magnetic field has a retarding
effect on the velocity while skin friction at the plate increases with it. Das et al (2001)
described the three–dimensional fluctuating free convective flow through a porous
medium bounded by an infinite vertical porous plate. Sharma et al (2002) and Chaudhary
and Chand (2002) have solved the problems of three–dimensional viscous flow and heat
transfer along a porous plate in the presence of sinusoidal suction.
The study of porous media together with the magneto - hydrodynamics has been
studied by many researchers. The flow past a vertical plate embedded in a saturated
porous media has been studied by Na and Pop (1983). They prescribed non-uniform wall
temperature and a non-uniform wall heat flux. The governing equations were solved
numerically by using two-point finite-difference method. Singh et al (2003) have
investigated Hydromagnetic heat and mass transfer in MHD flow of an incompressible,
electrically conducting, viscous fluid past an infinite vertical porous plate embedded with
porous medium of time dependent permeability under oscillatory suction velocity normal
to the plate. Chaudhary and Jain (2007) studied the MHD flow past an infinite vertical
oscillating plate embedded in porous medium. The governing equations were solved by
using Laplace transform technique. They observed that the skin-friction increased with an
increase in Schmidt number, Prandtl number, magnetic parameter, while it decreases with
an increase in the value of Grashof number, modified Grashof number, permeability
parameter and time. Nusselt number increases with an increase in Prandtl number while
temperature decreased with an increase in the value Prandtl number.
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Zueco and Ahmed (2010) analyzed MHD flow of an incompressible viscous
electrically conducting fluid past an infinite vertical porous plate with combined heat and
mass transfer. The porous plate is subjected to a constant suction velocity as well as a
uniform stream velocity. The governing equations are solved by the perturbation
technique and a numerical method. An increase in the heat source/sink or the Eckert
number is found to strongly enhance the fluid velocity. It is found that the velocity, the
fluid temperature, and the induced magnetic field decrease with the increase in the
destructive chemical reaction. Das and Jana (2010) investigated heat and mass transfer on
an infinite vertical plate embedded in porous medium, which moves with time dependent
velocity in a viscous, electrically conducting incompressible fluid. A uniform magnetic
field is applied normal to the plate. They solved the problem analytically by Laplace
transform technique. They obtained the expressions for velocity, temperature,
concentration, skin friction, rate of heat and mass transfer. They observed that skin
friction decreases with increase in permeability of the porous medium and increases with
increase in magnetic parameter.
In this study, the effects of permeability variation and oscillatory suction velocity
on free convection and mass transfer in MHD flow of a viscous incompressible
electrically conducting fluid past on infinite vertical porous non–conducting plate
embedded in a porous medium are presented. The plate is subjected to oscillatory suction
velocity normal to the plate in the presence of a uniform transverse magnetic field. The
rate of change of temperature and concentration species is prescribed on the boundary. In
certain situations these may be more convenient than prescribing the temperature and
concentration of species. The expressions for the velocity, temperature, concentration of
species and skin-friction are obtained. The effect of different parameters entering into the
expressions is shown graphically.
7.2 CO-ORDINATE SYSTEM AND EQUATIONS
Unsteady hydromagnetic flow past an infinite vertical porous plate bounded by a
porous medium of time dependent permeability and suction velocity is considered. The
fluid is assumed to be incompressible, viscous and electrically conducting. The x'-axis is
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taken along the direction in which fluid is flowing and y'-axis is normal to it. A uniform
magnetic field is applied normal to the direction of flow. The magnetic Reynolds number
is taken to be very small so that the induced magnetic field is small in comparison to the
applied magnetic field and hence can be neglected. The fluid properties are assumed to be
constant. The temperature difference between the wall and the medium develops
buoyancy force which induces the basic flow. Initially, the fluid as well as the plate are
taken to be at the same temperature. It is further assumed that the concentration of the
species is very low. Following Gebhart and Pera (1971), the Soret and Dofour effects are
neglected. For 0*t , the temperature of the plate and the concentration of species are
changed to *
wT and *
wC respectively. Taking usual Boussinesq approximation into
account, the governing equations for conservation of mass, momentum, energy and
concentration are
0
y
v (7.2.1)
t
u v
y
u= g )(
**
TT )(*
CCg +
2
2
y
u
uBK
u 2
0
(7.2.2)
v
t
T
y
T =
pC
2
2
y
T (7.2.3)
2
2
*
**
*
y
CD
y
Cv
t
C (7.2.4)
where *
v is the constant suction velocity, *u the fluid velocity, *T the fluid
temperature, *
T the fluid temperature in free stream, *C the species concentration in the
fluid, *
C the species concentration in free stream, the coefficient of thermal
expansion, the coefficient of thermal expansion with concentration, g the
acceleration due to gravity, D the chemical molecular diffusivity, pC the specific heat
at constant pressure, *K the permeability of porous medium, the thermal
conductivity, the kinematic viscosity, *
the fluid density, the electric
permeability and 0B the magnetic field intensity.
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The suction velocity on the vertical plate is imposed in the form
)1(***
0
tievv
(7.2.5)
The permeability of the porous medium is taken in the form
)1( **
0
tieKK (7.2.6)
where *
0v is the constant suction velocity of the fluid through the porous surface, 0K is
the constant permeability of the porous medium and is a constant )10( .
The following non-dimensional quantities are introduced
y =
0vy, t =
4
2*
0vt
, 2*
0
4
v
, *
0v
uu
, **
**
CC
CCC
w
, **
**
TT
TT
w
,
Grashof Number Gr = g3*
0
**)(
v
TTw ,
Modified Grashof Number Gc =
g3*
0
**)(
v
CCw ,
Prandtl Number Pr = )/( pC
, Magnetic parameter
0
0
v
BM ,
Schmidt number Sc =D
, Permeability parameter K =
2
*2*
0
Kv ,
(7.2.7)
Using (7.2.5) - (7.2.7) into the equations (7.2.1) – (7.2.4), we have
)1(ti
ev
(7.2.8)
uMeK
u
y
uCGcGr
y
ue
t
uti
ti 2
0
2
2
)1()1(
4
1
(7.2.9)
2
2
Pr
1)1(
4
1
yye
t
ti
(7.2.10)
2
21
)1(4
1
y
C
Scy
Ce
t
C ti
(7.2.11)
such that u is the velocity along the x-axis, the fluid temperature, C the species
concentration, the frequency of oscillation, t the time.
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The boundary conditions in the non-dimensional form are
u = 0, titi
ey
Ce
y
1,1 at y = 0 , (7.2.12)
u 0, 0, C 0 as y . (7.2.13)
7.3 SOLUTION OF EQUATIONS
The solution of governing equations is obtained by separating the steady and
unsteady parts in the following manner
ti
eyCuyCutyCu
))(,,())(,,(),)(,,( 111000 (7.3.1)
Substituting (7.3.1) into the equations (7.2.9) to (7.2.11) and separating the steady and
unsteady components, we obtain
000100 CGcGruauu , (7.3.2)
0
0
0111211K
uuCGcGruauu , (7.3.3)
00 Pr = 0, (7.3.4)
0111 PrPr4
Pr
i
, (7.3.5)
000 CScC , (7.3.6)
01114
CScCSci
CScC
, (7.3.7)
where prime denotes differentiation with respect to y.
The corresponding boundary conditions are
1,,1,,0, 101010 CCuu at 0y ,
0,,,,, 101010 CCuu as y . (7.3.8)
The solutions of equations (7.3.2) to (7.3.7) under the boundary conditions (7.3.8) are
recorded as under
tiyymyee
ie
i
mety
Pr
1
Pr 41
Pr41
Pr
1),( 1 (7.3.9)
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tiScyymScyee
ie
iSc
me
SctyC
41
411),( 2
2
(7.3.10)
ymymyScyymeaeaeaeaeaatyu 143
10154
Pr
343 )(),(
tiyScyymymeeaeaeaea
14
Pr
13121132 (7.3.11)
where 1m to 4m and 1a to 15a are constants depending on physical parameters and are
recorded in the APPENDIX - IX.
It is the real part alone of the complex quantities which have physical significance
in the flow problems. The velocity, temperature and concentration fields are expressed in
the following form
u (y, t) = u0(y) + (Mr tcos – Mi tsin ) (7.3.12)
(y, t) = )(0 y + (Kr tcos – Ki tsin ) (7.3.13)
C (y, t) = C0(y) + (Lr tcos – Li tsin ) (7.3.14)
where Mr , Mi , Kr , Ki , Lr and Li are constants recorded in the APPENDIX - IX.
7.4. RESULTS AND DISCUSSION
In this section, velocity field, temperature field, concentration field and skin-
friction co-efficient at the plate are discussed by assigning numerical values to various
parameters appearing in the solution.
The values of Prandtl number Pr are taken for Air ( Pr = 0.71), Water ( Pr =7.0)
and water at freezing point ( Pr =11.4). The values of Schmidt number Sc are taken for
Hydrogen ( Sc =0.22), Helium ( Sc = 0.30), water-vapour ( Sc = 0.60), Oxygen
( Sc =0.66) and Ammonia ( Sc = 0.78). The values of Grashoff number are chosen to be
Gr = 10.0, 20.0 and values of modified Grashoff number are taken as Gc = 10.0, 20.0.
Magnetic parameter are selected as M = 0.5, 1.0, permeability parameter as K0 = 10.0,
20.0 and frequency parameter = 5.0. The perturbation parameter is taken as 0.005.
The various profiles have been drawn for 2/ t .
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Velocity Field
The transient velocity for 2
t is given by
iMyuyu
)(
2, 0 (7.4.1)
Fig.7.1 shows transient velocity field for Gr = 10.0 and 20.0 which correspond to
the cooling of the plate. It is observed that an increase in Gc or Gr or K0 increases
numerically the transient velocity, while an increase in Sc and M decreases numerically
the transient velocity field. It is further noted that the transient velocity field first
increases numerically and then the curves fall gradually after attaining a maximum value.
The maxima occur in the near vicinity of the plate.
Temperature Field
The transient temperature field for 2
t is given by
iKyTyT
)(
2, 0 (7.4.2)
Fig. 7.2 shows transient temperature field due to variation in Prandtl number Pr
for air, water and water at freezing point at = 0.005 and 2/ t . It is observed that
an increase in Pr , increases numerically the transient temperature. The temperature
approaches zero for small values of y in case of air and water. In the present case the
slope of the curves does not indicate much variation.
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Concentration Field
The transient concentration field at 2
t is given by
iLyCyC
)(
2, 0 (7.4.3)
Fig. 7.3 shows transient concentration field due to variation in Schmidt number
Sc for Hydrogen, Helium, Water-vapour, Oxygen and Ammonia at = 0.005 and
2/ t . It is observed that the magnitude of transient concentration field is greater for
Hydrogen and Helium. With increase in Schimdt number the magnitude decreases. It is
seen that for Ammonia, it is least. With increase in y the magnitude goes on decreasing.
Out of the cases considered here Hydrogen maintains larger concentration level.
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Skin-Friction
Skin-friction coefficient () at the plate in terms of amplitude and phase is
)(cos||0
0
tNy
u
y
, (7.4.4)
where
0 = m3 (a3 + a4) – a3 Pr – a4 Sc , N = Nr + i Ni = 0
1
yy
u, tan =
r
i
N
N,
Ni = – B1N1 – A1N2 – B2N3 – A2N4 – m3N6 – PrN8 – ScN10 – B3N11 – A3N12,
Nr = –A1N1 + B1N2 – A2N3 + B2N4 – m3N5 – PrN7 – ScN9 – A3N11 + B3N12.
Table-1 represents the numerical values of skin-friction co-efficient () in terms of
amplitude N and phase angle () for variation in Gr , Gc , Sc , M, K0, and Pr
respectively for cooling of the plate. It is observed that increase in Gr , Gc , K0 leads to
an increase in the value of N while an increase in Sc , M, , Pr leads to decrease in
the value of N . However, the increase in Gr brings greater changes in N . It is
observed that there is a phase lag in skin-friction due to increase in Gr , Gc , Sc , K0, ,
Pr . The magnitude of phase lag varies for variation of different parameters as shown in
Table–1. Out of the cases which are considered in this study Prandtl number affects the
phase lag most.
Table – 2 shows the effect of Gr , Gc , Sc , K0, , M and Pr on skin –friction
coefficient (0) due to steady part of velocity and skin–friction coefficient () due to
cooling of the plate at = 0.005 and 2/ t . It is observed that the magnitude of skin-
friction coefficients 0 and increase due to increase in Gr , Gc , K0 while decreases due
to increase in Sc , M, Pr . However, with variation in , there is slight increase in
magnitude of .
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Table – 1 Values of Amplitude N and phase angle ( ) for Skin–friction
coefficient due to cooling of the plate.
Gr Gm Sc M 0K ω Pr N (Degree)
10.0 4.0 0.22 0.5 10.0 5.0 0.71 50.3271 –81.81
20.0 4.0 0.22 0.5 10.0 5.0 0.71 73.9358 –79.86
10.0 8.0 0.22 0.5 10.0 5.0 0.71 77.1288 –83.67
10.0 4.0 0.66 0.5 10.0 5.0 0.71 33.3457 –76.26
10.0 4.0 0.22 1.0 10.0 5.0 0.71 38.8981 –81.87
10.0 4.0 0.22 0.5 10.0 10.0 0.71 27.4392 –80.81
10.0 4.0 0.22 0.5 10.0 5.0 7.00 26.9494 –86.83
10.0 4.0 0.22 0.5 10.0 5.0 7.00 26.9494 –86.82
Table – 2 Values of 0 and due to cooling of the plate at 2/ t .
Gr Gm Sc M 0K ω Pr
0
10.0 4.0 0.22 0.5 10.0 5.0 0.71 51.0658 50.8167
20.0 4.0 0.22 0.5 10.0 5.0 0.71 65.3706 65.0067
10.0 8.0 0.22 0.5 10.0 5.0 0.71 87.8267 87.4434
10.0 4.0 0.66 0.5 10.0 5.0 0.71 20.7896 20.6276
10.0 4.0 0.22 1.0 10.0 5.0 0.71 30.8832 30.6907
10.0 4.0 0.22 0.5 10.0 10.0 0.71 54.1876 53.9197
10.0 4.0 0.22 0.5 10.0 5.0 7.00 51.0658 50.9303
10.0 4.0 0.22 0.5 10.0 5.0 7.00 36.9573 36.8227
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7.5 CONCLUSION
From this study, it is concluded that Grashoff number increases the transient
velocity, Schmidt number decreases the same. Grashoff number also shows greater
influence on skin-friction in comparison to other parameters. Schmidt number is also
responsible for decreasing the magnitude of concentration field .The effect of Prandtl
number is to increase the transient temperature.