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International Journal of Computational and Applied Mathematics.
ISSN 1819-4966 Volume 12, Number 1 (2017), pp. 53-64
© Research India Publications
http://www.ripublication.com
Finite Element Study of Soret and Radiation Effects
on Mass Transfer Flow through a Highly Porous
Medium with Heat Generation and Chemical
Reaction
B. Shankar Goud * and M.N. Raja Shekar
Department of Mathematics, JNTUH College of Engineering Kukatpally,
Hyderabad- 500085, TS, India.
Department of Mathematics, JNTUH College of Engineering Nachupally,
Karimnagar -505501, TS, India.
Abstract
The problem of Soret and Radiation effects on mass transfer flow through a highly
porous medium with heat generation and chemical reaction has been analyzed
numerically. Exact solutions of the governing equations are solved by Galerkin finite
element technique depending on the physical parameters including the Prandtl number
(Pr), Thermal Grasof number (Gr), mass Grashof number (Gc), the Schmidt number
(Sc), the Soret number (So), chemical reaction parameter (Kr) and radiation
parameter (R), The effects of physical parameters are discussed with the help graphs.
Keywords: Soret effects, Radiation effect, Chemical reaction, FEM, Heat Generation,
MHD.
INTRODUCTION
Free convection flow is an important factor in several practical applications that
include cooling of electronic components, in designs related to thermal insulation,
material processing, and geothermal systems etc. Magnetohydrodynamics has
attracted attention of a large number of scholars due to its various applications. N.G
Kafoussias and E.W Williams [1] investigated thermal-diffusion and diffusion -
54 B. Shankar Goud and M.N. Raja Shekar
thermo effects on mixed free forced convective and mass transfer boundary layer flow
with temperature dependent viscosity. Ahmed M. Salem and Mohamed Abd El-Aziz
[2] have studied the effect of Hall currents and chemical reaction on hydromagnetic
flow of a stretching vertical surface with internal heat generation/absorption. Magdy
A.Ezzat et.al [3] have studied free convection effects on a viscoelastic boundary layer
with one relaxation time through a porous medium. Mohamed AbdEl-Azziz [4] has
studied Thermo - diffusion and diffusion effects on combined heat and mass transfer
by hydromagentic three- dimensional free convection over a permeable stretching
surface with radiation. A. Rapti et.al [5] studied effect of thermal radiation on MHD
flow. J.H Merkin and I.Pop [6] investigated the forced convection flow of a uniform
stream over a flat surface with a convective surface boundary condition. Orhan Aydin
and Ahmet kaya [7] observed mixed convection of a viscous dissipating fluid about a
vertical plate. Finite element study of radaitive free convection flow over a linearly
moving permeable vertical surface in the presence of magnetic field was studied by
S.Rawat and S. Kapoor [8]. M.S Alam et.al [9] carried out a research on Dufour and
Soret effects on steady free convection and mass transfer flow past a semi – infinite
vertical plate in a porous medium. S. Shuteye [10] presented thermal radiation and
Buoyancy effects on heat and mass transfer over a semi – infinite stretching surface
with suction and blowing. K Vajravelu et.al [11] reported unsteady convective
boundary layer flow of a viscous fluid at a vertical surface with variable fluid
properties. M.Turkyimazogulu and I.Pop [12] presented the Soret and heat source
effects on the unsteady radiative MHD free convection flow from an impulsively
started infinite vertical plate. P.A Lakshmi Narayana and P.Sibanda [13] considered
the influence of the Soret effect and double dispersion on MHD mixed convection
along a vertical plate in non – Darcy porous medium. Effects of chemical reaction and
radiation on MHD free convection flow of Kuvshinshiki fluid through a vertical
porous plate with heat source have been studied by P.Mohan Krishna et.al [14]. G
Palani et.al [15] have analyzed the effect of viscous dissipation on an MHD free
convective flow past a semi – infinite vertical cone with a variable surface heat flux.
G.Seth et. al [16] presented the effects of hall current, radiation and rotation on
natural convection heat and mass transfer flow past a moving vertical plate. MHD
flow and heat transfer of a viscous fluid over a radially stretching power - law sheet
with suction/ injection in a porous medium has been studied by M.Khan et.al [17].
S.Mohammed Ibrahim and K.Suneetha [18] presented heat source and chemical
effects on MHD convection flow embedded in porous medium with Soret, viscous
and Joule’s dissipation. A.G Vijay Kumar and S. Vijay Kumar Varma [19] studied
thermal radiation and mass transfer effects on MHD flow past an impulsively started
exponentially accelerated vertical plate with variable temperature and mass diffusion.
G.S. Seth et.al [20] investigated MHD natural convection flow with radiative heat
Finite Element Study of Soret and Radiation Effects on Mass Transfer Flow…. 55
transfer past an impulsively moving vertical plate with ramped temperature in the
presence of hall current and thermal diffusion.
In this paper the unsteady MHD free convection and mass transfer flow past a vertical
porous plate has been investigated analytically by using Galerkin finite element
technique. The effects of the flow parameters on the temperature, concentration and
velocity have been studied graphically.
MATHEMATICAL ANALYSIS
A two dimensional flow of an incompressible and electrically taking viscous fluid
along an infinite vertical plate that is embedded in a porous medium in the presence of
thermal radiation, heat generation, and chemical reaction is considered. It is assumed
that there is radiation only in fluid. The 𝑥∗ - axis is taken along the infinite plate and
𝑦∗ - axis perpendicular to it and all the variables are functions of 𝑦∗ and 𝑡∗. Under
these conditions and assuming variation of density in the body force term
(Boussinesq’s approximation), the problem can be governed by the following set of
equations:
Equation of continuity:
*
*0
v
y --- (1)
Momentum equation:
2
* * * 2 ** * * * * * *
* * * **
1( ) ( )
u u p v vv v g T T g C C u
t y x Ky --- (2)
Energy equation:
2
2* * 2 * *
* * *
* * * **( )r
p p p p
qT T k T Q v uv T T
t y c c y c c yy --- (3)
Diffusion equation:
2 2
* * 2 * 2 ** * * *
* * * *( ) m T
r
m
D kC C C Tv D K C C
t y Ty y --- (4)
Where the Rosseland approximation is used, which leads to
4*
*
4
3
sr
e
Tq
K y --- (5)
56 B. Shankar Goud and M.N. Raja Shekar
With the appropriate initial and boundary conditions are given by
* * * ** * * * * * * * * * *
* * * * * * *
, ( ) , ( ) 0
, ,
n t n t
p w w w wu U T T T T e C C C C e at y
u U T T C C at y --- (6)
Assuming that the temperature difference within the flow is such that 4*T may be
expanded in Taylor’s series and expanding 4*T about *T
, the free stream temperature
and neglecting higher order terms we get 4* 3 44 4T T T T --- (7)
From the continuity equation (1), it is clear that the suction velocity normal to the
plate is either a constant or a function of time. Hence the suction velocity normal to
the plate is taken as *
0v v --- (8)
where 0v is scale of suction velocity which is a nonzero positive constant. The
negative sign indicates that suction is towards the plate. Outside the boundary layer,
(2) gives *
*
* *
1 p vU
x K --- (9)
Introducing the following non – dimensional quantities,
** 2 * * * * ** *
0 0
* * * * * *
0
* * * * * * 2*
0 0
* 2 * 2 2 2 2
0 0 0 0
**
02 3
0
, , , , , , ,
, , , , , ,
, , Pr ,4
p
p
w w
w w
p
m t wper
s
Uy t T T C Cuy u U t C
U U T T C C
g T T g C C vQ K vv vnGr Gc Sc Q n K
DU v U v C v v v
D K Tv CK kK vKr R S
kv T
* 2
0
* * * *,
m w P w
T vEc
T v C C c T T
--- (10)
In the view of above equation, the basic flow field equations can be expressed in the
following form:
2
2
11
u u uGr GcC u
t y Ky
--- (11)
22
2
ud Q Ec
t y yy
--- (12)
2 2
2 2
1C C CKrC So
t y Sc y y
--- (13)
Where 4 3
3 Pr
Rd
R
and , , Pr, , , ,Gr Gc Sc Kr R Q ,and K are the thermal Grashof
number, Solutal Grashof number, Prandtl number, Schmidt number, chemical reaction
Finite Element Study of Soret and Radiation Effects on Mass Transfer Flow…. 57
parameter ,radiation parameter, heat generation parameter, and permeability of the
porous medium respectively.
, 1 , 1 0
1, 0, 0
nt nt
pu U e C e at y
u C as y --- (14)
SOLUTION OF THE PROBLEM
By applying the Galerkin element method with Crank – Nikolson discretization,
taking 0.1, 0.01h k and 2
kr
h for equation (11) over the two noded linear
element ( )e , ( )j ky y y is
2
120
k
j
y ( e ) ( e ) ( e )( e )T ( e )
y
u u uNu R dy
y y t
--- (15)
Here 1
1R Gr GcC N,N
K
Integrating the first term in equation (15), by parts, one obtains
1 0
k k k k k
j j j jj
y y y y y( e ) ( e )T ( e ) ( e )( e )T ( e )T ( e ) ( e )T
y y y yy
u u udy dy N u dy R dy
y y y t
--- (16)
Since the derivative u
y
is not specified at either ends of the element ( )e ,
( )j ky y y we neglecting the first term in equation (16) to obtain
1 0
k k k k
j j j j
y y y y( e )T ( e ) ( e )( e )T ( e ) ( e )T
y y y y
u udy dy N u dy R dy
y y t
--- (17)
Finite element model may be obtained from equation (17) by substituting finite
element approximation over the two noded linear elements ( )e , ( )j ky y y of the
form:
( e ) ( e ) ( e )u N Here
T( e ) ( e )
j k j k, u u --- (18)
Where ju , ku are the velocity components atthj and thk nodes of the typical element )(e
( kj yyy ) and j , k are the basis functions defined as follows.
58 B. Shankar Goud and M.N. Raja Shekar
jkj k
k j k j
y yy y,
y y y y
. Substituting equation (18) into (17), the following is
obtained:
( )( ) ( )
1
( )
1 1 2 1 1 1 2 1 11 1
1 1 1 2 1 1 1 2 16 2 6 2
ee ejj j j
e
k k kk
u u uu R ll Nl
l u u uu
Where ‘ ’ denote the differentiation with respect to time, ( )e
k jl y y is the length of
the element. Assembling the element equations by inter-element connectivity for two
consecutive elements 1i iy y y and 1i iy y y .we get
2
1
1 1 1
1
( )
1 1 11
1 1 0 1 1 0 2 1 0 2 1 0 11 1 1
1 2 1 1 0 1 1 4 1 1 4 1 26 6 22
0 1 1 0 1 1 0 1 2 0 1 2 1
i
i i i
ii i iee
i i ii
uu u u
N Ru u u u
llu u u
u
--- (19)
On equating row corresponding to the node i to zero, the following difference
schemes with ( )el h is obtained:
1 11 1 1 1 1 1 12
1 1 12 4 4
2 6 6i i ii i i i i i i i
Nu u u u u u u u u u u R
h h
--- (20)
Applying the trapezoidal rule and from the equation (18), following system of
equations in Crank – Nicholson method are obtained:
1 1 1
1 1 1
1 1
2 6 3 8 12 4 2 6 3 2 6 3
8 12 4 2 6 3 12
j j j j
i i i ij j
i i
r rh Nk u r Nk u r rh Nk u r rh Nk u
r kN u r rh Nk u R k
1 1 1 *
1 1 2 3 1 4 1 5 6 1
n n n n n n
i i i i i iAu A u A u A u A u A u R --- (21)
Where
1 2 3 4
*
5 6 1
2 6 3 , 8 12 4 , 2 6 3 , 2 6 3
8 12 4 , 2 6 3 , 12 12 (( ) ( ) )j j
i i
A r rh Nk A r Nk A r rh Nk A r rh Nk
A r Nk A r rh Nk R R k k Gr Gm C
Similarly applying the Galerkin finite element method for equation (12) – (13) the
following equations are obtained:
1 1 1 **
1 1 2 3 1 4 1 5 6 1
j j j j j j
i i i i i iB B B B B B R
--- (22)
1 1 1
1 1 2 3 1 4 1 5 6 1
j j j j j j
i i i i i iC C C C C C C C C C C C
--- (23)
Finite Element Study of Soret and Radiation Effects on Mass Transfer Flow…. 59
Where
1 2 1 32
***
4 5 1 6 0 2
1 2 3
2 6 3 , 8 12 4 , 2 6 3 ,
2 6 3 , 8 12 4 , 2 6 3 , 12
2 6 3 . , 8 12 4 , 2 6 3 . ,
i
i
B rd rh kQ B rd k Q B rd rh kQ
B rd rh kQ B rd k Q B rd rh kQ R ScS ky
C Sc r rh Sc kScKr C Sc r kScKr C Sc r rh Sc kScKr
2
**
4 5 62 6 3 , 8 12 4 , 2 6 3 . , 12 i
i
uC Sc r rhSc kScKr C Sc r kScKr C Sc r rh Sc kScKr R kEc
y
Here 2
kr
h and ,h k are mesh sizes along y -direction and t – direction respectively.
Index ,i j refers to the space and time. In the equations (21), (22) and (23), taking
1(1)i n using initial and boundary conditions (14), the following system of equations
are obtained:
, 1(1)3i i iA X B i --- (22)
Where iA ’s are matrices of order n and iX , iB ’s column matrices having n -
components. The solutions of above system of equations are obtained by Thomas
algorithm for velocity, temperature, concentration. For various parameters the results
are computed and presented graphically.
The skin friction, Nusselt number and Sherwood number are important physical
parameters for this type of boundary later flow.
With known values of velocity, temperature and concentration fields, the Skin-friction
at the plate is given by non-dimensional form 0y
u
y .
The rate of heat transfer coefficient can be obtained in terms the Nusselt number in
non-dimensional, given by 0
u
y
TN
y .
The rate of mass transfer coefficient can be obtained terms of the Sherwood number
in non-dimensional form, given by 0
b
y
CS
y .
RESULTS AND DISCUSSION
We have analyzed the effects of the various parameters such as Prandtl number
(𝑃𝑟), thermal Grashof number(Gr), mass Grashof number(Gc),Schmidt number(𝑆𝑐),
radiation parameter (R), permeability of the porous medium (𝐾), chemical reaction
60 B. Shankar Goud and M.N. Raja Shekar
parameter (𝐾𝑟), heat generation parameter (Q), Eckert number (Ec) and are presented
graphically.
The influence of the mass Grashof number on the velocity is presented in figure 1. It
is observed that the velocity increases as the mass Grashof number increases. The
effect of thermal Grashof number on the velocity profiles is shown in figure 2. As the
value of 𝐺𝑟 increases, the velocity increases. Figure 3 displays the effect the
permeability of the porous medium on velocity profiles. It is observed that the
permeability of the porous medium is increases the velocity increases. Figures 4(a)
and 4(b) show the influence of the radiation parameter on the velocity and
temperature profiles. It is observed that the velocity and temperature decrease with
increasing radiation parameter. The effects of the Prandtl number on velocity and
temperature profiles are presented in figures 5(a) and 5(b). The numerical results
show that the effect of increasing value of Prandtl number results in decreasing
velocity and temperature. Figures 6(a) and 6(b) show the effects of Schmidt number
on velocity and concentration profiles respectively. From these figures it is observed
that an increase in Prandtl number decreases both velocity and concentration profiles.
Figures 7(a) and 7(b) illustrate the velocity and concentration profiles for different
values of the chemical reaction parameter; it is observed that an increase in the Kr
values results in increasing velocity and decreasing in concentration. The effect of the
Soret number on the velocity and concentration profiles is depicted in figures 8(a) and
8(b). It is observed that velocity and concentration increases with increase in Soret
number. The effect of the heat generation parameter Q on velocity and temperature
are shown in figures 9(a) and 9(b). It is noticed that an increase in the heat generation
parameter Q results in an increase in the velocity and temperature.
CONCLUSION
In this article a mathematical pattern has been presented for the Soret and radiation
effects on mass transfer flow through a highly porous medium with heat generation
and chemical reaction. The non-dimensional governing equations are solved by
Galerkin finite element method. The conclusions of the model are as follows:
The velocity increases with an increase in thermal Grashof number (Gr), Solutal
Grashof number (Gc), chemical reaction parameter (Kr).
The velocity and temperature decreases with an increase in Prandtl number
(Pr), radiation parameter (R),
The velocity and concentration decreases with an increase in Schmidt number
(Sc).
Finite Element Study of Soret and Radiation Effects on Mass Transfer Flow…. 61
The velocity and temperature increases with an increase in heat generation
parameter.
An increase in the Soret number (So) extends to an increase in velocity and
concentration.
An increase in the chemical reaction parameter (Kr) induces to decrease in the
velocity.
62 B. Shankar Goud and M.N. Raja Shekar
Finite Element Study of Soret and Radiation Effects on Mass Transfer Flow…. 63
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