Finite Element Study of Soret and Radiation Effects on ...

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)


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


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.


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)


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


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).


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.


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