IJMA

THERMAL RADIATION AND MASS TRANSFEREFFECTS ON FREE CONVECTION FLOW PAST A

SEMI-INFINITE VERTICAL MOVING POROUSPLATE WITH CHEMICAL REACTION

T. Sankar Reddy1, S. Mohammed Ibrahim2 and N. Bhaskar Reddy3

ABSTRACT: The combined free convection boundary layer flow with

radiation and mass transfer past a semi-infinite vertical moving porous plate

in the presence of chemical reaction is studied when the plate moves in its

own plane. The Rosseland approximation is used to describe radiative heat

transfer in the limit of optically thick fluids. The dimensionless governing

equations for this investigation are solved analytically using two-term

harmonic and non-harmonic functions. Numerical evaluation of the analytical

results is performed and some graphical results for the results for the velocity,

temperature and concentration profiles within the boundary layer and

tabulated results for the skin-friction coefficient, Nusselt number and the

Sherwood number are presented and discussed.

Keywords: Free convection; Radiation; Heat and Mass transfer; Chemical

reaction.

1. INTRODUCTION

The study of convection with heat and mass is very useful in fields such aschemistry, agriculture and oceanography. A few representative fields of interest

in which combined heat and mass transfer play an important role are the design

of chemical processing equipment, formation and dispersion of fog, distribution

of temperature and moisture over agriculture fields and groves of fruit trees,

damage of crops due to freezing, and pollution of the environment. This technique

is used in the cooling processes of plastic sheets, polymer fibres, glass materials,

and in drying processes of paper. Gebhart and Pera [1] studied the nature of

vertical natural convection flows resulting from the combined buoyancy effects

of thermal and mass diffusion. Radiative flows are encountered in countless

industrial and environment processes e.g. heating and cooling chambers fossil

fuel and combustion energy processes, evaporation from large open water

reservoirs, astrophysical flows, and solar power technology and space vehiclere-entry. Abdus Sattar and Hamid Kalim [2] investigated the unsteady free

convection interaction with thermal radiation in a boundary layer flow past a

vertical plate. Makinde [3] examined the transient free convection interaction

with thermal radiation of an absorbing -emitting fluid along moving vertical

permeable plate. Muthucumaraswamy and Senthil Kumar [4] presented heat and

mass transfer effects on moving vertical plate in the presence of thermal radiation.

Cogley et al. [5] showed that in the optically thin limit, the fluid does not absorb

its own emitted radiation, but the fluid does absorb radiation emitted by the

boundaries. Recently, Ibrahim et al. [6] have studied nonclassical thermal effects

in strokes second problem for micropolar fluids. Ghaly and Elbarbary [7] reported

the effect of radiation on free convection flow on MHD along a stretching surface

with uniform free stream. Sattar and Hamid [8] investigated the unsteady free

convection interaction with thermal radiation in a boundary layer flow past avertical porous plate.

Combined hat and mass transfer with chemical reaction are of importance in

many processes and have, therefore, received a considerable amount of attention

in recent years. In processes such as drying, evaporation at the surface of a water

body, energy transfer in a wet cooling tower and the flow in a desert cooler, heat

and mass transfer occur simultaneously. We are particularly interested in cases inwhich diffusion and chemical reaction occur at the roughly the same speed. When

diffusion is much faster than chemical reaction, then only chemical factors

influence the chemical reaction rat; when diffusion is not much faster than reaction,

the diffusion and kinematics and kinetics interact to produce very different effects.

Chambre and Young [9] have presented a first order chemical reaction in the

neighbourhood of a horizontal plate. Deka et al. [10] analyzed the effect of the

first order homogeneous chemical reaction on the process of an unsteady flow

past an infinite vertical plate with a constant heat and mass transfer.

Muthucumaraswamy and Ganesan [11] studied effect of the chemical reaction

and injection on the flow characteristics in an unsteady upward motion of an

isothermal plate.

The objective of this paper is to consider unsteady simultaneous convective

heat and mass transfer flow along a vertical moving porous plate in the presence

of thermal radiation and chemical reaction. It is assumed that the plate moves

with constant velocity in the flow direction. In addition, it is assumed temperature

and concentration at the wall as well as the suction velocity are exponentially

varying with time.

2. MATHEMATICAL ANALYSIS

An unsteady two-dimensional laminar free convection with mass transfer flow

of a viscous incompressible fluid past a semi-infinite vertical moving porous

plate in the presence of thermal radiation and chemical reaction is considered.

Let x* -axis be taken along the plate in the vertically upward direction and

y* -axis be taken normal to it. Since the semi-infinite plate in length, the flow

variables are functions of y* and t* only. Hence, under the usual Boussinesq’s

approximation, the equations of mass, linear momentum, micro-rotation, energyand diffusion can be written as follows:

continuity:*

*

v

y = 0 (1)

linear momentum:* *

** *

u uv

t y =

2 ** *

*2 ( ) ( )f c

ug T T g C C

y(2)

energy: ** *

T Tv

t y =

2 *

2 **

1 r

p p

k T q

c c yy(3)

diffusion:* *

** *

C Cv

t y =

2 ** *

*2 ( ),r

uD K C C

y(4)

where x*, y* and t* are the dimensional distances along and perpendicular to the

plate and dimensional time, respectively, u* and v* are the components of

dimensional velocities along x* and y* directions, respectively, C* and T are the

dimensional concentration and temperature, respectively, is the fluid density,

is the kinematic viscosity, cp is the specific heat at constant pressure, is fluid

electrical conductivity, g is the acceleration due to gravity, f is the volumetric-expansion coefficient due to temperature, c is the volumetric-expansion

coefficient due to concentration, D* is the chemical molecular diffusivity, Kr is

the chemical reaction parameter, and k is the fluid thermal conductivity. The

magnetic and viscous dissipations are neglected in this study. The magnetic and

viscous dissipations are neglected in this study. The second and third terms on

the right hand side of the momentum equation (2) denote the thermal and

concentration buoyancy effects, respectively. Also, the second on the right hand

side of the energy equation (3) represents thermal radiation. It is assumed that

the porous plate moves with a variable velocity in the direction of fluid flow. In

addition, it is also assumed that the temperature and the concentration at the

wall as well as the suction velocity are exponentially varying with time. The

radiative heat flux term is simplified by making use of the Rosseland

approximation as [12]

gr =4

*

4

3s

e

T

k y(5)

where s is the Stefan-Boltzman constant and ke is the mean absorption coefficient,

respectively. It should be noted that by Rosseland approximation we limit our

analysis to optically thick fluids. If the temperature differences with in the flow

are sufficiently small then Equation (5) can be linearized by expanding T4 into

the Taylor series about T and neglecting higher terms takes the form

T4 3 44 3T T T (6)

It is assumed that the permeable plate moves with constant velocity in the

direction of fluid flow. We also assume that the plate temperature and concentrationand suction velocity vary exponentially with time. Under these assumptions, the

appropriate boundary conditions for the velocity, microrotation, temperature and

concentration fields are

u* =* ** , ( ) ,n t

p w wu T T T T e

C* =* ** * *( ) n t

w wC C C e , at y* = 0,

u* T T , C* C* as y*

where u*p, C*

w and Tw are the wall dimensional velocity, concentration and

temperature, respectively. C* and T are the free stream dimensional concentration

and temperature, respectively, n* is constant.

It is clear from Equation (1) that the suction velocity normal to the plate iseither a constant or a function of time. Assuming that takes the following

exponential form:

v* =* *

0(1 )n tV Ae (8)

where A is a real positive constant, and A are small less than unity, and V0 is a

scale of suction velocity which has a non-zero positive constant. The negative

sign indicates that the suction is towards the plate.

In order to write the governing equations and the boundary conditions

dimensionless form, the following non-dimensional quantities are introduced.

u =*

0

,u

U

** * *0

20 0 0

, , , ,pp

uv V y nv y U n

V U V

t =* 2 * *

0* * 3, , ,

4e

w w s

t V T T C C kkC R

T T C C T

Sc = * 20

Pr p prr

C CKK

D V k k(9)

Gr = 20 0

( )f wg T T

U V,

* *

20 0

( )c wg C CGc

U V.

In view of Eqs. (5), (6), (8) and (9) the governing Eqs. (2)-(4) reduce to the

following dimensional form:

1 ntu uAe

t y =

2

2 r c

uG G C

y(10)

1 n tAet y

=2

2

1,

y(11)

1 n tC CAe

t y =

2

2

1r

CK C

Sc y

where =4

1 Pr3 4R

and Gc, Gr, Pr, R, Kr and Sc are denote the solutal Grashof number, thermal

Grashof number, Prandtl number, radiation parameter, chemical reaction parameter

and the Schmidt number, respectively.

The boundary conditions (7) are than given by the following dimensionless

equations:

u = 1 , 1nt nte C e , at y = 0

u 0 0, C 0 at y (13)

3. SOLUTION OF THE PROBLEM

In order to reduce the above system of partial differential equations to a system

of ordinary differential equations in dimensionless form, we may represent the

translational velocity, microrotation, temperature and concentration in the

neighbourhood of the plate as

u = 20 1( ) ( ) ( ) ...ntu y e u y O

= 20 1( ) ( ) ( ) ...nty e y O (14)

C = 20 1( ) ( ) ( ) ...ntC y e C y O

Substituting Eq. (14) into Eqs. (10)-(12), and equating the harmonic and

non-harmonic terms, and neglecting the higher-order terms of O( 2), we obtain

the following pairs of equations for (u0, 0, C0) and (u1, 1, C1).

" '0 0u u = 0 0r cG G C (15)

" '1 1 1u u nu = '

0 1 1r cAu G G C (16)

'' '0 0 = 0 (17)

'' '1 1 1n = '

0A (18)

'' '0 0 0rC ScC ScK C = 0, (19)

'' '1 1 1rC ScC Sc n K = '

0AScC (20)

where the primes denote differentiation with respect to y only. The corresponding

boundary conditions can be written as

u0 = Up, u1 = 0, 0 = 1, 1 = 1, C0 = 1, C1 = 1, at y = 0

u0 = 0, u1 = 0, 0 0, 1 0, C0 0, C1 0, as y (21)

Without going into detail, the solutions of Eqs (15)-(20) and subjected to

Eq. (21) can be shown to be

u(y, t) = 21 2 3

m yy ya e a e a e

31 2 41 2 3 4 5 6( )m ym y m y m ynt y ye b e b e b e b e b e b e (22)

(y, t) = 1m yy nt y yAe e e e e

n(23)

C(y, t) = 3 32 22m y m ym y m yn t Ame e e e e

n(24)

where m1 =4

1 12

n,

m2 =4

1 12

rSc K

Sc,

m3 =4

1 12

rn KSc

Sc, m4 =

11 1 4

2n

and a1 = 2 3( )pU a a , a2 = 22 2

Gr

m m

a3 = 2

Gc,

b1 = 21 1

1A

Grn

m m n, b2 =

22 2

22 2

GcmA a m

nm m n

b3 =

2

23 3

1Am

Gcn

m m n, b4 = 1 2 3 5 6( )b b b b b

b5 =3

24 4

GrA a

nm m n

, b6 = 1Aa

n

The skin-friction, the Nusselt number and the Sherwood number are important

physical quantities for this type of boundary-layer flow. These parameters can be

defined and determined as follows:

Cf =*

*

0 0 0

w

y

u

U V y

= 1 2 2 3 1 1 2 2 3 3 4 4 5 6n ta a m a e b m b m b m b m b b

Nu =*

*

0 1 1

0

Re 1y n tx

w y

T y A A mx Nu e

T T y n n

Sh =*

*

0 1 2 2 32 2

0

Re 1y n tx

w y

C y C Am Am mx Sh m e m

C C y n n

Where Rex = V0x/v is the local Reynolds number.

4. RESULTS AND DISCUSSION

Numerical evaluation of the analytical results reported in the previous section

was performed and a representative set of results is reported graphically in

Figure. 1-5. These results are obtained to illustrate the influence of the Grashof

number and solutal Grashof number, Radiation parameter, Schmidt number and

chemical reaction parameter on the velocity, temperature and concentration

profiles, while the values of physical parameters are fixed at A = 0.5, = 0.01,

n = 0.1, Pr = 0.7 and t = 1.0.

Figure 1 displays the effect of the plate moving velocity Up in the direction of

the fluid flow on the velocity profiles across the boundary layer. It is observed

that the values of the velocity on the porous plate increase, as Up increases.

Figure 1: Velocity Profiles Against Coordinate y for Different Up

In Figure 2, the velocity profiles are shown for different values of the Grashof

number and solutal Grashof number, we conclude that there is rise in the velocity,

when Gr or Gc increases. Physically, this is possible because as the Grashof

number increases, the contribution from the buoyancy near the plate becomes

significant, and hence a short rise in the velocity near the plate is observed. Here

the positive values of Gr correspond to a cooling of the surface by natural

convection.

Figure 2: Velocity Profiles Against Coordinate y for Different Gr and Gc

For different values of the radiation parameter R, the velocity and temperatureare plotted in Figure 3. It is obvious that an increase in R results in a decreasing

velocity and temperature within the boundary layer.

Figure 4. Shows the velocity profiles for different values of the Schmidt

number Sc and chemical reaction parameter Kr. It is clear that the velocity

decreases with increasing values of Sc and Kr.

The concentration profiles for different values of the Schmidt number Sc and

chemical reaction parameter Kr are shown Figure 5. It is observed that the

concentration decreases with increasing Sc and Kr.

Figure 3: Velocity and Temperature Profiles Against Coordinate y for Different R.

Figure 4: Velocity Profiles Against Coordinate y for Different Sc and Kr

Figure 5: Concentration Profiles Against Coordinate y for Different Sc and Kr

5. CONCLUSIONS

The dimensionless governing equations are solved by the perturbation technique.The effect of different parameters like plate moving velocity, Grashof number,solutal Grashof number, Radiation parameter, Schmidt number and chemicalreaction parameter

It was found the velocity profiles increased due to increases in plate movingvelocity. Conclusions of the study are as follows.

(1) The velocity as well as concentration decreases with increasingparameters Sc and Kr.

(2) The velocity as well as temperature decreases with increasing parameter R.

(3) The velocity increases with increasing parameters Gr or Gc.

Nomenclature

A suction velocity parameter

C dimensionless concentration

Cf skin friction coefficient.

Cp specific heat at constant pressure .

D mass diffusion coefficient.

g acceleration due to gravity.

Gr Grashof number.

Gc solutal Grashof number

k thermal conductivity

Kr chemical reaction parameter

n scalar constant

Nu Nusselt number.

p pressure.

Pr Prandtl number.

R Radiation parameter

Rex local Reynolds number

Sc Schmidt number

Sh Sherwood number

T temperature.

t time

u,v components of velocities along and perpendicular to the plate.

U0 scale of free stream velocity.

V0 scale of suction velocity.

x,y distances of along and perpendicular to the plate.

Greek Symbols

fluid thermal diffusivity

c coefficient of volumetric concentration expansion of the working fluid

f coefficient of volumetric thermal expansion of the working fluid

scalar constant (<<1)

dimensionless temperature

fluid dynamic viscosity

fluid density

electrical conductivity.

fluid kinematic viscosity

friction coefficient

angular velocity vector

Subscripts

w Wall condition

Free steam condition

Superscripts

() differentiation with respect to y.

* dimensional properties

REFERENCES

[1] Gebhart B and Pera L (1971), “The Nature of Vertical Natural Convection Flows

Resulting from the Combined Buoyancy Effects of Thermal and Mass Diffusion”,Int. J. Heat Mass Transfer, 14, pp. 2025-2050.

[2] Abdus Sattar M.D. and Hamid Kalim M.D. (1996), “Unsteady Free-ConvectionInteraction with Thermal Radiation in a Boundary Layer Flow Past a Vertical PorousPlate”, J. Math. Phys. Sci., 30(1), pp. 25-37.

[3] Makinde O.D. (2005), “Free Convection Flow with Thermal Radiation and MassTransfer Past a Moving Vertical Plate”, 32, pp. 1411-1419.

[4] Muthucumaraswamy R and Senthil Kumar G (2004), “Heat and Mass Transfer Effectson Moving Vertical Plate in the Presence of Thermal Radiation”, Theoretical Applied

Mechanics, 31(1), pp. 35-46.

[5] Cogley A.C., Vincenty W.E., Gilles S.E., (1968) “Differential Approximation forRadiation in a Non-Gray Gas near Equilibrium”, AIAA J, 6 pp. 551-563.

[6] Ibrihem F.S, Hassanien I.A, Bakr AA. (2005), “Nonclssical Thermal Effects in StokesSecond Problem for Micropolar Fluids”. ASME J. Appl.Mech. 72, pp. 468-474.

[7] Ghaly A. Y., and. Elbarbary E. M. E., (2002) “Radiation Effect on MHD FreeConvective Flow of a Gas Stretching Surface with a Uniform Free Steam”, Journal

of Applied Mathematics, 2(2), pp. 93-103.

[8] Abdus Sattar M.D. and Hamid K.M.D.(1996), “Unsteady Free-Convection Interactionwith Thermal Radiation in a Boundary Layer Flow Past a Vertical Porous Plate”,J. Math. Phys. Sci., 30, pp. 25-37.

[9] Chambre P.L and Young J.D (1958), “On the Diffusion of a Chemically ReactiveSpecies in a Laminar Boundary Layer Flow”, Phys. Fluids Flow, 1, pp. 48-54.

[10] Dekha R., Das U.N and Soundalgekar V.M (1994), “Effects on Mass Transfer onFlow Past an Impulsively Started Infinite Vertical Plate with Constant Heat Flux and

Chemical Reaction”, Forschungim Ingenieurwesen, 60, pp. 284-209.

[11] Muthucumaraswamy R and Ganesan (2001), “Effect of the Chemical Reaction and

Injection on the Flow Characteristics in an Unsteady Upward Motion of an IsothermalPlate”, J Appl. Mech. Tech. Phys., 42, pp. 665-671.

[12] Brewster M.Q. (1992), “Thermal Radiative Transfer and Properties”, John Wileyand Sons, New York.

T. Sankar Reddy

Department of Mathematics, Annamacharya Instituteof Technology and Sciences, C.K. Dinne (Village & mandal),Kadapa, YSR- 516003.

E-mail: tsthummalamaths@gmail.com

S. Mohammed Ibrahimb

Department of Mathematics, Priyadarshini Collegeof Engineering & Technology, NELLORE- 524004

N. Bhaskar Reddy

Dept. of Mathematics, Sri Venkateswara University,TIRUPATI-517502.